mm-3469 === > http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp27924 essay: But what manner of people are these circle-squarers, when > examined > by the light? Almost always they will be found to be imperfectly > educated persons, whose mathematical knowledge does not exceed > that > of a modern high-school student. It is seldom that they know > accurately what the requirements of the problem are and what its > nature; they are totally ignorant of the two and a half thousand > years' history of the problem; and they have no idea whatever of > the important investigations which have been made with regard to > it by great and real mathematicians in every century down to our > own time. I wonder why circle-squaring lost its appeal -- circle-squarers are > conspicuous by absence on sci.math, and some quality time with > Google just now didn't turn up a single mad claim. Although the > truth can be even stranger: http://en.wikipedia.org/wiki/Tarski's_circle-squaring_problem ~10^50 pieces! Very strange... -- Clive Tooth http://www.shutterstock.com/cat.mhtml?gallery_id=61771 === Subject: Hex to Float conversion of GPS Latitude I would really appreciate if someone could give me a hint how these floats are encoded: 0B7D2F70 41.8713 0B7D232F 41.8687 0B7D220F 41.8684 Vadim === Subject: Re: Hex to Float conversion of GPS Latitude Nope, that would be too easy: http://babbage.cs.qc.edu/IEEE-754/Decimal.html === Subject: Re: An inequality involving sin , > ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists a positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are infinitely many > integers n such that |sin n|<10^{-k}, but I was unable to prove that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is dense > in the unit circle. So infinitely many exp(in) land in the arc 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in (0,1). There could still be holes in (0,1) that {k * pi} does not fill. Density requires more work. The argument I gave can be used to show that there are infinitely many rationals, m/n, such that |pi - m/n| < 1/n^2. But it is not practicable to get to Hurwitz's theorem on rational approximations from here. Continued fractions is the better route. What I proved is well short of Hurwitz or density. That is the point. I proved a school book problem bare handed showing how little is necessary to get the result. -- Michael Press === Subject: Re: An inequality involving sin , > , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is dense > in the unit circle. So infinitely many exp(in) land in the arc 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is 1-1. Given eps > 0, it is then obvious that there exist m, n such that |exp(im) - exp(in)| < eps. This implies |1 - exp(i(n-m))| < eps. If we set E = {exp(ik(n-m)) : k = 1, 2, ...}, then d(z,E) < < eps for every z on the unit circle. This shows {exp(in) : n in Z} is dense in the unit circle. > The argument I gave can be used to show that there are > infinitely many rationals, m/n, such that > |pi - m/n| < 1/n^2. But it is not practicable to get to > Hurwitz's theorem on rational approximations from here. No one said it did. Hurwitz density is not density. > Continued fractions is the better route. What I proved is well short of Hurwitz or density. Hurwitz yes, density no. In fact, dealing with the unit circle directly gives a simpler proof, both of density and the original problem. > That > is the point. I proved a school book problem bare > handed showing how little is necessary to get the > result. === Subject: Re: An inequality involving sin , The World Wide Wade > , , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is dense > in the unit circle. So infinitely many exp(in) land in the arc 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. > Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this assumes the conclusion of the original problem. > This implies |1 - exp(i(n-m))| < > eps. If we set E = {exp(ik(n-m)) : k = 1, 2, ...}, then d(z,E) < > < eps for every z on the unit circle. This shows {exp(in) : n in > Z} is dense in the unit circle. The argument I gave can be used to show that there are > infinitely many rationals, m/n, such that > |pi - m/n| < 1/n^2. But it is not practicable to get to > Hurwitz's theorem on rational approximations from here. No one said it did. Hurwitz density is not density. Yes, that is so. The original poster used Hurwitz's theorem to solve the problem. I reckon that his solution is as long as mine, hence my remark on the route to Hurwitz. The problem is of the sort that is posed to give a student an opportunity to demonstrate that he knows the basics of rational approximation. > Continued fractions is the better route. What I proved is well short of Hurwitz or density. Hurwitz yes, density no. In fact, dealing with the unit circle > directly gives a simpler proof, both of density and the original > problem. I disagree that I have shown that {k * pi} is dense in (0,1). In Rose, A Course in Number Theory, it takes half a page to go from Hurwitz's Theorem to Theorem 7.4.1 If alpha is an irrational number and beta is a real number, then there are infinitely many pairs of integers x and y such that | x * alpha - y - beta | < 3/x. > That > is the point. I proved a school book problem bare > handed showing how little is necessary to get the > result. -- Michael Press === Subject: Re: An inequality involving sin ws.com>, > The World Wide Wade , , [...] > wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this > assumes the conclusion of the original problem. This implies |1 - exp(i(n-m))| < > eps. If we set E = {exp(ik(n-m)) : k = 1, 2, ...}, then d(z,E) < > < eps for every z on the unit circle. This shows {exp(in) : n in > Z} is dense in the unit circle. The argument I gave can be used to show that there are > infinitely many rationals, m/n, such that > |pi - m/n| < 1/n^2. But it is not practicable to get to > Hurwitz's theorem on rational approximations from here. No one said it did. Hurwitz density is not density. [...] > What I proved is well short of Hurwitz or density. Hurwitz yes, density no. In fact, dealing with the unit circle > directly gives a simpler proof, both of density and the original > problem. I disagree that I have shown that {k * pi} is dense in > (0,1). In Rose, A Course in Number Theory, it takes > half a page to go from Hurwitz's Theorem to Theorem 7.4.1 > If alpha is an irrational number and beta is a real > number, then there are infinitely many pairs of > integers x and y such that > | x * alpha - y - beta | < 3/x. again i think you are missing the point kronecker's approximation theorem in its strongest form (as above) is not needed it is not the route to density one would take if one was looking for simplicity you still end up explicitly invoking the archimedean theorem either implicitly or explicitly and density is much more directly derivable from just that in fact the first half of your proof is proving density around 0 and the second half is mapping the circle projection but there is no obstacle to proving density in general because you make an arbitrary choice of 0 coordinate it is not more difficult to make a different arbitrary choice the proof steps are the same obviously there is some limit point in the interval all you need is the standard chaining operation to get to zero or any other point -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: An inequality involving sin , > ws.com>, > The World Wide Wade , , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are > infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to > prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is > dense > in the unit circle. So infinitely many exp(in) land in the arc > 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this > assumes the conclusion of the original problem. Let C = unit circle, E = {exp(in) : n in Z}. College proof: C is compact, E is infinite, hence E has an accumulation point on C. The conclusion is immediate. HIgh school proof: Cover C by finitely many arcs A_1, ..., A_N of length < eps. The N+1 points 1, e^i, ..., e^iN are distinct, so by pigeon hole two of them lie in the same A_j. If p and q lie in A_j, then |p-q| < eps. > This implies |1 - exp(i(n-m))| < > eps. If we set E = {exp(ik(n-m)) : k = 1, 2, ...}, then d(z,E) < > < eps for every z on the unit circle. This shows {exp(in) : n in > Z} is dense in the unit circle. The argument I gave can be used to show that there are > infinitely many rationals, m/n, such that > |pi - m/n| < 1/n^2. But it is not practicable to get to > Hurwitz's theorem on rational approximations from here. No one said it did. Hurwitz density is not density. Yes, that is so. The original poster used Hurwitz's > theorem to solve the problem. I reckon that his > solution is as long as mine, hence my remark on the > route to Hurwitz. The problem is of the sort that is > posed to give a student an opportunity to demonstrate > that he knows the basics of rational approximation. Continued fractions is the better route. What I proved is well short of Hurwitz or density. Hurwitz yes, density no. In fact, dealing with the unit circle > directly gives a simpler proof, both of density and the original > problem. I disagree that I have shown that {k * pi} is dense in > (0,1). In Rose, A Course in Number Theory, it takes > half a page to go from Hurwitz's Theorem to Theorem 7.4.1 > If alpha is an irrational number and beta is a real > number, then there are infinitely many pairs of > integers x and y such that > | x * alpha - y - beta | < 3/x. That > is the point. I proved a school book problem bare > handed showing how little is necessary to get the > result. === Subject: Re: An inequality involving sin , The World Wide Wade > , ws.com>, > The World Wide Wade , , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are > infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to > prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is > dense > in the unit circle. So infinitely many exp(in) land in the arc > 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this > assumes the conclusion of the original problem. Let C = unit circle, E = {exp(in) : n in Z}. College proof: C is > compact, E is infinite, hence E has an accumulation point on C. > The conclusion is immediate. HIgh school proof: Cover C by > finitely many arcs A_1, ..., A_N of length < eps. The N+1 points > 1, e^i, ..., e^iN are distinct, so by pigeon hole two of them lie > in the same A_j. If p and q lie in A_j, then |p-q| < eps. Yes, that is much like the proof I gave. The proof should fit the situation. The original problem is a school book question. In pursuit of bigger game one would simply say `there are infinitely many n such that sin n < eps' and move on. That is why I have dragged out this discussion. There is no need to invoke any theorems on rational approximation, since the first steps solve the problem. -- Michael Press === Subject: Re: An inequality involving sin , > ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade , , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists > a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are > infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to > prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is > dense > in the unit circle. So infinitely many exp(in) land in the > arc > 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this > assumes the conclusion of the original problem. Let C = unit circle, E = {exp(in) : n in Z}. College proof: C is > compact, E is infinite, hence E has an accumulation point on C. > The conclusion is immediate. HIgh school proof: Cover C by > finitely many arcs A_1, ..., A_N of length < eps. The N+1 points > 1, e^i, ..., e^iN are distinct, so by pigeon hole two of them lie > in the same A_j. If p and q lie in A_j, then |p-q| < eps. Yes, that is much like the proof I gave. The proof > should fit the situation. The original problem is a > school book question. In pursuit of bigger game one > would simply say `there are infinitely many n such that > sin n < eps' and move on. That is why I have dragged > out this discussion. There is no need to invoke any > theorems on rational approximation, since the first > steps solve the problem. That's a bit disingeneous, because I did not discuss rational approximation, yet you responded to me We do not need the result on dense sets. But that result is proved using essentially the same ideas you used, except it's easier and gives a more powerful result. If you wish to argue with someone that Hurwitz and the like is not needed, argue with someone who invoked it, not someone who claimed it was simpler than Hurwitz. === Subject: Re: An inequality involving sin , The World Wide Wade > , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade , , ws.com>, > The World Wide Wade , ws.com>, > The World Wide Wade > How to prove that for any positive integer k there exists > a > positive > integer n such that 0 < sin n < 10^{-k}? I found, by using Hurwitz's theorem, that there are > infinitely > many > integers n such that |sin n|<10^{-k}, but I was unable to > prove > that > sin n >0 for infinitely many of those n. This is simpler than Hurwitz. The set {exp(in) : n in Z} is > dense > in the unit circle. So infinitely many exp(in) land in the > arc > 0 > < theta < 10^{-k}, and for such n, 0 < sin n < 10^{-k}. We do not need the result on dense sets. True, but we don't need your solution either. Would you give a proof? it would be exactly the same wwwade only said not to use hurwitz > which is a strict bound that is overly strong all you need is the density result > of the fractional part of irrational * N which you have stated No, I have not proved that the {k * pi} is dense in > (0,1). There could still be holes in (0,1) that > {k * pi} does not fill. Density requires more work. No it doesn't. Because Pi is irrational, the map n -> exp(in) is > 1-1. OK. exp(in) = exp(im) -> n-m = 2 k pi -> n-m = k = 0. Given eps > 0, it is then obvious that there exist m, n such > that |exp(im) - exp(in)| < eps. Still not obvious to me. To me it looks like this > assumes the conclusion of the original problem. Let C = unit circle, E = {exp(in) : n in Z}. College proof: C is > compact, E is infinite, hence E has an accumulation point on C. > The conclusion is immediate. HIgh school proof: Cover C by > finitely many arcs A_1, ..., A_N of length < eps. The N+1 points > 1, e^i, ..., e^iN are distinct, so by pigeon hole two of them lie > in the same A_j. If p and q lie in A_j, then |p-q| < eps. Yes, that is much like the proof I gave. The proof > should fit the situation. The original problem is a > school book question. In pursuit of bigger game one > would simply say `there are infinitely many n such that > sin n < eps' and move on. That is why I have dragged > out this discussion. There is no need to invoke any > theorems on rational approximation, since the first > steps solve the problem. That's a bit disingeneous, because I did not discuss rational > approximation, yet you responded to me We do not need the > result on dense sets. But that result is proved using > essentially the same ideas you used, except it's easier and gives > a more powerful result. If you wish to argue with someone that > Hurwitz and the like is not needed, argue with someone who > invoked it, not someone who claimed it was simpler than Hurwitz. Does your argument work for any irrational, alpha? Also the proof using exp requires complex analysis. As for rational approximations, this is exactly the topic of the original problem. Taking this further, how do you prove that for any irrational, alpha, {k * alpha} is uniformly distributed on (0,1) or S_1? -- Michael Press === Subject: Re: sorry guys!!! > i am so sorry guys > i found it in a book and you are right .... > .... > That was nice but provides absolutely no context.... > .... > Not sorry enough to learn proper nettiquette, though.... Well, I think you're being a bit hard on him/her. The reply should indeed have been within the thread Re: a silly question from algebra, but can't we be gentle with new users who don't know the ropes? Ken Pledger. === Subject: Re: sorry guys!!! i am so sorry guys > i found it in a book and you are right .... .... > That was nice but provides absolutely no context.... .... > Not sorry enough to learn proper nettiquette, though.... > Well, I think you're being a bit hard on him/her. The reply > should indeed have been within the thread Re: a silly question from > algebra, but can't we be gentle with new users who don't know the ropes? Yes, we can. And I was. The first TWO times he/she did this. This post is the ->third<- time he/she starts a new thread, fails to quote, and insists on getting his/her finger stuck on the exclamation point. Arturo Magidin === Subject: Re: ? intersect of two subspaces Hi: Suppose we have a vector space V with dimension N, and there are several subspaces U_1, U_2, ..., U_M, each of dimension Q ( < N ). Now how do we find the intersect of these M subspaces of V? To concentrate on the problem, let's suppose we know bases of U_1 to U_M already. by Cheng Cosine Nov/10/2k6 NC === Subject: Re: Sekilas Tentang Keberadaan Surga & Neraka - www.mediamuslim.info semoga bermanfaat dan berkenan langsung ke www.mediamuslim.info > Surga dan neraka adalah dua makhluk Alloh Ta'ala. Sebagai seorang > muslim kita harus mengimaninya. Kita yakin bahwasanya surga dan neraka > itu ada dan tidak akan pernah binasa. Setiap insan yang beriman tentu > ingin menikmati keindahan surga dan takut terjerumus dalam siksa > neraka. Oleh sebab itulah sudah seyogyanya kita kenali lebih dekat apa > itu surga dan neraka dan bagaimanakah cara agar kita bisa masuk ke > dalam surga dan terhindar dari siksa neraka. coba selengkapnya di www.mediamuslim.info === Subject: Help on Pell's equation. Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. a+b*sqrt(d) is minimal. Let (a n,b n) s.t. a n + b n * sqrt(D) = (a + b*sqrt(D))^n Then, (a 2, b 2) it«s a solution of x^2-Dy^2= 1, with a 2+b 2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got stuck... Any help would be appreciated!! === Subject: Re: Help on Pell's equation. > Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. > a+b*sqrt(d) is minimal. Let (a_n,b_n) s.t. a_n + b_n * sqrt(D) = (a + b*sqrt(D))^n Then, (a_2, b_2) it«s a solution of x^2-Dy^2= 1, with > a_2+b_2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got > stuck... Any help would be appreciated!! Suppose (with an obvious notation) e = a + b sqrt(d) is a smaller solution than e_2 of the +1 equation. If e < e_1 then e_1/e would be a smaller solution of the -1 equation while if e_1 < e then e/e_1 would be a smaller solution of this equation; since e/e_1 > e_1 => e > e_2. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Help on Pell's equation. > Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. > a+b*sqrt(d) is minimal. Let (a n,b n) s.t. a n + b n * sqrt(D) = (a + b*sqrt(D))^n Then, (a 2, b 2) it«s a solution of x^2-Dy^2= 1, with > a 2+b 2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got > stuck... Any help would be appreciated!! I believe you have stated something incorrectly in your problem. Take, for example, D = 2. Then, (a, b) = (1, 1). We have (a 2, b 2) = (3, 2), and 3 ^ 2 - 2 * 2 ^ 2 = 1. However, if you consider (a n, b n) for an odd n, you will have (a n) ^ 2 - D (b n) ^ 2 = -1. Did you mean to say that a 3 + b 3 * sqrt(d) is minimal among other solutions to the same equation? === Subject: Re: Help on Pell's equation. Jules ha escrito: > Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. > a+b*sqrt(d) is minimal. Let (a n,b n) s.t. a n + b n * sqrt(D) = (a + b*sqrt(D))^n Then, (a 2, b 2) it«s a solution of x^2-Dy^2= 1, with > a 2+b 2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got > stuck... Any help would be appreciated!! I believe you have stated something incorrectly in your problem. Take, > for example, D = 2. Then, (a, b) = (1, 1). We have (a 2, b 2) = (3, > 2), and 3 ^ 2 - 2 * 2 ^ 2 = 1. However, if you consider (a n, b n) for > an odd n, you will have (a n) ^ 2 - D (b n) ^ 2 = -1. Did you mean to > say that a 3 + b 3 * sqrt(d) is minimal among other solutions to the > same equation? My fault. It wasn't clear at all. I meant a 2 + b 2 * sqrt(d) is minimal among other solutions of x^2-Dy^2= 1. Actually, the problem has a second part, which is related solutions to the -1 equation and the even ones will generate the solutions to the equation with 1. === Subject: Re: Help on Pell's equation. > Jules ha escrito: Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. > a+b*sqrt(d) is minimal. Let (a n,b n) s.t. a n + b n * sqrt(D) = (a + b*sqrt(D))^n Then, (a 2, b 2) it«s a solution of x^2-Dy^2= 1, with > a 2+b 2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got > stuck... Any help would be appreciated!! I believe you have stated something incorrectly in your problem. Take, > for example, D = 2. Then, (a, b) = (1, 1). We have (a 2, b 2) = (3, > 2), and 3 ^ 2 - 2 * 2 ^ 2 = 1. However, if you consider (a n, b n) for > an odd n, you will have (a n) ^ 2 - D (b n) ^ 2 = -1. Did you mean to > say that a 3 + b 3 * sqrt(d) is minimal among other solutions to the > same equation? > My fault. It wasn't clear at all. I meant a 2 + b 2 * sqrt(d) is > minimal among other solutions of > x^2-Dy^2= 1. Actually, the problem has a second part, which is related > solutions to the -1 equation and the even ones will generate the > solutions to the equation with 1. > Sorry. Looking back over your original post, I see that I misread what you had written. === Subject: Help on Pell's equation. Hi. I«m having trouble proving this: Let (a,b) be a positive solution of the equation x^2-Dy^2= -1 s.t. a+b*sqrt(d) is minimal. Let (a n,b n) s.t. a n + b n * sqrt(D) = (a + b*sqrt(D))^n Then, (a 2, b 2) it«s a solution of x^2-Dy^2= 1, with a 2+b 2*sqrt(d) minimum (among other solutions of the same equation) I don«t know how to show that it is a minimal solution... I got stuck... Any help would be appreciated!! === Subject: Iman dan Amal Sholeh, Faktor Menggapai Kehidupan Bahagia www.mediamuslim.info -- Ketenangan hati, kebahagiaannya dan hilangnya kegundahan adalah keinginan setiap orang. Dengan itulah kehidupan yang baik, perasaan senang dan tenteram dapat dicapai. Dan untuk mendapatkan itu semua ada beberapa faktor yang harus dipenuhi. Ada faktor diniyah (keagamaan), faktor alami dan faktor amaliah (amal, pekerjaan). Hanya orang-orang mu'min saja yang mampu memenuhi tiga faktor tersebut. Adapun selain orang-orang mu'min, maka, kalaupun dari satu segi, sebagian dari faktor-faktor tersebut dapat dicapai dengan jasa dan usaha para cendekiawan mereka; akan tetapi banyak segi-segi lain yang lebih bermanfaat, lebih kuat dan lebih baik -baik jangka pendek atau jangka panjang- yang tidak mampu mereka dapatkan Faktor paling penting dan paling mendasar untuk menggapai bahagia adalah: Iman dan amal shalih. Alloh Subhanahu wa Ta'ala berfirman, yang artinya: Barangsiapa yang mengerjakan amal shalih, baik laki-laki maupun perempuan dalam keadaan beriman, maka sesungguh-nya akan Kami berikan kepadanya kehidupan yang baik dan sesungguhnya akan Kami beri balasan kepada mereka dengan pahala yang lebih baik dari apa yang telah mereka kerjakan. (QS: An-Nahl: 97) Dalam ayat ini Alloh Subhanahu wa Ta'ala memberitakan dan menjanjikan bagi orang yang dapat mengumpulkan antara iman dan amal shalih untuk mendapatkan kehidupan yang baik di dunia ini dan balasan yang baik pula di dunia dan akhirat. Sebabnya sudah jelas, karena orang yang beriman kepada Alloh Subhanahu wa Ta'ala dengan iman yang benar yang dapat membuahkan amal shalih dan dapat memperbaiki kondisi hati, moral (tingkah lakunya), atau urusan keduniaan dan akhiratnya, berarti dia sudah mem-punyai pondasi dan dasar yang kuat untuk menghadapi segala kemungkinan. Kemungkinan baik yang mendatang-kan kebahagiaan dan kesenangan atau kemungkinan bu-ruk yang dapat mendatangkan kegoncangan, kesumpekan dan kesedihan. Kebahagiaan dan kesenangan mereka sambut dengan menerimanya, mensyukurinya dan mempergunakannya untuk hal-hal yang bermanfaat. Dan bila mereka berhasil menerima dan mempergunakannya dengan cara semacam itu, maka akan timbullah sebagai buahnya --dari akumulasi suka cita dan keinginan untuk mempertahankan kebera-daan dan keberkahan nikmat tersebut serta harapan untuk memperoleh pahala syukur-- hal-hal besar lainnya yang kebaikan dan keberkahannya melebihi kebahagiaan dan kesenangan yang pertama. Begitu pula dengan cobaan, kemudharatan, kesempitan dan keruwetan. Yang mampu dia atasi dia pecahkan, yang hanya dapat dia minimalisasi dia lakukan dan yang tidak boleh tidak harus dia hadapi dia hadapi dengan kesabaran. Dan sebagai dampak dari akumulasi 'kemampuan meng-hadang ujian plus percobaan dan kekuatan' juga akumulasi dari 'kesabaran plus pengharapan akan pahala' maka mereka akan mendapatkan hal-hal besar lainnya yang dengan hal-hal tersebut semua ujian dan cobaan apapun tidak akan terasa bahkan akan berubah menjadi kese-nangan dan harapan-harapan baik serta keinginan untuk mendapatkan karunia dan pahala dari Alloh Subhanahu wa Ta'ala. Seperti yang diungkapkan oleh Nabi shallallahu 'alaihi wa sallam dalam sebuah hadits shahih, beliau bersabda, yang artinya: Sungguh luar biasa urusan seorang mu'min itu. Sesungguh-nya setiap urusannya (akan mendatangkan) kebaikan. Bila dia mendapatkan kesenangan, dia bersyukur dan (syukur) itu adalah kebaikan untuknya. Bila dia mendapatkan musibah, dia bersabar dan (sabar) itu adalah kebaikan untuknya. Hal itu tidak (diberikan) untuk siapa pun kecuali untuk seorang mu'min. (HR: Muslim) Dalam hadits ini Rasululloh shallallahu 'alaihi wa sallam memberitahukan bahwa seorang mu'min akan dilipat-gandakan kebaikannya dan buah amal-amalnya dalam kondisi yang dia hadapi, dalam kondisi nikmat atau musibah. Oleh karena itu, anda bisa mendapati dua orang yang mendapatkan ujian yang sama atau nikmat yang sama, tetapi ternyata, keduanya berbeda dalam cara mengha-dapinya. Hal itu kembali pada perbedaan keduanya dalam kualitas iman dan amal shalihnya. Yang satu dapat menghadapi kondisi nikmat atau musibah dengan syukur dan sabar, sehingga dia merasa senang dan suka cita. Sementara kesumpekan, keruwetan, kegundahan, perasaan sempit dada dan kesulitan hidup juga akan hilang, dan akhirnya dia bisa mendapatkan kehidupan yang baik di dunia ini. Adapun orang satunya lagi, dia sambut kondisi nikmat dengan keangkuhan, menolak kebenaran dengan kezha-liman, sehingga moral dan tingkah lakunya menjadi melenceng. Dia sambut kondisi nikmat itu seperti hewan, dengan penuh tamak dan loba. Walaupun demikian, hatinya tetap tidak merasa tenang bahkan terasa seperti dicabik-cabik dari segala penjuru. Dia khawatir kalau apa yang dia nikmati hilang, dia khawatir akan banyaknya tantangan-tantangan yang timbul menghadangnya, dia khawatir dan tidak tenang. Karena hawa nafsu itu tidak akan berhenti pada batas tertentu, tapi dia akan terus ingin mendapatkan yang lainnya lagi yang barangkali bisa dia raih, bisa juga tidak. Kalau berhasil diraih, kekhawatiran-kekhawatiran yang pertama tadi akan menghampirinya. Dia juga akan sambut musibah yang menghadangnya dengan kegoncangan, kegundahan, rasa takut dan jengkel. Bila sudah demikian, jangan tanyakan lagi bagaimana dia akan ditimpa kesulitan hidup, ditimpa penyakit-penyakit saraf dan perasaan takut yang mengkhawatirkan. Karena dia saat itu tidak mengharapkan pahala dari Allah dan tidak punya kesabaran yang dapat menghibur dan membuat penderitaannya berkurang. Hal di atas dapat kita saksikan sendiri dalam kenyataan. Bila anda renungi kondisi orang-orang sekarang ini, anda akan melihat bahwa perbedaan yang besar antara seorang mu'min yang bekerja dan bertindak dengan konsekwensi keimanannya dengan yang tidak demikian, yaitu bahwa agama itu sangat mendorong dan menganjurkan agar orang bersifat qona'ah (menerima) dengan rezeki Allah Subhanahu wa Ta'ala, karunia dan kemurahanNya yang bermacam ragam. Seorang mu'min --bila ditimpa penyakit, kefakiran dan berbagai musibah yang dapat menimpa setiap orang-- dengan keimanannya, juga dengan sifat qona'ah dan kerelaannya atas apa yang diberikan Allah kepadanya, dia akan tetap terlihat tenang. Hatinya tidak menuntut men-capai sesuatu yang tidak ditakdirkan baginya dan tidak melirik kepada orang yang berada di atasnya. Dan barangkali kebahagiaan, kesenangan dan ketenangannya melebihi orang yang berhasil meraih tuntutan-tuntutan duniawinya tetapi tidak qana'ah. Sebagaimana anda juga dapat menyaksikan orang yang bertindak dan beramal tidak sesuai dengan konsekwensi keimanan, bila ditimpa sedikit kekurangan atau tidak ber-hasil meraih sebagian tuntutan duniawinya, dia merasa di puncak kesengsaraan dan kesusahan. Contoh lain, apabila terjadi hal yang menakutkan atau hal-hal yang mengganggu lainnya, anda akan lihat bahwa orang yang benar iman-nya, hatinya kuat, jiwanya tenang, dia mampu mengurus dan menjalani apa yang menimpanya dengan kemampuan pikiran, perkataan dan amalnya. Semua itu akan memper-kuat dirinya bila berhadapan dengan gangguan atau musibah yang menimpanya. Kondisi semacam inilah yang dapat menenangkan manusia dan menguatkan hatinya. Sebaliknya kondisi orang yang tidak mempunyai iman, bila terjadi suatu hal yang menakutkan, hatinya gundah, urat sarafnya menegang, pikirannya kacau, rasa takut dan khawatir masuk ke dalam dirinya. Berkumpullah pada diri-nya perasaan takut dari luar dengan kegoncangan batinnya yang sulit untuk diketahui hakikatnya. Orang dengan tipe semacam itu --bila tidak didukung faktor-faktor alamiah dengan banyak latihan-- akan kehilangan semangat dan stres. Sebab dia tidak mempunyai iman yang dapat mendorongnya bersikap sabar, khususnya dalam kondisi-kondisi tegang dan menyedihkan. Orang baik dan orang jahat juga orang mu'min dan orang kafir, sama-sama berpotensi untuk belajar dan bisa berani. Juga sama-sama mempunyai potensi kejiwaan yang dapat melunakkan dan meringankan hal-hal yang menakut-kan. Hanya saja, seorang mu'min mempunyai keunggulan dengan imannya, kesabaran dan tawakkalnya kepada Allah serta harapannya untuk mendapatkan pahala dari Allah Subhanahu wa Ta'ala. Hal-hal inilah yang menambah rasa keberaniannya, memperingan beban takutnya juga me-ringankan musibah yang menimpanya. Seperti difirman-kan Alloh Subhanahu wa Ta'ala, yang artinya: Jika kamu menderita kesakitan, maka sesungguhnya mereka pun menderita kesakitan (pula) sebagaimana kamu menderita-nya, sedang kamu mengharap dari Allah apa yang tidak mereka harapkan. (QS: An-Nisa': 104) Selain itu dia akan mendapatkan pertolongan Alloh Subhanahu wa Ta'ala dan 'kebersamaanNya'. Dan hal itu dapat menghancurkan perasaan takutnya. Alloh Subhanahu wa Ta'ala berfirman, yang artinya: Dan bersabarlah kamu, sesungguhnya Alloh bersama orang-orang yang sabar. (QS: Al-Anfal: 46) Termasuk di antara faktor-faktor yang dapat menghilangkan kesedihan, musibah dan kegoncangan hati adalah: Berbuat baik kepada makhluk, baik dengan per-kataan, perbuatan dan berbagai macam perbuatan baik lainnya. Alloh Subhanahu wa Ta'ala akan menolak kesedihan dan musibah dari orang shalih dan orang yang jahat sesuai dengan perbuatan baik yang dilakukan. Hanya saja bagi seorang mu'min akan mendapatkan porsi yang lebih sempurna. Dan yang membedakan seorang mu'min dari yang lainnya, bahwa kebaikan yang dia lakukan didorong oleh keikhlasan dan harapan mendapatkan pahala dari Alloh Subhanahu wa Ta'ala. Dan hal itu memudahkan baginya mendapatkan kebaikan yang dia inginkan. Alloh Subhanahu wa Ta'ala juga akan menolak hal-hal yang tidak dia sukai karena berkah keikhlasan dan harapan mereka akan pahalaNya. Alloh Subhanahu wa Ta'ala berfirman, yang artinya: Tidak ada kebaikan pada kebanyakan bisikan-bisikan mereka, kecuali bisikan-bisikan dari orang yang menyuruh (manusia) memberi sedekah, atau berbuat ma'ruf, atau mengadakan perdamaian di antara manusia. Dan barangsiapa yang berbuat demikian karena mencari keridhaan Allah, maka kelak Kami memberi kepadanya pahala yang besar. (QS: An-Nisa': 114) Dalam ayat ini, Alloh Subhanahu wa Ta'ala menginformasikan bahwa hal-hal yang disebutkan tadi semuanya akan bernilai kebaikan bagi orang yang melakukannya. Dan sebuah kebaikan biasanya mendatangkan kebaikan serta menolak keburukan. Seorang mu'min yang hanya mengharapkan pahala Alloh Subhanahu wa Ta'ala akan mendapatkan balasan yang besar yang di antaranya adalah dalam bentuk hilangnya kesedihan, musibah, dan hal-hal yang mengganggu lainnya. (Sumber Rujukan: MENGGAPAI KEHIDUPAN BAHAGIA, Oleh: SYAIKH ABDURRAHMAN BIN NASHIR AS-SA'DY) www.mediamuslim.info === Subject: 'Ilaaju Harril Mushiybati wa Huznihaa [www.mediamuslim.info] www.mediamuslim.info -- Buku ini akan memberikan gambaran tentang kiat-kiat meredam duka dengan berupaya memaparkan cara-cara yang lebih baik untuk mengekspresikan rasa duka. Cara ini bukan saja membuat antum bisa mengurangi tekanan kesedihan dan lebih nyaman dalam menerima terpaan musibah, tapi lebih dari itu juga akan menjadi kunci yang membuka jalan bagi antum untuk keluar dari musibah dengan keberuntungan. Bahkan keberuntungan besar yang dimimpikan oleh banyak orang. Semua itu akan terasa sebagai penerang bagi antum yang tidak tahu, dan penguat hati bagi antum yang pernah mengetahui, harus berbuat apa saat duka melanda. Meski demikian, untuk meraih manfaat sempurna dari buku ini, diperlukan perjuangan keras untuk melaksanakannya. Tanpa itu, semua saran di buku ini tidak lebih dari barisan kata tanpa makna. www.mediamuslim.info === Subject: Re: 'Ilaaju Harril Mushiybati wa Huznihaa [www.mediamuslim.info] > www.mediamuslim.info -- Buku ini akan memberikan gambaran tentang > kiat-kiat meredam duka dengan berupaya memaparkan cara-cara yang lebih > baik untuk mengekspresikan rasa duka. Cara ini bukan saja membuat antum > bisa mengurangi tekanan kesedihan dan lebih nyaman dalam menerima > terpaan musibah, tapi lebih dari itu juga akan menjadi kunci yang > membuka jalan bagi antum untuk keluar dari musibah dengan > keberuntungan. Bahkan keberuntungan besar yang dimimpikan oleh banyak > orang. Semua itu akan terasa sebagai penerang bagi antum yang tidak tahu, dan > penguat hati bagi antum yang pernah mengetahui, harus berbuat apa saat > duka melanda. Meski demikian, untuk meraih manfaat sempurna dari buku > ini, diperlukan perjuangan keras untuk melaksanakannya. Tanpa itu, > semua saran di buku ini tidak lebih dari barisan kata tanpa makna. www.mediamuslim.info === Subject: Re: hamiltonian path = |V| - 1, then G has a hamiltonian path iff G+uv does. One direction is easy ( => ; do you see why?). Now suppose G+uv has a hamiltonian path P. If this P doesn't use the edge uv, then G has a hamiltonian path (namely P). So uv must be an edge of P. Now comes the tricky part. You know that deg(u) + deg(v) is at least |V| - 1. If you add certain pairs of edges to P, you get a hamiltonian path which does not use the edge uv. Let P be the path v(0), v(1), ..., v(n) with v(i) = u and v(i+1) = v for some i. If you have an edge from v(i) to v(n), then v(0), v(1), ..., v(i-1), v(i), v(n), v(n-1), v(n-2), ..., v(i+1) is a hamiltonian path in G. To get a hamiltonian path, you may need to add two edges (but that should be it). You'll have to draw out a hamiltonian path for, say, 8 vertices, and think about what the degree condition means; it means if u is adjacent to d other vertices (which does not include v), then v is adjacent to at least n-1-d vertices. (And you can assume that v is adjacent to exactly this many; if you can solve it in that case, you can solve it for any other number of neighbors of v.) So draw an edge from u to another vertex, and figure out which edge you would need to add to get a hamiltonian path. You're looking for patterns, and when you find one, write down what it means in the general case. > I am a graduate student in Mathematics, but this is my first time doing > graph theory. My advice is: Draw pictures! The right picture will lead to a proof of introductory Graph Theory results. --- > I would appreciate if you could explain as simple as possible. > Although these exercises appear to me as standard in graph theory, I > you in advance for your help. === Subject: Re: hamiltonian path so I'll have to go on memory. (In the future, when replying to a post, > you should include the post you're replying to.) You want to show that: If u and v are nonadjacent vertices such that > deg(u) + deg(v) >= |V| - 1, then G has a hamiltonian path iff G+uv > does. One direction is easy ( => ; do you see why?). Now suppose G+uv has a > hamiltonian path P. If this P doesn't use the edge uv, then G has a > hamiltonian path (namely P). So uv must be an edge of P. Now comes the tricky part. You know that deg(u) + deg(v) is at least > |V| - 1. If you add certain pairs of edges to P, you get a hamiltonian > path which does not use the edge uv. Let P be the path v(0), v(1), ..., > v(n) with v(i) = u and v(i+1) = v for some i. If you have an edge from > v(i) to v(n), then v(0), v(1), ..., v(i-1), v(i), v(n), v(n-1), v(n-2), > ..., v(i+1) is a hamiltonian path in G. To get a hamiltonian path, you may need to add two edges (but that > should be it). You'll have to draw out a hamiltonian path for, say, 8 > vertices, and think about what the degree condition means; it means if > u is adjacent to d other vertices (which does not include v), then v is > adjacent to at least n-1-d vertices. (And you can assume that v is > adjacent to exactly this many; if you can solve it in that case, you > can solve it for any other number of neighbors of v.) So draw an edge > from u to another vertex, and figure out which edge you would need to > add to get a hamiltonian path. You're looking for patterns, and when > you find one, write down what it means in the general case. I am a graduate student in Mathematics, but this is my first time doing > graph theory. My advice is: Draw pictures! The right picture will lead to a proof of > introductory Graph Theory results. --- I would appreciate if you could explain as simple as possible. > Although these exercises appear to me as standard in graph theory, I > you in advance for your help. === Subject: Re: To simplify a product of 16 sines in 5 variables I did not get the case for 7 variables as Maple keeps blowing up memory wise. But the 6 case seems to have 2341 symmetric terms in it. Chris schrieb CW : >> Given values a1, a2, a3, a4, a5, >> I am searching for a way >> to express the product of all terms of the form >> sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), >> as a polynomial in the values sin(ai)^2, i=1,..,5, >> where the variables eps1, eps2, eps3, eps4, eps5 >> take on values (+1) and (-1), resp., >> but not more than half of these five values are (-1). >> Let SQRs.i denote sin(a.i)^2, > let m[n1,n2,...] denote sum of all monomials in > SQRs1,SQRs2,SQRs3,SQRs4,SQRs5 with degree n1,n2,... then >> the product can be expressed as ANS. ANS:=-384*m[4,3,3]-5152*m[4,2,2,2]-3072*m[3,3,3,2]-36*m[4,2,2]+736*m[5,3,1,1] +4096*m[4,4,2,2,1]+208*m[4,3,1,1]+32*m[7,1,1,1]-1984*m[5,3,1,1,1]-16000*m[3,3 ,3,1,1]+56*m[5,3]+384*m[5,3,2]+1520*m[2,2,2,1,1]-2304*m[5,2,2,1,1]-2880*m[5,1 ,1,1,1]-12288*m[3,3,3,3,1]-4096*m[4,4,2,2,2]+512*m[4,4,3]-8192*m[4,3,2,2,1]-5 76*m[4,4,2]+416*m[3,3,1,1]+192*m[4,3,2]-8832*m[2,2,2,2,1]+752*m[4,1,1,1,1]-40 96*m[4,4,2,1,1]+2112*m[3,2,2,2]-38400*m[3,2,2,2,2]-2016*m[3,3,2,1]+256*m[6,2, 2,1]+512*m[5,3,3,1]+4352*m[4,3,2,1,1]+768*m[5,2,2,2]-m[8]-768*m[5,3,2,1]-256* m[6,2,2,2]+18176*m[4,2,2,2,1]+5056*m[3,3,2,1,1]-344*m[4,2,1,1]+72*m[5,2,1]-40 *m[6,1,1]+272*m[3,2,2,1]+1536*m[5,3,2,1,1]-1280*m[4,3,3,1]+176*m[5,1,1,1]+256 0*m[4,3,3,1,1]+2048*m[3,3,3,3]+144*m[6,1,1,1]+2048*m[4,3,2,2]-256*m[6,2,1,1]+ 5120*m[3,2,2,2,1]+38912*m[3,3,2,2,2]-16384*m[3,3,3,3,3]-70*m[4,4]+1024*m[4,4, 2,1]-16*m[7,1,1]+18432*m[3,3,3,2,1]-1024*m[4,4,1,1]-11264*m[3,3,2,2,1]-8320*m [4,2,2,1,1]-256*m[5,3,3]+224*m[6,1,1,1,1]+48768*m[2,2,2,2,2]-256*m[4,4,4]-222 72*m[4,2,2,2,2]+4928*m[3,3,3,1]-40*m[4,3,1]-336*m[5,2,1,1]+384*m[5,2,2,1]+512 *m[3,3,2,2]-1024*m[4,4,2,2]-240*m[5,3,1]-96*m[6,2,2]-36864*m[3,3,3,2,2]+1248* m[4,2,2,1]+2976*m[5,2,1,1,1]+8*m[7,1]-128*m[4,3,2,1]+96*m[6,2,1]-160*m[3,2,2, 1,1]-928*m[3,2,1,1,1]+384*m[3,3,1,1,1]-288*m[5,2,2]-28*m[6,2]-64*m[7,1,1,1,1] -2008*m[2,2,2,2]-3232*m[4,3,1,1,1]-16*m[3,3,2]+24576*m[3,3,3,3,2]-1024*m[5,3, 3,1,1]+320*m[4,4,1]+2544*m[4,2,1,1,1]+4096*m[4,4,1,1,1]+96*m[3,3,3]+8192*m[4, 3,2,2,2]; Mmm! - Same result as the one I already posted, > but much more compact notation. What about the Newton poyltope of this polynomial, > or of its homogeneous parts? - > I know these are five-dimensional; but they are also symmetric. - > Anyone got an idea what they look like? I also would like to see whether you can do the same > calculation for seven terms a1,...,a7 instead of five: sin(a1-a2+a3-a4+a5-a6-a7)*sin(a1-a2+a3-a4-a5+a6-a7) > *sin(a1-a2+a3-a4-a5-a6+a7)*sin(a1-a2+a3-a4-a5-a6-a7) > *sin(a1-a2+a3+a4-a5-a6-a7)*sin(a1-a2-a3+a4+a5-a6-a7) > *sin(a1-a2-a3+a4-a5+a6-a7)*sin(a1-a2-a3+a4-a5-a6+a7) > *sin(a1-a2-a3+a4-a5-a6-a7)*sin(a1-a2-a3-a4+a5+a6-a7) > *sin(a1-a2-a3-a4+a5-a6+a7)*sin(a1-a2-a3-a4+a5-a6-a7) > *sin(a1-a2-a3-a4-a5+a6+a7)*sin(a1-a2-a3-a4-a5+a6-a7) > *sin(a1-a2-a3-a4-a5-a6+a7)*sin(a1-a2-a3-a4-a5-a6-a7) > *sin(a1+a2-a3-a4+a5-a6-a7)*sin(a1+a2-a3-a4-a5+a6-a7) > *sin(a1+a2-a3-a4-a5-a6+a7)*sin(a1+a2-a3-a4-a5-a6-a7) > *sin(a1+a2-a3+a4-a5-a6-a7)*sin(a1+a2+a3-a4-a5-a6-a7) > *sin(a1-a2-a3+a4-a5+a6+a7)*sin(a1-a2-a3+a4+a5-a6+a7) > *sin(a1-a2-a3+a4+a5+a6-a7)*sin(a1-a2-a3+a4+a5+a6+a7) > *sin(a1-a2-a3-a4+a5+a6+a7)*sin(a1-a2+a3-a4-a5+a6+a7) > *sin(a1-a2+a3-a4+a5-a6+a7)*sin(a1-a2+a3-a4+a5+a6-a7) > *sin(a1-a2+a3-a4+a5+a6+a7)*sin(a1-a2+a3+a4-a5-a6+a7) > *sin(a1-a2+a3+a4-a5+a6-a7)*sin(a1-a2+a3+a4-a5+a6+a7) > *sin(a1-a2+a3+a4+a5-a6-a7)*sin(a1-a2+a3+a4+a5-a6+a7) > *sin(a1-a2+a3+a4+a5+a6-a7)*sin(a1-a2+a3+a4+a5+a6+a7) > *sin(a1+a2-a3-a4-a5+a6+a7)*sin(a1+a2-a3-a4+a5-a6+a7) > *sin(a1+a2-a3-a4+a5+a6-a7)*sin(a1+a2-a3-a4+a5+a6+a7) > *sin(a1+a2-a3+a4-a5-a6+a7)*sin(a1+a2-a3+a4-a5+a6-a7) > *sin(a1+a2-a3+a4-a5+a6+a7)*sin(a1+a2-a3+a4+a5-a6-a7) > *sin(a1+a2-a3+a4+a5-a6+a7)*sin(a1+a2-a3+a4+a5+a6-a7) > *sin(a1+a2-a3+a4+a5+a6+a7)*sin(a1+a2+a3-a4-a5-a6+a7) > *sin(a1+a2+a3-a4-a5+a6-a7)*sin(a1+a2+a3-a4-a5+a6+a7) > *sin(a1+a2+a3-a4+a5-a6-a7)*sin(a1+a2+a3-a4+a5-a6+a7) > *sin(a1+a2+a3-a4+a5+a6-a7)*sin(a1+a2+a3-a4+a5+a6+a7) > *sin(a1+a2+a3+a4-a5-a6-a7)*sin(a1+a2+a3+a4-a5-a6+a7) > *sin(a1+a2+a3+a4-a5+a6-a7)*sin(a1+a2+a3+a4-a5+a6+a7) > *sin(a1+a2+a3+a4+a5-a6-a7)*sin(a1+a2+a3+a4+a5-a6+a7) > *sin(a1+a2+a3+a4+a5+a6-a7)*sin(a1+a2+a3+a4+a5+a6+a7) = P(sin(a1)^2,sin(a2)^2,sin(a3)^2,sin(a4)^2,sin(a5)^2,sin(a6)^2,sin(a7)^2) ? > Thomas === Subject: Re: To simplify a product of 16 sines in 5 variables Follow link : http://tech.groups.yahoo.com/group/Solved_inMapleFiles/files/expanded_polyno m6.txt Chris I did not get the case for 7 variables as Maple keeps blowing up memory > wise. But the 6 case seems > to have 2341 symmetric terms in it. Chris > schrieb CW : >> Given values a1, a2, a3, a4, a5, >> I am searching for a way >> to express the product of all terms of the form >> sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), >> as a polynomial in the values sin(ai)^2, i=1,..,5, >> where the variables eps1, eps2, eps3, eps4, eps5 >> take on values (+1) and (-1), resp., >> but not more than half of these five values are (-1). >> Let SQRs.i denote sin(a.i)^2, > let m[n1,n2,...] denote sum of all monomials in > SQRs1,SQRs2,SQRs3,SQRs4,SQRs5 with degree n1,n2,... then >> the product can be expressed as ANS. ANS:=-384*m[4,3,3]-5152*m[4,2,2,2]-3072*m[3,3,3,2]-36*m[4,2,2]+736*m[5,3,1,1] +4096*m[4,4,2,2,1]+208*m[4,3,1,1]+32*m[7,1,1,1]-1984*m[5,3,1,1,1]-16000*m[3,3 ,3,1,1]+56*m[5,3]+384*m[5,3,2]+1520*m[2,2,2,1,1]-2304*m[5,2,2,1,1]-2880*m[5,1 ,1,1,1]-12288*m[3,3,3,3,1]-4096*m[4,4,2,2,2]+512*m[4,4,3]-8192*m[4,3,2,2,1]-5 76*m[4,4,2]+416*m[3,3,1,1]+192*m[4,3,2]-8832*m[2,2,2,2,1]+752*m[4,1,1,1,1]-40 96*m[4,4,2,1,1]+2112*m[3,2,2,2]-38400*m[3,2,2,2,2]-2016*m[3,3,2,1]+256*m[6,2, 2,1]+512*m[5,3,3,1]+4352*m[4,3,2,1,1]+768*m[5,2,2,2]-m[8]-768*m[5,3,2,1]-256* m[6,2,2,2]+18176*m[4,2,2,2,1]+5056*m[3,3,2,1,1]-344*m[4,2,1,1]+72*m[5,2,1]-40 *m[6,1,1]+272*m[3,2,2,1]+1536*m[5,3,2,1,1]-1280*m[4,3,3,1]+176*m[5,1,1,1]+256 0*m[4,3,3,1,1]+2048*m[3,3,3,3]+144*m[6,1,1,1]+2048*m[4,3,2,2]-256*m[6,2,1,1]+ 5120*m[3,2,2,2,1]+38912*m[3,3,2,2,2]-16384*m[3,3,3,3,3]-70*m[4,4]+1024*m[4,4, 2,1]-16*m[7,1,1]+18432*m[3,3,3,2,1]-1024*m[4,4,1,1]-11264*m[3,3,2,2,1]-8320*m [4,2,2,1,1]-256*m[5,3,3]+224*m[6,1,1,1,1]+48768*m[2,2,2,2,2]-256*m[4,4,4]-222 72*m[4,2,2,2,2]+4928*m[3,3,3,1]-40*m[4,3,1]-336*m[5,2,1,1]+384*m[5,2,2,1]+512 *m[3,3,2,2]-1024*m[4,4,2,2]-240*m[5,3,1]-96*m[6,2,2]-36864*m[3,3,3,2,2]+1248* m[4,2,2,1]+2976*m[5,2,1,1,1]+8*m[7,1]-128*m[4,3,2,1]+96*m[6,2,1]-160*m[3,2,2, 1,1]-928*m[3,2,1,1,1]+384*m[3,3,1,1,1]-288*m[5,2,2]-28*m[6,2]-64*m[7,1,1,1,1] -2008*m[2,2,2,2]-3232*m[4,3,1,1,1]-16*m[3,3,2]+24576*m[3,3,3,3,2]-1024*m[5,3, 3,1,1]+320*m[4,4,1]+2544*m[4,2,1,1,1]+4096*m[4,4,1,1,1]+96*m[3,3,3]+8192*m[4, 3,2,2,2]; Mmm! - Same result as the one I already posted, > but much more compact notation. What about the Newton poyltope of this polynomial, > or of its homogeneous parts? - > I know these are five-dimensional; but they are also symmetric. - > Anyone got an idea what they look like? I also would like to see whether you can do the same > calculation for seven terms a1,...,a7 instead of five: sin(a1-a2+a3-a4+a5-a6-a7)*sin(a1-a2+a3-a4-a5+a6-a7) > *sin(a1-a2+a3-a4-a5-a6+a7)*sin(a1-a2+a3-a4-a5-a6-a7) > *sin(a1-a2+a3+a4-a5-a6-a7)*sin(a1-a2-a3+a4+a5-a6-a7) > *sin(a1-a2-a3+a4-a5+a6-a7)*sin(a1-a2-a3+a4-a5-a6+a7) > *sin(a1-a2-a3+a4-a5-a6-a7)*sin(a1-a2-a3-a4+a5+a6-a7) > *sin(a1-a2-a3-a4+a5-a6+a7)*sin(a1-a2-a3-a4+a5-a6-a7) > *sin(a1-a2-a3-a4-a5+a6+a7)*sin(a1-a2-a3-a4-a5+a6-a7) > *sin(a1-a2-a3-a4-a5-a6+a7)*sin(a1-a2-a3-a4-a5-a6-a7) > *sin(a1+a2-a3-a4+a5-a6-a7)*sin(a1+a2-a3-a4-a5+a6-a7) > *sin(a1+a2-a3-a4-a5-a6+a7)*sin(a1+a2-a3-a4-a5-a6-a7) > *sin(a1+a2-a3+a4-a5-a6-a7)*sin(a1+a2+a3-a4-a5-a6-a7) > *sin(a1-a2-a3+a4-a5+a6+a7)*sin(a1-a2-a3+a4+a5-a6+a7) > *sin(a1-a2-a3+a4+a5+a6-a7)*sin(a1-a2-a3+a4+a5+a6+a7) > *sin(a1-a2-a3-a4+a5+a6+a7)*sin(a1-a2+a3-a4-a5+a6+a7) > *sin(a1-a2+a3-a4+a5-a6+a7)*sin(a1-a2+a3-a4+a5+a6-a7) > *sin(a1-a2+a3-a4+a5+a6+a7)*sin(a1-a2+a3+a4-a5-a6+a7) > *sin(a1-a2+a3+a4-a5+a6-a7)*sin(a1-a2+a3+a4-a5+a6+a7) > *sin(a1-a2+a3+a4+a5-a6-a7)*sin(a1-a2+a3+a4+a5-a6+a7) > *sin(a1-a2+a3+a4+a5+a6-a7)*sin(a1-a2+a3+a4+a5+a6+a7) > *sin(a1+a2-a3-a4-a5+a6+a7)*sin(a1+a2-a3-a4+a5-a6+a7) > *sin(a1+a2-a3-a4+a5+a6-a7)*sin(a1+a2-a3-a4+a5+a6+a7) > *sin(a1+a2-a3+a4-a5-a6+a7)*sin(a1+a2-a3+a4-a5+a6-a7) > *sin(a1+a2-a3+a4-a5+a6+a7)*sin(a1+a2-a3+a4+a5-a6-a7) > *sin(a1+a2-a3+a4+a5-a6+a7)*sin(a1+a2-a3+a4+a5+a6-a7) > *sin(a1+a2-a3+a4+a5+a6+a7)*sin(a1+a2+a3-a4-a5-a6+a7) > *sin(a1+a2+a3-a4-a5+a6-a7)*sin(a1+a2+a3-a4-a5+a6+a7) > *sin(a1+a2+a3-a4+a5-a6-a7)*sin(a1+a2+a3-a4+a5-a6+a7) > *sin(a1+a2+a3-a4+a5+a6-a7)*sin(a1+a2+a3-a4+a5+a6+a7) > *sin(a1+a2+a3+a4-a5-a6-a7)*sin(a1+a2+a3+a4-a5-a6+a7) > *sin(a1+a2+a3+a4-a5+a6-a7)*sin(a1+a2+a3+a4-a5+a6+a7) > *sin(a1+a2+a3+a4+a5-a6-a7)*sin(a1+a2+a3+a4+a5-a6+a7) > *sin(a1+a2+a3+a4+a5+a6-a7)*sin(a1+a2+a3+a4+a5+a6+a7) = P(sin(a1)^2,sin(a2)^2,sin(a3)^2,sin(a4)^2,sin(a5)^2,sin(a6)^2,sin(a7)^2) ? > Thomas === Subject: Re: To simplify a product of 16 sines in 5 variables schrieb CW : > Follow link : http://tech.groups.yahoo.com/group/Solved_inMapleFiles/files/expanded_polynom 6.txt > I would have liked to, but I don't have a Yahoo! ID and don't want to Sign Up for one. :-( Any other chance of obtaining your file? >> >> I did not get the case for 7 variables as Maple keeps blowing up memory >> wise. But the 6 case seems to have 2341 symmetric terms in it. So, its a total of ca. 2341 * 700 = ca. 1.600.000 coefficients in this polynomial. - No wonder, the case of 7 variables seems out of reach. - There might be arounf 1000^2 * 7! = ca. 5.000.000.000 coefficients in this polynomial. Still, if we could circumvent the expansion of all factors, and get directly to the expansion into symmetric terms, we might have a chance (there is *only* about 1 - 4 million of these). I am thinking here of something along the line However, at the moment I am not sure any more what I want to get out of these calculations. :-( The sequence 3, 10, 94, 2341 of numbers of symmetric terms in these polynomials does not seem to be contained in the O.E.I.S. :-( Neither does the sequence 7, 71, 2922 of the total number of terms in these polynomials for 3, 4, resp. 5 variables. By the way: How many hours (days?) did your system take to calculate the polynomial for 6 variables? >>>> Given values a1, a2, a3, a4, a5, >>> I am searching for a way >>> to express the product of all terms of the form >>>> sin( eps1*a1 + eps2*a2 + eps3*a3 + eps4*a4 + eps5*a5 ), >>>> as a polynomial in the values sin(ai)^2, i=1,..,5, >>> where the variables eps1, eps2, eps3, eps4, eps5 >>> take on values (+1) and (-1), resp., >>> but not more than half of these five values are (-1). >>> I also would like to see whether you can do the same >> calculation for seven terms a1,...,a7 instead of five: [ ... ] Oh, and did you notice that the product for 6 variables splits into two factors? - This happens for every even number of variables; and it is described in Moebius paper: E.g., for 4 variables, -sin(a1+a2+a3+a4)*sin(a1+a2+a3-a4)*sin(a1+a2-a3+a4)*sin(a1+a2-a3-a4) *sin(a1-a2+a3+a4)*sin(a1-a2+a3-a4)*sin(a1-a2-a3+a4)*sin(a1-a2-a3-a4) splits according to the formula sin(x) = 2 sin(x/2) cos(x/2) up to a constant into the product of sin(1/2*a1+1/2*a2+1/2*a3+1/2*a4)*cos(1/2*a1+1/2*a2+1/2*a3-1/2*a4) *cos(1/2*a1+1/2*a2-1/2*a3+1/2*a4)*sin(1/2*a1+1/2*a2-1/2*a3-1/2*a4) *cos(1/2*a1-1/2*a2+1/2*a3+1/2*a4)*sin(1/2*a1-1/2*a2+1/2*a3-1/2*a4) *sin(1/2*a1-1/2*a2-1/2*a3+1/2*a4)*cos(1/2*a1-1/2*a2-1/2*a3-1/2*a4) with -cos(1/2*a1+1/2*a2+1/2*a3+1/2*a4)*sin(1/2*a1+1/2*a2+1/2*a3-1/2*a4) *sin(1/2*a1+1/2*a2-1/2*a3+1/2*a4)*cos(1/2*a1+1/2*a2-1/2*a3-1/2*a4) *sin(1/2*a1-1/2*a2+1/2*a3+1/2*a4)*cos(1/2*a1-1/2*a2+1/2*a3-1/2*a4) *cos(1/2*a1-1/2*a2-1/2*a3+1/2*a4)*sin(1/2*a1-1/2*a2-1/2*a3-1/2*a4); and both these factors yield polynomials in sin(ai), even in sin(ai)^2: (a1-a2-a3+a4)*(a1+a2+a3+a4)*(a1+a2-a3-a4)*(a1-a2+a3-a4)*r^2 -4*(a1*a2-a3*a4)*(-a2*a3+a1*a4)*(-a2*a4+a1*a3), resp. -(a1+a2+a3-a4)*(a1+a2-a3+a4)*(a1-a2+a3+a4)*(a1-a2-a3-a4)*r^2 -4*(a1*a2+a3*a4)*(a2*a3+a1*a4)*(a2*a4+a1*a3). === Subject: Re: Knights combinatorics. > On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each other]? Can > you generalize this problem to k knights? If the number of knights is small, you might be able to use Inclusion-Exclusion. I wouldn't recommend it, because things get messy even when you're considering 2 knights. If you have a board B, there is a polynomials called the Rook Polynomial, where the coefficient of x^k is the number of ways to put k _rooks_ on a board without any attacking each other. Formulas exist to calculate the Rook Polynomial (see http://mathworld.wolfram.com/RookPolynomial.html for instance), and with a bit of cleverness, you can calculate the Bishop Polynomial of a board. More difficult formulas exist to calculate the polynomials for the other pieces, but the process can get long and hairy. I have submitted a recursive formula for the general problem to MathWorld, but they haven't posted it yet. I can convert the document I submitted and post in on my webpage, if no one else has any better ideas. --- === Subject: Re: Knights combinatorics. > On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each other]? Can > you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. For m and n at least 2, the number of ways of placing 2 nonattacking knights on the mxn board is C(mn,2) - 2(m-2)(n-1) - 2(m-1)(n-2). I wouldn't call that messy. For m and n at least 4, the number of ways of placing 3 nonattacking knights seems to be C(mn,3) - 2(m-2)(n-1)(mn-2) - 2(m-1)(n-2)(mn-2) + 2(m-1)(n-4) + 2(m-4)(n-1) + 8(m-2)(n-1) + 4(m-2)(n-3) + 4(m-3)(n-2) + 2(m-2)(n-4) + 2(m-4)(n-2) + 4(m-3)(n-3) which is messy enough for me. === Subject: Re: Knights combinatorics. > On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each other]? Can > you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. For m and n at least 2, the number of ways of placing 2 nonattacking > knights on the mxn board is C(mn,2) - 2(m-2)(n-1) - 2(m-1)(n-2). > I wouldn't call that messy. I wouldn't call it correct, unless I see a derivation. (Clearly your claim is that the number of ways to put 2 knights on an mxn board so that they _are_ attacking each other is 2(m-2)(n-1) + 2(m-1)(n-2), and this is what I'm asking for. It does look correct asymptotically (when n, m are big, the answer should be ~ 4mn).) --- > For m and n at least 4, the number of ways of placing 3 nonattacking > knights seems to be C(mn,3) > - 2(m-2)(n-1)(mn-2) - 2(m-1)(n-2)(mn-2) > + 2(m-1)(n-4) + 2(m-4)(n-1) > + 8(m-2)(n-1) > + 4(m-2)(n-3) + 4(m-3)(n-2) > + 2(m-2)(n-4) + 2(m-4)(n-2) > + 4(m-3)(n-3) which is messy enough for me. === Subject: Re: Knights combinatorics. > For m and n at least 2, the number of ways of placing 2 nonattacking > knights on the mxn board is C(mn,2) - 2(m-2)(n-1) - 2(m-1)(n-2). > I wouldn't call that messy. I wouldn't call it correct, unless I see a derivation. (Clearly your > claim is that the number of ways to put 2 knights on an mxn board so > that they _are_ attacking each other is 2(m-2)(n-1) + 2(m-1)(n-2), The mxn chessboard contains (m-2)(n-1) rectangular subboards of size 3x2, and (m-1)(n-2) of size 2x3. In each of those subboards, there are 2 ways to place a pair of attacking knights. Clearly, this defines a 2-to-1 correspondence between the edge-set of the knight's graph and the set of all subboards of size 3x2 or 2x3. === Subject: Re: Knights combinatorics. > For m and n at least 2, the number of ways of placing 2 nonattacking > knights on the mxn board is C(mn,2) - 2(m-2)(n-1) - 2(m-1)(n-2). > I wouldn't call that messy. I wouldn't call it correct, unless I see a derivation. (Clearly your > claim is that the number of ways to put 2 knights on an mxn board so > that they _are_ attacking each other is 2(m-2)(n-1) + 2(m-1)(n-2), The mxn chessboard contains (m-2)(n-1) rectangular subboards of size > 3x2, and (m-1)(n-2) of size 2x3. In each of those subboards, there are > 2 ways to place a pair of attacking knights. Clearly, this defines a > 2-to-1 correspondence between the edge-set of the knight's graph and > the set of all subboards of size 3x2 or 2x3. Nice. --- === Subject: Re: Knights combinatorics. > On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each > other]? Can you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. > Do note that knights of the same color cannot capture each other. ;-) > If you have a board B, there is a polynomials called the Rook > Polynomial, where the coefficient of x^k is the number of ways to put > k _rooks_ on a board without any attacking each other. Formulas exist > to calculate the Rook Polynomial (see > http://mathworld.wolfram.com/RookPolynomial.html for instance), and > with a bit of cleverness, you can calculate the Bishop Polynomial of > a board. More difficult formulas exist to calculate the polynomials for the > other pieces, but the process can get long and hairy. I have submitted > a recursive formula for the general problem to MathWorld, but they > haven't posted it yet. I can convert the document I submitted and post > in on my webpage, if no one else has any better ideas. --- === Subject: Re: Knights combinatorics. On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each > other]? Can you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. Do note that knights of the same color cannot capture each other. ;-) I don't think that makes the calculation that much easier. > If you have a board B, there is a polynomials called the Rook > Polynomial, where the coefficient of x^k is the number of ways to put > k _rooks_ on a board without any attacking each other. Formulas exist > to calculate the Rook Polynomial (see > http://mathworld.wolfram.com/RookPolynomial.html for instance), and > with a bit of cleverness, you can calculate the Bishop Polynomial of > a board. More difficult formulas exist to calculate the polynomials for the > other pieces, but the process can get long and hairy. I have submitted > a recursive formula for the general problem to MathWorld, but they > haven't posted it yet. I can convert the document I submitted and post > in on my webpage, if no one else has any better ideas. The Rook Polynomial paper is now at http://math.asu.edu/~checkman/RookPolys.pdf . --- === Subject: Re: Knights combinatorics. On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each > other]? Can you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. Do note that knights of the same color cannot capture each other. ;-) I don't think that makes the calculation that much easier. > There are only two colors and only two knights of each color. Thus the problem generalizes non-trivially to two knights of different colors, to three knights and to four knights. In addition 1 x n and m x 1 boards are trivial. Also trivial is 2 x 2 board. Now if we look at the 2 x 2 size board for 2,3 and 4 knights, we are led to question all ways or all isomorphic ways. This later circumstance generates a new non-trivial problem of defining isomorphic positions. A thought I leave for you to sleep on over knight. Now let's expand the board to 8x8 or oo x oo if necessary. How many isomorphic positions are there for one knight attact, for two knight attacts? === Subject: Re: Knights combinatorics. On mxn chessboard, how many ways there are to put three knight on the > board such that no two of knights are threatening to [capture] [each > other]? Can you generalize this problem to k knights? If the number of knights is small, you might be able to use > Inclusion-Exclusion. I wouldn't recommend it, because things get messy > even when you're considering 2 knights. Do note that knights of the same color cannot capture each other. ;-) I don't think that makes the calculation that much easier. There are only two colors and only two knights of each color. I think knights can capture knights of either color here. --- > Thus the problem generalizes non-trivially to two knights of different > colors, to three knights and to four knights. In addition > 1 x n and m x 1 boards are trivial. Also trivial is 2 x 2 board. Now if we look at the 2 x 2 size board for 2,3 and 4 knights, we are led > to question all ways or all isomorphic ways. This later circumstance > generates a new non-trivial problem of defining isomorphic positions. A > thought I leave for you to sleep on over knight. Now let's expand the board to 8x8 or oo x oo if necessary. How many > isomorphic positions are there for one knight attact, for two knight > attacts? === Subject: Re: Funny integral Numerically, the integral >> integral(0..1) x^4(1-x)^4/(1+x^2) dx >>is approximately zero. >>Don't tell me. I know why. Do you know why? Because the area between the curve and the x-axis is more correctly > 0.00126448926734961 and that is approximately = 1. Huh? This is like asking why the Sqrt(*2) is approximately = 1.41421. The maximum value of F(X) is .0.00315543135322952491 > at x = 0.47575. Therefore the maximum value of the integral must be < > 0.00316 > which might be considered approximately zero. So what are you asking? So what are you saying? Regardless of what I said, you should be able to explain your question? BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. I don't think this follows from the definition of approximately. --- > Therefore, .00126 + is approximately 1. Bill J. Han de Bruijn === Subject: Re: Funny integral Originator: pouya@localhost [Bill] [...] >> BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. [Proginoskes ] >I don't think this follows from the definition of approximately. Does `approximately' have a single universal definition? If so, I don't seem to know it. Could you please elucidate? -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Re: Funny integral > [Bill] > [...] >> BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. [Proginoskes ] >I don't think this follows from the definition of approximately. Does `approximately' have a single universal definition? If > so, I don't seem to know it. Could you please elucidate? Arnold Ross used to ask his students, rhetorically, What is an approximation to 5? and answer it, Any number other than 5. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Funny integral <1163237862_411@sicinfo3.epfl.ch [Bill] > [...] >> BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. [Proginoskes ] >I don't think this follows from the definition of approximately. Does `approximately' have a single universal definition? If > so, I don't seem to know it. I don't, either; I was hoping that bill could supply it and show why a is approximately b implies a+c is approximately b+c. --- > Could you please elucidate? === Subject: Re: Funny integral <1163237862_411@sicinfo3.epfl.ch [Bill] > [...] >> BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. [Proginoskes ] >I don't think this follows from the definition of approximately. Does `approximately' have a single universal definition? If > so, I don't seem to know it. I don't, either; I was hoping that bill could supply it and show why a > is approximately b implies a+c is approximately b+c. What If a = 0, b = 1 and c = 10^8? If we know that the numerical value of integral in question is 'approximately' 0.00126, why would we say that it is approximately zero? Bill J. --- Could you please elucidate? === Subject: Re: Funny integral <1163237862_411@sicinfo3.epfl.ch [Bill] > [...] >> BTW: If 10^8 +1 is approximately 10^8; then 0 is approximately 1. [Proginoskes ] >I don't think this follows from the definition of approximately. Does `approximately' have a single universal definition? If > so, I don't seem to know it. I don't, either; I was hoping that bill could supply it and show why a > is approximately b implies a+c is approximately b+c. What If a = 0, b = 1 and c = 10^8? If we know that the numerical value of integral in question is > 'approximately' 0.00126, why would we say that it is > approximately zero? Without a formal definition for approximately, such questions cannot be logically answered. --- > Could you please elucidate? === Subject: Re: Deformations in Topology > Are the following true? A smooth 2D parametric curve (x(u), y(u)) is homeomorphable to itself > between arbitrary limits of parameter u.It entails arc length > elongation (or compression) and consequently curvature changes > associated with partial derivatives of x and y with respect to u. Example: (cos(u),sin(u)) ; 0 < u < pi for unit semicircle, 0 < u < 2pi > for elongated unit full circle. A smooth analytic 2-surface embedded in 3-space (x(u,v), y(u,v),z(u,v)) > is homeomorphable to itself between arbitrary limits of parameters u > and v. It entails area elongation (or compression or shear which is a > combination of elongation and compression) and consequently double > curvature changes associated with partial derivatives of x, y, and z > with respect to u and v. Example: (u cos(v), u sin(v),v) ; (( 0 < u < 1), ( 0 < v < pi)) for > small helicoid, (( -1 < u < 1), ( 0 < v < 2 pi)) for stretched larger > helicoid. Small deformations (strain, curvature) in elasticity theory do not deal > exactly enough with large deformations that topology can deal with. What are the differential relations in each case? > Narasimham Is the following mapping a valid homeomorphism for two areas between > given latitude and longitude limits on the same unit sphere? ( cos(ph) cos(th) , cos(ph) sin(th), sin(ph) ), > ( (ph1,th1),(ph2,th2) ) -> ( (ph3,th3),(ph4,th4) ). I don't quite understand the notation. However, iff the map is one-to- one, onto, continuous, and its inverse is continuous, then it is a homeomorphism. Is there some reason you can't check these properties for your maps? -- Marcus === Subject: How to Solve a Gridlock in a Two Parties Game? People invest in company stocks with their real money. You invest yours and other people invest theirs. Invested in lower price will get you more money if the stock price goes up. We base on some public and private information to invest our money, and we assume the flow of information into each party is fair and strongly efficient. I think there should have some rules between major investors. Small cash vs small cash. Real money vs real money, and virtual money (margin) vs virtual money. The virtual money size may be much larger than the real money size, but the actual effect should be caused by real money. We are representing an upward force, but there is another party or a group who bought shares in lower price. I refer this party as lower party. They have a price advantage over us. They try to pull your leg so that you shrug off your stash of money. How can handle this gridlock? Hui === Subject: Paraordinals&Metaordinals x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} } -> m is not an ordinal. I think this will lead to the following: { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { {} },{ { {} } } , {{{{}}}} },...................... ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } },...........} Now we can symbolize this as: { } = 0 { {} } = 1 { {} , {{}} } =2 { {} , {{}}, {{{}}} }=3 { {}, {{}},{{{}}},{{{{}}}} } =4 . . . . { {}, {{}},{{{}}},{{{{}}}},............ } = w { {}, {{}},{{{}}},{{{{}}}},............, { {}, {{}},{{{}}},{{{{}}}},............ } } = w+1 . . .. . . we can produce a system parallel to Von Neumann's. Also another system that is parallel to Von Neumann's can be contemplated. {} = 0 {{}} = 1 {{{}}}=2 {{{{}}}} = 3 . . .. Ua =w , a = { {} , {{}} , {{{}}} , {{{{}}}} ,............. } {Ua} = w+1 {{Ua}} = w+2 . . . . These are called metaordinals. Zuhair === Subject: Re: Paraordinals&Metaordinals x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} > } -> m is not an ordinal. I think this will lead to the following: { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { > {} },{ { {} } } , {{{{}}}} },...................... ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} > } } },...........} This does not express any clear thought to me. I honestly don't know what you're trying to convey here. Are you working in ZF here? With regularity? If so, what makes you think that the paraordinals are totally ordered? Do you have a proof? -- Jesse F. Hughes [Iota]'s the smallest infinitesimal, Russell, there are smaller infinitesimals. -- Ross Finlayson === Subject: Re: Paraordinals&Metaordinals <87lkmhx60h.fsf@phiwumbda.org x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} > } -> m is not an ordinal. I think this will lead to the following: { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { > {} },{ { {} } } , {{{{}}}} },...................... ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} > } } },...........} > This does not express any clear thought to me. I honestly don't know > what you're trying to convey here. Are you working in ZF here? With regularity? If so, what makes you think that the paraordinals are totally ordered? > Do you have a proof? -- > Jesse F. Hughes [Iota]'s the smallest infinitesimal, Russell, there are smaller > infinitesimals. -- Ross Finlayson You , read me wrong again as you usually always do. Not every paraordinal is well ordered, in reality I defined them so that some will be well ordered and some are not well ordered. But I can tell you for sure for every ordinal there is a paraordinal that have order isomorphism to. But of course not every paraordinal is well ordered. Therefore you can say that there are well ordered paraordinals, and there are not well ordere paraordinals. It is these paraordinals that I will use to define cardinality with, in a matter that do not depend on chioce. Zuhair === Subject: Re: Paraordinals&Metaordinals x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} > } -> m is not an ordinal. I think this will lead to the following: { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { > {} },{ { {} } } , {{{{}}}} },...................... ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} > } } },...........} > This does not express any clear thought to me. I honestly don't know > what you're trying to convey here. Are you working in ZF here? With regularity? If so, what makes you think that the paraordinals are totally ordered? > Do you have a proof? You , read me wrong again as you usually always do. If you are often misunderstood, there are (at least) two possible explanations. But, which explanation is most likely? -- Marcus === Subject: Re: Paraordinals&Metaordinals >> x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} >> } -> m is not an ordinal. >> I think this will lead to the following: >> { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { >> {} },{ { {} } } , {{{{}}}} },...................... >> ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} >> } } },...........} >> This does not express any clear thought to me. I honestly don't know >> what you're trying to convey here. >> Are you working in ZF here? With regularity? >> If so, what makes you think that the paraordinals are totally ordered? >> Do you have a proof? You , read me wrong again as you usually always do. Not every paraordinal is well ordered, in reality I defined them so > that some will be well ordered and some are not well ordered. I didn't even ask about whether each paraordinal is well-ordered. I asked whether the class of paraordinals is totally ordered. > But I can tell you for sure for every ordinal there is a paraordinal > that have order isomorphism to. But of course not every paraordinal is > well ordered. Huh? Are you sure you know what order isomorphism is? Anyway, where's your proof? Therefore you can say that there are well ordered paraordinals, and > there are not well ordere paraordinals. Great. Whatever. Can you prove that the class of paraordinals is totally ordered? -- Jesse F. Hughes The sole cause of all human misery is the inability of people to sit quietly in their rooms. -- Blaise Pascal === Subject: Re: Paraordinals&Metaordinals <87lkmhx60h.fsf@phiwumbda.org> <87fycpwpze.fsf@phiwumbda.org >> x is a paraordinal <-> x is a transitive set / Am:mex,m=/={},m=/={ {} >> } -> m is not an ordinal. >> I think this will lead to the following: >> { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} } } }, { {}, { >> {} },{ { {} } } , {{{{}}}} },...................... >> ...............,{ { } , { {} } , { {}, { {} } }, { {}, { {} },{ { {} >> } } },...........} >> This does not express any clear thought to me. I honestly don't know >> what you're trying to convey here. >> Are you working in ZF here? With regularity? >> If so, what makes you think that the paraordinals are totally ordered? >> Do you have a proof? You , read me wrong again as you usually always do. Not every paraordinal is well ordered, in reality I defined them so > that some will be well ordered and some are not well ordered. I didn't even ask about whether each paraordinal is well-ordered. I > asked whether the class of paraordinals is totally ordered. But I can tell you for sure for every ordinal there is a paraordinal > that have order isomorphism to. But of course not every paraordinal is > well ordered. Huh? Are you sure you know what order isomorphism is? Anyway, > where's your proof? > Therefore you can say that there are well ordered paraordinals, and > there are not well ordere paraordinals. Great. Whatever. Can you prove that the class of paraordinals is > totally ordered? -- > Jesse F. Hughes The sole cause of all human misery is the inability of people > to sit quietly in their rooms. -- Blaise Pascal They are not totally ordered. The aim behind using these paraordinals is not the same as the ordinals, the aim for which I have constructed these paraordinals is so that they will not be totally ordered. I want it to be the cardinality of every set without depending on the axiom of choice, but my trials till now failed. (mind you I didn't mention these trials in this forum). But there is a subset of these paraordinals that is totally ordered. and where each paraordinal have order isomorphism with one ordinal. Zuhair Zuhair === Subject: Re: Paraordinals&Metaordinals >> Great. Whatever. Can you prove that the class of paraordinals is >> totally ordered? > They are not totally ordered. The aim behind using these paraordinals > is not the same as the ordinals, the aim for which I have constructed > these paraordinals is so that they will not be totally ordered. I want > it to be the cardinality of every set without depending on the axiom of > choice, but my trials till now failed. (mind you I didn't mention these > trials in this forum). Weird. A moment ago you said there was an order-isomorphism between the ordinals and these guys. Now you say that these guys aren't totally ordered. Do you have any idea what order-isomorphism means? > But there is a subset of these paraordinals that is totally ordered. > and where each paraordinal have order isomorphism with one ordinal. Er, what? -- Reality has a fascinating ability to check us when we get a little too big for our britches... Make no mistake. There isn't a mathematician alive today that I can't now touch, and not a mathematical career on the planet that I can't now affect. --James Harris, render of worlds === Subject: Re: Paraordinals&Metaordinals <87lkmhx60h.fsf@phiwumbda.org> <87fycpwpze.fsf@phiwumbda.org> <87k621v0h3.fsf@phiwumbda.org >> Great. Whatever. Can you prove that the class of paraordinals is >> totally ordered? > They are not totally ordered. The aim behind using these paraordinals > is not the same as the ordinals, the aim for which I have constructed > these paraordinals is so that they will not be totally ordered. I want > it to be the cardinality of every set without depending on the axiom of > choice, but my trials till now failed. (mind you I didn't mention these > trials in this forum). Weird. A moment ago you said there was an order-isomorphism between > the ordinals and these guys. Now you say that these guys aren't > totally ordered. Do you have any idea what order-isomorphism means? Yes , I know. I am telling you, a subset of these guys is well ordered, but not all of them. for example P(w) is not well ordered. Anyhow this is another subject, I will present it when I complete it. Zuhair But there is a subset of these paraordinals that is totally ordered. > and where each paraordinal have order isomorphism with one ordinal. Er, what? > -- > Reality has a fascinating ability to check us when we get a little too > big for our britches... Make no mistake. There isn't a mathematician alive > today that I can't now touch, and not a mathematical career on the planet > that I can't now affect. --James Harris, render of worlds === Subject: Re: Is g=0 a.e.? > Let F be a (Lebesgue) measurable set in R with positive measure. Suppose g > is > a nonnegative integrable function on R such that g(x+y)=c(x) for all y in > F, i.e. > for each fixed x, g is constant on x+F. > Prove or disprove that g=0 a.e. You can do this with the Fourier transform. Let's assume merely that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g in L^1 for all h in F, where T_h(g)(x) = g(x+h). If g is not 0 in L^1, then by the continuity of g^, there exists a nonzero y such that g^(y) is nonzero. But (T_h(g))^(y) = exp(-ihy)g^(y), so we have the latter equal to g^(y) for all h in F. This gives exp(-ihy) = 1 for all h in F, and since y is nonzero, this is a contradiction. === Subject: Re: Is g=0 a.e.? > Let F be a (Lebesgue) measurable set in R with positive measure. Suppose >> g >> is >> a nonnegative integrable function on R such that g(x+y)=c(x) for all y >> in >> F, i.e. >> for each fixed x, g is constant on x+F. >> Prove or disprove that g=0 a.e. You can do this with the Fourier transform. Let's assume merely > that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g > in L^1 for all h in F, where T_h(g)(x) = g(x+h). But the problem stipulates that g(x+y)=c(x) for all y in F, where c(x) need not be g(x). Maybe a modification of your proof is needed? === Subject: Re: Is g=0 a.e.? > Let F be a (Lebesgue) measurable set in R with positive measure. Suppose >> g >> is >> a nonnegative integrable function on R such that g(x+y)=c(x) for all y >> in >> F, i.e. >> for each fixed x, g is constant on x+F. >> Prove or disprove that g=0 a.e. You can do this with the Fourier transform. Let's assume merely > that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g > in L^1 for all h in F, where T_h(g)(x) = g(x+h). But the problem stipulates that g(x+y)=c(x) for all y in > F, where c(x) need not be g(x). Maybe a modification of your proof is > needed? That can be fixed easily, but I missed the real result here: If g is in L^1 and T_h(g) = g for one nonzero h, then g = 0 a.e. Proof: If not, then (T_h(g))^(y) = exp(-ihy)g^(y) = g^(y) is nonzero for all y in some interval. Thus exp(-ihy) = 1 for all y in some interval, contradiction. Back to your original problem: Suppose F contains two distinct values, h1 and h2. Then g(x+h1) = g(x+h2) a.e. Define f(x) = g(x+h1). Then T_(h2-h1)(f) = f. Now apply the above. === Subject: Re: Is g=0 a.e.? , Let F be a (Lebesgue) measurable set in R with positive measure. Suppose >> g >> is >> a nonnegative integrable function on R such that g(x+y)=c(x) for all y >> in >> F, i.e. >> for each fixed x, g is constant on x+F. >> Prove or disprove that g=0 a.e. You can do this with the Fourier transform. Let's assume merely > that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g > in L^1 for all h in F, where T_h(g)(x) = g(x+h). But the problem stipulates that g(x+y)=c(x) for all y in > F, where c(x) need not be g(x). Maybe a modification of your proof is > needed? That can be fixed easily, but I missed the real result here: If g > is in L^1 and T_h(g) = g for one nonzero h, then g = 0 a.e. Actually this is trivial. T_h(g) = g => g is periodic with period h. A periodic L^1 function is 0 a.e., because int_R |f| = sum_{n in Z} int_[nh,(n+1)h] |f| and the latter integrals are all the same. The original problem is a bit weird and kind of got me sidetracked. Did it arise in solving another problem? > Proof: If not, then (T_h(g))^(y) = exp(-ihy)g^(y) = g^(y) is > nonzero for all y in some interval. Thus exp(-ihy) = 1 for all y > in some interval, contradiction. Back to your original problem: Suppose F contains two distinct > values, h1 and h2. Then g(x+h1) = g(x+h2) a.e. Define f(x) = > g(x+h1). Then T_(h2-h1)(f) = f. Now apply the above. === Subject: Re: Is g=0 a.e.? > , > message >>> Let F be a (Lebesgue) measurable set in R with positive measure. >>> Suppose >>> g >>> is >>> a nonnegative integrable function on R such that g(x+y)=c(x) for >>> all y >>> in >>> F, i.e. >>> for each fixed x, g is constant on x+F. >>> Prove or disprove that g=0 a.e. >> You can do this with the Fourier transform. Let's assume merely >> that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g >> in L^1 for all h in F, where T_h(g)(x) = g(x+h). >> But the problem stipulates that g(x+y)=c(x) for all y in >> F, where c(x) need not be g(x). Maybe a modification of your proof is >> needed? >> That can be fixed easily, but I missed the real result here: If g >> is in L^1 and T_h(g) = g for one nonzero h, then g = 0 a.e. Actually this is trivial. T_h(g) = g => g is periodic with period > h. A periodic L^1 function is 0 a.e., because int_R |f| = sum_{n > in Z} int_[nh,(n+1)h] |f| and the latter integrals are all the > same. > Your are right. That is what I thought. > The original problem is a bit weird and kind of got me > sidetracked. Did it arise in solving another problem? > Yes. I was trying to prove that if 1 Let F be a (Lebesgue) measurable set in R with positive measure. Suppose > g > is > a nonnegative integrable function on R such that g(x+y)=c(x) for all y > in > F, i.e. > for each fixed x, g is constant on x+F. > Prove or disprove that g=0 a.e. >> You can do this with the Fourier transform. Let's assume merely >> that F has a limit point. Suppose g is in L^1(R) and T_h(g) = g >> in L^1 for all h in F, where T_h(g)(x) = g(x+h). But the problem stipulates that g(x+y)=c(x) for all y in > F, where c(x) need not be g(x). Maybe a modification of your proof is > needed? I see the neccessary modification. Instead of F, let G=F-F. Then g(x+h)=g(x) for all h in G. So your proof works. === Subject: Re: Is g=0 a.e.? > Let F be a (Lebesgue) measurable set in R with positive measure. Suppose g > is > a nonnegative integrable function on R such that g(x+y)=c(x) for all y in > F, i.e. > for each fixed x, g is constant on x+F. > Prove or disprove that g=0 a.e. > F+F contains an open interval. The given condtion implies g is constant c on F+F. A little work also shows that g is =c on F+F+F, F+F+F+F, ...., whose union is R. So g=c on R. Since g is integrable, c=0. So in fact g is identically 0. Hopefully this argument is correct. === Subject: Re: diagonal argument on ordered array of reals > You appear to be trying to show that the reals are countable. Please confirm. > >> I've just realized what my mistake is, and it is a very stupid one, >> though not the one some people seem to think I'm making. It is that of >> failing to read Cantor's argument properly. I simply missed the condition >> that the list should be of non-terminating decimals (or binaries). I >> thought I was just reprising his argument but with an ordering imposed on >> the list. If you include all the reals (rationals and all), you produce a >> diagonal about which it is completely missing the point to say its inverse >> is not on the list. It's a common mistake seen quite often on sci.math. Someone produces >a method to generate all finite-length numbers (or fractions, or >strings, whatever), then proceeds to show that the diagonalization of >the list is a member of the list. Assuming that the procedure does >indeed produce all finite-length numbers, this will always be the case. >Which only proves that you can exhaustively list all finite-length >numbers. Which, as you discovered, is not what Cantor was talking about. I made the same mistake myself in my younger years, after I found >what I thought was a mapping of the binary naturals to the reals >in [0,1). Alas, that fails since there are no infinite naturals. >It seems that many people have to make these kinds of mistakes >on their own before they aquire a true understanding. >> and others time. At least now I understand the need for that condition on >> the reals. As for the brainless know-it-alls who think you can dismiss what >> someone says without even thinking about it - well, never mind. It's a common problem, trying to communicate clearly to someone >what the flaw in their thinking is. Ten Letters === Subject: Re: diagonal argument on ordered array of reals > These three posts have been a great help. At last I begin to >> understand what you are saying, and clearly there is a good deal of muddle >> and ignorance in my thinking.I understood that it was part of the >> hypothesis that the reals would be numbered, but hadn't thought through >> what the consequences of this were, thinking of it in fact as something >> that could just be tagged on at the end of constructing the array. I also >> had been puzzled as to how, if what you were all saying is true, could >> there be any need for Cantor's proof. But you have helped me see that just >> because the reals have infinite subsets doesn't mean that there could not >> be some special arrangement of the reals which could be put in 1:1 C with >> N. So I understand I cannot just add things on the end of an infinte list >> and still call it a list. I would have to somehow weave them into the >> array, but then the diagonal would be involved in a way which would allow >> Canor's proof to bite. >> There is something more, because there is something peculiarly >> about this combinatorial construction that I started out with that commends >> itself to us, falaciously or otherwise, as a genuine list. Not to us. You mean to you, I guess. If you still feel this way, >then somewhere in the argument below I suspect that >once again we'll find the concept of putting stuff after >the end of a thing that has no end. Let's see... > But before I go >> on I should say I am grateful to you for persevering. I know (did know, but >> temporarily forgot) how difficult it can be to unravel other people's >> muddles. Considering the general nature of usenet, and the fact that I >> started out here like an ignorant prick, this is all the more remarkable. You're welcome. > In what follows, I may be going beyond what was strictly in my >> original post, but it is something which I think I have been in two minds >> about all along. There may too be a connection with one or two comments in >> this thread which have gone so far over my head I could see a vapour trail. >> Let me write this down one more time: >> 1) .0 >> 2) .1 >> 3) .0 1 >> 4) .1 1 >> 5) .0 0 1 >> 6) .0 1 1 >> ----------- >> ----------- >> DON'T STOP >> It seems to me that what we have here is something which understood >> as carried out to infinity produces every real (in the interval). This is where your thinking gets muddled. You have to get >away from thinking about carried out to infinity. > The limit of this process is not something stuck somewhere in the >> middle of a table with finite reals above and infinite reals below. It is >> the whole set of reals. No, it really isn't. Every step in this process is numbered with >a natural number. You will never run out of natural numbers. >And every natural number is finite, and the string next >to that natural number is finite in length. The ill-defined concepts of taking to infinity or >the limit of this process do not allow you to leap to >steps which aren't numbered. > The point is not that you cannot add something to the end of it and >> still call it a list, it is that there isn't anything else to add to it. You're still muddled. > It is true that the process produces an infinite number of >> rationals. But that doesn't mean that that is all it produces. The very way it is defined restricts it. You have a >set of values a_1, a_2, ... all indexed with finite >natural numbers. There is nothing in this definition >that has defined a_pi, or a_sqrt(2), or a_sub anything >except natural numbers, unless you define those >things from the start, which you haven't. - Randy Ten Letters. === Subject: M(n,z) = Sum[G(m,z) / m^z ; m=0,...,n] = ? G(m,z) is the multiple Gamma function; G(m+1,z+1) = G(n,z) * G(n+1,z) G(m,1) = 1 {z in C; m,n in N} M is equal to the Zeta function for G=1. For example G(3,1)=6, G(3,2)=12, G(3,3)=24 with G(3,n)=3*2^n http://a-theory-of-nothingness.blogspot.com/2006/11/multifactorials.html Have you seen it analysed? My interest comes from it's simple combination of the two monsters. It is also related to my Mirror Set project. AB === Subject: Re: M(n,z) = Sum[G(m,z) / m^z ; m=0,...,n] = ? > G(m,z) is the multiple Gamma function; G(m+1,z+1) = G(n,z) * G(n+1,z) G(m+1,z+1) = G(m,z) * G(m+1,z) > G(m,1) = 1 > {z in C; m,n in N} M is equal to the Zeta function for G=1. For example G(3,1)=6, G(3,2)=12, G(3,3)=24 with G(3,n)=3*2^n > http://a-theory-of-nothingness.blogspot.com/2006/11/multifactorials.html Have you seen it analysed? My interest comes from it's simple > combination of the two monsters. It is also related to my Mirror Set > project. AB === Subject: Re: Annoying Puzzle In sci.math, faizankhan666@gmail.com on 10 Nov 2006 01:52:20 -0800 > 20 sheeps are to be scattered in 5 days....Only odd number of sheeps > are allowed to be scattered in any day....What is the arrangement of > sheeps to be scattered for each day? > First, only 19 sheep really need to be scattered over the 5-day period (the 20th will then be just as scattered), as someone has already pointed out. One now can model the problem as finding a combination of 5 odd numbers that add to 19. There are a fair number: 1+1+1+1+15 --- the hockey stick or last minute approach 5+5+5+3+1 9+7+1+1+1 Assuming this modeling is correct there is no unique solution, but there are a fair number of solutions. Was there supposed to be 1 unique solution? Is there more to the problem? -- #191, ewill3@earthlink.net Is it cheaper to learn Linux, or to hire someone to fix your Windows problems? -- === Subject: Re: Annoying Puzzle <7c8g24-j8j.ln1@sirius.tg00suus7038.net In sci.math, faizankhan666@gmail.com > 20 sheeps are to be scattered in 5 days....Only odd number of sheeps > are allowed to be scattered in any day....What is the arrangement of > sheeps to be scattered for each day? > First, only 19 sheep really need to be scattered over the 5-day period > (the 20th will then be just as scattered), as someone has already > pointed out. One now can model the problem as finding a combination of 5 odd numbers > that add to 19. There are a fair number: 1+1+1+1+15 --- the hockey stick or last minute approach > 5+5+5+3+1 > 9+7+1+1+1 Assuming this modeling is correct there is no unique solution, but there > are a fair number of solutions. Was there supposed to be 1 unique > solution? Is there more to the problem? satisfied with the 20th sheep scattered by default solution. But maybe the problem originally read how is this possible or words to that effect and the was somehow picked up in transmission. -- > #191, ewill3@earthlink.net > Is it cheaper to learn Linux, or to hire someone > to fix your Windows problems? -- === Subject: Re: Annoying Puzzle are allowed to be scattered in any day....What is the arrangement of > sheeps to be scattered for each day? > Five odd numbers add to an odd number. The puzzle has to be some word > gimmick. In which case an _exact_ formulation of the problem is needed. Otherwise, there is no resolution, like the GRY problem: Think of words ending in -GRY. Angry and hungry are two of them. There are only three words in the English language. What is the third word? The word is something that everyone uses every day. If you have listened carefully, I have already told you what it is. To keep the thread from growing exponentially, I'll give the answer. You have to read the problem as follows: Think of words ending in -GRY. 'Angry' and 'hungry' are two of them. There are only three words in 'the English language'. What is the third word? The word is something that everyone uses every day. If you have listened carefully, I have already told you what it is. Indeed, the word in question is the third word in the phrase 'the English language'; namely, language. If you don't keep the original wording, you get something like There are three words in the English language which end in GRY; 'angry' and 'hungry' are two of them; what is the third? which is, of course, asking something else. --- === Subject: Re: Parametric solutions of Euler Brick? Those who don't know what an Euler brick is > (a quader with integer sides and face diagonals - > one with additional space diagonal is still > sought and conjectured to be nonexistent) I thought nonexistence was proved a couple of years ago, by a student > from Georgia (Georgia Europe, not Georgia USA). Actually, it was only last year. > News to me. (I have a copy of that > paper at home, which has a parameterization of the rational solutions > to x^2 + y^2 + z^2 = 1, and which uses the fact that 0, 1, and 144 are the only perfect squares > in the Fibonacci sequence.) I hope you'll find that paper & supply some more details. I'll try to find it then, and post an outline sometime soon. First of all, a link: http://www.mualphatheta.org/Science_Fair/Science_Fair_Winners.html This shows a picture of Lasha Margishvili, along with his poster presenting his solution. Now for the first part of the proof. (It is longer than I remember.) If anyone can find a nice proof of Proposition 7, then the whole thing should go through. Proposition 7: The expression (m^2 + n^2)(P^4 - n^4 m^2)(P^4 - n^2 m^4) is a perfect square iff one of the following holds: (a) P^2 = n^2 m, (b) P^2 = n m^2, (c) m = a^2, n = t, P = a^3 (where a and t are integers) OR (d) m = t, n = a^2, P = a^3 (where a and t are integers). ---------- The Diophantine Rectangular Parallelepiped (A Perfect Cuboid) Lasha Margishvili Supervisor: Dr. Mamuka Meskhishvili Tbilisi, Georgia, 2005 [A proof that there is no Euler Brick, as summarized by C C Heckman.] [Section: Rectangular Parallelpiped of the First Class] Let a, b, c denote the sides of the parallelpiped, and p, q, r the diagonal, chosen so that a^2 + b^2 = p^2 a^2 + c^2 = r^2 b^2 + c^2 = q^2 The parameterization of the positive rational solutions to x^2 + y^2 = 1 yields equations: (1,2) a / p = 2 xi / (1 + xi^2), b / p = (1 - xi^2) / (1 + xi^2) (3,4) a / r = 2 zeta / (1 + zeta^2), c / r = (1 - zeta^2) / (1 + zeta^2) (5,6) b / q = 2 mu / (1 + mu^2), c / q = (1 - mu^2) / (1 + mu^2) for some positive rationals xi, zeta, mu. Multiplying these equations yields Boltianski's equation: 8 xi mu zeta = (1 - xi^2)(1 - zeta^2)(1 - mu^2) (also proven in Boltianski's Pythagorean Tetrahedrons, Journal Kvant 1986, #8). Proposition 3: If (xi, mu, zeta) are solutions to Boltianski's equation, and two are rational, then the third is as well. Proposition 4: If xi, mu, and zeta are rational solutions to B.'s equation, then there exists an Euler brick. [proof: write xi = m/n, mu = p/q. Then a = 4 m n p q b = 2 p q (n^2 - m^2) c = (n^2 - m^2) (q^2 - p^2) is an Euler brick.] Proposition 5: If (xi, mu, zeta) is a solution to B's equation, then so is xi' = (1 - xi) / (1 + xi) mu' = (1 - mu) / (1 + mu) zeta' = (1 - zeta) / (1 + zeta). [Section: Rectangular Parallelpiped of the Second Class] We also suppose that a^2 + b^2 + c^2 = L^2 for some integer L. Proposition: The positive rational solutions to x^2 + y^2 + z^2 = 1 are parameterized by x = abs(T1^2 + T2^2 - 1) / (T1^2 + T2^2 + 1) y = 2 T1 / (T1^2 + T2^2 + 1) z = 2 T2 / (T1^2 + T2^2 + 1) where T1 and T2 are positive rational numbers less than 1 such that T1^2 + T2^2 is not 1. Thus we may write (7) a / L = abs(T1^2 + T2^2 - 1) / (T1^2 + T2^2 + 1) (8) b / L = 2 T1 / (T1^2 + T2^2 + 1) (9) c / L = 2 T2 / (T1^2 + T2^2 + 1) [Section: Diophantine Rectangular Parallelpiped Does Not Exist] Consider all the equations involving a, b, c, xi, mu, zeta, and L above. WOLOG T1^2 + T2^2 > 1. Multiplying the equations (1) and (8), and multiplying the equations (2) and (7), yields (10) 4 xi T1 = (1 - xi^2) (T1^2 + T2^2 - 1) Multiplying the equations (3) and (9), and multiplying the equations (7) and (4), yields (11) (1 - zeta^2) T1 = zeta (T1^2 + T2^2 - 1) Multiplying the equations (5) and (9), and multiplying the equations (6) and (8), yields (12) 2 mu T2 = (1 - mu^2) T1 Solve for T2 in (12) and substitute it into (10). Write as a quadratic equation in T1. T1^2 [(1 - xi^2)(1 + mu^2)]^2 - T1 16 xi mu^2 - 4 mu^2 (1 - xi^2) = 0 In order for T1 to be rational, the discriminant must be a perfect square, so (B) 16 xi^2 mu^2 + (1 - xi^2)^2 (1 + mu^2)^2 must be a perfect square (rational). Proposition 3 states that (M) 16 xi^2 mu^2 + (1 - xi^2)^2 (1 - mu^2)^2 must also be a perfect square. Proposition 4 states that there is another solution, where xi is replaced by (1 - xi) / (1 + xi). This implies that (B') (1 - xi^2)^2 mu^2 + xi^2 (1 - mu^2)^2 and (M') (1 - xi^2)^2 mu^2 + xi^2 (1 + mu^2)^2 must be perfect (rational) squares. Now assume that xi = m/k and mu = n/k. (B') and (M') show that the following are perfect squares: (1 - m^2 / k^2)^2 n^2 / k^2 + m^2 / k^2 (1 - n^2 / k^2)^2 and (1 - m^2 / k^2)^2 n^2 / k^2 + m^2 / k^2 (1 + n^2 / k^2)^2 Consequently, (m^2 + n^2)(n^2 m^2 + k^4) and (m^2 + n^2)(n^2 m^2 + k^4) - 4k^2 m^2 n^2 are prefect squares. Hence, we have a Pythagorean triple; there exist P and Q such that (m^2 + n^2)(n^2 m^2 + k^4) = (P^2 + Q^2)^2 and 2 P Q = 2 n m k Write Q = n m k / P, substitute into the other equation above, rewrite as a quadratic equation in k^2: (k^2)^2 [P^4(m^2+n^2)-n^4 m^4] - k^2 (P^4 n^2 m^2) + (P^4 n^2 m^2 (m^2 + n^2) - P^8 = 0 Then solve for k^2. [If anyone is still reading this ... The next Proposition is where I feel the weak link of the chain of proof is.] Proposition 7: The expression (m^2 + n^2)(P^4 - n^4 m^2)(P^4 - n^2 m^4) is a perfect square iff one of the following holds: (a) P^2 = n^2 m, (b) P^2 = n m^2, (c) m = a^2, n = t, P = a^3 (where a and t are integers) OR (d) m = t, n = a^2, P = a^3 (where a and t are integers). [END OF PART ONE] ------------------- --- === Subject: Re: Parametric solutions of Euler Brick? Those who don't know what an Euler brick is > (a quader with integer sides and face diagonals - > one with additional space diagonal is still > sought and conjectured to be nonexistent) I thought nonexistence was proved a couple of years ago, by a student > from Georgia (Georgia Europe, not Georgia USA). Actually, Georgia, (Western) Asia. > Actually, it was only last year. ---------- The Diophantine Rectangular Parallelepiped (A Perfect Cuboid) Lasha Margishvili Supervisor: Dr. Mamuka Meskhishvili Tbilisi, Georgia, 2005 [A proof that there is no Euler Brick, as summarized by C C Heckman.] [BEGINNING OF PART TWO] We now consider cases (a)-(d). If (a) is true, then k^2 = m^2, so we do not have a solution, because of Proposition 6. If (b) is true, then k^2 = n^2, so this is not true. If (c) is true, then k = a^2 (which cannot happen, since then k = m) or k = a^2 * sqrt((a^4 t^2 - a^8 + t^4) / (a^4 t^2 + a^8 - t^4)) and since m = a^2 and n = t, k = m * sqrt((m^2 n^2 - m^4 + n^4) / (m^2 n^2 + m^4 - n^4)). If (d) is true, then k = a^2 (again; this can't happen, since k = n follows) or k = n * sqrt((m^2 n^2 - n^4 + m^4) / (m^2 n^2 + n^4 - m^4)). By symmetry, we only consider the value of k in case (c). Now, let d = gcd(m,n), and m_1 = m / d, n_1 = n / d. Proposition 8: The expression m * sqrt((m^2 n^2 - m^4 + n^4) / (m^2 n^2 + m^4 - n^4)) is integer iff the equation m_1^2 n_1^2 + m_1^4 - n_1^4 = 1 has a solution, with m = m_1 d and n = n_1 d, and gcd(m_1,n_1) = 1. Proof of Prop 8: Since k is an integer, either the expression under the square root is a perfect (integer) square, or m must be divided by the denominator under square root. Since gcd(m_1, m_1^2 n_1^2 + m_1^4 - n_1^4) = 1, d^2 must be a multiple of m_1^2 n_1^2 + m_1^4 - n_1^4; then d^4 is a multiple of m^2 n^2 + m^4 - n^4. Then we get m^2 n^2 + m^4 - n^4 = d^4, so (**) m_1^2 n_1^2 + m_1^4 - n_1^4 = 1. If the expression under the square root is a perfect square, then (***) N^2 = (m_1^2 n_1^2 - m_1^4 + n_1^4) / (m_1^2 n_1^2 + m_1^4 - n_1^4) m_1^2 n_1^2 (N^2 - 1) = (n_1^4 - m_1^4)(N^2 + 1) gcd (m_1,n_1) = 1 implies that gcd(m_1^2 n_1^2, n_1^4 - m_1^4) = 1, so P = (N^2 + 1)/(m_1^2 n_1^2) = (N^2 - 1)/(n_1^4 - m_1^4) for some positive integer P. This gives us a system of Diophantine equations: N^2 + 1 = P(m_1 n_1)^2 N^2 - 1 = P(n_1^4 - m_1^4) If we subtract the second expression from the first, we get 2 = P [(m_1 n_1)^2 - n_1^4 + m_1^4]. Hence P = 1 or 2. If P = 1, then m_1^2 n_1^2 + m_1^4 - n_1^4 = 2. If we view this equation as a quadratic equation in m_1^2, we have m_1^2 = (-n_1^2 + sqrt(5 n_1^4 + 8))/2 The last digit of 5 n_1^4 + 8 is 3 or 8 (as n_1^4 ends in 0, 1, 5, or 6), which means it can't be a perfect square. Hence P = 2. Then (**) is true. (End of proof of Prop 8.) [Section: Fibonacci Enters] The solutions to (**) are m_1^2 = F_(2k-1) and n_1^2 = F_(2k), where F_s is the s-th Fibonacci number. (Proof: _Mathemtical Gems III_, No 9, 1985, MAA, A Second Look at the Fibonacci and Lucas Numbers / K. R. S. Sastry, A Fermat-Fibonacci Collaboration, _Crux Mathematicorum No 5, Vol 23, 1997.) Furthermore, the only Fibonacci numbers which are perfect squares are F_0 = 0, F_1 = 1, F_2 = 1, and F_12 = 144. (J. H. E. Cohn, Square Fibonacci Numbers, Etc, _Fibonacci Quarterly_, vol 2, 1964, pp. 109-113. Online link: http://math.asu.edu/~checkman/SquareFibonacci.html ). Thus m_1 = 1, n_1 = 1, which means m = d = n, contradicting Proposition 6. Consequently, the Diophantine rectangular parallelepiped does not exist. [END OF PART TWO] ------------------- --- === Subject: Re: Parametric solutions of Euler Brick? > (Georgia Europe, not Georgia USA). Is the Georgia that is not the USA in Europe or in Asia? === Subject: Re: Parametric solutions of Euler Brick? (Georgia Europe, not Georgia USA). > Is the Georgia that is not the USA in Europe or in Asia? Georgia, Eurasia, then. 8-) Wikipedia actually puts it in Western Asia. --- === Subject: Re: Parametric solutions of Euler Brick? > [END OF PART ONE] OK, up to here everything looks trivial on the first glance, the snag invariably will come when trying to reduce the perfect square condition to something useful :-) Anxiously waiting for Part 2 (the inconceivable one that I *couldn't* have done 20 years ago) :-) -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn === Subject: Re: orthogonal compliment Originator: pouya@localhost [jennifer ] >If U is a subspace of V prove that U*= {0} iff U=V U*= {v in V: = 0; for all u in U} The necessary way: Let v be in U* and we know that : U/V = U and U*/V= U* U*/V= U*= {0} this implies that for some vector v in V, v = 0 [...] But you already know that 0 belongs to V, because V is a vector space. Hints: (1) What is U / U*? What does this tell you if U=V? (2) What do you know about U (+) U* (i.e. the direct sum)? -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Re: orthogonal compliment <1163236293_409@sicinfo3.epfl.ch [jennifer ] >If U is a subspace of V prove that U*= {0} iff U=V U*= {v in V: = 0; for all u in U} The necessary way: Let v be in U* and we know that : U/V = U and U*/V= U* U*/V= U*= {0} this implies that for some vector v in V, v = 0 > [...] But you already know that 0 belongs to V, because V is a > vector space. Hints: (1) What is U / U*? What does this tell you if U=V? > (2) What do you know about U (+) U* (i.e. the direct sum)? -- > Pouya D. Tafti > p dot d dot tafti at ieee dot o to start with U/U*= {0} or empty but in our case this means that it is {0} because U is a subspace of V and 0 is in U. U = V tells me that U/U*= V U(+)U* = V How does this help me in anyway? === Subject: Re: orthogonal compliment Originator: pouya@localhost [jennifer ] >>If U is a subspace of V prove that U*= {0} iff U=V [...] [Pouya D. Tafti] [...] >> Hints: (1) What is U / U*? What does this tell you if U=V? >> (2) What do you know about U (+) U* (i.e. the direct sum)? [Jennifer] >to start with U/U*= {0} or empty but in our case this means that it is >{0} because U is a subspace of V and 0 is in U. U = V tells me that U/U*= V It most certainly does _not_ tell us that U/U* = V. What it does tell us is that V/U* = {0}. Now, U* is a subspace of V, so... (Here we are trying to show that if U=V, then U*={0}.) U(+)U* = V How does this help me in anyway? > Now let us try to prove that if U*={0}, then U=V. Replacing U* by {0} in the above equation we get U(+){0} = V. What does this tell us about U? -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Re: orthogonal compliment <1163236293_409@sicinfo3.epfl.ch> <1163273407_415@sicinfo3.epfl.ch [jennifer ] >>If U is a subspace of V prove that U*= {0} iff U=V > [...] [Pouya D. Tafti] > [...] >> Hints: (1) What is U / U*? What does this tell you if U=V? >> (2) What do you know about U (+) U* (i.e. the direct sum)? [Jennifer] >to start with U/U*= {0} or empty but in our case this means that it is >{0} because U is a subspace of V and 0 is in U. U = V tells me that U/U*= V It most certainly does _not_ tell us that U/U* = V. What > it does tell us is that V/U* = {0}. Now, U* is a subspace > of V, so... (Here we are trying to show that if U=V, then > U*={0}.) >U(+)U* = V How does this help me in anyway? > Now let us try to prove that if U*={0}, then U=V. Replacing > U* by {0} in the above equation we get U(+){0} = V. What > does this tell us about U? -- > Pouya D. Tafti > p dot d dot tafti at ieee dot org Now i see, thank you. But would it be possible to prove the converse of this by using dimensions as follows. Since U= V this implies that dim U = dim V, since U is a subspace of V, we have that: Assume dimV< oo dimU*= dimV-dimU So, dim U* = dimV-dimV= 0 this implies that U*= {0} === Subject: Re: orthogonal compliment Originator: pouya@localhost [jennifer ] >If U is a subspace of V prove that U*= {0} iff U=V >> [...] >Now i see, thank you. But would it be possible to prove the converse of >this by using dimensions as follows. Since U= V this implies that dim U = dim V, since U is a subspace of V, >we have that: Assume dimV< oo dimU*= dimV-dimU So, dim U* = dimV-dimV= 0 this implies that U*= {0} > Well, yes; assuming that you have already proven the dim identity you are using, I think your proof would be correct for the finite-dimensional case. -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Re: orthogonal compliment <1163273407_415@sicinfo3.epfl.ch> <1163280511_419@sicinfo3.epfl.ch [jennifer ] >If U is a subspace of V prove that U*= {0} iff U=V >> [...] Now i see, thank you. But would it be possible to prove the converse of >this by using dimensions as follows. Since U= V this implies that dim U = dim V, since U is a subspace of V, >we have that: Assume dimV< oo dimU*= dimV-dimU So, dim U* = dimV-dimV= 0 this implies that U*= {0} > Well, yes; assuming that you have already proven the dim > identity you are using, I think your proof would be correct > for the finite-dimensional case. -- > Pouya D. Tafti > p dot d dot tafti at ieee dot org How do i find an orthonormal basis of P_2(R) s.t. the differentiation operator(the operator that takes p to p') on P_2(R) has an upper triangular matrix with respect to this basis. The inner product was defined like this: = integral from 0 to 1 f(x)g(x) dx How do i do this problem, i am having trouble understanding how to tackle this problem. === Subject: Re: orthogonal compliment Originator: pouya@localhost [jennifer ] >How do i find an orthonormal basis of P_2(R) s.t. the differentiation >operator(the operator that takes p to p') on P_2(R) has an upper >triangular matrix with respect to this basis. The inner product was defined like this: = integral from 0 >to 1 f(x)g(x) dx [...] All right. Do you know any basis for P_2(R) (need not be orthonormal)? Can you write the matrix corresponding to the differentiation operator with respect to this basis? (I am assuming that by P_2(R) you mean the space of polynomials up to degree 2.) What do you know about Gram-Schmidt orthogonalization? -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Re: orthogonal compliment <1163280511_419@sicinfo3.epfl.ch> <1163328169_423@sicinfo3.epfl.ch [jennifer ] >How do i find an orthonormal basis of P_2(R) s.t. the differentiation >operator(the operator that takes p to p') on P_2(R) has an upper >triangular matrix with respect to this basis. The inner product was defined like this: = integral from 0 >to 1 f(x)g(x) dx > [...] All right. Do you know any basis for P_2(R) (need not be > orthonormal)? Can you write the matrix corresponding to the > differentiation operator with respect to this basis? (I am > assuming that by P_2(R) you mean the space of polynomials up > to degree 2.) What do you know about Gram-Schmidt orthogonalization? -- > Pouya D. Tafti > p dot d dot tafti at ieee dot org === Subject: Re: orthogonal compliment <1163273407_415@sicinfo3.epfl.ch> <1163280511_419@sicinfo3.epfl.ch [jennifer ] >If U is a subspace of V prove that U*= {0} iff U=V >> [...] Now i see, thank you. But would it be possible to prove the converse of >this by using dimensions as follows. Since U= V this implies that dim U = dim V, since U is a subspace of V, >we have that: Assume dimV< oo dimU*= dimV-dimU So, dim U* = dimV-dimV= 0 this implies that U*= {0} > Well, yes; assuming that you have already proven the dim > identity you are using, I think your proof would be correct > for the finite-dimensional case. -- > Pouya D. Tafti > p dot d dot tafti at ieee dot org === Subject: Re: Infinity Again <26740153.1162902751682.JavaMail.jakarta@nitrogen.mathforum.org> <4551D6D5.2010409@et.uni-magdeburg.de> <4552E1E7.4030709@et.uni-magdeburg.de> <45545224.9000306@et.uni-magdeburg.de> Ross A. Finlayson ha escrito: And we have as least as good reason to consider those who cannot > distinguish different qualities of infiniteness both stupid and > ignorant. > Half of the integers are even. A proper superset is larger than the regular set. ************************************************************************ Unless you define what you mean by larger this is nonsense: the naturals are a subset of the integers...are the integers Larger than the naturals?? They both are equipotent. What do you mean? Common, be like-a-mathematician and define your stuff. The same applies to the following stuff. Tonio *************************************************************************** > The reals are obviously larger than the integers, rationals, or > irrationals, where each of those are proper subsets of the reals. So, Virgil, are you stupid, or ignorant? No: both. Virgil, whoever your we is: no, you don't. Ross === Subject: Re: Infinity Again <26740153.1162902751682.JavaMail.jakarta@nitrogen.mathforum.org> <4551D6D5.2010409@et.uni-magdeburg.de> <4552E1E7.4030709@et.uni-magdeburg.de> <45545224.9000306@et.uni-magdeburg.de Ross A. Finlayson ha escrito: > And we have as least as good reason to consider those who cannot > distinguish different qualities of infiniteness both stupid and > ignorant. > Half of the integers are even. A proper superset is larger than the regular set. > ************************************************************************ > Unless you define what you mean by larger this is nonsense: the > naturals are a subset of the integers...are the integers Larger than > the naturals?? They both are equipotent. What do you mean? Common, be > like-a-mathematician and define your stuff. > The same applies to the following stuff. > Tonio > *************************************************************************** > The reals are obviously larger than the integers, rationals, or > irrationals, where each of those are proper subsets of the reals. So, Virgil, are you stupid, or ignorant? No: both. Virgil, whoever your we is: no, you don't. > Hi Toni, I put forth a wide variety of arguments here to that effect. People said I was wrong. smaller than the set, quite some time ago before I had heard of it. So, that is not a contentious point anymore. I was right. Sets of numbers are sets of numbers and everything about them applies. Ross === Subject: Re: Infinity Again <26740153.1162902751682.JavaMail.jakarta@nitrogen.mathforum.org> <4551D6D5.2010409@et.uni-magdeburg.de> <4552E1E7.4030709@et.uni-magdeburg.de> <45545224.9000306@et.uni-magdeburg.de> Ross A. Finlayson ëúá: > Ross A. Finlayson ha escrito: > And we have as least as good reason to consider those who cannot > distinguish different qualities of infiniteness both stupid and > ignorant. > Half of the integers are even. A proper superset is larger than the regular set. > ************************************************************************ > Unless you define what you mean by larger this is nonsense: the > naturals are a subset of the integers...are the integers Larger than > the naturals?? They both are equipotent. What do you mean? Common, be > like-a-mathematician and define your stuff. > The same applies to the following stuff. > Tonio > *************************************************************************** > The reals are obviously larger than the integers, rationals, or > irrationals, where each of those are proper subsets of the reals. So, Virgil, are you stupid, or ignorant? No: both. Virgil, whoever your we is: no, you don't. > Hi Toni, I put forth a wide variety of arguments here to that effect. People said I was wrong. smaller than the set, quite some time ago before I had heard of it. So, that is not a contentious point anymore. I was right. Sets of numbers are sets of numbers and everything about them applies. Ross ************************************************************* Hi Ross: mean proved or showed or something? Or perhaps he just TRIED something??) that a proper subset is smaller than a set that contains is then I'm sure he MUST have given a comprehensive definition of smaller, otherwise his thesis would have been rejected even by the congress, leave alone by intelligent, academic professors. Again, I ask: what is the definition of smaller or larger WITH RESPECT to infinite sets? Tonio Ps. BTW, what did people tell you you were wrong about, and how did you come to the conclusion they all (who??) were wrong and you were right? And how does the existence of some mathematical thesis, even a VERY GOOD ONE, decide that some contentious point (which one) is over? Was that thesis ever peer-reviewed somewhere? I mean, if there was some then I bet some teams of mathematicians went thru that thesis and decided it was correct and thus the point was settled. Is this the case? Again, what the point is? Thanx === Subject: Surface Interpolation Hi I need some advice in selecting an appropriate surface interpolation method. I have an array of regularly spaced data containing well defined bands of artifacts. After excising these artifacts, the empty spaces are then interpolated. I'm doing the interpolation in TableCurve, and have tried the triangulation method (Renka I) and B-spline. But i'm not sure if there are better approaches and if the triangulation method is necessary/appropriate. I had the impression that TIN is used only for irregularly spaced data A figure of the surface plot and details can be found here: http://www.flickr.com/photos/ken22/294302869/ === Subject: Re: Middle school math, but forget how to solve it Ken Pledger emailed me to give me the answer. And I think it is better to post the answer here so that everyone can read and share. Euclid I.21. Euclid's rather neat proof produces BD to meet AC at E, then uses the triangle inequality (I.20) twice. Ken Pledger. > Given Triangle ABC. D is the center of the incircle. > Prove: AB+AC> DB+DC. Can anyone give a solution? === Subject: How to estimate parameters for Dirichlet distribution? There is a system with parameters V(v1,...,vn) where ·vi = 1 vi>=0 n>100 so V~Dir(a1,...,an) which give response data D=G(V) and G is linear or non-linear functions. When given data distribution D(d1...dm)~Norm(D0,delta) and the prior information is a1=a2=...=an, I make samples of V via, wi = gamma(ai,1) vi = wi/·wi I know the real vi could be rather skew, and the prior settings has make sampling with very small variation, so the chance of sampling the target area is small. How to make improvement as each simulation outputs a residual R=G(vi)-D and shape parameters a1...an approach the right set? Rebecca === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? Consider the image of p.Zp in the quotient Zp/G to show that, for some > n>0, the open subgroup p^n.Zp is contained in G. This implies that G is > open. HTH. J. This clearly works if Z_p/G is a p-group. But is that obvious? === Subject: Re: A question about p-adic integers > Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. This implies that G is >> open. >> HTH. >> J. This clearly works if Z_p/G is a p-group. But is that obvious? It works since Zp/G is finite, by assumption. This implies that the quotient is a p-group. But I do not see why the reasoning I have suggested above works easily for p-groups, but not obviously if the quotient is not a p-group. Or I am missing anything? === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <45575FEA.1080600@web.de Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. This implies that G is >> open. >> HTH. >> J. This clearly works if Z_p/G is a p-group. But is that obvious? It works since Zp/G is finite, by assumption. This implies that the > quotient is a p-group. But I do not see why the reasoning I have > suggested above works easily for p-groups, but not obviously if the > quotient is not a p-group. Or I am missing anything? You said that for sufficiently large n, p^n(Z_p/G) would be contained in G. Surely this would only be the case if Z_p/G is a finite p-group. === Subject: Re: A question about p-adic integers > Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. This implies that G is >> open. >> HTH. >> J. > This clearly works if Z_p/G is a p-group. But is that obvious? >> It works since Zp/G is finite, by assumption. This implies that the >> quotient is a p-group. But I do not see why the reasoning I have >> suggested above works easily for p-groups, but not obviously if the >> quotient is not a p-group. Or I am missing anything? You said that for sufficiently large n, p^n(Z_p/G) would be contained > in G. Surely this would only be the case if Z_p/G is a finite p-group. Zp/G is finite, hence a p-group. But if Lagrange's theorem is applied I do not think that we need the quotient to identify as p-group. So where do you need it in your proof? === Subject: Re: A question about p-adic integers >> Let Zp be the additive group of p-adic integers. Let G be a subgroup of >> Zp of finite index. Show that G is open. >> Can someone give me a hint please? > Consider the image of p.Zp in the quotient Zp/G to show that, for some > n>0, the open subgroup p^n.Zp is contained in G. This implies that G is > open. HTH. J. >> This clearly works if Z_p/G is a p-group. But is that obvious? > It works since Zp/G is finite, by assumption. This implies that the > quotient is a p-group. But I do not see why the reasoning I have > suggested above works easily for p-groups, but not obviously if the > quotient is not a p-group. Or I am missing anything? >> You said that for sufficiently large n, p^n(Z_p/G) would be contained >> in G. Surely this would only be the case if Z_p/G is a finite p-group. Zp/G is finite, hence a p-group. But if Lagrange's theorem is applied I > do not think that we need the quotient to identify as p-group. So where > do you need it in your proof? Well, Lagrange's theorem is perhaps not the right theorem (since I confused addition and multiplication in the quotient Zp/G). Instead consider the decreasing sequence of subgroups of Zp/G induced by p^nZp, n>0, which gets stationary finally. But I still do not see where we urgently need to know that the quotient is a p-group. === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <45575FEA.1080600@web.de> <4557A3CC.1040903@web.de> <4557A8E0.3040305@web.de> Let Zp be the additive group of p-adic integers. Let G be a subgroup of >> Zp of finite index. Show that G is open. >> Can someone give me a hint please? > Consider the image of p.Zp in the quotient Zp/G to show that, for some > n>0, the open subgroup p^n.Zp is contained in G. This implies that G is > open. HTH. J. >> This clearly works if Z_p/G is a p-group. But is that obvious? > It works since Zp/G is finite, by assumption. This implies that the > quotient is a p-group. But I do not see why the reasoning I have > suggested above works easily for p-groups, but not obviously if the > quotient is not a p-group. Or I am missing anything? >> You said that for sufficiently large n, p^n(Z_p/G) would be contained >> in G. Surely this would only be the case if Z_p/G is a finite p-group. Zp/G is finite, hence a p-group. But if Lagrange's theorem is applied I > do not think that we need the quotient to identify as p-group. So where > do you need it in your proof? Well, Lagrange's theorem is perhaps not the right theorem (since I > confused addition and multiplication in the quotient Zp/G). Instead > consider the decreasing sequence of subgroups of Zp/G induced by p^nZp, > n>0, which gets stationary finally. But you still need to prove that it stabilizes at the trivial group. But I still do not see where we urgently need to know that the quotient > is a p-group. It all depends on how you look at it, I guess. I was having some trouble seeing why there couldn't be a prime q distinct from p, and an element x of Z_p, with x not in G but qx in G. That's an immediate consequence of what you said, so something must have been obvious to you which immediately implied it. It took me just a little while to see why it was true. === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <45575FEA.1080600@web.de> <4557A3CC.1040903@web.de Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. This implies that G is >> open. >> HTH. >> J. > This clearly works if Z_p/G is a p-group. But is that obvious? >> It works since Zp/G is finite, by assumption. This implies that the >> quotient is a p-group. But I do not see why the reasoning I have >> suggested above works easily for p-groups, but not obviously if the >> quotient is not a p-group. Or I am missing anything? You said that for sufficiently large n, p^n(Z_p/G) would be contained > in G. Surely this would only be the case if Z_p/G is a finite p-group. Zp/G is finite, hence a p-group. Well, hold on, that needs a lot more elaboration. Here is how I would argue it. Z_p/G is finite, let its order be k, then kZ_p is contained in G, so G=p^n.Z_p for some n, so it is a p-group. > But if Lagrange's theorem is applied I > do not think that we need the quotient to identify as p-group. You're claiming that p^n.Z_p is contained in G for sufficiently large n. That's clearly only the case of Z_p/G is a p-group. I guess you could argue, let k be the order of Z_p/G, then kZ_p is contained in G, and kZ_p equals p^n.Z_p for some n. However, I didn't see that before. I didn't see that G had to be closed. > So where > do you need it in your proof? === Subject: Re: A question about p-adic integers > Let Zp be the additive group of p-adic integers. Let G be a subgroup of > Zp of finite index. Show that G is open. Can someone give me a hint please? >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. This implies that G is >> open. >> HTH. >> J. > This clearly works if Z_p/G is a p-group. But is that obvious? >> It works since Zp/G is finite, by assumption. This implies that the >> quotient is a p-group. But I do not see why the reasoning I have >> suggested above works easily for p-groups, but not obviously if the >> quotient is not a p-group. Or I am missing anything? > You said that for sufficiently large n, p^n(Z_p/G) would be contained > in G. Surely this would only be the case if Z_p/G is a finite p-group. >> Zp/G is finite, hence a p-group. Well, hold on, that needs a lot more elaboration. Here is how I would > argue it. Z_p/G is finite, let its order be k, then kZ_p is contained > in G, so G=p^n.Z_p for some n, so it is a p-group. >> But if Lagrange's theorem is applied I >> do not think that we need the quotient to identify as p-group. You're claiming that p^n.Z_p is contained in G for sufficiently large > n. That's clearly only the case of Z_p/G is a p-group. I guess you > could argue, let k be the order of Z_p/G, then kZ_p is contained in G, > and kZ_p equals p^n.Z_p for some n. However, I didn't see that before. > I didn't see that G had to be closed. Note that every open subgroup G of a topological group is closed - and vice versa if G is of finite index. >> So where >> do you need it in your proof? Best wishes, J. === Subject: Re: A question about p-adic integers > >> Consider the image of p.Zp in the quotient Zp/G to show that, for some >> n>0, the open subgroup p^n.Zp is contained in G. >> Hint: Lagrange for finite groups. in the right direction. With your hints, I have shown that there is an > n such that p^n.Zp+G=p^{n+i}.Zp+G for all i>0. But I am not sure what > to do next. Show, as stated above, that p^n.Zp is contained in G. BTW: I am not sure whether you applied the Lagrange theorem here. It seems to me that you got your result from a kind of descending chain condition. However it works in any case. === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <4554770F.7080701@web.de> <45557DD4.6040602@web.de Show, as stated above, that p^n.Zp is contained in G. BTW: I am not sure whether you applied the Lagrange theorem here. It > seems to me that you got your result from a kind of descending chain > condition. However it works in any case. I know it must sound really dumb, but I don't know how to show that. All I can show, from what I have done, is that p^n.Zp is inside the closure of G. Is it possible to show G is closed? I have used the third isomorphism theorem to get a quotient and apply Lagarange to that quotient. It might not be the best way to proceed. How did you do it? === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <4554770F.7080701@web.de> <45557DD4.6040602@web.de> Cao ha escrito: Show, as stated above, that p^n.Zp is contained in G. BTW: I am not sure whether you applied the Lagrange theorem here. It > seems to me that you got your result from a kind of descending chain > condition. However it works in any case. I know it must sound really dumb, but I don't know how to show that. > All I can show, from what I have done, is that p^n.Zp is inside the > closure of G. Is it possible to show G is closed? I have used the third isomorphism theorem to get a quotient and apply > Lagarange to that quotient. It might not be the best way to proceed. > How did you do it? > *********************************************************** Hi: You have an exercise that requires a minimal knowledge in group theory and in topology, and perhaps even a little topological groups. Jannick already solved the exercise for you, but if you don't know the minimal stuff you're suposed to know when dealing with this exercise then things get tough: (1) Show that if H is ANY topological sbgp in some bigger top. group G, and if H contains a non-empty open SET T, then automatically H is open (a further hint: for any sbgp. H in some group G, and any non-empty set B contained in H, we have that BH = H); (2) If H is any normal sbgp. of index n in a group G , then for any g in G we have that g^n is in H (a further hint: Look at the quotient group G/H of order n and use Lagrange's theorem, or perhaps one of its straightforward corollaries: the order of any element in a finite group divides the order of the group); (3) Now apply the above to the image of the sbgp. H:= pZp of Zp in the quotient group Zp/G, and be sure you know how to show that the sbgp. p^nZp is open...and contained in G for some natural n. Tonio === Subject: Re: A question about p-adic integers <4553A006.9080004@web.de> <4554770F.7080701@web.de> <45557DD4.6040602@web.de> I finally got it! === Subject: Re: Cantor's diagonalization argument > I do not understand. As I see things you say we can do > set theory using either: -formal set theory (ZF) or > -informal condsiderations (IC) I'm not saying anything of that kind. We do set theory the way we do mathematics in general, by producing all sorts of considerations, conjectures, intuitive guesses, mathematical proofs, devising new concepts, and what you have. Some part of our basic conception of sets, and all sorts of other stuff, is captured by various formal theories, which we might study for certain reasons - as we do in e.g. logic and in axiomatic set theory - and which to a fuller or lesser extent are taken as idealised ultimate arbitrators of rigour - e.g. when it is said than in principle all set theoretical theorems proved in ordinary mathematics have proofs in ZF, or even that all proofs of theorems of ordinary mathematics can be formalized in ZF. People often ask questions like do we know that ZF is consistent and so forth, and think that these have direct epistemological significance in the philosophy of mathematics. In reality they have only indirect significance, in that if e.g. ZF were shown inconsistent that would mean that our mathematical ideas about sets are somehow incoherent or inconsistent. It is these, our basic mathematical ideas, conceptions, pictures, that are important, even if we might approach them through results about formal theories. And when drawing conclusions concerning them from mathematical results special arguments are always needed to show how and why the mathematical and logical results are relevant to understanding the mathematics human mathematicians actually practice, to the concepts, assumptions and principles they in reality, implicitly or explicitly, embrace. > We also know that a lot of effort has been put into > showing that [ZF] is inconsistent, but to date it has not been shown to be > inconsistent. It's not the case that a lot of effort has been put into showing that [ZF] is inconsistent. I know but of a few serious attempts at that direction by anyone sane and competent. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument set theory using either: -formal set theory (ZF) or > -informal condsiderations (IC) I'm not saying anything of that kind. My appologies. A re-reading of your post does not support my precis. >We do set theory the way we do > mathematics in general, by producing all sorts of considerations, > conjectures, intuitive guesses, mathematical proofs, devising new > concepts, and what you have. Some part of our basic conception of sets, > and all sorts of other stuff, is captured by various formal theories, > which we might study for certain reasons - as we do in e.g. logic and in > axiomatic set theory - and which to a fuller or lesser extent are taken > as idealised ultimate arbitrators of rigour - e.g. when it is said than > in principle all set theoretical theorems proved in ordinary > mathematics have proofs in ZF, or even that all proofs of theorems of > ordinary mathematics can be formalized in ZF. People often ask questions like do we know that ZF is consistent and > so forth, and think that these have direct epistemological significance > in the philosophy of mathematics. In reality they have only indirect > significance, in that if e.g. ZF were shown inconsistent that would mean > that our mathematical ideas about sets are somehow incoherent or > inconsistent. It is these, our basic mathematical ideas, conceptions, > pictures, that are important, even if we might approach them through > results about formal theories. And when drawing conclusions concerning > them from mathematical results special arguments are always needed to > show how and why the mathematical and logical results are relevant to > understanding the mathematics human mathematicians actually practice, to > the concepts, assumptions and principles they in reality, implicitly or > explicitly, embrace. We also know that a lot of effort has been put into > showing that [ZF] is inconsistent, but to date it has not been shown to be > inconsistent. It's not the case that a lot of effort has been put into showing that > [ZF] is inconsistent. I know but of a few serious attempts at that > direction by anyone sane and competent. Well, I could try to weasel out of this by noting that I never mentioned sane and/or competent, however it is clear from context what I meant. Still even the efforts that cannot be clasified as sane and/or competent at least indicate that it is not easy. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus It is interesting to speculate about the effect on this newsgroup if schweigen had a good translation. - William Hughes === Subject: Re: Cantor's diagonalization argument >> It's not the case that a lot of effort has been put into showing that >> [ZF] is inconsistent. I know but of a few serious attempts at that >> direction by anyone sane and competent. Well, I could try to weasel out of this by noting that I never > mentioned sane and/or competent, however it is clear from > context what I meant. Still even the efforts that cannot > be clasified as sane and/or competent at least indicate that > it is not easy. All that it indicates is that it's not utterly trivial to find a contradiction in set theory. More interesting is that various competent (and probably at least relatively sane) people have actually tried to demonstrate the inconsistency of set theory, either directly - Randall Holmes has produced interesting, though flawed, proofs of inconsistency of set theory - or indirectly - Edward Nelson seems serious about finding a contradiction in PA. I'm not sure whether Eduard Wette is sane or competent - it's rather difficult to tell - and whether he tried to demonstrate the consistency or inconsistency of PA - possibly both -, but if we include him that pretty much exhausts the set of people I know of who have tried to find a contradiction in ZF or a subtheory of ZF. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument It's not the case that a lot of effort has been put into showing that >> [ZF] is inconsistent. I know but of a few serious attempts at that >> direction by anyone sane and competent. Well, I could try to weasel out of this by noting that I never > mentioned sane and/or competent, however it is clear from > context what I meant. Still even the efforts that cannot > be clasified as sane and/or competent at least indicate that > it is not easy. All that it indicates is that it's not utterly trivial to find a > contradiction in set theory. More interesting is that various competent > (and probably at least relatively sane) people have actually tried to > demonstrate the inconsistency of set theory, either directly - Randall > Holmes has produced interesting, though flawed, proofs of > inconsistency of set theory - or indirectly - Edward Nelson seems > serious about finding a contradiction in PA. I'm not sure whether Eduard > Wette is sane or competent - it's rather difficult to tell - and whether > he tried to demonstrate the consistency or inconsistency of PA - > possibly both -, but if we include him that pretty much exhausts the set > of people I know of who have tried to find a contradiction in ZF or a > subtheory of ZF. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus That excludes pretty much everybody who has considered discarding excluded middle, eg, intuitionists. When you say subtheory of ZF, do you mean a theory with: i) more axioms and a smaller universe, ii) less axioms and a larger universe, iii) more axioms and a larger universe, or iv) less axioms and a smaller universe? This is where axioms besides regularity (and infinity) increase comprehension, admit a larger universe, while regularity (and maybe infinity) decrease, restrict comprehension. So, is a subtheory just a subset of the axioms, or the collection of theorems about a smaller universe, or: both and neither and non-archimedean? There is no universe in ZF. The set of sets in ZF is a subset of sets that don't contain themselves: the Russell set. Then, Russell isn't a paradox, it's a theorem, and ZF is inconsistent. Ross === Subject: Re: Cantor's diagonalization argument >> More interesting is that various competent (and probably at least relatively >> sane) people have actually tried to demonstrate the inconsistency of set theory, >> either directly - Randall Holmes has produced interesting, though flawed, >> proofs of inconsistency of set theory - or indirectly - Edward Nelson seems >> serious about finding a contradiction in PA. I'm not sure whether Eduard >> Wette is sane or competent - it's rather difficult to tell - and whether >> he tried to demonstrate the consistency or inconsistency of PA - >> possibly both -, but if we include him that pretty much exhausts the set >> of people I know of who have tried to find a contradiction in ZF or a >> subtheory of ZF. That excludes pretty much everybody who has considered discarding > excluded middle, eg, intuitionists. Do you have in mind some intuitionist who has tried to find a contradiction in set theory? > When you say subtheory of ZF, do you mean a theory with: I mean a theory T such that every axiom of T is a theorem of ZF either directly or through interpretation, e.g. Zermelo set theory, Peano arithmetic, second order arithmetic. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument More interesting is that various competent (and probably at least relatively >> sane) people have actually tried to demonstrate the inconsistency of set theory, >> either directly - Randall Holmes has produced interesting, though flawed, >> proofs of inconsistency of set theory - or indirectly - Edward Nelson seems >> serious about finding a contradiction in PA. I'm not sure whether Eduard >> Wette is sane or competent - it's rather difficult to tell - and whether >> he tried to demonstrate the consistency or inconsistency of PA - >> possibly both -, but if we include him that pretty much exhausts the set >> of people I know of who have tried to find a contradiction in ZF or a >> subtheory of ZF. That excludes pretty much everybody who has considered discarding > excluded middle, eg, intuitionists. Do you have in mind some intuitionist who has tried to find a > contradiction in set theory? When you say subtheory of ZF, do you mean a theory with: I mean a theory T such that every axiom of T is a theorem of ZF either > directly or through interpretation, e.g. Zermelo set theory, Peano > arithmetic, second order arithmetic. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus I'd misinterpreted your definition of subtheory. A subtheory has every axiom T as a theorem of ZF. So, it must be ZF. You can't add more sets, there is no theorem of ZF to support them or they would already be in ZF. You can't remove sets, there are theorems of ZF that they exist. So when you say sub-theory you just mean reinterpretation. By your definition of subtheory T, only all sets in ZF are sets in T. It would seem a subtheory is a superset of the axioms of the theory, where all the theorems of the supertheory must hold true, and a supertheory would thus be a subset of the axioms of the theory, where to maintain consistency some of the theorems of the proper subtheory would be undecideable. A different definition of subtheory might be more intuitively that a subtheory has a subset of the axioms of the theory, in terms of set theory. In ZF, some of the axioms basically describe what _are_ sets, and then regularity goes about saying what of those _are not_ sets. I wonder about the axiom of choice: does it make more sets because it enables quantification over anything deemed a set so those elements can be enumerated by the well-ordering to make more sets, or make less sets because a non-well-orderable entity would not be a set. Are there any sets in ZF not in ZFC? Are there any sets in ZFC not in ZF? Are there both sets in ZF not in ZFC and sets in ZFC not in ZF? That's a rare question. An added axiom might be along the lines of the oft-suggested large cardinal axioms, basically that there would exist large cardinals that don't exist in ZF. They're too big, as for example, but in a different way, element of is too big to be a set. These might be seen to expand the scope of the universe of ZF, except there is no universe in/of ZF. That's not the point, those are examples of axioms that make more sets. An axiom along the lines of that there was no (proper) superset of the infinite set axiomatized into existence by the axiom of infinity is an example of some axiomatization that would reduce the perceived universe of ZF. Then, it might be obviously stated that via comprehension and infinite induction, that would contradict the other axioms. So, is something of that sort a false axiom? Where the subtheory has all the axioms of ZF, to introduce an axiom that reduces or makes less the number of sets that can be constructed, true statements of the theory are not true statements of the subtheory, of the theory. So, is ZF not a subtheory of ZF - Regularity? There is no universe in ZF. There's no model of ZF because there's no maximal ordinal of ZF to be the model. In a set theory with a universal set, people go directly to Russell's paradox and say there is a set of all sets that don't contain themselves thus it contains itself, that the specification is always overriden. There is no set of sets that don't contain themselves in ZF (because of regularity, well-foundedness), and that's all it could ever be. Infinity doesn't need an axiom, particularly not a false one. Proper classes are non-sets. The universe is an example that infinite sets are equivalent. If you accept that, then ZF has been shown inconsistent for years. Ross === Subject: Re: Cantor's diagonalization argument All that it indicates is that it's not utterly trivial to find a > contradiction in set theory. More interesting is that various competent > (and probably at least relatively sane) people have actually tried to > demonstrate the inconsistency of set theory, either directly - Randall > Holmes has produced interesting, though flawed, proofs of > inconsistency of set theory - or indirectly - Edward Nelson seems > serious about finding a contradiction in PA. I'm not sure whether Eduard > Wette is sane or competent - it's rather difficult to tell - and whether > he tried to demonstrate the consistency or inconsistency of PA - > possibly both -, but if we include him that pretty much exhausts the set > of people I know of who have tried to find a contradiction in ZF or a > subtheory of ZF. > Bryan Ford had a proof on fom a while back which purported to show that ZF was inconsistent. I believe it took Solovay to find the flaw. === Subject: Re: Cantor's diagonalization argument > Bryan Ford had a proof on fom a while back which purported to show that > ZF was inconsistent. Right. I was actually thinking of Holmes's proof of inconsistency of PA. > I believe it took Solovay to find the flaw. If I recall correctly it was Randall Holmes who identified the crucial flaw, Solovay having found a number of fixable errors earlier. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument set theory using either: -formal set theory (ZF) or > -informal condsiderations (IC) I'm not saying anything of that kind. We do set theory the way we do > mathematics in general, by producing all sorts of considerations, > conjectures, intuitive guesses, mathematical proofs, devising new > concepts, and what you have. Some part of our basic conception of sets, > and all sorts of other stuff, is captured by various formal theories, > which we might study for certain reasons - as we do in e.g. logic and in > axiomatic set theory - and which to a fuller or lesser extent are taken > as idealised ultimate arbitrators of rigour - e.g. when it is said than > in principle all set theoretical theorems proved in ordinary > mathematics have proofs in ZF, or even that all proofs of theorems of > ordinary mathematics can be formalized in ZF. People often ask questions like do we know that ZF is consistent and > so forth, and think that these have direct epistemological significance > in the philosophy of mathematics. In reality they have only indirect > significance, in that if e.g. ZF were shown inconsistent that would mean > that our mathematical ideas about sets are somehow incoherent or > inconsistent. It is these, our basic mathematical ideas, conceptions, > pictures, that are important, even if we might approach them through > results about formal theories. And when drawing conclusions concerning > them from mathematical results special arguments are always needed to > show how and why the mathematical and logical results are relevant to > understanding the mathematics human mathematicians actually practice, to > the concepts, assumptions and principles they in reality, implicitly or > explicitly, embrace. We also know that a lot of effort has been put into > showing that [ZF] is inconsistent, but to date it has not been shown to be > inconsistent. It's not the case that a lot of effort has been put into showing that > [ZF] is inconsistent. I know but of a few serious attempts at that > direction by anyone sane and competent. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus You'll notice nobody has proven ZF consistent. Assuredly more people have tried to seriously prove ZF consistent than inconsistent. This is in the context where a variety of people have presented reasonings why Goedel's famous incompleteness and some of his meta-theoretical statements are not seen to hold. Those systems aren't necessarily ZF. So, someone who valued consistency and completeness in a logical theory would necessarily have to have a system where basically Goedel's arguments/damnations are not seen to hold. Anyways that's very interesting, Aatu. Well said. There's only one theory with no axioms, it has meaningful theorems, and no Goedelian paradoxes get in the way. That's why it's intellectually satisfying to be post-Cantorian and post-Goedelian, because anything else is wrong, and provably so. Ross === Subject: Re: Cantor's diagonalization argument This is in the context where a variety of people have presented > reasonings why Goedel's famous incompleteness and some of his > meta-theoretical statements are not seen to hold. Those systems aren't > necessarily ZF. Why would you mean by Goedel's famous incompleteness ... [is] not seen to hold? What would be the statement he asserted that you think is incorrect? So, someone who valued consistency and completeness in a logical theory > would necessarily have to have a system where basically Goedel's > arguments/damnations are not seen to hold. One could come up with examples of (in)complete, (in)consistent theories, and value those as one wishes. What does all that have to do with GIT at all, let alone Goedel's damnation? Anyways that's very interesting, Aatu. Well said. There's only one theory with no axioms, it has meaningful theorems, and > no Goedelian paradoxes get in the way. That's why it's intellectually satisfying to be post-Cantorian and > post-Goedelian, because anything else is wrong, and provably so. Ross > -- ----------------------------------------------------- What we call 'I' is just a swinging door which moves when we inhale and exhale. Shunryu Suzuki ---------------------------------------------------- === Subject: Re: Cantor's diagonalization argument reasonings why Goedel's famous incompleteness and some of his > meta-theoretical statements are not seen to hold. Those systems aren't > necessarily ZF. Why would you mean by Goedel's famous incompleteness ... [is] not > seen to hold? What would be the statement he asserted that you > think is incorrect? > So, someone who valued consistency and completeness in a logical theory > would necessarily have to have a system where basically Goedel's > arguments/damnations are not seen to hold. One could come up with examples of (in)complete, (in)consistent > theories, and value those as one wishes. What does all that have > to do with GIT at all, let alone Goedel's damnation? > Anyways that's very interesting, Aatu. Well said. There's only one theory with no axioms, it has meaningful theorems, and > no Goedelian paradoxes get in the way. That's why it's intellectually satisfying to be post-Cantorian and > post-Goedelian, because anything else is wrong, and provably so. Ross -- > ----------------------------------------------------- What we call 'I' is just a swinging door which moves > when we inhale and exhale. > Shunryu Suzuki > ---------------------------------------------------- Basically: i) any theory able to formalize the natural numbers is incomplete, where there are true statements about the objects of the theory unprovable in the theory, ii) a complete theory is inconsistent I expect true theorems of a theory to be provable, as was consensus before Goedel used Cantor's antidiagonal argument to illustrate that no list of theorems contained all the Goedel numbers of those theorems. You've probably heard of the Hilbert program, which is the notion that all true mathematical statements should be formalizable. That has long been a goal of mathematics, although there are questions of the actual and potential in terms of which of all mathematical theorems will be ever expressly encoded, in a human or symbolic language or similar way. Still, the notion is to prove that mathematical statements are true or false, and the necessary conditions to so do so. The goal of foundations of mathematics, well, the goals vary, but a main goal of a foundation of mathematics is to rigorously prove only and all true statements, ie, be consistent, and complete. Where ZF is put forward as a candidate foundation for mathematics, and it can never be consistent nor complete, then it does not seem honest, genuine. It is often said that an inconsistent theory can prove true and false statements, consider instead: an inconsistent theory proves nothing. Then, all proofs of all mathematics of all time are contingent on there being a consistent and complete theory of everything, a framework for those definitions, of objects and their rules/axioms, among all of them. I see that as grandiose, that's what's good about it. Honestly, I don't have enough humility to not derive some satisfaction from saying that. When it comes down to it, what else is there in a mathematical conscience besides honesty? Ross === Subject: Re: Cantor's diagonalization argument > This is in the context where a variety of people have presented > reasonings why Goedel's famous incompleteness and some of his > meta-theoretical statements are not seen to hold. Those systems aren't > necessarily ZF. >> Why would you mean by Goedel's famous incompleteness ... [is] not >> seen to hold? What would be the statement he asserted that you >> think is incorrect? > So, someone who valued consistency and completeness in a logical theory > would necessarily have to have a system where basically Goedel's > arguments/damnations are not seen to hold. >> One could come up with examples of (in)complete, (in)consistent >> theories, and value those as one wishes. What does all that have >> to do with GIT at all, let alone Goedel's damnation? > Anyways that's very interesting, Aatu. Well said. There's only one theory with no axioms, it has meaningful theorems, and > no Goedelian paradoxes get in the way. That's why it's intellectually satisfying to be post-Cantorian and > post-Goedelian, because anything else is wrong, and provably so. Ross Basically: i) any theory able to formalize the natural numbers is incomplete, > where there are true statements about the objects of the theory > unprovable in the theory, Did Goedel really say this? ii) a complete theory is inconsistent Presburger arithmetic is both complete and consistent! I still don't see Goedel's damnation so far! I expect true theorems of a theory to be provable, as was consensus I thought theorems are always provable, by definition, whether they're are true nor not, whether before GIT or not. not so? > before Goedel used Cantor's antidiagonal argument to illustrate that no > list of theorems contained all the Goedel numbers of those theorems. You've probably heard of the Hilbert program, which is the notion that > all true mathematical statements should be formalizable. That has long > been a goal of mathematics, although there are questions of the actual > and potential in terms of which of all mathematical theorems will be > ever expressly encoded, in a human or symbolic language or similar way. > Still, the notion is to prove that mathematical statements are true or > false, and the necessary conditions to so do so. The goal of foundations of mathematics, well, the goals vary, but a > main goal of a foundation of mathematics is to rigorously prove only > and all true statements, ie, be consistent, and complete. Where ZF is > put forward as a candidate foundation for mathematics, and it can never > be consistent nor complete, then it does not seem honest, genuine. Have you heard of the phrase one-to-many? Imho, so long as a reasoning framework is based on finite knowledge and finite symbolism, contemplating infinity, especially if the infinity has a non-recursive portion, then the framework would be on the one side and be very limited. So if the goal of a foundation of mathematics is to rigorously prove only and all true statements, then that goal doesn't look attainable. In fact, imho, keep not recognizing the such limitation of the framework is very problematic, if not outright dis-genuine. > It is often said that an inconsistent theory can prove true and false > statements, consider instead: an inconsistent theory proves nothing. Which means you have to change the rules of inference. Would you care to be specific as what the changes be? > Then, all proofs of all mathematics of all time are contingent on there > being a consistent and complete theory of everything, a framework for > those definitions, of objects and their rules/axioms, among all of > them. I see that as grandiose, that's what's good about it. Honestly, I > don't have enough humility to not derive some satisfaction from saying > that. When it comes down to it, what else is there in a mathematical > conscience besides honesty? Ross > -- ----------------------------------------------------- What we call 'I' is just a swinging door which moves when we inhale and exhale. Shunryu Suzuki ---------------------------------------------------- === Subject: Re: Cantor's diagonalization argument reasonings why Goedel's famous incompleteness and some of his > meta-theoretical statements are not seen to hold. Those systems aren't > necessarily ZF. >> Why would you mean by Goedel's famous incompleteness ... [is] not >> seen to hold? What would be the statement he asserted that you >> think is incorrect? > So, someone who valued consistency and completeness in a logical theory > would necessarily have to have a system where basically Goedel's > arguments/damnations are not seen to hold. >> One could come up with examples of (in)complete, (in)consistent >> theories, and value those as one wishes. What does all that have >> to do with GIT at all, let alone Goedel's damnation? > Anyways that's very interesting, Aatu. Well said. There's only one theory with no axioms, it has meaningful theorems, and > no Goedelian paradoxes get in the way. That's why it's intellectually satisfying to be post-Cantorian and > post-Goedelian, because anything else is wrong, and provably so. Ross Basically: i) any theory able to formalize the natural numbers is incomplete, > where there are true statements about the objects of the theory > unprovable in the theory, Did Goedel really say this? > ii) a complete theory is inconsistent Presburger arithmetic is both complete and consistent! I still don't see Goedel's damnation so far! > I expect true theorems of a theory to be provable, as was consensus I thought theorems are always provable, by definition, whether they're > are true nor not, whether before GIT or not. not so? before Goedel used Cantor's antidiagonal argument to illustrate that no > list of theorems contained all the Goedel numbers of those theorems. You've probably heard of the Hilbert program, which is the notion that > all true mathematical statements should be formalizable. That has long > been a goal of mathematics, although there are questions of the actual > and potential in terms of which of all mathematical theorems will be > ever expressly encoded, in a human or symbolic language or similar way. > Still, the notion is to prove that mathematical statements are true or > false, and the necessary conditions to so do so. The goal of foundations of mathematics, well, the goals vary, but a > main goal of a foundation of mathematics is to rigorously prove only > and all true statements, ie, be consistent, and complete. Where ZF is > put forward as a candidate foundation for mathematics, and it can never > be consistent nor complete, then it does not seem honest, genuine. Have you heard of the phrase one-to-many? Imho, so long as a reasoning > framework is based on finite knowledge and finite symbolism, > contemplating infinity, especially if the infinity has a non-recursive > portion, then the framework would be on the one side and be very > limited. So if the goal of a foundation of mathematics is to rigorously > prove only and all true statements, then that goal doesn't look > attainable. In fact, imho, keep not recognizing the such limitation > of the framework is very problematic, if not outright dis-genuine. It is often said that an inconsistent theory can prove true and false > statements, consider instead: an inconsistent theory proves nothing. Which means you have to change the rules of inference. Would you care to > be specific as what the changes be? Then, all proofs of all mathematics of all time are contingent on there > being a consistent and complete theory of everything, a framework for > those definitions, of objects and their rules/axioms, among all of > them. I see that as grandiose, that's what's good about it. Honestly, I > don't have enough humility to not derive some satisfaction from saying > that. When it comes down to it, what else is there in a mathematical > conscience besides honesty? Ross -- > ----------------------------------------------------- What we call 'I' is just a swinging door which moves > when we inhale and exhale. > Shunryu Suzuki > ---------------------------------------------------- I should perhaps better have said I expect facts of a theory to be theorems, but where theorem actually means true statement, I still expect theorems of a theory to be theorems of that theory, if there are true statements about the objects of the theory unprovable in the theory, they are provable by some other, richer theory, coconsistent with that one. Then, of all theories where all those are true, as each would be having more and more, take their union. Finite proofs in PA, Peano Arithmetic, are finite proofs in PA, Presburger Arithmetic. If an inconsistent theory contains every statement and its negation, encoded their sum is zero. Ross === Subject: Re: Cantor's diagonalization argument > >> Sure. But if all we assume about ZF is that it is consistent, and, as >> you earlier suggested, all we do is show that this or that follows from >> the axioms of ZF, why should ZF proving that it is inconsistent trouble >> us at all? After all, that a theory proves of itself that it's >> inconsistent is no reason to think it actually is inconsistent. Oh? Yes. For example that PA + PA is inconsistent proves PA + PA is inconsistent is inconsistent is no reason to think PA + PA is inconsistent is inconsistent, which it, of course, isn't. All this is just belaboring the simple point that consistency is a trifling soundness condition, and in practice we assume much more of the theories formalizing what we take to be our fundamental mathematical principles. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument > P.S. If set theory proves set theory is inconsistent, then does that > entail that theory proves things that ain't true in the standard model > of first order PA? Yes. Although I would prefer to phrase that simply as: if ZF proves it is inconsistent, then ZF proves false arithmetical statements, since it either is inconsistent or falsely proves that it is. Often dragging in PA and models just gives a false impression of technicality to what are perfectly simple observations, though of course there is nothing technically wrong in it. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument > Even though I'm shakey on some of the techncials here, to the extent > that I do follow you, I find your argument fascinating and cogent. > However, I wonder if it is an overstatement to say equally > unacceptable and just as troubling. I don't see that it would be > unreasonable to be dissatisfied with ZF proving that ZF is inconsistent > but not find that to be every bit as unsatisfactory as ZF being > inconsistent. One can certainly imagine some purposes for which a consistent but 1-inconsistent ZF might suffice, but for all practical purposes finding that ZF is 1-inconsistent would be just as disastrous as finding that it proves a contradiction. Both would mean that our conception of the world of sets is utterly bonkers, even if a 1-inconsistent ZF might have some technical interest and all sorts of fascinating logical properties to the delight of the logicians. Of course, as Virgil said, the reaction to finding either that ZF is 1-inconsistent - or proves a false Sigma_100 sentence, say - or that it is outright inconsistent would depend on the exact nature of the contradiction, or 1-inconsistency. It could be that it is the replacement axiom that is at fault - in which case most of ordinary mathematics would survive -, or perhaps all talk of infinite sets is incoherent as some here would have it - in which case a much more drastic re-evaluation would be called for. All this is speculation, obviously. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument Aatu Koskensilta says... >> Even though I'm shakey on some of the techncials here, to the extent >> that I do follow you, I find your argument fascinating and cogent. >> However, I wonder if it is an overstatement to say equally >> unacceptable and just as troubling. I don't see that it would be >> unreasonable to be dissatisfied with ZF proving that ZF is inconsistent >> but not find that to be every bit as unsatisfactory as ZF being >> inconsistent. One can certainly imagine some purposes for which a consistent but >1-inconsistent ZF might suffice, but for all practical purposes finding >that ZF is 1-inconsistent would be just as disastrous as finding that it >proves a contradiction. Both would mean that our conception of the world >of sets is utterly bonkers, even if a 1-inconsistent ZF might have some >technical interest and all sorts of fascinating logical properties to >the delight of the logicians. Even though it is technically possible to have a theory that is consistent, but not 1-consistent (I thought the phrase was omega-consistent--what does the 1 mean?) it's hard to imagine a naturally occurring theory such as that. It's easy to slip up and make a theory inconsistent, such as set theory with unrestricted comprehension or the origin Martin-Lof type theory. But I haven't heard of a theory that someone might be interested in for other reasons that just *happens* to be 1-inconsistent. Another point is that if you actually *know* that a theory is 1-inconsistent, and you know a particular false theorem of the form exists x, natural(x) and Phi(x), then you could try to repair the theory by adding a new unary predicate standard(x), together with axioms along the lines of standard(0), forall x, standard(x) -> standard(x+1). Presumably you could extend your theory to one that *was* 1-consistent in the sense that it proved no false statements of the form exists x, standard(x) and Phi(x). -- Daryl McCullough Ithaca, NY === Subject: Re: Cantor's diagonalization argument > Even though it is technically possible to have a theory that > is consistent, but not 1-consistent (I thought the phrase was > omega-consistent--what does the 1 mean?) it's hard to imagine > a naturally occurring theory such as that. No one certainly knows any such theory. As to 1-consistency, it's a condition related to omega-consistency but technically more convenient in certain contexts. A theory T is 1-consistent if for no p.r. formula P(x) both T |- P(n) for every numeral n and T |- Ex~P(x). For sufficiently strong theories this is equivalent to Sigma-1-soundness (or, equivalently, to consistency of T + all true Pi-1 sentences). 1-consistency thus lies naturally in the hierarchy (consistency or Pi-1 soundness,) Sigma-1-soundness, Sigma-2-soundness, ... of stronger and stronger soundness conditions. Omega-consistency of T is equivalent, again for sufficiently strong theories, to consistency of T + all true Pi-2 sentences + uniform reflection for T. > Another point is that if you actually *know* that a theory is > 1-inconsistent, and you know a particular false theorem > of the form exists x, natural(x) and Phi(x), then you could try to repair > the theory by adding a new unary predicate standard(x), together > with axioms along the lines of standard(0), forall x, standard(x) -> standard(x+1). > Presumably you could extend your theory to one that *was* 1-consistent in the sense > that it proved no false statements of the form > exists x, standard(x) and Phi(x). I'm not sure exactly what extension you have in mind. Unless you extend induction in some suitable sense to cover formulae involving the predicate standard you will not be able to prove anything particularly interesting about standard numbers. In any case, it seems you then end up embedding some other theory to an extension of the 1-inconsistent theory, which doesn't sound particularly enlightening in itself. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument is consistent, but not 1-consistent (I thought the phrase was > omega-consistent--what does the 1 mean?) 1-inconsistency means you don't get (a name for) the one thing that has to witness the asserted existential. It's basically the contrapositive of omega-inconsistency. Omega-inconsistency was when you got P(0),P(1),P(...everything in omega) but DIDN'T get Ax[P(x)]. You therefore DO (at least possibly) get Ex[~P(x)], but DON'T get any one named n for which ~P(n). The n is out there in the domain of the model but it's not in the signature of the language. In provability terms, the n is non-standard; there is no natural numeral for it; it's not any of the numbers we were TRYING to quantify over; it's not [denoted by, under ANY interpretation] any term in the first-order language of the theory. > it's hard to imagine > a naturally occurring theory such as that. It is not HARD to imagine PA+~Con(PA). Whether it is NATURAL for PA +~Con(PA) to occur is a tougher question; it has certainly been occurring OFTEN in people's papers since Godel, whether naturally or otherwise. Natural is of course almost a bad pun there since we are speaking precisely about models that must contain numbers that are NOT natural. > It's easy to slip up > and make a theory inconsistent, such as set theory with unrestricted > comprehension or the original Martin-Lof type theory. But I haven't > heard of a theory that someone might be interested in for other > reasons that just *happens* to be 1-inconsistent. You have heard of PA+~Con(PA), but I suppose I can agree that people are interested in that primarily because it is neither omega- nor 1- consistent. This form of inconsistency is hard to *sustain*, in any case. Once you have discovered that your theory is 1-inconsistent, the arguably NATURAL thing to do is just ADD TERMS TO THE LANGUAGE for the domain elements that didn't already have them. > Another point is that if you actually *know* that a theory is > 1-inconsistent, and you know a particular false theorem > of the form exists x, natural(x) and Phi(x), then you could try to repair > the theory by adding a new unary predicate standard(x), No, you couldn't; that is GUARANTEED not to work. In the language of PA, standard(x) means intuitively that x is 0, or s0, or ss0, or sss0, etc. This is NOT first-orderizable. The problem arises from the lack of a name for the 1 thing that has to exist; the solution is just to name it. If you are constrained not to perturb the signature of the language then that is not a viable solution either. But in my subfield, we skolemized-away existentials ROUTINELY. > Presumably you could > extend your theory to one that *was* 1-consistent in the sense > that it proved no false statements of the form > exists x, standard(x) and Phi(x). The issue is not about proving or not proving false statements. Unless you have a standard model, there is no such thing as an inherently true or false statement. If we are talking about standard classical first-order theories, then this deprecation of ANY model as non-standard is not really defensible. PA is sort of a special case because one of its models occurs as a sub-model of ALL the others (that DOES entitle it to be standard). But I doubt that theories in general are like that. === Subject: Re: Cantor's diagonalization argument > 1-inconsistency means you don't get (a name for) the one thing > that has to witness the asserted existential. It's basically the > contrapositive of omega-inconsistency. No. > Omega-inconsistency was when you got P(0),P(1),P(...everything in > omega) but DIDN'T get Ax[P(x)]. No, that's omega-incompleteness. > It is not HARD to imagine PA+~Con(PA). > Whether it is NATURAL for PA +~Con(PA) to occur > is a tougher question; it has certainly been occurring OFTEN > in people's papers since Godel, whether naturally or otherwise. PA+~Con(PA) has only technical interest. It certainly is not a naturally occurring theory in the sense that anyone would come up with it as a formalization of some piece of our mathematical knowledge. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's diagonalization argument > Another point is that if you actually *know* that a theory is > 1-inconsistent, and you know a particular false theorem > of the form exists x, natural(x) and Phi(x), then you could try to repair > the theory by adding a new unary predicate standard(x), together > with axioms along the lines of standard(0), > forall x, standard(x) -> standard(x+1). Presumably you could > extend your theory to one that *was* 1-consistent in the sense > that it proved no false statements of the form > exists x, standard(x) and Phi(x). I guess you want 'standard' to be the *least* predicate containing 0 and closed under successor. But you didn't show how one does that, so it appears to me that you haven't solved the 1-inconsistency[1]. Footnotes: [1] I'm not sure if Daryl's definition of 1-inconsistent is the same as Aatu's. Like Daryl, I'm not sure what Aatu has in mind. -- Reality is that I've worked all over the United States in so many different jobs that I jokingly call myself The Pretender. I've been a bartender, worked at Equifax, sold new cars for Honda, and that's from about six months of my life. -- James S. Harris: a great Pretender. === Subject: Re: Cantor's diagonalization argument <45509568.3070309@et.uni-magdeburg.de> said: > Actually, if one reads carefully, one cannot have a really universal > quantifier in ZF but only quantify over the members of a set already > known to exist on other grounds. Why not? Every set fails to contain itself as a member. How does that > fail to quantify over all sets? What do you mean every set? Do you mean each of all? Is that: for each set, it does not contain itself (in ZF)? How does that meaningfully quantify? The only property you could ever > say holds is that. Basically you're saying that ZF does not contradict > its axioms, that sets in ZF are sets in ZF. There is no model of ZF because there's no maximal ordinal in ZF. In consideration of the cumulative hierarchy, which in basically a > constructivist manner is said to have within it a large enough set to > model any system of interest, there is no it, there is no cumulative > hierarchy, in ZF. I'm not talking about a model of the finite sets which you might claim > exists because of the axiom of infinity, I'm talking about a model of > ZF. Your universe is the collection of all sets that don't contain > themselves? 1) That doesn't exist in ZF so you'd not really be using > ZF but some other, richer system, were ZF not to have false axioms, and > 2) that collection is the Russell set, of Russell's paradox. You can't even say something's true for all sets in ZF, because via > set-building that would be a set, and a universal set, and, where there > is such a thing in ZF: ZF is inconsistent. > Does that makes sense to you? Do you understand those words in those sentences in that presentation? I'm not asking if you disagree because it contradicts your professed statements, I wonder if you get it. Ross === Subject: Re: irreducible polynomial x^{p^n} - a over a field of characteristic p doesn't have a p'th root in K. Prove that the polynomial f(x) = x^{p^n} - a is irreducible over K. > >Here is what i've done. Let b such that b^p^n = a in the splitting >field of f. Then f(x) = (x - b)^p^n. Suppose f isn't irreducible and let g(x) be it's irreducible factor. >Then g(x) = (x - b)^k for some 1<= k <= p^n. (Ooo! You're using k for more than one thing!) Okay... In fact, you can say more. Factoring f in k into irreducibles > would look like f(x) = (x-b)^{r1} * (x-b)^{r2} * ... * (x-b)^{rm} Clearly, all the ri must be equal, since we are assuming the factors > (x-b)^{ri} are irreducible (if ri < rj, then (x-b)^{ri} | (x-b)^{rj}, > contradicting the irreducibility of (x-b)^{rj}. So in fact what you have is that f(x) = g(x)^r for some r, with > rs=p^n. So r is more than just less than or equal to p^n, it is in > fact a divisor of p^n. So you really have f(x) = (x-b)^(p^n) = ( (x-b)^{p^m} )^{p^s} where 1<= m,s < n, m+s = n, and g(x) = (x-b)^{p^m}. This is where you got to below, by using the characterization of > nonseparable polynomials as polynomials in x^p. But as you can see, it > is too much work, and you can get it from first principles. >But now, since g isn't separable Well, I know g is not separable, but I deduced it from the above. It > was not immediately obvious to me, at any rate. , g(x) = h(x^p) for some h in K[x] >and so on ... let's choose e in >such a way that g(x) = h(x^p^e) but g(x) isn't a polynomial of >x^p^{e+1}. in x^{p^{e+1}}. Okay. Let h(y) = prod_{i=1}^m ( y - c_i) and (d_i)^p^e = c_i then g(x) = >prod{i=1}^m (x^e - c_i) = prod{i=1}^m (x - d_i)^p^e, thus all d_i = b >and if m>1 h(y) has repeated roots and thus is a polynomial of x^p - >contradiction with our choice of e, hence m = 1. This means that g(x) >= (x - b)^p^e for 1<=e < n. Way too much work to deduce this, as you can see above. But our g in K[x], thus b^p^e in K and >b^p^{e+1} = (b^p^e)^p in K and so on ... b^p^{n-1} in K, but >(b^p^{n-1})^p = a - contradiction with that a has no p^n'th roots of >unity. Is that right? Or there may be other solutions. > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin > magidin-at-member-ams-org === Subject: A typical equation Solve the equation sin (sin X)=(sin X)^2 ?? === Subject: Re: A typical equation > Solve the equation sin (sin X)=(sin X)^2 ?? > x = n.pi, n in Z === Subject: Re: A typical equation William Elliot ?rta: Solve the equation sin (sin X)=(sin X)^2 ?? x = n.pi, n in Z you mean there is only solution if sin(x)=0? === Subject: Re: A typical equation > Solve the equation sin (sin X)=(sin X)^2 ?? x = n.pi, n in Z The above are some solutions. In addition, x_{k_1} ~= 1.069013044 + k*Pi x_{k_2} ~= Pi - x_{k_1} + k*Pi, k in Z. -- Ioannis ------- The best way to predict reality, is to know exactly what you DON'T want. === Subject: Re: A typical equation vikraman.choudhury@gmail.com ?rta: > Solve the equation sin (sin X)=(sin X)^2 ?? let k = sin(x) then: sin(k) = k^2 then iterate k and approximate the x using arcus sinus === Subject: Sum this! What is the sum of all real numbers from 1 to 10? What is the sum of all rational numbers from 1 to 10? What is the sum of all irrational numbers from 1 to 10? === Subject: Re: Sum this! In sci.math, vikraman.choudhury@gmail.com on 11 Nov 2006 03:22:13 -0800 > What is the sum of all real numbers from 1 to 10? What is the sum of all rational numbers from 1 to 10? What is the sum of all irrational numbers from 1 to 10? > Others have already pointed out that these are infinity (there is some quibble about whether the infinity is uncountable or countable, but that's a detail that's probably fairly useless at this point). If you prefer I can stipulate integration of all three of these instead, which is probably more interesting. Using a variant of Lebesgue integration one can evaluate the following almost trivially: integral(x over real in [1,10]) 1 dx = 9 integral(x over rational in [1,10]) 1 dx = 0 integral(x over irrational in [1,10]) 1 dx = 9 since the rational numbers are a measure 0 set over the continuum. All these are, of course, are the measures of their respective sets. Other integrals are possible. The ones you're probably most interested in are integral(x over real in [1,10]) x dx = 99/2 integral(x over rational in [1,10]) x dx = 0 integral(x over irrational in [1,10]) x dx = 99/2 And of course the sum of all *integers* from 1 to 10 is 55. A reason this sum is greater than the integral is because the integrals stop at 10 whereas the sum of integers includes 10 and basically stops at 11. integral(x over real in [1,11]) x dx = 60 One can also get into some interesting debate about finding the sum of all real numbers from, say, -10 to +10. The integral, of course, is zero: integral(x over real in [-10,+10]) x dx = 0 The best I can do regarding the sum is treat it as undefined, as it's a form of infinity - infinity, which has no rigorous value unless one looks at limits. -- #191, ewill3@earthlink.net Useless C++ Programming Idea #12398234: void f(char *p) {char *q = strdup(p); strcpy(p,q);} -- === Subject: Re: Sum this! > In sci.math, vikraman.choudhury@gmail.com > >>What is the sum of all real numbers from 1 to 10? >>What is the sum of all rational numbers from 1 to 10? >>What is the sum of all irrational numbers from 1 to 10? > Others have already pointed out that these are infinity > (there is some quibble about whether the infinity is > uncountable or countable, but that's a detail that's > probably fairly useless at this point). If you prefer I > can stipulate integration of all three of these instead, > which is probably more interesting. Using a variant of Lebesgue integration one can evaluate > the following almost trivially: integral(x over real in [1,10]) 1 dx = 9 > integral(x over rational in [1,10]) 1 dx = 0 > integral(x over irrational in [1,10]) 1 dx = 9 since the rational numbers are a measure 0 set over the continuum. > All these are, of course, are the measures of their respective sets. Other integrals are possible. The ones you're probably most interested > in are integral(x over real in [1,10]) x dx = 99/2 > integral(x over rational in [1,10]) x dx = 0 > integral(x over irrational in [1,10]) x dx = 99/2 And of course the sum of all *integers* from 1 to 10 > is 55. A reason this sum is greater than the integral > is because the integrals stop at 10 whereas the sum of > integers includes 10 and basically stops at 11. integral(x over real in [1,11]) x dx = 60 One can also get into some interesting debate about > finding the sum of all real numbers from, say, -10 to +10. > The integral, of course, is zero: integral(x over real in [-10,+10]) x dx = 0 The best I can do regarding the sum is treat it as undefined, > as it's a form of infinity - infinity, which has no rigorous > value unless one looks at limits. > When he is asking for the sum, he is really asking for the integral with respect to the counting measure. This fits exactly with your last observation. === Subject: Re: Sum this! > What is the sum of all real numbers from 1 to 10? What is the sum of all rational numbers from 1 to 10? What is the sum of all irrational numbers from 1 to 10? You didn't specify the *series* you want summed. However, the natural way of defining an unordered sum would give 'infinity' for each of these. This follows easily from the fact that each set contains infinitely many numbers greater than 1 and no numbers less than 0. === Subject: Re: Sum this! > What is the sum of all real numbers from 1 to 10? What is the sum of all rational numbers from 1 to 10? What is the sum of all irrational numbers from 1 to 10? > The answer to all three questions is infinity. I disagree with replicator's reply that they are different types of infinity. The question you are asking is not an issue of cardinality of sets, but rather how to define infinite sums. To say that the sums are infinity is meaning that the sum of any finite subcollection of the summands can be made arbitrarily large. Or to put it another way, this infinity has meaning as an extended real number. === Subject: Re: Sum this! vikraman.choudhury@gmail.com ?rta: > What is the sum of all real numbers from 1 to 10? > infinite type B > What is the sum of all rational numbers from 1 to 10? infinite type A > What is the sum of all irrational numbers from 1 to 10? infinite type B type A = number of all rational numbers type B = number of all irrational numbers === Subject: Re: Sum this! > vikraman.choudh...@gmail.com .92rta:> What is the sum of all real numbers from 1 to 10?infinite type B What is the sum of all rational numbers from 1 to 10?infinite type A What is the sum of all irrational numbers from 1 to 10?infinite type B type A = number of all rational numbers > type B = number of all irrational numbers I need a clear explanation man! === Subject: Re: analysis with series! mina_world a .8ecrit : > hello sir~ sum{n=1 to 00} [(2n)!*{(1/4)^n}] / (n!)^2 i want to show that this series diverges. > ratio test is impossible.(result is 1) > maybe, i need your transformation with inequality. so, i need your advice. It is not a proof but just a nice formula... Let's define f by : f(x)= sum_{n>=0} binomial(2n,n) x^n you want f(1/4)-1 but f(x) is the expansion of 1/sqrt(1-4x) so... (http://mathworld.wolfram.com/CentralBinomialCoefficient.html) hoping it helped, Raymond === Subject: Re: analysis with series! Nntp-Posting-Host: apps.cwi.nl >mina_world a .8ecrit : >> hello sir~ >> >> sum{n=1 to 00} [(2n)!*{(1/4)^n}] / (n!)^2 >> >> i want to show that this series diverges. >> ratio test is impossible.(result is 1) >> maybe, i need your transformation with inequality. >> >> so, i need your advice. >> >> > It is not a proof but just a nice formula... Let's define f by : f(x)= sum_{n>=0} binomial(2n,n) x^n you want f(1/4)-1 but f(x) is the expansion of 1/sqrt(1-4x) so... (http://mathworld.wolfram.com/CentralBinomialCoefficient.html) > hoping it helped, > Raymond The OP mentioned the ratio test. Let a_n = [(2n)!*{(1/4)^n}] / (n!)^2 for n >= 0. All a_n are positive. We check that a_(n+1) (2n+1)(2n+2) 2n + 1 --------- = ------------ = -------- = 1 - 1/(2n+2). a_n 4(n+1)^2 2n + 2 We observe a_{n+1} / a_n < 1, so a_{n+1} < a_n and the sequence {a_n} is decreasing. How fast does it decrease? If b_n = a_n * n^k where the constant k remains unspecified, then b_{n+1} a_{n+1} ------- = -------- * (1 + 1/n)^k b_n a_n is about (1 - 1/2n)*(1 + k/n) for large n. Try k = 1/2 so the -1/2n and k/n terms cancel when we expand the product. Then we find b_n = a_n * sqrt(n) b_{n+1} / b_n = [ (2n + 1) / (2n+2) ] * sqrt(1 + 1/n) = (2n + 1) / (2*sqrt(n*(n+1)) The right side is larger than 1 (why)? So, while {a_n} was decreasing, we find {b_n} is increasing. That is b_1 < b_2 < b_3 < ... a_1 * sqrt(1) < a_2*sqrt(2) < a_3*sqrt(3) * ... If c = a_1 * sqrt(1), then a_n > c /sqrt(n) for all n > 1, What does this tell you about convergence or divergence of the original series? -- 44 months after Japan attacked Pearl Harbor, Japan surrendered. 43 months after US attacked Iraq, it's time for the US to surrender. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: get the collision point of a circle and a line <6737695.1162167415897.JavaMail.jakarta@nitrogen.mathforum.org> <4545ca66$0$8013$5a62ac22@per-qv1-newsreader-01.iinet.net.au> dl skrev: > Forgotten a point 2.5 (but it should be evident). Intersections in the > past should be discarded too (that is, if the circumference is going > away from the line). I'll post a link to my project here when im done. === Subject: Re: a question from algebra!!! thank you for your attention! i don't want to trick anyone for sure this question was for my mid-test and i tryed to solve it i myself suppose that (m,n)=1 and i proved [m,n]=o(ab) but for the second part (Suppose that G is a group and there is g belongs to G such that o(g)=a and for all x belongs to G o(x).b2 a. prove that x» =e) i think the question is wrong and i told it to my teacher . because:(for all x belongs to G o(x) can devides o(g) or not. if it does, x» =e ,and if it doesnot : there are p,r belong to Z(integer numbers) such that a=o(x)p+r and 0 i lost my eyes!!!!! this question is for abelian groups and the first part of the question > is(suppose > that a,b belongs to G and o(a)=n,o(b)=m. prove that o(ab)=[m,n] ) It has already been pointed out to you in another thread that this is > simply false (take a,b non-identity elements which are inverses of each > other). You need additional conditions on a,b,m,n. Are you *sure* you > read the problem correctly. Perhaps it is part of a series of problems > with some assumptions stated at the beginning. Suppose that G is a group and there is g belongs to G such that o(g)=a > and for all x belongs to G o(x).b2 a. prove that x» =e It doesn't help you to stipulate that a and b are not inverses, even in > the case of cyclic groups. In Z 12, o(2) = 6, o(6) = 2, but o(2+6) = > o(8) = 3 which isn't [2,6] = 6. It is almost trivial to show that o(ab) > *divides* [a,b]. Maybe that is what you are looking for. Or ... maybe you are being sly and the excercise said Prove or give a > counterexample to ... in which case you've tricked us into doing your > homework. -John Coleman === Subject: Re: a question from algebra!!! > thank you for your attention! > i don't want to trick anyone for sure > this question was for my mid-test > and i tryed to solve it > i myself suppose that (m,n)=1 and i proved [m,n]=o(ab) > but for the second part (Suppose that G is a group and there is g > belongs to G such that o(g)=a and for all x belongs to G o(x).b2 a. > prove that x» =e) i think the question is wrong and i told it to my > teacher . > because:(for all x belongs to G o(x) can devides o(g) or not. > if it does, x» =e ,and if it doesnot : > there are p,r belong to Z(integer numbers) such that a=o(x)p+r and > 0 a, in the first part is not equal to the a, in the second part > what is your opinion? > merci i lost my eyes!!!!! this question is for abelian groups and the first part of the question > is(suppose > that a,b belongs to G and o(a)=n,o(b)=m. prove that o(ab)=[m,n] ) It has already been pointed out to you in another thread that this is > simply false (take a,b non-identity elements which are inverses of each > other). You need additional conditions on a,b,m,n. Are you *sure* you > read the problem correctly. Perhaps it is part of a series of problems > with some assumptions stated at the beginning. Suppose that G is a group and there is g belongs to G such that o(g)=a > and for all x belongs to G o(x).b2 a. prove that x» =e It doesn't help you to stipulate that a and b are not inverses, even in > the case of cyclic groups. In Z 12, o(2) = 6, o(6) = 2, but o(2+6) = > o(8) = 3 which isn't [2,6] = 6. It is almost trivial to show that o(ab) > *divides* [a,b]. Maybe that is what you are looking for. Or ... maybe you are being sly and the excercise said Prove or give a > counterexample to ... in which case you've tricked us into doing your > homework. -John Coleman === Subject: Re: a question from algebra!!! > thank you for your attention! > i don't want to trick anyone for sure > this question was for my mid-test > and i tryed to solve it > i myself suppose that (m,n)=1 and i proved [m,n]=o(ab) > but for the second part (Suppose that G is a group and there is g > belongs to G such that o(g)=a and for all x belongs to G o(x).b2 a. > prove that x» =e) i think the question is wrong and i told it to my > teacher . > because:(for all x belongs to G o(x) can devides o(g) or not. > if it does, x» =e ,and if it doesnot : > there are p,r belong to Z(integer numbers) such that a=o(x)p+r and > 0 a, in the first part is not equal to the a, in the second part > what is your opinion? > merci i lost my eyes!!!!! this question is for abelian groups and the first part of the question > is(suppose > that a,b belongs to G and o(a)=n,o(b)=m. prove that o(ab)=[m,n] ) It has already been pointed out to you in another thread that this is > simply false (take a,b non-identity elements which are inverses of each > other). You need additional conditions on a,b,m,n. Are you *sure* you > read the problem correctly. Perhaps it is part of a series of problems > with some assumptions stated at the beginning. Suppose that G is a group and there is g belongs to G such that o(g)=a > and for all x belongs to G o(x).b2 a. prove that x» =e It doesn't help you to stipulate that a and b are not inverses, even in > the case of cyclic groups. In Z 12, o(2) = 6, o(6) = 2, but o(2+6) = > o(8) = 3 which isn't [2,6] = 6. It is almost trivial to show that o(ab) > *divides* [a,b]. Maybe that is what you are looking for. Or ... maybe you are being sly and the excercise said Prove or give a > counterexample to ... in which case you've tricked us into doing your > homework. -John Coleman === Subject: Re: counter example in analysis On Fri, 10 Nov 2006 14:38:26 -0500, Jesse F. Hughes > You don't seem to have noticed one of my posts. >> You said something the other day that I didn't believe. >> And I don't believe that _you_ believed it when you >> said it. [...] > People all over the world are anxiously awaiting your reply. Er, speaking of lies.... Ok, you caught me. ************************ === Subject: Re: counter example in analysis > How do we educate young people? What do you mean? -- Marcus === Subject: Re: counter example in analysis > How do we educate young people? What do you mean? EB is terrified that young people will learn things EB does not want known. More power to those young people! === Subject: Re: counter example in analysis Not incoherent but not adherent to something most likely wrong. What is likely wrong and what is your measure of likely? You have not made an iota of sense since you started posting. Bob Kolker === Subject: Cardinality without choice x is a Z-cardinal <-> x is an equivalence class of transitive sets , under equivalence relationbijection. Z-card(z) = { x| x is a transitive set / x has bijection to z } 1)Every set z has one and only one Z-card(z). ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y Proof: for a proof by negation Let EyEzAz y=Z-card(z) , x=Z-card(z) x=/=y Let a e y , b e x By definition of Z-card(z), it leads that a is bijective to z, and b is bijective to z. Accordingly a is bijective to b. But x=/=y if and only if every member of x is not bijective to every member of y ( see the definition of Z-card ), accordingly a is not bijective to b . which is contradictive. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. There is another way of proving this . for the same a,b defined above. x is the set of all transitive sets bijectable to z. Every a in y is a transitive set bijective to z. Then Aa:aey -> aex Then y is a subset of x. y is the set of all transitive sets bijectable to z. Every b in x is a transitive set bijective to z Then Ab:bex ->bey Then x is a subset of y. x is a subset of y / y is a subset of x -> x=y which contradicts y=/=x. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. Accordingly : ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y . 2) Z-card (x) = Z-card(y) <-> x is 1-1 with y. for a proof by negation , let ~ x is 1-1 with y. ~x is 1-1 with y <-> x is not bijective to y Let peZ-card(x) Since Z-card(x) = Z-card(y) Then p is bijective to x and p is bijective to y. therefore x is bijective to y. which contradicts ~x is 1-1 with y. Accordingly x is 1-1 with y. 3) Avery z-card(x) is a set. z-card(x) is the intersection of the set of all transitive sets ,T ,and the set of all sets bijective to x,Y. z-card(x) = T.Y Now we know for sure that Y is a proper class( when x=/={} ), and T is a proper class. There is no proof that T.Y is a proper class. Then T.Y is a set. so every z-card(x) is a set. I have an intuitive feeling of what a proper class is, I call it a flooding set, since every proper class in T.Y is subclassing the other, then there is no flooding. ( perhaps this means that T.Y is well founded?) 3) is till now not proved. the above is of course not a prove but a plan of a proof that T.Y is a set. The beauty of this definition of cardinality of a set, is that it doesn't require choice to define the cardinality of the power set of omega and powers of powers of omega, and also cardinality is a set, unlike frege's cardinalities which are not sets. Zuhair === Subject: Re: Cardinality without choice x is a Z-cardinal <-> x is an equivalence class of transitive sets , > under equivalence relationbijection. Z-card(z) = { x| x is a transitive set / x has bijection to z } 1)Every set z has one and only one Z-card(z). ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y Proof: for a proof by negation Let EyEzAz y=Z-card(z) , x=Z-card(z) x=/=y Let a e y , b e x By definition of Z-card(z), it leads that a is bijective to z, and b is > bijective to z. Accordingly a is bijective to b. But x=/=y if and only if every member of x is not bijective to every > member of y ( see the definition of Z-card ), accordingly a is not bijective to b . which is contradictive. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. There is another way of proving this . for the same a,b defined above. x is the set of all transitive sets bijectable to z. Every a in y is a transitive set bijective to z. Then Aa:aey -> aex Then y is a subset of x. y is the set of all transitive sets bijectable to z. Every b in x is a transitive set bijective to z Then Ab:bex ->bey Then x is a subset of y. x is a subset of y / y is a subset of x -> x=y which contradicts y=/=x. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. Accordingly : ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y . > 2) Z-card (x) = Z-card(y) <-> x is 1-1 with y. for a proof by negation , let ~ x is 1-1 with y. ~x is 1-1 with y <-> x is not bijective to y Let peZ-card(x) Since Z-card(x) = Z-card(y) Then p is bijective to x and p is bijective > to y. therefore x is bijective to y. which contradicts ~x is 1-1 with y. Accordingly x is 1-1 with y. 3) Avery z-card(x) is a set. z-card(x) is the intersection of the set of all transitive sets ,T > ,and the set of all sets bijective to x,Y. z-card(x) = T.Y Now we know for sure that Y is a proper class( when x=/={} ), and T is > a proper class. There is no proof that T.Y is a proper class. Then T.Y is a set. so every z-card(x) is a set. I have an intuitive feeling of what a proper class is, I call it a > flooding set, since every proper class in T.Y is subclassing the other, > then there is no flooding. ( perhaps this means that T.Y is well > founded?) 3) is till now not proved. the above is of course not a prove but a > plan of a proof that T.Y is a set. > The beauty of this definition of cardinality of a set, is that it > doesn't require choice to define the cardinality of the power set of > omega and powers of powers of omega, and also cardinality is a set, > unlike frege's cardinalities which are not sets. Zuhair The reason why this definition of cardinality do not require choice ( but requires Regularity ). is that because the power set of every transitive set is a transitive set. Ax:x is transitive , m=P(x) -> m is transitive. This bridges the gap in Von Neumann's definition of ordinals which do not work outside choice , since it cannot describe the cardinality not well orderable sets like P(w), P(P(w)), ...etc,because the power set of an ordinal is not an ordinal, therefore this definition of cardinality of Von Neumann's require's choice to extend it to such power sets. While the definition I have just presented above, do not require that. This definition depends on the assumption that for every set x every set y , if x is bijective to y, then P(x) is bijective to P(y). This doesn't require choice to prove. Zuhair === Subject: Re: Cardinality without choice x is a Z-cardinal <-> x is an equivalence class of transitive sets , > under equivalence relationbijection. Z-card(z) = { x| x is a transitive set / x has bijection to z } 1)Every set z has one and only one Z-card(z). ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y Proof: for a proof by negation Let EyEzAz y=Z-card(z) , x=Z-card(z) x=/=y Let a e y , b e x By definition of Z-card(z), it leads that a is bijective to z, and b is > bijective to z. Accordingly a is bijective to b. But x=/=y if and only if every member of x is not bijective to every > member of y ( see the definition of Z-card ), accordingly a is not bijective to b . which is contradictive. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. There is another way of proving this . for the same a,b defined above. x is the set of all transitive sets bijectable to z. Every a in y is a transitive set bijective to z. Then Aa:aey -> aex Then y is a subset of x. y is the set of all transitive sets bijectable to z. Every b in x is a transitive set bijective to z Then Ab:bex ->bey Then x is a subset of y. x is a subset of y / y is a subset of x -> x=y which contradicts y=/=x. Therefore EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y is contradictive. Accordingly : ~EyExAz y=Z-card(z) , x = Z-card(z) , x=/=y . > 2) Z-card (x) = Z-card(y) <-> x is 1-1 with y. for a proof by negation , let ~ x is 1-1 with y. ~x is 1-1 with y <-> x is not bijective to y Let peZ-card(x) Since Z-card(x) = Z-card(y) Then p is bijective to x and p is bijective > to y. therefore x is bijective to y. which contradicts ~x is 1-1 with y. Accordingly x is 1-1 with y. 3) Avery z-card(x) is a set. z-card(x) is the intersection of the set of all transitive sets ,T > ,and the set of all sets bijective to x,Y. z-card(x) = T.Y Now we know for sure that Y is a proper class( when x=/={} ), and T is > a proper class. There is no proof that T.Y is a proper class. Then T.Y is a set. so every z-card(x) is a set. I have an intuitive feeling of what a proper class is, I call it a > flooding set, since every proper class in T.Y is subclassing the other, > then there is no flooding. ( perhaps this means that T.Y is well > founded?) 3) is till now not proved. the above is of course not a prove but a > plan of a proof that T.Y is a set. The argument is that if T is a set, then T.Y is a set ( axiom of seperation). But if T is a proper class then this possibility is discussed above. > The beauty of this definition of cardinality of a set, is that it > doesn't require choice to define the cardinality of the power set of > omega and powers of powers of omega, and also cardinality is a set, > unlike frege's cardinalities which are not sets. Zuhair === Subject: x(ln(x)) What is the linear approximation of x(ln(x)) ? You dont have to be perfectly correct. === Subject: Re: x(ln(x)) > What is the linear approximation of x(ln(x)) ? > You dont have to be perfectly correct. There is a linear approximation for each x > 0. Which one do you want? === Subject: Re: x(ln(x)) Well I was reading a thermodynamic book where they did as follows:- 247T - 14.2TlnT is approximately equal to 141.3T in the temperature range 298K to 1200K. can you guys explain how it was done and how should i approximate in this temperature range? === Subject: Re: x(ln(x)) > Well I was reading a thermodynamic book where they > did as follows:- > 247T - 14.2TlnT is approximately equal to > 141.3T in the temperature range 298K to 1200K. can you guys explain how it was done and how should > i approximate in this temperature range? Beats me. This looks like one of those instances where a one-sentence signal to the reader (perhaps in a footnote) as to how this was obtained would have been appropriate. Maybe this type of approximation is well known in numerical analysis, a subject I know nothing about. However, I suspect if you look at a numerical analysis text, you'll find there are several types of approximation on an interval (maximum deviation on the interval, some type of averaged-out deviation over the interval, etc.), and therefore it doesn't even make sense to say that 247T - 14.2TlnT is approximately equal to 141.3T in the temperature range 298K to 1200K unless one specifies which type of approximation over an interval is being used. === Subject: Re: x(ln(x)) Calculated values Aproximated values T 298K 49498 42107 T 1200K 175585 169560 The aproximation is not too good. kunzmilan === Subject: Convergence factors of series What are convergence factors for the following series a) sum_{j=0}^{n} frac{(-1)^j}{2j+1} (limit = pi/4) b) sum_{j=1}^{n} j^{-frac{3}{2}} c) sum_{j=1}^{n} frac{1}{j^2} (limit = pi^2/6) d) sum_{j=1}^{n} frac{1}{j^4} (limit = pi^4/90) e) sum_{j=1}^{n} frac{(-1}^{j+1}}{j} (limit = ln(2)) f) sum_{j=1}^{n} frac{(-1)^j}{ln{j}} Any ideas how to count convergence factors for the series? Best, Damian Sobota. === Subject: Re: Convergence factors of series What are convergence factors for the following series a) sum_{j=0}^{n} frac{(-1)^j}{2j+1} (limit = pi/4) > b) sum_{j=1}^{n} j^{-frac{3}{2}} > c) sum_{j=1}^{n} frac{1}{j^2} (limit = pi^2/6) > d) sum_{j=1}^{n} frac{1}{j^4} (limit = pi^4/90) > e) sum_{j=1}^{n} frac{(-1}^{j+1}}{j} (limit = ln(2)) > f) sum_{j=1}^{n} frac{(-1)^j}{ln{j}} > Any ideas how to count convergence factors for the series? > Best, > Damian Sobota. Define convergence factor. === Subject: Re: Convergence factors of series > Define convergence factor. OK. Let (x_n) be a sequence convergent to limit g. Then p is a convergence factor of (x_n), if: lim |g-x_{n+1}|/|g-x_n|^p = C with n->infty and where C is positive real constant. If p=1,2,3 then convergence is linear, quadratic, cubic (respectively). I don't know how to count this factor for series from my first post. Best, DS. === Subject: Re: Convergence factors of series > Define convergence factor. OK. Let (x_n) be a sequence convergent to limit g. > Then p is a convergence factor of (x_n), if: lim |g-x_{n+1}|/|g-x_n|^p = C with n->infty and where C is positive real constant. If p=1,2,3 then convergence is linear, quadratic, cubic (respectively). I don't know how to count this factor for series from my first post. Best, > DS. Take for example sum(m=1, oo) 1/m^2. You are then looking at [sum(m=n+2, oo) 1/m^2] vs [sum(m=n+1, oo) 1/m^2]. Both are asymptotic to positive multiples of 1/n. So it would seem that p = 1 is the convergence factor here. === Subject: Re: Proper class.Proper class ? Under some simple class theories, ALWAYS, because > EVERYthing is a class How about the classes that are not elements of themselves - That depends. Some theories have a foundation axiom that prevents that. Other theories allow it. > is that a class? There is a class of all SETS that are not elements of themselves. No proper class is an element of ANY class, so, no, there is not a class of all classes. That would be a hyperclass. Yes, this regress is silly. > Likewise for those loons who say everything is a set - how > about the sets that are not elements of themself - is that a set? There is no set of all&only such sets; however, there is a class of them. There are no loons alleging that everything is a set. That is alleged by certain THEORETICAL TREATMENTS of this stuff (like ZFC). People's opinions are NOT involved. > C-B > (author of CBL which answers all of these questions formally and > automatically) Every individual set or class theory also answers them. The problem is that different theories answer them in different ways. Yours is just another raindrop in the hurricane. === Subject: Re: Proper class.Proper class ? > still my question about what is A.B? when A is the set of all sets that > subsets their power sets, and B is the set of all sets that has > bijection to S is till now not solved. First of all, the class of all sets is a proper class, not a set. > Secondly, what do you mean by the predicate that subsets their > powersets? And what is this S that you speak of in the second > sentence? You have undefined (or at least badly defined) terms here. In any case, I believe the question you asked was answered definitely: > The intersection of two non-disjoint proper classes, where one is not a > subclass of the other, can either result in a set or another proper > class. Jonathan Hoyle > Eastman Kodak Classes are non-sets. Non-sets in a set theory are non-sense. There can be only one proper class or none, in a set theory it would be a set. There is no class of classes in set theory with classes. There is no universe in ZF, ZF is inconsistent. Ross === Subject: Re: Proper class.Proper class ? > still my question about what is A.B? when A is the set of all sets that > subsets their power sets, and B is the set of all sets that has > bijection to S is till now not solved. First of all, the class of all sets is a proper class, not a set. > Secondly, what do you mean by the predicate that subsets their > powersets? And what is this S that you speak of in the second > sentence? You have undefined (or at least badly defined) terms here. In any case, I believe the question you asked was answered definitely: > The intersection of two non-disjoint proper classes, where one is not a > subclass of the other, can either result in a set or another proper > class. Jonathan Hoyle > Eastman Kodak > Classes are non-sets. Non-sets in a set theory are non-sense. There can be only one proper class or none, in a set theory it would be > a set. There is no class of classes in set theory with classes. There is no universe in ZF, ZF is inconsistent. Ross ZFC is inconsistent because it doesn't have a universe, that's why I am trying to develop a set theorum having a universe. In reality the whole idea of axiomatization is silly really. Since e is not defined and a set is not defined then the whole set theory mounts to nothing. And those people seems to think that they can produce things from nothing, which is philosophically absurd. Zuhair === Subject: Re: Proper class.Proper class ? > still my question about what is A.B? when A is the set of all sets that > subsets their power sets, and B is the set of all sets that has > bijection to S is till now not solved. First of all, the class of all sets is a proper class, not a set. > Secondly, what do you mean by the predicate that subsets their > powersets? And what is this S that you speak of in the second > sentence? You have undefined (or at least badly defined) terms here. In any case, I believe the question you asked was answered definitely: > The intersection of two non-disjoint proper classes, where one is not a > subclass of the other, can either result in a set or another proper > class. Jonathan Hoyle > Eastman Kodak > Classes are non-sets. Non-sets in a set theory are non-sense. There can be only one proper class or none, in a set theory it would be > a set. There is no class of classes in set theory with classes. There is no universe in ZF, ZF is inconsistent. Ross ZFC is inconsistent because it doesn't have a universe, that's why I am > trying to develop a set theorum having a universe. In reality the whole idea of axiomatization is silly really. Since e > is not defined and a set is not defined then the whole set theory > mounts to nothing. Anny system must have primitives, i.e., things not defined within the system, to start with or one reall is making something out of nothing. > And those people seems to think that they can > produce things from nothing, which is philosophically absurd. It is those who think they can produce something without even axioms and primitives who trying to produce something from nothing. === Subject: Re: Proper class.Proper class ? still my question about what is A.B? when A is the set of all sets that > subsets their power sets, and B is the set of all sets that has > bijection to S is till now not solved. First of all, the class of all sets is a proper class, not a set. > Secondly, what do you mean by the predicate that subsets their > powersets? And what is this S that you speak of in the second > sentence? You have undefined (or at least badly defined) terms here. In any case, I believe the question you asked was answered definitely: > The intersection of two non-disjoint proper classes, where one is not a > subclass of the other, can either result in a set or another proper > class. Jonathan Hoyle > Eastman Kodak > Classes are non-sets. Non-sets in a set theory are non-sense. There can be only one proper class or none, in a set theory it would be > a set. There is no class of classes in set theory with classes. There is no universe in ZF, ZF is inconsistent. Ross ZFC is inconsistent because it doesn't have a universe, that's why I am > trying to develop a set theorum having a universe. In reality the whole idea of axiomatization is silly really. Since e > is not defined and a set is not defined then the whole set theory > mounts to nothing. > Anny system must have primitives, i.e., things not defined within the > system, to start with or one reall is making something out of nothing. And those people seems to think that they can > produce things from nothing, which is philosophically absurd. It is those who think they can produce something without even axioms and > primitives who trying to produce something from nothing. Virgil ZF can't produce anything without everything. Ross === Subject: Re: Proper class.Proper class ? > still my question about what is A.B? when A is the set of all sets that > subsets their power sets, and B is the set of all sets that has > bijection to S is till now not solved. First of all, the class of all sets is a proper class, not a set. > Secondly, what do you mean by the predicate that subsets their > powersets? And what is this S that you speak of in the second > sentence? You have undefined (or at least badly defined) terms here. In any case, I believe the question you asked was answered definitely: > The intersection of two non-disjoint proper classes, where one is not a > subclass of the other, can either result in a set or another proper > class. Jonathan Hoyle > Eastman Kodak > Classes are non-sets. Non-sets in a set theory are non-sense. There can be only one proper class or none, in a set theory it would be > a set. There is no class of classes in set theory with classes. There is no universe in ZF, ZF is inconsistent. Ross ZFC is inconsistent because it doesn't have a universe, that's why I am > trying to develop a set theorum having a universe. In reality the whole idea of axiomatization is silly really. Since e > is not defined and a set is not defined then the whole set theory > mounts to nothing. And those people seems to think that they can > produce things from nothing, which is philosophically absurd. Zuhair Actually, to produce things from nothing is the requirement. That's because there is obviously not nothing. There is existence, E, as I recently was illustrating, in a technical way. The point of deaxiomatization and deriving truths from that process, reverse mathematics of a sort, is to attempt to gain insight from the most primitive truths and how they could be at all. It resolves basically to what looks like a paradox, on reexamination always true because its opposite is the same. That can be stated in the terms of mathematical logic, conveniently. Ross === Subject: Re: Looking for information on Correlation Matrices sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? The sample variance has to be nonnegative definite. -- This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each sample does not have have the same number of observations at the same time... Eg some samples may have only a few data points and you have to calculate the correlation from these few data points... I have in practice seen the negative eigenvalues. But you are saying if all samples have the same number of points and all have same observation times then we are okay - matrix is non-negative definite ? === Subject: Re: Looking for information on Correlation Matrices > The problem we run into with the samples is that we may not have enough > sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? The sample variance has to be nonnegative definite. This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each > sample > does not have have the same number of observations at the same time... > Eg some > samples may have only a few data points and you have to calculate the > correlation > from these few data points... I have in practice seen the negative > eigenvalues. So, you are calculating correlations from separate samples, then combining these to form a matrix. In that case, all bets are off. > But you are saying if all samples have the same number of points and all > have same observation times then we are okay - matrix is non-negative definite ? If you have samples x_i, i=1,...,N, where each x_i is a vector, then the sample variance of the x_i is nonnegative definite. In fact, it is the same as the variance of the probability measure which assigns weight 1/N to each point x_i. -- Marcus === Subject: Re: Looking for information on Correlation Matrices I'm collecting the samples at the same time.. I have n variables that I'm sampling all at the end of each hour - I record all their values, wait another hour and get all their values etc... Then I'm making a correlation matrix using pearson correlation.. You are saying this should lead to non negative eigenvalues for the correlation matrix ? ( I have in the past had cases where some of the variables missed a measurement at the end of the hour.. OR more commonly part way through the experiment another random variable was added in.. so we started with n and part way through had n+1.. at the end the n+1 * n+1 correlation matrix had a negative eigenvalue... ) > The problem we run into with the samples is that we may not have enough > sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? The sample variance has to be nonnegative definite. This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each > sample > does not have have the same number of observations at the same time... > Eg some > samples may have only a few data points and you have to calculate the > correlation > from these few data points... I have in practice seen the negative > eigenvalues. So, you are calculating correlations from separate samples, then > combining these to form a matrix. In that case, all bets are off. But you are saying if all samples have the same number of points and all > have same observation times then we are okay - matrix is non-negative definite ? If you have samples x_i, i=1,...,N, where each x_i is a vector, then the > sample variance of the x_i is nonnegative definite. In fact, it is the > same as the variance of the probability measure which assigns weight 1/N > to each point x_i. -- > Marcus === Subject: Re: Looking for information on Correlation Matrices > I'm collecting the samples at the same time.. I have n variables that > I'm sampling all at the end of each hour - I record all their values, > wait another hour and get all their values etc... Then I'm making a > correlation matrix using pearson correlation.. What is Pearson correlation? Can you tell us the formulas that you are using? Why aren't you just calculating the correlation matrix in the standard way? > You are saying this > should lead to non negative eigenvalues for the correlation matrix ? ( I have in the past had cases where some of the variables missed a > measurement at the end of the hour.. OR more commonly part way through > the experiment another random variable was added in.. so we started > with n and part way through had n+1.. at the end the n+1 * n+1 > correlation matrix had a negative eigenvalue... ) -- Marcus === Subject: Re: Looking for information on Correlation Matrices This is the formula ( Best to give web page since fonts look better than ascii ).. http://en.wikipedia.org/wiki/Correlation The page says it is also know as the sample correlation coefficient > I'm collecting the samples at the same time.. I have n variables that > I'm sampling all at the end of each hour - I record all their values, > wait another hour and get all their values etc... Then I'm making a > correlation matrix using pearson correlation.. What is Pearson correlation? Can you tell us the formulas that you are > using? Why aren't you just calculating the correlation matrix in the standard > way? You are saying this > should lead to non negative eigenvalues for the correlation matrix ? ( I have in the past had cases where some of the variables missed a > measurement at the end of the hour.. OR more commonly part way through > the experiment another random variable was added in.. so we started > with n and part way through had n+1.. at the end the n+1 * n+1 > correlation matrix had a negative eigenvalue... ) -- > Marcus === Subject: Re: Looking for information on Correlation Matrices On 2006-11-11 13:34:31 -0400, asdf said: The problem we run into with the samples is that we may not have enough > sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? >> >> The sample variance has to be nonnegative definite. >> >> -- This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each > sample > does not have have the same number of observations at the same time... > Eg some > samples may have only a few data points and you have to calculate the > correlation > from these few data points... I have in practice seen the negative > eigenvalues. But > you are saying if all samples have the same number of points and all > have same observation > times then we are okay - matrix is non-negative definite ? > If the matrix is a collection of pairwise present correlations then the fact that it does not have the Grammian property of being positive semidefinite should be well known. I seem to I expect that all the responses have made the very sensible assumption that you were asking about a correlation matrix and not a collection of pairwise correlation coefficients. === Subject: Re: Looking for information on Correlation Matrices <2006111114123716807-gsande@worldnetattnet> Does anybody know of sample data that produces a correlation matrix with a negative eigenvalue - where the matrix is produced using pearson correlations on the sample data ( pair wise ).. I think it exists just want a simple case... > On 2006-11-11 13:34:31 -0400, asdf said: > The problem we run into with the samples is that we may not have enough > sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? >> The sample variance has to be nonnegative definite. >> -- This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each > sample > does not have have the same number of observations at the same time... > Eg some > samples may have only a few data points and you have to calculate the > correlation > from these few data points... I have in practice seen the negative > eigenvalues. But > you are saying if all samples have the same number of points and all > have same observation > times then we are okay - matrix is non-negative definite ? > If the matrix is a collection of pairwise present correlations > then the fact that it does not have the Grammian property of > being positive semidefinite should be well known. I seem to I expect that all the responses have made the very sensible assumption > that you were asking about a correlation matrix and not a collection > of pairwise correlation coefficients. === Subject: Re: Looking for information on Correlation Matrices On 2006-11-11 15:04:40 -0400, asdf said: > Does anybody know of sample data that produces a correlation matrix > with a negative eigenvalue - where the matrix is produced using pearson > correlations on the sample data ( pair wise ).. I think it exists just > want a simple case... Examples were in the paper by Richard Heiberger of Temple University if I recall correctly. Unfortunately Google does not seem to provide anything relevant on a couple tries. Judging from comments in other related posting there is a need for explicit formulae to make sure that your understanding of correlation coefficient and pairwise present matches the common technical usage. >> On 2006-11-11 13:34:31 -0400, asdf said: > > The problem we run into with the samples is that we may not have enough > sample points to get the matrix to become semi positive definite. If > we had infinite samples it would approach a semi positive definite > matrix ? >> >> The sample variance has to be nonnegative definite. >> >> -- This is what I was confused about .. So this is true ? The only time I have seen negative eigenvalues in practice is when each > sample > does not have have the same number of observations at the same time... > Eg some > samples may have only a few data points and you have to calculate the > correlation > from these few data points... I have in practice seen the negative > eigenvalues. But > you are saying if all samples have the same number of points and all > have same observation > times then we are okay - matrix is non-negative definite ? > >> >> If the matrix is a collection of pairwise present correlations >> then the fact that it does not have the Grammian property of >> being positive semidefinite should be well known. I seem to >> >> I expect that all the responses have made the very sensible assumption >> that you were asking about a correlation matrix and not a collection >> of pairwise correlation coefficients. === Subject: Re: Looking for information on Correlation Matrices <2006111114123716807-gsande@worldnetattnet> <2006111115470916807-gsande@worldnetattnet > Examples were in the paper by Richard Heiberger of Temple University > if I recall correctly. Unfortunately Google does not seem to provide > anything relevant on a couple tries. Judging from comments in other related posting there is a need for explicit > formulae to make sure that your understanding of correlation coefficient > and pairwise present matches the common technical usage. > Here is in a nutshell my calculations: A B C hour1 10 2 7 hour2 -4 4 -3 hour3 5 10 3 hour4 4 1 4 correlation matrix is 3x3 symmetric matrix and entry for A,B correlation is -0.082046112 This is value excel gives for pearson and correl functions that it has. Actually I just looked at the help and CORREL and PEARSON in excel give same formula. which matches formula on this page: http://en.wikipedia.org/wiki/Correlation So it appears pearson correlation is same thing as standard way of calculating sample correlation. Also diagonals will have value 1 === Subject: Re: Looking for information on Correlation Matrices On 2006-11-11 16:29:40 -0400, asdf said: > >> >> Examples were in the paper by Richard Heiberger of Temple University >> if I recall correctly. Unfortunately Google does not seem to provide >> anything relevant on a couple tries. > >> Judging from comments in other related posting there is a need for explicit >> formulae to make sure that your understanding of correlation coefficient >> and pairwise present matches the common technical usage. > > Here is in a nutshell my calculations: > A B C > hour1 10 2 7 > hour2 -4 4 -3 > hour3 5 10 3 > hour4 4 1 4 correlation matrix is 3x3 symmetric matrix and entry for A,B > correlation is -0.082046112 This is value excel gives for pearson and correl functions that it has. Actually I just looked at the help and CORREL and PEARSON in excel give > same formula. which matches formula on this page: http://en.wikipedia.org/wiki/Correlation So it appears pearson correlation is same thing as standard way of > calculating sample correlation. Also diagonals will have value 1 Good! Everything in these examples is as expected. Then there is your story about adding variables on the fly or something. And maybe there were observations missed during the runs. Or Whatever. So not all correlations were based on the same number of observations. Your example does not show such things. That sure sounded like having pairwise present with differing missing patterns. In that case there is no reason to expect the correlation matrix to be Garammian. Negative eigenvalues may well be present as such a matrix is not a cross product matrix and may well not be positive (semi)definite. === Subject: Re: Looking for information on Correlation Matrices <2006111114123716807-gsande@worldnetattnet> <2006111115470916807-gsande@worldnetattnet> <2006111116524516807-gsande@worldnetattnet > Good! Everything in these examples is as expected. Then there is your story about adding variables on the fly or something. > And maybe there were observations missed during the runs. Or Whatever. > So not all correlations were based on the same number of observations. > Your example does not show such things. Okay this is good then.. I should not have to worry about getting negative eigenvalues so long as I don't miss samples and don't add variables etc... ? For example if we have: A B C D .... hour1 a1 b1 c1 d1 ... hour2 a2 b2 c2 d2 ... hour3 c3 b3 c3 d3 ... ... No matter what the values above are ( eg a1,b1,c1 etc can by any numbers ). No matter how many random vars we have ( A,B,C,D, E etc... ) No matter how many hours... The sample correlation matrix will always have non-negative eigenvalues ? This is true yes ? That sure sounded like having pairwise present with differing missing > patterns. In that case there is no reason to expect the correlation > matrix to be Garammian. Negative eigenvalues may well be present as > such a matrix is not a cross product matrix and may well not be > positive (semi)definite. === Subject: Re: Looking for information on Correlation Matrices On 2006-11-11 17:20:35 -0400, asdf said: >> >> Good! Everything in these examples is as expected. >> >> Then there is your story about adding variables on the fly or something. >> And maybe there were observations missed during the runs. Or Whatever. >> So not all correlations were based on the same number of observations. >> Your example does not show such things. Okay this is good then.. I should not have to worry about getting > negative eigenvalues so long as I don't miss samples and don't add > variables etc... ? For example if we have: > A B C D .... hour1 a1 b1 c1 d1 ... > hour2 a2 b2 c2 d2 ... > hour3 c3 b3 c3 d3 ... > ... No matter what the values above are ( eg a1,b1,c1 etc can by any > numbers ). > No matter how many random vars we have ( A,B,C,D, E etc... ) > No matter how many hours... The sample correlation matrix will always have non-negative eigenvalues > ? This is true yes ? IF things are as shown AND you manage to avoid numerical problems. When there is an unlucky sample or if you have linear dependencies or too small of sample then you may well have numerical problems. In exact arithmetic you would only have a zero eigenvalue but a computer that does exact arithmetic will be bit hard to come by. In the real world you will have to deal with numerical issues and that may result in the zero eigenvalue being reported as negative but close to zero. The closeness would be within the bounds that that are given numerical analysis text books. All pretty standard, even classical, numerical analysis. Eigenvalues are not rational numbers (except for special cases) so you have to deal with numerical issues. If you are really out to play games then you could do rational arithmetic to evaluate the determinant, by Cholesky for example, to see if any eigenvalue is zero. Then the numerical error still might make a small eigenvalue appear to be negative. >> That sure sounded like having pairwise present with differing missing >> patterns. In that case there is no reason to expect the correlation >> matrix to be Garammian. Negative eigenvalues may well be present as >> such a matrix is not a cross product matrix and may well not be >> positive (semi)definite. === Subject: Re: Looking for information on Correlation Matrices Examples were in the paper by Richard Heiberger of Temple University > if I recall correctly. Unfortunately Google does not seem to provide > anything relevant on a couple tries. > Judging from comments in other related posting there is a need for explicit > formulae to make sure that your understanding of correlation coefficient > and pairwise present matches the common technical usage. > Here is in a nutshell my calculations: > A B C > hour1 10 2 7 > hour2 -4 4 -3 > hour3 5 10 3 > hour4 4 1 4 correlation matrix is 3x3 symmetric matrix and entry for A,B > correlation is -0.082046112 So, what is wrong with that? > This is value excel gives for pearson and correl functions that it has. Actually I just looked at the help and CORREL and PEARSON in excel give > same formula. which matches formula on this page: http://en.wikipedia.org/wiki/Correlation So it appears pearson correlation is same thing as standard way of > calculating sample correlation. I don't like the 1/(n-1): unbiasedness is a bad way of choosing an estimator. But, that shouldn't affect whether your matrix is nonnegative definite. > Also diagonals will have value 1 -- Marcus === Subject: Re: Looking for information on Correlation Matrices <2006111114123716807-gsande@worldnetattnet> <2006111115470916807-gsande@worldnetattnet> Examples were in the paper by Richard Heiberger of Temple University > if I recall correctly. Unfortunately Google does not seem to provide > anything relevant on a couple tries. > Judging from comments in other related posting there is a need for explicit > formulae to make sure that your understanding of correlation coefficient > and pairwise present matches the common technical usage. > Here is in a nutshell my calculations: > A B C > hour1 10 2 7 > hour2 -4 4 -3 > hour3 5 10 3 > hour4 4 1 4 correlation matrix is 3x3 symmetric matrix and entry for A,B > correlation is -0.082046112 So, what is wrong with that? > Nothing is wrong ... I'm just wondering about cases where we could get negative eigenvalue is all.. === Subject: Re: Looking for information on Correlation Matrices > Examples were in the paper by Richard Heiberger of Temple University > if I recall correctly. Unfortunately Google does not seem to provide > anything relevant on a couple tries. > Judging from comments in other related posting there is a need for explicit > formulae to make sure that your understanding of correlation coefficient > and pairwise present matches the common technical usage. > Here is in a nutshell my calculations: > A B C > hour1 10 2 7 > hour2 -4 4 -3 > hour3 5 10 3 > hour4 4 1 4 correlation matrix is 3x3 symmetric matrix and entry for A,B > correlation is -0.082046112 So, what is wrong with that? > Nothing is wrong ... I'm just wondering about cases where we could get > negative eigenvalue is > all.. variables aren't linearly related, you won't get negative eigenvalues in practice (i.e., numerically). If the three variables are linearly related, then you will have an eigenvalue of zero which numerically might come out as slightly negative, but just change it to zero. It depends on how you find the eigenvalues. I think there are algorithms you can use that assume the matrix is nonnegative definite and so know that all eigenvalues are nonnegative. -- Marcus === Subject: Re: Looking for information on Correlation Matrices <2006111114123716807-gsande@worldnetattnet> <2006111115470916807-gsande@worldnetattnet> variables aren't linearly related, you won't get negative eigenvalues in > practice (i.e., numerically). If the three variables are linearly > related, then you will have an eigenvalue of zero which numerically > might come out as slightly negative, but just change it to zero. It > depends on how you find the eigenvalues. I think there are algorithms > you can use that assume the matrix is nonnegative definite and so know > that all eigenvalues are nonnegative. -- > Marcus === Subject: Re: Moebius Band is not homeomorphic with a Torus > NarasimhamContinued.. The source of my discomfiture with what to me always appeared as some > sort of a Moebius Trick is simple and I shall make an encore.. A half Torus between the main two points of its meridian(with tangents > parallel to the original uncut Torus axis) can be homeo - compressed to > a flat ring meridionally. When the fibers are twisted by a toroidal > co-ordinate rotation double that of the polar angle, the twisted torus > can likewise be homeo - compressed to a Moebius Band. But the topology goes awry and way off !! I am unable to accept any > topological difference between a flat concentric ring and a its > twisted counterpart, the MB.The justification to the contrary I cannot > understand in simple terms. And I think it ought to be expressed in > simple terms starting from the postulates. > Narasimham 2 things... 1: The width of a Moebius band is about as thick as your desire to understand its true existense. I do not mean this to be mean, but most people who study this type of thing have devoted their entire lives to understanding the subtle points which can't be just thought of without training. 2: If the Moebius and and an annulus were the same things would be VERY bad. Lets say for example you go jogging on track with 2 sides divided by a center line. Now in the normal world this is an annulus with a core circle drawn on it. You can run day in and day out and never cross that line. However, if you were to do this on a Moebius band (as you think it may be the same), on each lap you would return to the start line but be on the OTHER side of the dividing line and thus be crashing into oncoming traffic. A HA! you might say that you are on the other side of the line but also upside down on the bottom. Now notice that this ONLY fixes the dilemma if I can't now argue that you were running inside the thickness of the band. This idea can be cutely summed up in the classic trick: take a Moebius band, draw the core circle, and cut along it. This will result in a longer band with 2 twists. Doing this with an annulus will obviously result in 2 annuli being produced. If that isn't a good enough difference between the Moebius band and an Annulus then I don't know what else you could want. === Subject: Re: Moebius Band is not homeomorphic with a Torus mappings, conformal, isometric etc among surfaces depending what is > held invariant between the differential surface elements (du,dv)) > between patches of a surface generally according to sketch indicated: http://img297.imageshack.us/img297/5171/surfacehomeomorphismqe0.jpg It could be a small extension, compression or large surface > translations or even gliding 'parallel to' parameter lines (a > partial modification of my response to Hero) accompanied by extension > and compression. In all cases it is a simple change of parameters (u,v) > limits on a parameterized surface. Can't say that answers the question. Are you familiar with the standard > meanings of the words homeomorphism, homotopy, continuous map, > surjection, injection? e.g.,Wikipedia and Mathworld, .Hope an adequate and intuitive grasp is > there although I am not a full student of topology.You can see my past > posts in general,I have helped several students although I am not a > teacher in a university. In continuum mechanics of materials I am very familiar with homotopy > although it does not go by that name. There geometrical strain/stress is > further subdivided into in-plane (tension, compression) and out- of > -plane (bending), dealt with in classical laminate theory but for small > deformations only. In topology large deformations are possible. Other > concepts are relatively easy. Please indicate where exactly I am making an error in simple terms if > indeed I am. I appreciate improved objectivity,avoiding subjective > remarks without reasons stated. I don't think anyone here knows what point you are making. You may not think so. Respectfully I would say you cannot speak on everyone's behalf without addressing the central question. Any one going through this thread would suppose you and Victor constitute a subset of topologists who cared to respond here. God, OK ! if it is still not clear, I shall once again repeat it. Is change of (u,v) parameter domain of 2D surface in 3-space (x,y,z) of following parameterization in my first post a valid homeomorphism ? b=3 ; a=1 ; x = ( a Cos[th/2+ k Pi/ m ] + b ) Cos[th] ; y = ( a Cos[th/2+ k Pi/m] + b ) Sin[th] ; z = a Sin[th/2+ k Pi/m] ; Tori = ParametricPlot3D[{x,y,z},{th,0,4 Pi},{k,1,2 m },PlotPoints ->{101,2m},Boxed->False]; If so, you should accept m = 1 MB case as homeomorphic to the other tori set members in my above mentioned first posting.If not, please state why an exception has to be made in this case... What all axoims, definitions, theorems, lemmas, tenets or whatever are brought into fore for direct or implied mathematical/ topological logic to invalidate an apparently simple and straight-forward embeddings of these surfaces in 3D. It may be classical and so deeply ingrained. But an enquiring student looks to a pointed answer while unlearning his errors. > A Moebius Band is not homeomorphic to a torus (as these words are normally defined). I know it very well that this is (that MB and torus are not homeomorphic) emphatically stated in all the textbooks. I myself quoted from 'Geometry of Surfaces' in the very first post as Definition 5. This appears to me (of course I have to ignore Victor's objections to the reference to the first person as there is no other possibility) as a contradiction based on my parameterization and that is why I at all started this thread as OP. The point I ask here is how it deviates from such a normally defined situation. I suppose definitive words need not per se matter at all, really what matters is what they should be connoting. New words can be always coined to convey a concept at a later stage when it becomes an accepted rule or tenet. > The picture you showed of gluing a strip back to back to create a Moebius band is pretty. great new kind of mathematical help. I used it to look at the m = 1 case un-orientable (Quotes until it is properly explained) MB using RealTime 3D` with pressed mouse from all orientations. I shall upload a few more pictures out of the same program, the MB and torus sit neatly back to back obvoiusly a small distance that can be compressed as a rubber membrane.Only the unifying parameterization in above first post is due to me. > but so what? Your call to answer. ----- > ----- > Marcus Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus > God, OK ! if it is still not clear, I shall once again repeat it. Is change of (u,v) parameter domain of 2D surface in 3-space (x,y,z) > of following parameterization in my first post a valid homeomorphism ? b=3 ; a=1 ; > x = ( a Cos[th/2+ k Pi/ m ] + b ) Cos[th] ; y = ( a Cos[th/2+ k > Pi/m] + b ) Sin[th] ; z = a Sin[th/2+ k Pi/m] ; > Tori = ParametricPlot3D[{x,y,z},{th,0,4 Pi},{k,1,2 m },PlotPoints > ->{101,2m},Boxed->False]; I don't use Mathematica, but tell me if the following is correct. Define a function f:R^2 -> R^3 by f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), ( cos(t/2 + k Pi/m) + 3 ) sin(t), sin(t/2 + k Pi/m ) ). You are asking whether if we restrict f to {(t,k)| 0 < t < 4 Pi, 1 < k < 2m}, it is a homeomorphism. Is that your question? -- Marcus === Subject: Re: Moebius Band is not homeomorphic with a Torus of following parameterization in my first post a valid homeomorphism ? b=3 ; a=1 ; > x = ( a Cos[th/2+ k Pi/ m ] + b ) Cos[th] ; y = ( a Cos[th/2+ k > Pi/m] + b ) Sin[th] ; z = a Sin[th/2+ k Pi/m] ; > Tori = ParametricPlot3D[{x,y,z},{th,0,4 Pi},{k,1,2 m },PlotPoints > ->{101,2m},Boxed->False]; I don't use Mathematica, but tell me if the following is correct. Define > a function f:R^2 -> R^3 by f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), > ( cos(t/2 + k Pi/m) + 3 ) sin(t), > sin(t/2 + k Pi/m ) ). You are asking whether if we restrict f to {(t,k)| 0 < t < 4 Pi, 1 < k < 2m}, it is a homeomorphism. Is that your question? > -- > Marcus No, my question is not about the character of functional dependence f(t, k) but about the flexibility of the two parameter domain, the rubber sheet area parameter domain, if you will. The question again: For f (t, k) and for all arbitrary parameter domain sets (in your notation) : {(t, k) | tmin < t < tmax, kmin < k < kmax}, where t's are real and k's are integers, do we have a set of surfaces with valid homeomorphism among them? Only in the present tori set we have k interal,in general they also can be real. f(t,k) is Monge's form of surface where we are varying the projected window border limits of the surface whose topography is morphed in homeomorphism as a rubber membrane. Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus > I don't use Mathematica, but tell me if the following is correct. Define > a function f:R^2 -> R^3 by f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), > ( cos(t/2 + k Pi/m) + 3 ) sin(t), > sin(t/2 + k Pi/m ) ). You are asking whether if we restrict f to {(t,k)| 0 < t < 4 Pi, 1 < k < 2m}, it is a homeomorphism. Is that your question? No, my question is not about the character of functional dependence > f(t, k) but about the flexibility of the two parameter domain, the > rubber sheet area parameter domain, if you will. The question again: For f (t, k) and for all arbitrary parameter domain sets (in your > notation) : {(t, k) | tmin < t < tmax, kmin < k < kmax}, where t's are real and k's are integers, do we have a set of surfaces > with valid homeomorphism among them? I don't understand what you are asking. I seem to be asking if something is a homeomorphism. A homeomorphism is a map between two surfaces. So, what are your two surfaces and what is the map? And, what does f have to do with it? > Only in the present tori set we > have k interal,in general they also can be real. -- Marcus === Subject: Re: Moebius Band is not homeomorphic with a Torus Continued... Three Monge patches are shown,plan rectangles are cut out of the same classic buckled surface z = sin(x)*sin(y). Two patches are overlapping/coalscing/occupying same 3D and 2D space/interferng in 2D and 3D space. Two patches are disjoint: http://img58.imageshack.us/img58/1059/mongepatchhomeomorphismpu6.jpg It is easy to see that between surfaces contained above any two plan projections, membrane areas are inter-stretchable or homeomorphable. Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus Continued... Three Monge patches are shown,plan rectangles are cut out of the classic buckled surface z = sin(x)*sin(y). Two are overlapping/coalsced/occupy same 3D/interfere in space even as 2D surfaces and two are disjoint: http://img58.imageshack.us/img58/1059/mongepatchhomeomorphismpu6.jpg It is easy to see that surfaces contained above any two plan projections areas are inter-stretchable or homeomorphable. Narasimham === Subject: Re: Moebius Band is not homeomorphic with a Torus > Continued... Three Monge patches are shown,plan rectangles are cut out of the > classic buckled surface z = sin(x)*sin(y). Two are overlapping/coalsced/occupy same 3D/interfere in space even as > 2D surfaces and two are disjoint: http://img58.imageshack.us/img58/1059/mongepatchhomeomorphismpu6.jpg It is easy to see that surfaces contained above any two plan > projections areas are inter-stretchable or homeomorphable. What does homeomorphable mean? Not sure what point you are making. -- Marcus === Subject: Re: General Arithmetic > General Arithmetic is the second-order theory consisting of induction > on a successor function. Normal arithmetic, in the axiomatization > called Peano Arithmetic, makes certain demands on the successor > function S. First, that S be total. Secondly, that it be one-to-one. > And thirdly, that there exist a special element, usually called 0, > which is not in the successor function's image. General Arithmetic > foregoes all three of these further assumptions, yet is still strong > enough to prove many meaningful arithmetic truths, Well, surely that is important. Has anybody on fom gotten around to saying so, yet? What kind of response are you getting? One reason why I personally care about this kind of question is that I suffered a disaster in grad.school around the proof of a simple lemma from group theory, idempotence implies identity, i.e., the only element a in a group for which a*a=a must be the identity. I was trying to generate an automated proof of this (with no luck, from an inference engine largely developed by another student) and my advisor insisted for weeks that the failure was due to my mismanagement of axioms for = . It turns out that the proof of this theorem does not depend on equality or any of the special properties conferred by axioms or inference rules defining it; ANY old binary predicate would do. If I had figured this out a month sooner than I actually did, I probably would've graduated. So, as if Occam's Razor weren't enough motivation already, I have since cared a great deal about proving more from less. I especially prefer FOL without = to with. === Subject: Zeta Distribution and Pareto Distribution I am studying statistics and probability. One of the distributions I am studying is the Pareto Distribution. I read on the internet that the Pareto Distribution is the continuous version of the Zeta Distribution. I have been unable to make this link. If you could refer me to some good resources or if you have some hints, please let me know. === Subject: Re: Zeta Distribution and Pareto Distribution >I am studying statistics and probability. One of the distributions I >am studying is the Pareto Distribution. I read on the internet that >the Pareto Distribution is the continuous version of the Zeta >Distribution. I have been unable to make this link. If you could >refer me to some good resources or if you have some hints, please let >me know. If one considers the Hurwitz Zeta function instead of just the Riemann Zeta function the connection is clear. The Hurwitz Zeta function is defined by Z(q, a) = sum_0 1/(a+j)^q where q > 1 and j runs through the positive integers, and the corresponding distribution has the probabilities equal to the terms divided by the sum. As a approaches infinity, the probabilities approach the density function of the Pareto distribution with density proportional to 1/(h+x)^q, x > 0. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Zeta Distribution and Pareto Distribution The result is not so clear to me. I think that I am probably lacking the math background to see what you are proposing. I have only taken the calculus sequence and it has been awhile since I have had a calculus class. Please let me know if there is more that you could expand on to get the desired result without going into advanced mathematics. >I am studying statistics and probability. One of the distributions I >am studying is the Pareto Distribution. I read on the internet that >the Pareto Distribution is the continuous version of the Zeta >Distribution. I have been unable to make this link. If you could >refer me to some good resources or if you have some hints, please let >me know. > If one considers the Hurwitz Zeta function instead of > just the Riemann Zeta function the connection is clear. The Hurwitz Zeta function is defined by Z(q, a) = sum_0 1/(a+j)^q where q > 1 and j runs through the positive integers, > and the corresponding distribution has the probabilities > equal to the terms divided by the sum. As a approaches > infinity, the probabilities approach the density function > of the Pareto distribution with density proportional to > 1/(h+x)^q, x > 0. > -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Zeta Distribution and Pareto Distribution It should be easier to see graphically than show mathematically. Good resources which seem to be good enough for a Professor at Reading University (pronounced Redding - not reeding!) (UK) to refer to them http://www.personal.rdg.ac.uk/~sis04itd/Cources/Stochastic%20Methods/student s/Definitions.doc http://en.wikipedia.org/wiki/Pareto_distribution and http://en.wikipedia.org/wiki/Zipf%27s_law Note that the according to http://en.wikipedia.org/wiki/Pareto_distribution the Zipf distribution is also known as the Zeta distribution. but with respect if it was that easy to derive mathematically do you think that I would have done a year's postgraduate course in statistics. See http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html where it says: Although the literature surrounding both the Zipf and Pareto distributions is vast, there are very few direct connections made between Zipf and Pareto, and when they exist, it is by way of a vague reference [1] or an overly complicated mathematical analysis[2,3]. Here I show a simple and direct relationship between the two by walking through an example using real data. Nick > The result is not so clear to me. I think that I am probably lacking > the math background to see what you are proposing. I have only taken > the calculus sequence and it has been awhile since I have had a > calculus class. Please let me know if there is more that you could > expand on to get the desired result without going into advanced > mathematics. >>I am studying statistics and probability. One of the distributions I >>am studying is the Pareto Distribution. I read on the internet that >>the Pareto Distribution is the continuous version of the Zeta >>Distribution. I have been unable to make this link. If you could >>refer me to some good resources or if you have some hints, please let >>me know. >> If one considers the Hurwitz Zeta function instead of >> just the Riemann Zeta function the connection is clear. >> The Hurwitz Zeta function is defined by >> Z(q, a) = sum_0 1/(a+j)^q >> where q > 1 and j runs through the positive integers, >> and the corresponding distribution has the probabilities >> equal to the terms divided by the sum. As a approaches >> infinity, the probabilities approach the density function >> of the Pareto distribution with density proportional to >> 1/(h+x)^q, x > 0. >> -- >> This address is for information only. I do not claim that these views >> are those of the Statistics Department or of Purdue University. >> Herman Rubin, Department of Statistics, Purdue University >> hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > === Subject: Re: Zeta Distribution and Pareto Distribution >I am studying statistics and probability. One of the distributions I > am studying is the Pareto Distribution. I read on the internet that > the Pareto Distribution is the continuous version of the Zeta > Distribution. I have been unable to make this link. If you could > refer me to some good resources or if you have some hints, please let > me know. See http://en.wikipedia.org/wiki/Pareto_distribution Pareto distributions are continuous probability distributions. Zipf's law, also sometimes called the zeta distribution, may be thought of as a discrete counterpart of the Pareto distribution. Joe === Subject: Re: Zeta Distribution and Pareto Distribution <-IKdnQY6msMzjsvYnZ2dnUVZ8tudnZ2d@bt.com> Joe, That is what I read but how is the Pareto Distribution derived from the zeta distribution? Here is my initial thought: First try to derive the cummulative Pareto Distribution and then take the derivative of it with respect to k Proposed cummulative function. limit as the change in k goes to zero of the sum from 1 to n of 1/[((1+(i)(change k))^s]*zeta fuction Do you think that I am heading down the right path? >I am studying statistics and probability. One of the distributions I > am studying is the Pareto Distribution. I read on the internet that > the Pareto Distribution is the continuous version of the Zeta > Distribution. I have been unable to make this link. If you could > refer me to some good resources or if you have some hints, please let > me know. See http://en.wikipedia.org/wiki/Pareto_distribution > Pareto distributions are continuous probability distributions. Zipf's law, > also sometimes called the zeta distribution, may be thought of as a discrete > counterpart of the Pareto distribution. > Joe === Subject: Re: Zeta Distribution and Pareto Distribution > Joe, That is what I read but how is the Pareto Distribution derived from the > zeta distribution? Here is my initial thought: First try to derive the cummulative Pareto Distribution and then take > the derivative of it with respect to k Proposed cummulative function. limit as the change in k goes to zero of the sum from 1 to n of > 1/[((1+(i)(change k))^s]*zeta fuction Do you think that I am heading down the right path? > >I am studying statistics and probability. One of the distributions I > am studying is the Pareto Distribution. I read on the internet that > the Pareto Distribution is the continuous version of the Zeta > Distribution. I have been unable to make this link. If you could > refer me to some good resources or if you have some hints, please let > me know. See http://en.wikipedia.org/wiki/Pareto_distribution > Pareto distributions are continuous probability distributions. Zipf's law, > also sometimes called the zeta distribution, may be thought of as a discrete > counterpart of the Pareto distribution. > Joe > Note: counterpart or version, not derived from. === Subject: Re: Zeta Distribution and Pareto Distribution <-IKdnQY6msMzjsvYnZ2dnUVZ8tudnZ2d@bt.com> <111120061750098437%anniel@nym.alias.net.invalid> Extremely skewed distributions are known in as Pareto, Lotka, Zipf, Bradford and other laws. They can be formulated as continuous or discontinuous functions. The most important distinction is, if they are frequency or rank distribution functions, which end of relations they consider as the starting point of the correlation. They were studied systematically by scientometrics. kunzmilan === Subject: Re: Zeta Distribution and Pareto Distribution <-IKdnQY6msMzjsvYnZ2dnUVZ8tudnZ2d@bt.com> <111120061750098437%anniel@nym.alias.net.invalid> I still don't understand how the Pareto Distribution is derived. I know the following: -The first 20% of the observations account for 80% of the results. -The definition of the failure function is alpha/x -x>=beta -alpha>2 formula for density function=(alpha/beta)*(beta/x)^(alpha+1) Can the density fuction be derived from the conditions noted above? If so, how can it be done? If not, what other conditions am I missing? > Extremely skewed distributions are known in as Pareto, Lotka, Zipf, > Bradford and other laws. They can be formulated as continuous or > discontinuous functions. The most important distinction is, if they are > frequency or rank distribution functions, which end of relations they > consider as the starting point of the correlation. They were studied > systematically by scientometrics. > kunzmilan === Subject: Re: Zeta Distribution and Pareto Distribution >I still don't understand how the Pareto Distribution is derived. I > know the following: -The first 20% of the observations account for 80% of the results. -The definition of the failure function is alpha/x -x>=beta -alpha>2 formula for density function=(alpha/beta)*(beta/x)^(alpha+1) Can the density fuction be derived from the conditions noted above? If > so, how can it be done? If not, what other conditions am I missing? > Statistical distributions are descriptions that are found to fit certain data. So it will be observed that particular data has characteristics of certain distributions (eg the Normal Distribution). The properties of these distributions can be derived (see http://en.wikipedia.org/wiki/Pareto_distribution) but the distribution itself cannot be derived. Indeed if the Pareto distribution can be derived from the Zipf distribution under certain conditions, how is the Zip distribution. The Pareto distribution can no more be derived than can the straight line y=x. It can be said to be equivalent to other distributions. But it can't be derived from first principles. I am a statistician but probability and distribution theory is not my specialism. Nick === Subject: Re: Zeta Distribution and Pareto Distribution <-IKdnQY6msMzjsvYnZ2dnUVZ8tudnZ2d@bt.com> <111120061750098437%anniel@nym.alias.net.invalid> Jonas napsal: > I still don't understand how the Pareto Distribution is derived. The recipe can be found in Bible. I have not the English Bible at hand. The algorithm is in Mathew, approximately as: To them, who have, it will be added, and to them, who have not, it will taken even what they have. I am sorry, it is too long ago when I studied the topics, it would be too long search in my notes. kunzmilan === Subject: Re: Zeta Distribution and Pareto Distribution <-IKdnQY6msMzjsvYnZ2dnUVZ8tudnZ2d@bt.com> <111120061750098437%anniel@nym.alias.net.invalid> I sympathetic to your situation. I'm pretty determined to figure this out. Even if I don't find a source that shows me how it is done, I might have to learn a bit about some subjects that I am not familiar with right now. Learning more math can't hurt. Once I get the answer, I'll post it here. > Jonas napsal: > I still don't understand how the Pareto Distribution is derived. > The recipe can be found in Bible. > I have not the English Bible at hand. The algorithm is in Mathew, > approximately as: > To them, who have, it will be added, and to them, who have not, it will > taken even what they have. > I am sorry, it is too long ago when I studied the topics, it would be > too long search in my notes. > kunzmilan === Subject: A bright note... Why not escape from business staff for a second and go for a little fun...? http://www.FreeFaces.com JOIN us now FREE and start DATING girls and boys COMPLETELY FREE. Join the International FREE Dating Site now... http://www.FreeFaces.com The Webmaster webmaster@freefaces.com === Subject: Re: An uncountable countable set <4543543d@news2.lightlink.com> <45437ec0@news2.lightlink.com> <4543a3fa@news2.lightlink.com> <4543b56f@news2.lightlink.com> <45462ba0@news2.lightlink.com> <45479064@news2.lightlink.com> <454b7dd6@news2.lightlink.com> <4554dca0$1@news2.lightlink.com I don't know where to >> start with your topologically distinct polyhedra. Hmm, not polyhedra, I see. > Why don't you give me a rundown of how YOU compare those two sets? OK; how would I think about counting the topologically distinct > polyhedra? First I'd observe that I could attack from the faces, the > edges, or the vertices (F, E, V; and I could remember that there's an > obvious vertex-face duality), so I might choose the faces. I'd assemble > a list of the first few, out of curiosity (which has to be the driving > force here: what are the conditions for a polyhedron to be its own V-F > dual, for example?) The minimum number of faces is obviously 4; not less than 3 faces must > meet each vertex, and there must be more than one vertex. I make a > little list of the number of possibilities: 4: 1 (tetrahedron) > 5: 2 (square pyramid, triangular prism) > 6: (pentagonal prism, cube, etc. at this point, cheat > http://www.research.att.com/~njas/sequences/table?a=944&fmt=4 ) > 7: and so on Now I would notice that by topologically distinct polyhedron I am > referring to a bounded geometrical object; not only bounded, but also > discrete, in the sense that if I have one in my hand I know I can count > the vertices. A bit of minor handwaving, and I would see that for any > number of vertices there will be a limited number of possibilities for > arranging that number of vertices into a polyhedron. So I know that I > can pick an ordering scheme, and put all of the polyhedra in it. So I > can count them, in the sense that I know that with my counting scheme > there will not be a polyhedron that escapes it. I also notice that this > counting sequence will never end, because there is no maximum to the > number of vertices. After all, if there were, then given a polyhedron > with that number of vertices I could simply pick any face, construct a > pyramid on that face, and get a polyhedron with more than the supposed > maximum number of vertices, which proves (by contradiction) that my > sequence of polyhedra never ends. (This is all incredibly obvious, but > I'm afraid I never know which incredibly obvious bits you still haven't > grokked.) So with the constraints of the present audience, I would express this > by saying that I can see I can count the polyhedra, a process which > will account for every one of them, given enough time, but I can also > see that the process of counting will never end. It's much messier, but I can do a similar thing for the polygons with > vertices having integral x-y coordinates. How could I compare the two sets? I don't know - in both cases it's > true that I can find a way of counting them, and that the counting will > never end. Since it never ends, it's hard to see how I might think that > the counting of the polygons was going to be over before the counting > of the polyhedra, or vice versa. It's not even possible to say of a > process that never ends that after so many counts the processs is an > appreciable way to completion, because there _is_ no completion. So my answer is rather limited: both sets are (ok, dammit, in normal > words) countably infinite; there's no way obvious to me that I could > regard either as less endlessly endless than the other. What's your answer? You appear to be the one claiming to have numbers > for counting things when the counting never ends: do you have any here? > Or are you happy to accept that these two sets, and the set of pofnats > can all be put in 1-1 correspondence, or (in normal words again) have > the same cardinality? Perhaps this particular fact raises no objections > from your intuition module? Notice that of course there are other things one can say about these > sets. For example, the number of polyhedrons with n edges [E(n)] is > always less than the number with n vertices [V(n)] (= the number with n > faces); I can see that E(n) < V(n-2), but on average these are > close(?). Does the ratio V(n)/E(n) approach a particular value? I don't > know. Brian Chandler > http://imaginatorium.org countability for the set, though that depends on there being only a > finite n umber of vertices, edges or faces for each polyhedron in your > set. Oh, you mean you agree that it would be possible to arrange all the different polyhedra in a (one-ended) line so as to count them, reaching any one eventually, but never getting to the nonexistent other end? Good. Of course my treatment of polyhedra depends on them being members of the class of objects normally referred to in mathematics as polyhedra. You still haven't grasped the utility of having standard terms for things so everyone knows what they are talking about. But you might ought to be aware that of course mathematicians have thought about unbounded, infinite pseudopolyhedra - for example in Wells' Geometry dictionary (Penguin) there's a section under honeycombs with some nice pictures in. > I notice that the infinite topologically distinct polyhedra known > as tiling systems are not included in your set. I'll ignore that point > for now. No, that's right. I also excluded Borges 17 Chinese categories of animal, strawberry, vanilla, and pistacchio ice cream, the King of Peru, and a box of rotten carrots. (This is only a representative sample, to which the exclusions are not restricted. This is also not medical advice. (Hope this helps.)) > In order for me to formulate this question in terms of Bigulosity > Theory, I'm going to have to formulate the relationships between > vertices, edges and faces in general, and their relation to the number > of polyhedra possible. I've played with these numbers in relation to > vertex angle for regular polyhedrons with some interesting results, but > not in the general case for irregular polyhedra. So, let me mull this > over, and see if I can concoct some numbers for you, formulaic > expressions of the size of this set, in terms of Big'un vertices, edges, Wonderful. Another of many things you have not grasped about mathematics is that it is not about schemes in which the author of the scheme promises to try to concoct some numbers. But are you hoping to provide (for example) expressions that state how many polyhedra in this set _if_ this set includes a total of Big'un vertices, and a presumably different expression _if_ it contains Big'un edges? We can be fairly sure that even with your calculations these will be different expressions - so how, in any sense at all, could either/which be called the size of the set? And suppose you arrive at these expressions, and suppose you also arrive at expressions for the size of the set of Gaussian polygons (the other set I mentioned), _if_ the set of integer coordinates (x,y) has [some Big'un' expression] you've concocted size, then what sort of mulling or concocting are you going to use to approach how to mangle these expressions together? (You might remember you never actually managed to avoid the obvious problems with the set of two-ended strings from the alphabet {0, 1}...) Brian Chandler http://imaginatorium.org === Subject: Re: An uncountable countable set > It holds for all finite naturals, but if there are an infinite number of > naturals generating using increment, then there are naturals which are > the result of infinite increments, which must have infinite value. Can you show us one of those infinite naturals? And while you're at it, show us the finite natural that is its > predecessor. Of course, you first have to define what you mean > by infinite increments. > >> I meant an infinite number of increments, each being a successive >> difference of +1 in measure. >> Here is an infinite natural, Big'un: 100...000. That's in binary, and >> there are log2(Big'un) 0's after the 1, so the most significant bit is >> at log2(Big'un). Its predecessor, 0111...111, with the 0 also at >> log2(Big'un), is also infinite, as are countably, or evenly uncountably >> many, in the chain of predecession. Your BigUn notational is meaningless. I'm assuming that you're > building on the existing positional digit notation we are all familiar > with, which is: > for all x in R, > x = (sum{i=0 to oo} d_i x B^i) x B^p, > for some base B and some integer p, and 0 <= d_i < B. First of all, you're summing digits only to the right of the digital point, and why are you multiplying them by some power of the base after that? I don't see any fractional component. I'd say: for all x in R x = (sum{i=-oo to oo} d_i x B^i) for some natural base B and some natural d_i such that 0<=d_i digits (negative powers of B): > for all n in N, > n = sum{i=0 to oo} d_i x B^i. Sure. Your notation looks to be, in more formal terms: > for all n in T, > n = (sum{i=0 to oo} d_i x B^i) + (sum{j=? to ?} d_j x B^j) > You mean the two ends of the string (and there may be extra limit points within the string) each constitute a countable neighborhood of digits within the string? That's correct. The uncountable portions oft he string are filled in with a repeating pattern that equates to a rational portion of an infinite quantity. It does end up needing to be represented as a multi-part formula, as you suggest. That's the whole idea, to represent such notions as oo^2-3*oo-log2(oo)+sqrt(oo), as a digital string. :) > I put '?' for the limits of the right sum, because I can't figure out > what j is supposed to be once you've used up all the powers of B > with B^i in the left sum. It is a declared formulaic infinity based on Big'un. I mean it looks like you're saying that there are digits beyond > an infinite number of digits, but that creates a problem for you, > because you've run out of indexes for those extra digit positions. I don't see any more problem with that than saying you have traversed and infinite number of points from the origin, but you are at a specific point, with an infinite number of more points ahead if you continue. Can you index each of those points with a finite string? No, of course not, unless you have some infinite alphabet. It's uncountable. But, that doesn't mean it can't also be viewed as residing along the same linear sequence. > Which puts you back at square one, because in order to define > some meaning for those supra-infinite digit positions, you need > some kind of supra-infinite indexing numbers, which is what > you're trying to define the T-numbers to be in the first place. > Which is just circular reasoning. > Not circular at all. I declare Big'un to be the number of reals per unit interval, and the length of the hyperreal line in unit intervals. Then I employ this value in formulas which can be compared using infinite-case induction. It's very straightforward. === Subject: Re: An uncountable countable set > Your BigUn notational is meaningless. I'm assuming that you're > building on the existing positional digit notation we are all familiar > with, which is: > for all x in R, > x = (sum{i=0 to oo} d_i x B^i) x B^p, > for some base B and some integer p, and 0 <= d_i < B. First of all, you're summing digits only to the right of the digital > point, and why are you multiplying them by some power of the base after > that? I don't see any fractional component. 6/37 = 0.162162162... repeating 162 forever. But this only works for fractions less than 1 in absolute value. For improper fractions like 6000/37 it must be modified to allow representation of the integer part. 6000/37 = (162.1621621622...) = (sum{i=0 to oo} d_i / 10^i) x 10^3 where d_i = 1 for i ==1 mod 3 and d_i = 6 for i ==2 mod 3 and d_i = 2 for i == 0 mod 3 I'd say: > for all x in R > x = (sum{i=-oo to oo} d_i x B^i) > for some natural base B and some natural d_i such that 0<=d_i For naturals, we can simplify that a bit by eliminating fractional > digits (negative powers of B): > for all n in N, > n = sum{i=0 to oo} d_i x B^i. Even here there is no arithmetical operation valid for arbitrary TOintegers. Sure. > Your notation looks to be, in more formal terms: > for all n in T, > n = (sum{i=0 to oo} d_i x B^i) + (sum{j=? to ?} d_j x B^j) > You mean the two ends of the string (and there may be extra limit points > within the string) each constitute a countable neighborhood of digits > within the string? That's correct. The uncountable portions oft he > string are filled in with a repeating pattern that equates to a rational > portion of an infinite quantity. It does end up needing to be > represented as a multi-part formula, as you suggest. That's the whole > idea, to represent such notions as oo^2-3*oo-log2(oo)+sqrt(oo), as a > digital string. :) Garbage. I put '?' for the limits of the right sum, because I can't figure out > what j is supposed to be once you've used up all the powers of B > with B^i in the left sum. It is a declared formulaic infinity based on Big'un. it is a declared nonsense based only on TO's mental short circuits. > I mean it looks like you're saying that there are digits beyond > an infinite number of digits, but that creates a problem for you, > because you've run out of indexes for those extra digit positions. I don't see any more problem with that than saying you have traversed > and infinite number of points from the origin, but you are at a specific > point, with an infinite number of more points ahead if you continue. Except that WE make no such claims for discretely ordered index sets (in which every index but a first must have a unique predecessor and all have a unique successor) like N. WE never claim to have reached an index with infinitely many predecessors. WE leave that sort of nonsense to TO. > Can > you index each of those points with a finite string? We can index any one of them with a finite string, and we do. > No, of course not, > unless you have some infinite alphabet. It's uncountable. What is uncountable, TO's alphabet? > But, that > doesn't mean it can't also be viewed as residing along the same linear > sequence. Definition of ordinal limit points ( as distinct from topological limit points): P is a left limit point of an ordered set S if P is a member of S and for every Q in S with P < Q there is an R with P < R < Q. P is a right limit point of an ordered S if P is a member of S and for every Q in S with Q < P, there is an R in S with Q > R > P. In standard notation, index sets have no limit points of either type. In the standard set of natural numbers, N, there are no limit points of either type. In any standard ordinal which contains a limit ordinal as a member, those members which are limit ordinals are right limit points. but there are no left limit points in any ordinal. Which puts you back at square one, because in order to define > some meaning for those supra-infinite digit positions, you need > some kind of supra-infinite indexing numbers, which is what > you're trying to define the T-numbers to be in the first place. > Which is just circular reasoning. > Not circular at all. Then spiralling and into the oblivion of a central black hole. > I declare Big'un to be the number of reals per unit > interval, and the length of the hyperreal line in unit intervals. TO can declare anything he wants to, but declaring things does not make them so. Then I > employ this value in formulas which can be compared using infinite-case > induction. It's very straightforward. === Subject: Re: An uncountable countable set > What part of my definition says that? For the positives: > 1 e H > x e H -> 2^x e H > x e H -> 2^-x e H No, there are no successors in your tree for that number p, > because there is no last L/R branch in the path defining that > H2-number. So there can be no different or next branching for > the last node for either successor of p, because there is no last > node in the path. I'm talking over your head here, because you obviously do not > follow what I'm saying. > >> Geeze, David, flatter yourself a little, why don't you? >> You are upset because what is clearly an uncountably long string might >> have a successor. That's understandable. But, allow me to elaborate a >> tad. I'll break into pieces so you can respond bit by bit: >> The function 2^x, as a real function, is continuous over R. It goes from >> an asymptotic y of 0 as x->-oo to an asymptotic y of oo as x->oo, and no >> discontinuities in between, monotonically increasing from -oo to oo. Right? >> Since the function is monotonically increasing, one can always find an >> intermediate value between a and b, which is 2^((log(a)+log(b))/2), that >> is, f((g(a)+g(b))/2), where f(x)=2^x, f(g(x))=g(f(x)), xeR. Right? >> Does that not make this set continuous over the range of reals? Sure. >> So, we're left with a little conundrum. >> Every unique real x has a unique real y such that 2^x=y, and such that >> -2^x=-y. So each real has two successors. We can't pick a point where >> that's not true. We can start anywhere (of course, zero's a good spot to >> start), but in any case, in enumerating this uncountable set, we're >> going to get to uncountable strings, which have successors. You're confusing things here. You're going to get countably infinite > long digits strings (I suspect that most of your H-riffic numbers are > irrational). However, you will only be able to derive a countably > infinite number of them. Most of the reals will not be derived by > your rules, regardless of the starting point. I've given a simple example many times. Starting with 0, your > rules will never generate 3, 1/3, or any integer multiple or > power of 3. You still have not responded to that little flaw, > which would go a long way towards convincing others that > you might be on to something. > Why is that a flaw in the H-riffics, and not in the digital reals? You cannot express 1/3 as a decimal fraction either. Do the decimal fractions constitute a countable set? That's counter to Cantor's Diagonal Argument for the uncountability of the reals. The set of all digital fractions is uncountable. The set of all H-riffics is uncountable. Every digital fraction has successor and predecessor as well, when the bits are mirrored to the other side of the digital point, and ordered as naturals. Ah, you say, but 0.333... does not have a natural mirror, since ...333 is not a natural. And yet, I say to you, that is every digit in that string is in a finite position with respect to the digital point, then there is no point in that string where it achieves an infinite value, and so it represents some sort of finite but unbounded quantity. Since any such unending strings also has a successor and a predecessor (except perhaps for ...000 and ...999), it would seem that in two significant ways such numbers are very like the terminated finite strings you call naturals. You might be reassured by keeping in mind that the set of such finite string representations of infinite strings is still countable, since they depend on a repetition of a finite string of digits. > >> - If every element in the reals has at least one successor, what >> does that say to you? Not much, apart from the fact that some countable subset of the reals > is covered by your generating rules. Your rules produce a countable set by their very nature. For any > H-riffic H_n, I can produce H_2n and H_2n+1 from it. Which means > that I can map every natural k to some H-riffic, and vice versa. > Hence the H-riffics are a countable set. > But there is no r in R for which you can say there is no successor. How do you partition R into countable subsets? Which r in R does not have two successors? === Subject: Re: An uncountable countable set > I've given a simple example many times. Starting with 0, your > rules will never generate 3, 1/3, or any integer multiple or > power of 3. You still have not responded to that little flaw, > which would go a long way towards convincing others that > you might be on to something. > Why is that a flaw in the H-riffics, and not in the digital reals? You > cannot express 1/3 as a decimal fraction either. Not as a terminating decimal fraction, but every rational can be represented by either a terminating or an eventually repeating decimal fraction. > Do the decimal > fractions constitute a countable set? That's counter to Cantor's > Diagonal Argument for the uncountability of the reals. TO, as usual, puts his mind into neutral and his typing into high gear before posting. The terminating decimals, and even the terminating together with repeating decimals (which is no more than the rationals), form countable sets. If TO want a set of decimals which is uncountable, he must include non-terminating and nonrepeating ones. And the set of all such represents the reals but no countable subset of them does. The set of all > digital fractions is uncountable. The set of all H-riffics is > uncountable. Every digital fraction has successor and predecessor as > well, when the bits are mirrored to the other side of the digital point, > and ordered as naturals. Ah, you say, but 0.333... does not have a > natural mirror, since ...333 is not a natural. And yet, I say to you, > that is every digit in that string is in a finite position with respect > to the digital point, then there is no point in that string where it > achieves an infinite value, and so it represents some sort of finite but > unbounded quantity. Since any such unending strings also has a successor > and a predecessor (except perhaps for ...000 and ...999), it would seem > that in two significant ways such numbers are very like the terminated > finite strings you call naturals. You might be reassured by keeping in > mind that the set of such finite string representations of infinite > strings is still countable, since they depend on a repetition of a > finite string of digits. >> - If every element in the reals has at least one successor, what >> does that say to you? Not much, apart from the fact that some countable subset of the reals > is covered by your generating rules. Your rules produce a countable set by their very nature. For any > H-riffic H_n, I can produce H_2n and H_2n+1 from it. Which means > that I can map every natural k to some H-riffic, and vice versa. > Hence the H-riffics are a countable set. > But there is no r in R for which you can say there is no successor. How > do you partition R into countable subsets? Which r in R does not have > two successors? === Subject: Re: An uncountable countable set > , > >> I went round and round and round and round with regarding the >> staircase problem, in which, to make the point that the lim f(g(n)) >> is not always the same as f(lim g(n)), I gave an absurd proof that >> diagonal in sci.math brings you to a random but representative section >> of the discussion under the thread Calculus XOR Probability, if >> you're feeling interested or even just excessively bored). >> Part of the argument was an (increasingly painstakingly detailed) >> definition of the limit of a sequence of curves, which gave the limit >> of a sequence of progressively finer staircases in R^2 traversing >> opposing corners of the unit square as being the diagonal of the unit >> square. Each staircase has length 2; but of course (I said, spreading >> my arms and smiling encouragingly) the length of the diagonal of the >> unit square obviously /cannot/ be 2, so therefore... I cannot resolve this problem, and have not located the > thread you cite. If we define a curve in R^2 as a > continuous map from [0,1] to R^2, how do we exclude the > staircase sequence as a converging to the curve? > Possibly by saying that a piecewise linear > approximation must have the terminal points of the line > segments on the curve. But how do we know that other > pathologies do not lurk? What exactly is necessary for > a sequence of curves to converge to a curve, and what > curve is it? > Hi Michael. Excellent questions. This arose in, I think, Calculus XOR Probability, but then I started a thread to discuss the topic itself, The link to my paper in the post is expired. Here's a new link to my paper, which I just uploaded: http://www.lightlink.com//Limits.htm In short, my contention is that inductive proof works, not only for finite n, but also for infinite n, when proving equalities between expressions, or inequalities based on differences which do not have a limit of 0 as x->oo. Chas' example was meant to be a counterexample to such use of infinite-case induction, by showing that such inductive arguments do not hold in the infinite case, since the infinite-case staircase clearly has a length of sqrt(2), not 2 as inductively proven. However, my immediate response to this was that the staircase in the limit is not the same object as the diagonal, because it still maintains those right angles, and is therefore not a straight line exactly, but rather a sort of fractal, whose length actually is 2. While the two objects seem to be the same from the point-set perspective, since points can only be distinguished if there is measurable distance between them, and the distance between corresponding points has a lower bound of 0, point-set is not the only perspective. Since points have no measure other than location, it is not surprising that measure would be lost in the translation of a curve into points. So, I reformulated the problem as a segment-sequence topology, which clearly shows a difference between the two objects. Then Chas asked for an example of another curve that WAS the same as the diagonal in the limit, so I concocted such a thing, a saw tooth that went from vertical to diagonal as n increased, in segment-sequence form, which gave an answer that differed from the original staircase by some infinitesimal amount, which could be discarded according to the notion that the square of an infinitesimal may be considered as nothing. Anyway, if you care to read the thing, it's short, and now you have the links. :) === Subject: Combinatorics and functions W is a set that contains only 3 distinct elements. S is the set of all W -> W functions. o is S^2 -> S function such that for every x in W and every f, g in S: (f o g)(x) = f(g(x)) What is the number of all S^2 -> S functions # such that for all f, g, h in S: f o (g # h) = (f o g) # (f o h) (1) f # (g # h) = (f # g) # h (2) i.e. # is associative and o is left distributive over #. right distributive over them? If W contains only 2 distinct elements a and b, then there are 2^2 distinct W -> W functions: t = {(a, a), (b, a)} => t(x) = a r = {(a, a), (b, b)} = identity function on W u = {(a, b), (b, a)} v = {(a, b), (b, b)} => v(x) = b So the number of all S^2 -> S functions in this case is: (2^2)^(2^2 * 2^2) = 4294967296. 40 is commutative and for only 2, o is right distributive over them: x # y = x x # y = y Here is another #: t # t = t t # r = v t # u = t t # v = v r # t = u r # r = r r # u = u r # v = r u # t = u u # r = r u # u = u u # v = r v # t = t v # r = v v # u = t v # v = v Theron === Subject: Re: Combinatorics and functions > W is a set that contains only 3 distinct elements. > S is the set of all W -> W functions. > o is S^2 -> S function such that for every x in W and every f, g in S: > (f o g)(x) = f(g(x)) What is the number of all S^2 -> S functions # such that for all f, g, > h in S: f o (g # h) = (f o g) # (f o h) (1) > f # (g # h) = (f # g) # h (2) i.e. # is associative and o is left distributive over #. right distributive over them? > (Note: This is certainly all just re-inventing the wheel; but I'll post it as I haven't seen it elsewhere.) I think a good way to approach this problem is by starting with S_W, the constant functions in S. Certainly, for all A, B, C in S_W, we already require from # that A # (B # C) = (A # B) # C and for all f in S and A, B in S_W, we also require from # that f o (A # B) = (f o A) # (f o B) Note that if A in S_W, then (f o A) is itself a constant function for all f in S; and so for any f, g in S; (f # g) o A = (f o A) # (g o A) is therefore /also/ a constant function. This implies that given a # satisfying our requirements, it follows that for all A, B in S_W, A # B is in S_W; i.e., S_W is closed w.r.t. #. Conversely, Theorem (1): Suppose we can first define # : (S_W)^2 -> S_W so that it satisfies the two constraints. Then there is a /unique/ extension of # to S^2 -> S, which also satisfies (1) and (2). Proof: Let f, g be in S. Let A(x) = a for some A in W be a constant function in S_W. Then (f # g)(a) = ((f # g) o A)(a) = ((f o A) # (g o A))(a) which is uniquely determined by # since (f o A) and (g o A) are in S_W. Thus, for each a in W, (f # g)(a) is defined when we define # for S_W; and thus (f # g) is uniquely defined. Associativity follows: ((f # g) # h)(A) = ((f # g) # h) o A)(a) = (((f # g) o A) # (h o A))(a) = (((f o A) # (g o A)) # (h o A))(a) [*] = ( (f o A) # ((g o A)) # (h o A))(a) = ( (f o A) # ( (g # h) o A))(a) = ( (f # (g # h)) o A)(a) = (f # (g # h))(a) again because at step [*], f o A, g o A, and h o A are all in S_W. Finally, (f o (g # h))(a) = ( (f o (g # h)) o A)(a) = (f o ((g # h) o A))(a) // by assoc. of o = (f o ((g o A) # (h o A))) (a) = ((f o (g o A)) # (f o (h o A))) (a) // g o A, h o A in S_W = ( ((f o g) o A) # ((f o h) o A) ) (a) // by assoc. of o = ( ((f o g) # (f o h)) o A) (a) = ( (f o g) # (f o h) ) (a) So the extension satisfies (1) and (2). Therefore, we don't need to examine the |S|^|S^2| different binary functions on S; we only need to look at the |W|^|W^2| binary functions on S_W. This is an enourmous help; for |W| = 3, the former is (3^3)^(3^6) which is over 1000 digits long base 10; versus the later which is 3^9 or only 19,683 possibilities. But we can do better than this with a little more analysis... We require from # that, for all f in S and A in S_W that A = A o (f # g) = (A o f) # (A o g) so we have (3) for all A in S_W, A = A # A. So we (at least) require that S_W be an /idempotent semigroup/ (which seems to be a common area of enquiry). Also, suppose that #: S^2 -> S satisfies our requirements. Then if we define a @ b = b # a, then @ : S^2 -> S also satisfies our requirements. So we can consider anti-isomorphisms as well as isomorphisms to cut down on the number of multiplication tables to consider. For the problem of |W| = 2, W = {a,b}, you previously defined the four functions S, which I will relabel here for my convenience: A = {(a, a), (b, a)} => A(x) = a e = {(a, a), (b, b)} = identity function on W s = {(a, b), (b, a)} = swap function on W B = {(a, b), (b, b)} => B(x) = b In this case, S_W = {A, B}, and there are only 4 possible binary operations satisfying the constraint (3), which represent only two (up to isomorphism/anti-isomorphism) cases: # A B # A B A A A A A B A B B B Each of these easily satisfy f o (x # y) = (f o x) # (f o y) for all f in {A, B, e}. But the first fails for s: s o (A # B) = s o A = B (s o A) # (s o B) = B # A = A while the second satisfies: s o (A # B) = s o A = B = B # A = (s o A) # (s o B) s o (B # A) = s o B = A = A # B = (s o B) # (s o A) s o (A # A) = B = (s o A) # (s o A) s o (B # B) = A = (s o B) # (s o B). Thus, there exists (up to isomorphism/anti-isomorphism) one # meeting the requirements for W = {a,b}; and it is non-commutative. Applying the same type of logic to W = {a, b, c}, let f = (xyz) denote the function f(a) = x, f(b) = y, and f(c) = z. Let A = (aaa), B = (bbb), C = (ccc). Let I = (abc). By inspection (waves hands to avoid endless calculations...), the only idempotent semigroups on the 3 elements of S_W = {A, B, C} are (up to isomorphism/anti-isomorphism): # A B C A A A A B A B A C A A C This fails: (bac) o (A # B) = (bac) o A = B; ((bac) o A) # ((bac) o B) = B # A = A # A B C A A A A B A B B C A B C This also fails: (bac) o (A # B) = (bac) o A = B; ((bac) o A) # ((bac) o B) = B # A = A # A B C A A A A B B B B C C C C This works! It also suggests yet another trivial solution to the general non-commutative problem: define, for all x, y in S_W, x # y = x. Then extend as in theorem 1 to get (f # g) = f for all f, g in S. Does this exclusively trivial behavior continue for |W| > 3? I don't know... === Subject: Re: Combinatorics and functions > W is a set that contains only 3 distinct elements. > S is the set of all W -> W functions. > o is S^2 -> S function such that for every x in W and every f, g in S: > (f o g)(x) = f(g(x)) What is the number of all S^2 -> S functions # such that for all f, g, > h in S: f o (g # h) = (f o g) # (f o h) (1) > f # (g # h) = (f # g) # h (2) i.e. # is associative and o is left distributive over #. right distributive over them? > the constant functions in S. > S_W so that it > satisfies the two constraints. Then there is a /unique/ extension of # > to S^2 -> S, which also satisfies (1) and (2). > A A A A > B B B B > C C C C This works! It also suggests yet another trivial solution to the > general non-commutative problem: define, for all x, y in S_W, x # y = > x. Then extend as in theorem 1 to get (f # g) = f for all f, g in S. Does this exclusively trivial behavior continue for |W| > 3? I don't > know... Given the above, we can state two more things, regardless of the cardinality of W (4) For all A, B in S_W, either A # B = A or A # B = B. Suppose otherwise; A # B = C. Then (aac) o (A # B) = (aac) o C = C; but ((aac) o A) # ((aac) o B) = A # A = A. (5) For all A, B in S_W, A # B = A implies B # A = B; and A # B = B implies B # A = A A # B = A implies (ba) o (A #B) = (ba) o A = B = ((ba) o A) # ((ba) o B) = B # A; A # B = B implies (ba) o (A #B) = (ba) o B = A = ((ba) o A) # ((ba) o B) = B # A; (4) and (5) together say: there is /never/ a commutative # for n > 1 (and in particular, this puts the kibosh on your original question regarding W = R). At any rate, the above description implies that the number of multiplication tables for # as a function of (finite) n, up to isomorphism and anti-isomorphism, has an upper bound of... um... I'll have to think about it a bit more! :) === Subject: Re: Binary operator, (x + y)^2 = x^2 + x*y + y*x + y^2 > S is the set of all R -> R functions. > Is it possible to exist S^2 -> S function # such that for every f, g, h > in S: T(f # g) = T(f) # (f o g) # (g o f) # T(g) > f # (g # h) = (f # g) # h > f # g = g # f > Assuming left and right distributivity of o over #: h o (f # g) = (h o f) # (h o g) > (f # g) o h = (f o h) # (g o h) > This is certainly stronger than your original constraint; and is > related to the occassionally asked question: given a monoid (G, *), can > we define a semigroup operator + so that (G, +, *) is sort of like a > ring? The latter of your two distributive equalities seems to be much easier > to satisfy; for example define (f # g)(x) = f(x) + g(x). More generally o is right distributive over any &: S^2 -> S such that: (f & g)(x) = u(f(x), g(x)), where u is R^2 -> R function. This suggests that the problem could be simplified by placing further > restrictions on S. Can the functions in S be assumed to be infinitely > differentiable? Infinitely differentiable except at a finite number of > discontinuities? Bijections? Can we change the domain of functions in S > from R to the dyadic rationals? Etc. If there are solutions to a simplified form of the problem, maybe some > of them can be extended to address the general problem (where S > contains all R -> R functions). Do you think that this is possible? > To be honest, I'm skeptical. I've seen this sort of question come up > semi-frequently on sci.math, and the answer doesn't seem to be a well > known consequence of semigroup theory. My guess is that any solutions > are going to be impossible to construct; sort of like there being a > well-ordering of the real numbers, except we can't actually construct > one. Maybe it is too hard to find constructible solutions, but not impossible. > Perhaps someone who has studied this further can offer other > suggestions? If you are interested in this problem check my post Combinatorics and functions. Theron === Subject: PCA/PLS related question Hi there, Hope someone can help me with this. I have some signals (time-profiles of variables): Y(t) , X(t), Z(t). I also have many signals which correspond to measurements performed at different frequencies: F1(t) F2(t) F3(t) ... F20(t) These F signals are highly correlated (not the same magnitude, but the overall shapes are quite similar). Still, there are little differences in the patterns which I try to exploit Basically, I would like to use this data set to determine if the Y(t), X(t) or Z(t) profiles could be reconstructed using a combination of (some of) the F signals. I'm not too sure how to approach this. by fewer variables: PC1, PC2, PC3 etc Then, should I try to perform a regression between my Y, X, Z signals and the PC1, PC2, etc ? Is PLS the way to go ? === Subject: Verification: Sequences in the cofinite topology Kelley (which I read in self-study) poses the following exercise (Problem 2.A): Consider a countable space with the cofinite topology. What sequences converge to what points? It seems to me that the countability of the space is irrelevant. My answer: [S P O I L E R S P A C E] A sequence converges if and only if there is at most point which appears infinitely often. If there is one such point, the sequence converges to that point only. Otherwise, the sequence converges to every point in the space. In particular, if the space is finite, only eventually constant sequences converge. (In this case, the topology is discrete.) True? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Verification: Sequences in the cofinite topology > Kelley (which I read in self-study) poses the following exercise > (Problem 2.A): Consider a countable space with the cofinite topology. > What sequences converge to what points? It seems to me that the countability of the space is irrelevant. My > answer: > Finite cofinite spaces are discrete, hence only eventually constant sequences converge. For infinite cofinite spaces .... > [S > P > O > Please do not spoil your posts! It makes much inconvenience reading your post and the replies. > A sequence converges if and only if there is at most point which appears > infinitely often. If there is one such point, the sequence converges to > that point only. Otherwise, the sequence converges to every point in > the space. In particular, if the space is finite, only eventually > constant sequences converge. (In this case, the topology is discrete.) True? > If a point p appears infinitely often, then if the sequence converges, it can converge only at p. In addition, if p is the only infinitely recurrent point, then the sequence converges at p. However for cofinite N 1 2 3 4 5 ... converges to every point in N. So we have additional situation of no infinitely recurring points. Such sequences converge to every point. === Subject: Re: Verification: Sequences in the cofinite topology > Kelley (which I read in self-study) poses the following exercise > (Problem 2.A): Consider a countable space with the cofinite topology. > What sequences converge to what points? It seems to me that the countability of the space is irrelevant. My answer: [S P O I L E R S P A C E] > A sequence converges if and only if there is at most point which appears > infinitely often. If there is one such point, the sequence converges to > that point only. Otherwise, the sequence converges to every point in > the space. In particular, if the space is finite, only eventually > constant sequences converge. (In this case, the topology is discrete.) True? -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor on the Internet and in Central New Jersey and Manhattan Wrong. If a sequence actually has infinitely many points in its range, then it converges to every point. If a sequence has only a finite range than one or more points is repeated infinitely often and it converges to that or those point(s) and to no other. Remember that a non-Hausdorff space will necessarily have an ultrafilter that converges to more than one point. A countable space will have a sequence with that property. === Subject: Re: Verification: Sequences in the cofinite topology > Wrong. If a sequence actually has infinitely many points in its range, > then it converges to every point. If a sequence has only a finite > range than one or more points is repeated infinitely often and it > converges to that or those point(s) and to no other. Wrong. The original poster is completely correct, except perhaps in assuming that a classic textbook would not pose an exercise with a superfluous assumption. > Remember that a non-Hausdorff space will necessarily have an > ultrafilter that converges to more than one point. Right. > A countable space > will have a sequence with that property. Wrong again. Let N be the set of all natural numbers. Let U be some nonprincipal ultrafilter on N. Define a topology on N in which the open sets are the elements of U and the empty set. With this topology, N is a countable non-Hausdorff space in which no sequence converges to more than one point. Indeed, the only convergent sequences are the eventually-constant sequences, and each of those converges to only one point. Of course, the ultrafilter U converges to every point. === Subject: Re: Verification: Sequences in the cofinite topology > Kelley (which I read in self-study) poses the following exercise > (Problem 2.A): Consider a countable space with the cofinite topology. > What sequences converge to what points? It seems to me that the countability of the space is irrelevant. My answer: [S P O I L E R S P A C E] > A sequence converges if and only if there is at most point which appears > infinitely often. If there is one such point, the sequence converges to > that point only. Otherwise, the sequence converges to every point in > the space. In particular, if the space is finite, only eventually > constant sequences converge. (In this case, the topology is discrete.) True? -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor on the Internet and in Central New Jersey and Manhattan Wrong. If a sequence actually has infinitely many points in its range, > then it converges to every point. False. Take N as the space. Then 1, 2, 3, 1, 2, 4, 1, 2, 5, ... does not converge. > If a sequence has only a finite > range than one or more points is repeated infinitely often and it > converges to that or those point(s) and to no other. False. 1, 2, 1, 2, 1, ... does not converge. I believe Stephen's answer is correct. > Remember that a non-Hausdorff space will necessarily have an > ultrafilter that converges to more than one point. A countable space > will have a sequence with that property. === Subject: Re: Verification: Sequences in the cofinite topology > > <>Kelley (which I read in self-study) poses the following exercise >> (Problem 2.A): Consider a countable space with the cofinite topology. >> What sequences converge to what points? >> It seems to me that the countability of the space is irrelevant. My >> answer: >> [S >> P >> O >> I >> L >> E >> R >> S >> P >> A >> C >> E] >> A sequence converges if and only if there is at most point which appears >> infinitely often. If there is one such point, the sequence converges to >> that point only. Otherwise, the sequence converges to every point in >> the space. In particular, if the space is finite, only eventually >> constant sequences converge. (In this case, the topology is discrete.) >> True? >> Wrong. If a sequence actually has infinitely many points in its range, >> then it converges to every point. If a sequence has only a finite >> range than one or more points is repeated infinitely often and it >> converges to that or those point(s) and to no other. > Hmm. What is wrong with this argument? Suppose x appears infinitely often, and y ! = x. Then U = X {x} (where X is the space) is a neighborhood of y, and the sequence if not eventually in U. >Remember that a non-Hausdorff space will necessarily have an >ultrafilter that converges to more than one point. A countable space >will have a sequence with that property. > > I will relearn filters later. Stay tuned, maybe. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: CBL (The system your professor doesn't want you to see.) 4. If an axiomatizable system is consistent and decidable, then its refutable sentences coincide with its unprovable sentences, but the former is recursively enumerable while the latter is not. Really?? Take the theory which is built in a propositional language with P, Q the only atoms, and the usual connectives, and whose sole axiom is P. This theory is (i) consistent, (ii) decidable [since X will be a theorem just if P --> X is a tautology], but (iii) the sentence Q is evidently unprovable but not refutable. So it is NOT the case that if an axiomatizable system is consistent and decidable, then its refutable sentences coincide with its unprovable sentences. === Subject: Re: CBL (The system your professor doesn't want you to see.) > 4. If an axiomatizable system is consistent and decidable, then its > refutable sentences coincide with its unprovable sentences, but the > former is recursively enumerable while the latter is not. This is just bull on its face; if two collections of sentences of sentences, if ONE is r.e, then the other, must also be r.e. More to the point, if the system is decidable, then BY DEFINITION (REGARDLESS of whether it is consistent OR NOT), its provable sentences are just exactly the denials of its refutable ones, and EVERY sentence in the language is one or the other, and BOTH classes are recursively enumerable. BY DEFINITION. Implying that you don't know the definitions of the terms you are using. Can't you just go back to school for a year and get grounding in the basic concepts before presuming to pontificate further? === Subject: New Account Sign up for a PayPal Business or Premier account and join a Network of buyers. One in three online buyers in the U.S. has a PayPal account, and over 58,000 users worldwide sign up for PayPal each day. join today https://www.paypal.com/us/mrb/pal=PC2EPEREFB7A6 === Subject: Re: Cantor Confusion <45548ff2$0$97253$892e7fe2@authen.yellow.readfreenews.net> <45550b9a$0$97245$892e7fe2@authen.yellow.readfreenews.net> a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. The potentially infinite set N does not contain an infinite number. In this case (b) is correct. An actually infinite set is a set which has a cardinal number like omega. In set theory every set is actually infinite. So N is actually infinite. If its cardinal number omega is claimed to count the members of N, then there is a contradiction, because omega counts all n in N but does not belong to N while the initial segments of N are counted by natural numbers. This statement remains true as far as the natural numbers reach, i.e., for all natural numbers , i.e., for N. The contradiction becomes obvious in the matrix discussed or in the following few lines: Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a sequence with ordinal number omega + n. Here omega does not appear in the sequence. omega is nothing but the sequence n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number alike the finite ordinals. Therefore, omega must appear in the sequence: n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n What is the ordinal number of this sequence? === Subject: Re: Cantor Confusion > a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. > You will not understand it. But I'll try. The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. Sets cannot be merely potential. If not actual, then not sets. Whatever it is that WM is talking about when he misspeaks of potential sets, it is not sets. An actually infinite set is a set which has a cardinal number like > omega. In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. There is nothing in mathematics that says that the number of elements in a set must be a member of that set. WM wants to have the number of members of a natural to be a member of itself, which does not happen in ZF or NBG. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. The contradiction becomes obvious in the matrix discussed or in the > following few lines: Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a > sequence with ordinal number omega + n. > Here omega does not appear in the sequence. omega is nothing but the > sequence > n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number > alike the finite ordinals. Therefore, omega must appear in the > sequence: > n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n > What is the ordinal number of this sequence? Union( omega, { omega+1, omega+2, omega+3, ... omega+n+1}) === Subject: Re: Cantor Confusion a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. Sets cannot be merely potential. If not actual, then not sets. Whatever it is that WM is talking about when he misspeaks of potential > sets, it is not sets. Probably nothing that WM talks about is a set (where the word set has its standard mathematical meaning), but I wonder whether WM's notion of potential set isn't closer to the mathematical concept than is his notion of actual set. After all, in mathematics (b) is true, and WM says (b) is true if we change set to potential set. Not sure if WM says (b) is true if we change set to actual set. > An actually infinite set is a set which has a cardinal number like > omega. In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. There is nothing in mathematics that says that the number of elements > in a set must be a member of that set. > WM wants to have the number of members of a natural to be a member of > itself, which does not happen in ZF or NBG. Kind of an odd thing to want, since 5 isn't in {0,1,2,3,4}. -- Marcus === Subject: Re: Cantor Confusion <45548ff2$0$97253$892e7fe2@authen.yellow.readfreenews.net> <45550b9a$0$97245$892e7fe2@authen.yellow.readfreenews.net> not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. > You will not understand it. But I'll try. Not a very clear attempt. Try again, starting your answer with something like a is true and b is false or b is true and a is false or it depends on which assumptions you make. The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. An actually infinite set is a set which has a cardinal number like > omega. You are confusing cardinals and ordinals. For finite sets they are the same. For infinite sets they are different. You want ordinals. This cardinal (actually ordinal) number is not an element of the set > In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. BZZZ The fact that something is true for all sets of the form {1,2,3,...n} where n is a finite natural number, does not mean that it is true for N. For example: For every set B of the form {1,2,3,...,n} there exists a finite natural number m such that B={1,2,3,...,m}. However there is no finite natural number m such that N={1,2,3,...,m} [This is true even if we assume that N consists of exaclty those natural numbers that will be named during the lifetime of the universe]. The contradiction becomes obvious in the matrix discussed or in the > following few lines: Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a > sequence with ordinal number omega + n. > Here omega does not appear in the sequence. omega is nothing but the > sequence > n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number > alike the finite ordinals. Therefore, omega must appear in the > sequence: > n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n > What is the ordinal number of this sequence? > omega + n + 1 This is the same as the ordinal number of 1,2,3,...,omega,omega+1,omega+2,...,omega+n You assume that there is only one representation of an ordinal and get an immediate contradiction because there is in fact more than one representation of an ordinal. - William Hughes === Subject: Re: Cantor Confusion a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. Not a very clear attempt. It was clearer than most of WM's attempts. Be encouraging! > Try again, starting your answer with something like a is true and b > is false > or b is true and a is false or it depends on which assumptions > you make. -- Marcus === Subject: Re: Cantor Confusion <45548ff2$0$97253$892e7fe2@authen.yellow.readfreenews.net> <45550b9a$0$97245$892e7fe2@authen.yellow.readfreenews.net> a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. Not a very clear attempt. It was clearer than most of WM's attempts. Now that's what I call damning with faint praise! > Be encouraging! Ahw! do I havta? Sigh! Ok. - William Hughes === Subject: Re: Cantor Confusion a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. An actually infinite set is a set which has a cardinal number like > omega. In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. The contradiction becomes obvious in the matrix discussed or in the > following few lines: Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a > sequence with ordinal number omega + n. > Here omega does not appear in the sequence. omega is nothing but the > sequence > n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number > alike the finite ordinals. Therefore, omega must appear in the > sequence: > n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n > What is the ordinal number of this sequence? It appears that you are saying that the terms infinite set and infinite number are not meaningful. There are only potentially infinite sets and actually infinite sets. Is that correct? Do you agree with the following statements? c. It is possible to have a potentially infinite set of numbers that does not contain an infinite number. d. The set of all natural numbers, i.e., {1,2,3,...}, is actually infinite. e. An infinite number is a number other than the natural numbers. f. An actually infinite set must contain an infinite number. -- Marcus === Subject: Re: Cantor Confusion as many philosophers, and others, who will claim that mathematics > belongs to philosophy. Did I say that *all* philosophy belongs to mathematics? The philosophy of mathematics belongs to mathematics. === Subject: Re: Cantor Confusion They were discussing mathematics. Philosophy belongs to mathematics. There may be some benighted mathematicians who claim so, but there are > as many philosophers, and others, who will claim that mathematics > belongs to philosophy. Did I say that *all* philosophy belongs to mathematics? The philosophy > of mathematics belongs to mathematics. > A statement like Philosophy belongs to mathematics does not seem to leave much room for any part of philosophy to belong anywhere else. Besides which, there are philosophers, and busybodies like WM, who seem to believe that mathematics is too important to be left to mathematicians. === Subject: Re: Cantor Confusion mathematicians who know what a mathematician should understand by > limit, number, grow. Please post their definitions of the words. Try to get a clue of Cantor's second creation principle. Then you will know these things. Your changing finished entity to finished infinity is a bit of a > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > comment, not discussing mathematics. They were discussing mathematics. Philosophy belongs to mathematics. That's nonsense. That's ignorance. So, it remains true that finished > infinity does not have a mathematical meaning. Ironically you are even right, for once, though you don't know it. > term in a mathematical discussion, it is incumbent on you to define it. This term has been used in a mathematical discussion by Hrbacek and > Jech. I don't see why I shouldn't do the same. Several reasons: They were discussing philosophy of mathematics, not > mathematics itself. You don't understand what they were saying. They said: Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process (such as N) should not be treated as a finished entity. > And, you > haven't given a definition of the term. I thought you'd know transitivity: Take the expression finshed entity where entity is a variable like the set X in set theory. Now replace this variable by a fixed set like N, which in mathematics, is an infinite process. This leads to finished infinite process, abbreviated by finished infinity. Was this simple enough for you to understand? >You can use any term you wish, > but only if you define it. That is one of the rules of the game. Another rule is that every player should at least know 10 terms, including the application of transitivity. > If you > don't want to play, then don't, but you can't unilaterally change the > rules. Which mathematics do you allude to? That which is taught in school, explained in textbooks, published in > journals, and discussed by some in this newsgroup. === Subject: Re: Cantor Confusion > You carried out a survey? Or is that your guess? I know some > mathematicians who know what a mathematician should understand by > limit, number, grow. Please post their definitions of the words. Try to get a clue of Cantor's second creation principle. Then you will > know these things. We already know what current mathematics says about limit and number, What we do not know, and cannot find out from anybody but WM, what WM means by them. Which mathematics do you allude to? That which is taught in school, explained in textbooks, published in > journals, and discussed by some in this newsgroup. > What you teach in school, write in textbooks and publish in journals, is not, by any evidence produced here, liable to be relevant to mathematics. === Subject: Re: Cantor Confusion You carried out a survey? Or is that your guess? I know some > mathematicians who know what a mathematician should understand by > limit, number, grow. Please post their definitions of the words. Try to get a clue of Cantor's second creation principle. Then you will > know these things. I suppose that means you are unable to define them. > Your changing finished entity to finished infinity is a bit of a > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > comment, not discussing mathematics. They were discussing mathematics. Philosophy belongs to mathematics. That's nonsense. That's ignorance. If your that's refers to the same thing as my that's, then I agree. > So, it remains true that finished > infinity does not have a mathematical meaning. Ironically you are even right, for once, though you don't know it. Notice the quotes. You really should learn to read more carefully. > This term has been used in a mathematical discussion by Hrbacek and > Jech. I don't see why I shouldn't do the same. Several reasons: They were discussing philosophy of mathematics, not > mathematics itself. You don't understand what they were saying. They said: Some mathematicians object to the Axiom of Infinity on the > grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity. Indeed. If people *object* to an axiom, that is philosophy. Everyone is welcome to choose their own axioms. > And, you haven't given a definition of the term. I thought you'd know transitivity: Take the expression finshed entity > where entity is a variable like the set X in set theory. Now replace > this variable by a fixed set like N, which in mathematics, is an > infinite process. This leads to finished infinite process, > abbreviated by finished infinity. Was this simple enough for you to > understand? Amusing, but that just shows how you can make up new terms. You still haven't provided a defintion. It appears you are saying that finished infinite process and finished infinty are synonyms. Fine. But, you haven't defined either one. Please define them. > Which mathematics do you allude to? That which is taught in school, explained in textbooks, published in > journals, and discussed by some in this newsgroup. > Irrelevant since that isn't what my sentence means in English. -- Marcus === Subject: Re: Cantor Confusion argument can be expressed in the language of set theory and uses only > first order logic applied to the axioms of set theory? I need no belief. If all consistent mathematics can be expressed in ZFC, then my argument can be expressed in ZFC too. === Subject: Re: Cantor Confusion argument can be expressed in the language of set theory and uses only > first order logic applied to the axioms of set theory? I need no belief. If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. First you said that your argument is about set theory. Then said your argument has nothing to do with set theory. Now you're saying that if all consistent mathematics can be expressed in ZFC then your argument can be put in ZFC. If you would at least say clearly just what your argument about trees is supposed to be about - to what theory it is supposed to apply - then that would help. MoeBlee === Subject: Re: Cantor Confusion > So let's be definite here. Do you have a strong belief that your > argument can be expressed in the language of set theory and uses only > first order logic applied to the axioms of set theory? I need no belief. But you have all sorts of beliefs for which your only justification is other beliefs. > If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. That presumes, contrary to present evidence, that your argument is consistent with all consistent mathematics. === Subject: Re: Cantor Confusion So let's be definite here. Do you have a strong belief that your > argument can be expressed in the language of set theory and uses only > first order logic applied to the axioms of set theory? I need no belief. If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. The most dubious assumption in that statement is that your argument is consistent mathematics. -- Marcus === Subject: Re: Cantor Confusion write: a reasonable way to make this conform to a Platonistic point of > view is to look at the universe of all sets not as a fixed entity but > as an entity capable of growing. Why should they write reasonable if they found it unreasonable? Because, as YOU SKIPPED RESPONDING to my explanation for you, and as > would be clear had you bothered to read and UNDERSTAND the remarks in > that section of the book, theyr'e NOT claiming that sets in general > grow or are indeterminate. They are pointing out that if we adopt > different axioms, then the entire universe of all sets is something > different depending on which axioms we adopt. I will not discuss this topic further. Only a last remark for the lurkers: In fact, there are axiom systems which tolerate the set of all sets. Therefore, if different axiomatic systems were discussed by Fraenkel et al., no capability of growing was required at all. Further there is no talk about something different sets but about growing sets. Therefore exactly this growing of sets is meant as a reasonable view. Unfortunately MoeBlee is far from understanding anything behind his formalism. The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works. (Bertrand Russell) . === Subject: Re: Cantor Confusion p. 118, considering that the set of all sets does not exist, they > write: a reasonable way to make this conform to a Platonistic point of > view is to look at the universe of all sets not as a fixed entity but > as an entity capable of growing. Why should they write reasonable if they found it unreasonable? Because, as YOU SKIPPED RESPONDING to my explanation for you, and as > would be clear had you bothered to read and UNDERSTAND the remarks in > that section of the book, theyr'e NOT claiming that sets in general > grow or are indeterminate. They are pointing out that if we adopt > different axioms, then the entire universe of all sets is something > different depending on which axioms we adopt. I will not discuss this topic further. Only a last remark for the > lurkers: In fact, there are axiom systems which tolerate the set of all > sets. Therefore, if different axiomatic systems were discussed by > Fraenkel et al., no capability of growing was required at all. That's a non sequitur. They discussed the options of different axioms and mentioned that the results are that we can prove the existence of certain sets in some theories and in other theories we can't prove the existence of those sets. > Further there is no talk about something different sets but about > growing sets. No, growing in the AUTHOR'S own scare quotes to convey a notion you ignore as you ignore the entire context of the section of that book. > Therefore exactly this growing of sets is meant as a > reasonable view. The authors expound no endorsement that sets themselves grow or indeterminate. The axiom of extensionality is used throughout the book and the author's comments in that particular section do not contradict that a set is compelely determined by its members; rather, the authors remark that we get a different UNIVERSE of sets depending on what axioms we adopt, so that the UNIVERSE (not just sets in general) grows (notice scare quotes) depending on what axioms we adopt. Grow is used in scare quotes by the authors to convey that we get different universes from different axioms; and there is NOTHING in the book to suggest that the authors endorse a notion that sets themselves grow over time or through any process or that sets do not have a determinate membership as per the axiom of extensionality. > Unfortunately MoeBlee is far from understanding > anything behind his formalism. The formalists are like a watchmaker > who is so absorbed in making his watches look pretty that he has > forgotten their purpose of telling the time, and has therefore omitted > to insert any works. (Bertrand Russell) I've never posted a committment to an UNDEFINED formalism, and not to any such extreme formalism that holds that mathematics is only symbol manipulation. MoeBlee === Subject: Re: Cantor Confusion On Fri, 10 Nov 2006 18:19:05 -0500, Marcus > WM's problem is that he does not, perhaps cannot, understand that there >> is a difference in the way infinite sets and infinite processes work >> from the way that finite sets and finite processes work. I think he has a bigger problem. He doesn't seem to agree that there are >infinite sets. It is very strange. You mean if the editorial we agree that there are infinite sets there are infinite sets? You have a very curious sense of words in others but not in yourself. You claim to be able to prove things without being able to prove they're true. And what if one doesn't agree that there are infinite sets? Are you going to prove they're true? Clearly like most trained in the modern mathematical arts you don't take words seriously enough to form critical thoughts. ~v~~ === Subject: Re: Cantor Confusion > On Fri, 10 Nov 2006 18:19:05 -0500, Marcus > >I think he has a bigger problem. He doesn't seem to agree that there are >infinite sets. It is very strange. You mean if the editorial we agree that there are infinite sets > there are infinite sets? I don't know what you mean. What does there are mean in this context? > You have a very curious sense of words in > others but not in yourself. You claim to be able to prove things > without being able to prove they're true. I'm using prove in its mathematical sense. I don't know what you mean by prove they're true. I suspect the meaning of the word prove is different in the two senses. > And what if one doesn't agree that there are infinite sets? If you mean you want to use different axioms for your mathematics, then you are welcome to. It that's not what you mean, then I don't know what you mean. What does there are mean in your sentence? > Are you going to prove they're true? I don't understand the question. > Clearly like most trained in the modern mathematical arts > you don't take words seriously enough to form critical thoughts. I take words seriously enough to be sure that I and the person I am conversing with are using the words with the same meaning before I jump to any conclusions. Mathematics is a language. People who learn the language communicate using that language. People who are not fluent in the language may misunderstand what is being said, since many of the same words are used as in English. However, the meaning of many words is different in the two languages. -- Marcus === Subject: Re: Cantor Confusion On Fri, 10 Nov 2006 17:42:34 -0500, Marcus >> >> WM's logic (if we can call it that) >> >> Please do not! >> You have no balls at noon. >> You have Tristram Shandy complete his diary. >> You have a diagonal longer than every line. >> You can make two balls of one. >> You have a countable model of an uncountable theory. >> >> I would be dismayed if you found any parallel between my thinking and >> your logic. You need not be dismayed, since so far we haven't found any evidence >that you think at all. Look who's talking. Your thinking is no evidence one way or the other. ~v~~ === Subject: Re: Cantor Confusion On Fri, 10 Nov 2006 17:33:16 -0500, Marcus >> Exactly this is done by Cantor's definition given above: It is allowed >> to understand the new number omega as limit to which the (natural) >> numbers n grow. This is a definition, at least for those >> mathematicians who know what limit, number, grow means. >> I seriously doubt that Cantor considered it to be a definition. If he >> did, he wouldn't have said, It is allowed to understand. Regardless, >> it is not a definition by modern standards. >> >> Who judges what modern standards are, in your opinion? A silly question. You haven't even bothered to learn the current meaning >of the word definition. If you would actually read a textbook or take >a math course at a university, you might learn something. Curious coming from one whose definitions aren't demonstrably true. ~v~~ === Subject: Re: Cantor Confusion > On Fri, 10 Nov 2006 17:33:16 -0500, Marcus > >> Exactly this is done by Cantor's definition given above: It is allowed >> to understand the new number omega as limit to which the (natural) >> numbers n grow. This is a definition, at least for those >> mathematicians who know what limit, number, grow means. >> I seriously doubt that Cantor considered it to be a definition. If he >> did, he wouldn't have said, It is allowed to understand. Regardless, >> it is not a definition by modern standards. >> >> Who judges what modern standards are, in your opinion? A silly question. You haven't even bothered to learn the current meaning >of the word definition. If you would actually read a textbook or take >a math course at a university, you might learn something. Curious coming from one whose definitions aren't demonstrably true. Care to define demonstrably true? -- Marcus === Subject: Re: Cantor Confusion On Sat, 11 Nov 2006 15:41:39 -0500, Marcus >> On Fri, 10 Nov 2006 17:33:16 -0500, Marcus >> Exactly this is done by Cantor's definition given above: It is allowed > to understand the new number omega as limit to which the (natural) > numbers n grow. This is a definition, at least for those > mathematicians who know what limit, number, grow means. I seriously doubt that Cantor considered it to be a definition. If he > did, he wouldn't have said, It is allowed to understand. Regardless, > it is not a definition by modern standards. Who judges what modern standards are, in your opinion? >>A silly question. You haven't even bothered to learn the current meaning >>of the word definition. If you would actually read a textbook or take >>a math course at a university, you might learn something. >> >> Curious coming from one whose definitions aren't demonstrably true. Care to define demonstrably true? Already have in the root post to the thread Epistemology 201: The Science of Science. Your main problem seems to be a willingness to assume the truth of whatever you're talking without demonstration. You repeatedly use the term true in a collateral reply to Gene Ward Smith while unable to demonstrate the truth of what you claim to prove. 'Tis a puzzlement indeed for one trained in precise and exact meanings of modern mathematics. ~v~~ === Subject: Re: Cantor Confusion > On Sat, 11 Nov 2006 15:41:39 -0500, Marcus >> On Fri, 10 Nov 2006 17:33:16 -0500, Marcus >>A silly question. You haven't even bothered to learn the current meaning >>of the word definition. If you would actually read a textbook or take >>a math course at a university, you might learn something. >> >> Curious coming from one whose definitions aren't demonstrably true. Care to define demonstrably true? Already have in the root post to the thread Epistemology 201: The > Science of Science. Your main problem seems to be a willingness to > assume the truth of whatever you're talking without demonstration. You > repeatedly use the term true in a collateral reply to Gene Ward > Smith while unable to demonstrate the truth of what you claim to > prove. 'Tis a puzzlement indeed for one trained in precise and exact > meanings of modern mathematics. The word true has different meanings in different contexts. Its meaning in mathematics is somewhat unusual. Usually in mathematics, it just means provable. With this meaning, it makes no sense to say something is true, but not provable. It also makes no sense to ask whether a definition is true, since we don't prove definitions. They are simply abbreviations. In logic, true has a technical meaning. Outside mathematics, true has an entirely different meaning. I don't know what you mean by demonstrably true, but I am reasonably sure you don't mean what the word true means in mathematics. I'm not particularly interested in reading old threads. If you care to give a concise explanation again, I'll read it. Mathematics is a language. If you take Mathematics and assume it is English, you will misunderstand what the writer means. -- Marcus === Subject: Re: Cantor Confusion On Fri, 10 Nov 2006 17:48:43 -0500, Marcus >> It's amazing that 21st century mathematikers cannot analyze the truth >> of contentions without playing cards:the race card, the religion card, >> the homeland card, etc. My last name is Zick. I'm American. Period. >> When it comes to mathematics I'm universalist. A pity you don't know any mathematics, though. I don't know any mathematics you know. And vice versa. I try to stick to the truth and let math fall where it may. Others prefer to assume the truth of what they can't demonstrate. It's exactly how they got into playing cards instead of mathematics in the first place. A pity. >> Patriotism and religion >> among others are the last resort of scoundrels. Perhaps I should add set theory and modern math to the list. > In mathematics these >> kinds of issues are historical anachronisms unless you can't establish >> truth in universal terms which it would seem mathematikers can't. ~v~~ === Subject: Re: Cantor Confusion >> Your way or the highway huh, Moe(x). >If he has an argument that he thinks can be put in set theory, then I'm >interested in his argument; If he doesn't think his argument can be put >in set theory, then I'm not interested. He can post about his argument >all he wants, but I'm not obligated to study his argument. No one suggests you are. The problem I see is that one might cast an argument in such terms as are acceptable to you and still not satisfy exactly the same criteria on the part of others. I mean unless you are the generally acknowledged expert in the field. Otherwise it would look to me like you're just trying to take control of the discussion in terms you find acceptable whether or not others do. Let me see if I can simplify how the issue is or ought to be argued. You posit certain properties of an infinite however you define it. So the question then becomes whether your claim is or can be true. Now one way to show it's actually true would be to produce some entity with the properties you posit of an infinite. Otherwise you'd have to find some other way to get at the truth of what you claim unless you just intend to claim it's true because you or others say so. Now as I understand WM's argument he suggests you can never actually produce any physical infinite because the physical universe is finite. However he then apparently concludes from this that there can be no infinites at all because there can be no physical infinites if the universe is finite. Now personally I find most of the arguments disingenuous on both sides.And I see no special merit to your definition for the properties of infinites you recommend to the exclusion of others. But they are specific properties you can't demonstrate through exemplification so if you wish to show that the characteristics you assign to infinites can be true you have to approach the proof some other way than empirical exemplification. ~v~~ === Subject: Re: Cantor Confusion <1o7cl2phksk2q7ng3dibc0ogfg25vf04v1@4ax.com >> Your way or the highway huh, Moe(x). >If he has an argument that he thinks can be put in set theory, then I'm >interested in his argument; If he doesn't think his argument can be put >in set theory, then I'm not interested. He can post about his argument >all he wants, but I'm not obligated to study his argument. No one suggests you are. The problem I see is that one might cast an > argument in such terms as are acceptable to you and still not satisfy > exactly the same criteria on the part of others. I mean unless you are > the generally acknowledged expert in the field. Otherwise it would > look to me like you're just trying to take control of the discussion > in terms you find acceptable whether or not others do. Nothing of the kind. HE suggested to ME that I can see if his argument can be put into set theory. If I am take MY time and effort to do that, then I have every prerogative to set my own terms for doing it. > Let me see if I can simplify how the issue is or ought to be argued. > You posit certain properties of an infinite however you define it. So > the question then becomes whether your claim is or can be true. Now > one way to show it's actually true would be to produce some entity > with the properties you posit of an infinite. Otherwise you'd have to > find some other way to get at the truth of what you claim unless you > just intend to claim it's true because you or others say so. Now as I understand WM's argument he suggests you can never actually > produce any physical infinite because the physical universe is finite. WM and were talking about his tree argument. The finititude of the physical universe is a separate subject. > However he then apparently concludes from this that there can be no > infinites at all because there can be no physical infinites if the > universe is finite. Now personally I find most of the arguments disingenuous on both > sides.And I see no special merit to your definition for the properties > of infinites you recommend to the exclusion of others. But they are > specific properties you can't demonstrate through exemplification so > if you wish to show that the characteristics you assign to infinites > can be true you have to approach the proof some other way than > empirical exemplification. MoeBlee === Subject: Re: Cantor Confusion > >> Your way or the highway huh, Moe(x). >If he has an argument that he thinks can be put in set theory, then I'm >interested in his argument; If he doesn't think his argument can be put >in set theory, then I'm not interested. He can post about his argument >all he wants, but I'm not obligated to study his argument. No one suggests you are. The problem I see is that one might cast an > argument in such terms as are acceptable to you and still not satisfy > exactly the same criteria on the part of others. I mean unless you are > the generally acknowledged expert in the field. Otherwise it would > look to me like you're just trying to take control of the discussion > in terms you find acceptable whether or not others do. Let me see if I can simplify how the issue is or ought to be argued. > You posit certain properties of an infinite however you define it. So > the question then becomes whether your claim is or can be true. What does is or can be true mean? In mathematics, we are normally only concerned with provability (unless discussing philosphy). > Now > one way to show it's actually true would be to produce some entity > with the properties you posit of an infinite. Otherwise you'd have to > find some other way to get at the truth of what you claim unless you > just intend to claim it's true because you or others say so. Now as I understand WM's argument he suggests you can never actually > produce any physical infinite because the physical universe is finite. > However he then apparently concludes from this that there can be no > infinites at all because there can be no physical infinites if the > universe is finite. That doesn't seem to be what WM is saying. He seems to be saying that the notion of a completed infinity leads to either absurdities or contradictions. Perhaps he thinks the way to avoid these absurdities is to only consider things that can be physically produced. > Now personally I find most of the arguments disingenuous on both > sides. what is a standard mathematical argument that you find disingenuous? > And I see no special merit to your definition for the properties > of infinites you recommend to the exclusion of others. But they are > specific properties you can't demonstrate through exemplification so > if you wish to show that the characteristics you assign to infinites > can be true you have to approach the proof some other way than > empirical exemplification. -- Marcus === Subject: Re: Cantor Confusion > What surprises me is that opposition against it in this way comes from > especially the German posters in this group. They all want to stress > the difference between actual infinity and potential infinity, that > are indeed not mathematical terms, and most other posted do not know > the difference. Is it still the old religious issue? I think so, > based on the postings of at least one of the German posters. If Cantor's last name had been Shickelgrueber or von Richthoffen, none > of this would be written. > Perhaps. Though I do not think so. But what I am wondering about is the > preoccupation with actual infinity and potential infinity with some > of the posters. I think you are objecting to my German posters. > I can shrink this to German physicists. In this newsgroup there are > actually three posters that refer to these terms. You may try to find out > who those three are. I think a lot of this opposition would go away if the word transfinite instead of infinite had been used to describe a set that can be put into a one-to-one correspondence with a proper subset of itself. The word infinite sends people down strange philosophical paths, as does the word infinity despite the fact that it is not really even used in set theory. Noone would argue about transfinity. Stephen === Subject: Re: Cantor Confusion >> What surprises me is that opposition against it in this way comes from >> especially the German posters in this group. They all want to stress >> the difference between actual infinity and potential infinity, that >> are indeed not mathematical terms, and most other posted do not know >> the difference. Is it still the old religious issue? I think so, >> based on the postings of at least one of the German posters. >> >> If Cantor's last name had been Shickelgrueber or von Richthoffen, none >> of this would be written. >> Perhaps. Though I do not think so. But what I am wondering about is the >> preoccupation with actual infinity and potential infinity with some >> of the posters. I think you are objecting to my German posters. >> I can shrink this to German physicists. In this newsgroup there are >> actually three posters that refer to these terms. You may try to find out >> who those three are. > I think a lot of this opposition would go away if the word > transfinite instead of infinite had been used to describe > a set that can be put into a one-to-one correspondence with > a proper subset of itself. The word infinite sends people > down strange philosophical paths, as does the word infinity > despite the fact that it is not really even used in set theory. > Noone would argue about transfinity. > Stephen By the way, what is the German word for transfinite? I had thought that native German would have less trouble with infinite given that Unendlich obviously means unending whereas infinite is derived from Latin and that gives it mysterious properties from the get go. :) I always thought it was rather sensible of the Germans to construct new German words from old German words, instead of the English tradition of plundering other languages. :) Stephen === Subject: Re: Cantor Confusion >> What surprises me is that opposition against it in this way comes from >> especially the German posters in this group. They all want to stress >> the difference between actual infinity and potential infinity, that >> are indeed not mathematical terms, and most other posted do not know >> the difference. Is it still the old religious issue? I think so, >> based on the postings of at least one of the German posters. >> >> If Cantor's last name had been Shickelgrueber or von Richthoffen, none >> of this would be written. > Perhaps. Though I do not think so. But what I am wondering about is the >> preoccupation with actual infinity and potential infinity with some >> of the posters. I think you are objecting to my German posters. >> I can shrink this to German physicists. In this newsgroup there are >> actually three posters that refer to these terms. You may try to find out >> who those three are. I think a lot of this opposition would go away if the word >transfinite instead of infinite had been used to describe >a set that can be put into a one-to-one correspondence with >a proper subset of itself. The word infinite sends people >down strange philosophical paths, as does the word infinity >despite the fact that it is not really even used in set theory. >Noone would argue about transfinity. Oh I dunno, Stephen. I certainly might. A rose by any other name. Of course you might duck the issues here with such a strategem of verbal regression. It seems every time modern mathematikers get into trouble conceptually they just adopt a different name and pretend the problem has gone away. On another thread I call it The Transfinite Zen Abacus. The only reason mathematikers don't define infinity is that they can't and prefer to sublimate the problem by referring to infinites instead. ~v~~ === Subject: Re: Cantor Confusion > I think a lot of this opposition would go away if the word > transfinite instead of infinite had been used to describe > a set that can be put into a one-to-one correspondence with > a proper subset of itself. The word infinite sends people > down strange philosophical paths, as does the word infinity > despite the fact that it is not really even used in set theory. > Noone would argue about transfinity. You could be right. Although, it seems unfair of the cranks to dictate what words mathematicians can appropriate. It is hard to make up good names. We have enough names like second category as it is. -- Marcus === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > I think a lot of this opposition would go away if the word > transfinite instead of infinite had been used to describe > a set that can be put into a one-to-one correspondence with > a proper subset of itself. The word infinite sends people > down strange philosophical paths, as does the word infinity > despite the fact that it is not really even used in set theory. > Noone would argue about transfinity. > > You could be right. Although, it seems unfair of the cranks to dictate > what words mathematicians can appropriate. It is hard to make up good > names. We have enough names like second category as it is. I think stephen is right. But as you say, there are so many other words that mathematicians do use to which there is opposition. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion numbers. And this definition, given by Cantor and translated by myself > is given above. There is no point to draw but only to understand > Cantor's *definition* (or not). Okay, so for you there is no point to draw about your quote other than > understanding Cantor's own writings. And, I'll add that in this > particular instance, Cantor, as you translated, is not in conflict with > the theorem of current set theory that omega is a limit ordinal and the > first oridinal that is greater than all natural numbers. That is what I said. omega is a limit. In modern set theory there are > limits. Moe said that omega is a limit ordinal. He did not say that omega is > a limit. The two statements are not the same. Moe said Cantor, as you translated, is not in conflict with the theorem of current set theory. Cantor said omega can be understood as a limit. Is transitivity unknown in your personal branch of mathematics? > Do you really believe the > two statements are the same? Or, are you trolling? A bottle of beer is not a beer bottle. But both notions have to do with beer and with bottles. Why is a limit ordinal called so if it is not the limit of some set of ordinals? > Further this (Cantor's) definition supports my definition of > definition. Regardless of whether Cantor meant his remark to be a definition, you > can't change what the word definition means in modern mathematics. It Again: What you think or have learned or think to have learned from some books concerning modern mathematics need not be a law enacted by God or by a majority of mathematicians. Further mathematics has nothing to do with democracy. > is rude to use a common word with a personal meaning and not point this > out. If someone asks you for a definition, etiquette and honesty > requires you to say, I'm sorry, but I do not know what you mean. In mathematics it is practical to give names to various particular properties and objects, i.e., to define new properties. Mathematics without definitions would be possible, but exceedingly clumsy. > The phrases actually existing and cannot exist are not defined. Eetiquette and honesty requires you to say, I'm sorry, but I do not know what the definitions of these words are. And then you should attach a list of words you know. It can't be too long. So I will look whether there are words which could be used to explain actually existing and cannot exist. === Subject: Re: Cantor Confusion > It is now to be shown how one is lead to the definition of the new > numbers. And this definition, given by Cantor and translated by myself > is given above. There is no point to draw but only to understand > Cantor's *definition* (or not). Okay, so for you there is no point to draw about your quote other than > understanding Cantor's own writings. And, I'll add that in this > particular instance, Cantor, as you translated, is not in conflict with > the theorem of current set theory that omega is a limit ordinal and the > first oridinal that is greater than all natural numbers. That is what I said. omega is a limit. In modern set theory there are > limits. Moe said that omega is a limit ordinal. He did not say that omega is > a limit. The two statements are not the same. Moe said Cantor, as you translated, is not in conflict with the > theorem of current set theory. Cantor said omega can be understood as > a limit. Cantor said understood. That is not the same as saying is. Regardless, Cantor's papers are very old and predate the modern formulations. > Do you really believe the > two statements are the same? Or, are you trolling? A bottle of beer is not a beer bottle. But both notions have to do with > beer and with bottles. Why is a limit ordinal called so if it is not > the limit of some set of ordinals? Because the notions of limit ordinal and of limit are related to intuitive ideas that are similar enough that people decided to use similar terminology to name them. However, you can't deduce anything from the name. You must use the precise, rigorous definition. Thinking that you can use the name is a common fallacy. The fact remains that a limit ordinal is not the limit of a set of ordinals. A limit ordinal is an ordinal other than zero that is not a successor ordinal. This definition is given by both Halmos and Kunen in their books on set theory. > is rude to use a common word with a personal meaning and not point this > out. If someone asks you for a definition, etiquette and honesty > requires you to say, I'm sorry, but I do not know what you mean. In mathematics it is practical to give names to various particular > properties and objects, i.e., to define new properties. Mathematics > without definitions would be possible, but exceedingly clumsy. Yes, but the problem is that you fail to state what definitions you are using. > The phrases actually existing and cannot exist are not defined. > Eetiquette and honesty requires you to say, I'm sorry, but I do not > know what the definitions of these words are. And then you should > attach a list of words you know. It can't be too long. So I will look > whether there are words which could be used to explain actually > existing and cannot exist. Amusing. I (and others) have repeatedly asked you what you mean. I have suggested that you use the terminology in Halmos's Naive Set Theory or in Kunen's Set Theory. If you don't like these books, then pick a different one. However, Cantor's papers are much too old to be used for a discussion today. -- Marcus === Subject: Re: Cantor Confusion numbers. And this definition, given by Cantor and translated by myself > is given above. There is no point to draw but only to understand > Cantor's *definition* (or not). Okay, so for you there is no point to draw about your quote other than > understanding Cantor's own writings. And, I'll add that in this > particular instance, Cantor, as you translated, is not in conflict with > the theorem of current set theory that omega is a limit ordinal and the > first oridinal that is greater than all natural numbers. That is what I said. omega is a limit. In modern set theory there are > limits. Moe said that omega is a limit ordinal. He did not say that omega is > a limit. The two statements are not the same. Moe said Cantor, as you translated, is not in conflict with the > theorem of current set theory. Cantor said omega can be understood as > a limit. Cantor said understood. That is not the same as saying is. Cantor also said: is. Therefore it is clear that here he meant is. > Regardless, Cantor's papers are very old and predate the modern > formulations. That does not mean that the modern formulations are better. Do you really believe the > two statements are the same? Or, are you trolling? A bottle of beer is not a beer bottle. But both notions have to do with > beer and with bottles. Why is a limit ordinal called so if it is not > the limit of some set of ordinals? Because the notions of limit ordinal and of limit are related to > intuitive ideas that are similar enough that people decided to use > similar terminology to name them. However, you can't deduce anything > from the name. You must use the precise, rigorous definition. Thinking > that you can use the name is a common fallacy. The fact remains that a limit ordinal is not the limit of a set of > ordinals. A limit ordinal is an ordinal other than zero that is not a > successor ordinal. This definition is given by both Halmos and Kunen in > their books on set theory. is rude to use a common word with a personal meaning and not point this > out. If someone asks you for a definition, etiquette and honesty > requires you to say, I'm sorry, but I do not know what you mean. In mathematics it is practical to give names to various particular > properties and objects, i.e., to define new properties. Mathematics > without definitions would be possible, but exceedingly clumsy. Yes, but the problem is that you fail to state what definitions you are > using. The phrases actually existing and cannot exist are not defined. Eetiquette and honesty requires you to say, I'm sorry, but I do not > know what the definitions of these words are. And then you should > attach a list of words you know. It can't be too long. So I will look > whether there are words which could be used to explain actually > existing and cannot exist. Amusing. I (and others) have repeatedly asked you what you mean. I have > suggested that you use the terminology in Halmos's Naive Set Theory or > in Kunen's Set Theory. If you don't like these books, then pick a > different one. However, Cantor's papers are much too old to be used for > a discussion today. No. The hidden errors can better be recognized at the roots. === Subject: Re: Cantor Confusion It is now to be shown how one is lead to the definition of the > new > numbers. And this definition, given by Cantor and translated by > myself > is given above. There is no point to draw but only to understand > Cantor's *definition* (or not). Okay, so for you there is no point to draw about your quote other > than > understanding Cantor's own writings. And, I'll add that in this > particular instance, Cantor, as you translated, is not in conflict > with > the theorem of current set theory that omega is a limit ordinal and > the > first oridinal that is greater than all natural numbers. That is what I said. omega is a limit. In modern set theory there are > limits. Moe said that omega is a limit ordinal. He did not say that omega is > a limit. The two statements are not the same. Moe said Cantor, as you translated, is not in conflict with the > theorem of current set theory. Cantor said omega can be understood as > a limit. Cantor said understood. That is not the same as saying is. Cantor also said: is. Therefore it is clear that here he meant is. Regardless, Cantor's papers are very old and predate the modern > formulations. That does not mean that the modern formulations are better. Nor does it mean that they are worse. But since mathematics generally retains what is found good, and dumps what is found wanting, the odds certainly favor the modern formulations as being better. Do you really believe the > two statements are the same? Or, are you trolling? A bottle of beer is not a beer bottle. But both notions have to do with > beer and with bottles. Why is a limit ordinal called so if it is not > the limit of some set of ordinals? Why is the field of rationals so different from a field of oats? Because the notions of limit ordinal and of limit are related to > intuitive ideas that are similar enough that people decided to use > similar terminology to name them. However, you can't deduce anything > from the name. You must use the precise, rigorous definition. Thinking > that you can use the name is a common fallacy. The fact remains that a limit ordinal is not the limit of a set of > ordinals. A limit ordinal is an ordinal other than zero that is not a > successor ordinal. This definition is given by both Halmos and Kunen in > their books on set theory. is rude to use a common word with a personal meaning and not point this > out. If someone asks you for a definition, etiquette and honesty > requires you to say, I'm sorry, but I do not know what you mean. In mathematics it is practical to give names to various particular > properties and objects, i.e., to define new properties. Mathematics > without definitions would be possible, but exceedingly clumsy. Yes, but the problem is that you fail to state what definitions you are > using. The phrases actually existing and cannot exist are not defined. Eetiquette and honesty requires you to say, I'm sorry, but I do not > know what the definitions of these words are. And then you should > attach a list of words you know. It can't be too long. So I will look > whether there are words which could be used to explain actually > existing and cannot exist. Amusing. I (and others) have repeatedly asked you what you mean. I have > suggested that you use the terminology in Halmos's Naive Set Theory or > in Kunen's Set Theory. If you don't like these books, then pick a > different one. However, Cantor's papers are much too old to be used for > a discussion today. No. The hidden errors can better be recognized at the roots. Then by all means let us pull WM up out of his pot and examine his roots to find the sources of all his errors. === Subject: Re: Cantor Confusion > Okay, so for you there is no point to draw about your quote other than > understanding Cantor's own writings. And, I'll add that in this > particular instance, Cantor, as you translated, is not in conflict with > the theorem of current set theory that omega is a limit ordinal and the > first oridinal that is greater than all natural numbers. That is what I said. omega is a limit. In modern set theory there are > limits. Moe said that omega is a limit ordinal. He did not say that omega is > a limit. The two statements are not the same. Moe said Cantor, as you translated, is not in conflict with the > theorem of current set theory. Cantor said omega can be understood as > a limit. Cantor said understood. That is not the same as saying is. Cantor also said: is. Therefore it is clear that here he meant is. meant is? Since the two words don't mean the same, please give your > Regardless, Cantor's papers are very old and predate the modern > formulations. That does not mean that the modern formulations are better. No, but the two formulations are different. If your point is that the old formulations have problems, then I doubt anyone will disagree. But, if your point is that the modern formulations have problems, then (obviously) you need to stick to the modern formulations in your discussion. > The phrases actually existing and cannot exist are not defined. Eetiquette and honesty requires you to say, I'm sorry, but I do not > know what the definitions of these words are. And then you should > attach a list of words you know. It can't be too long. So I will look > whether there are words which could be used to explain actually > existing and cannot exist. Amusing. I (and others) have repeatedly asked you what you mean. I have > suggested that you use the terminology in Halmos's Naive Set Theory or > in Kunen's Set Theory. If you don't like these books, then pick a > different one. However, Cantor's papers are much too old to be used for > a discussion today. No. The hidden errors can better be recognized at the roots. Perhaps, but irrelevant. If you can't find the hidden errors in the modern formulations, a likely explanation is that the hidden errors have been removed in the process of changing the formulations. Regardless, if the hidden errors are still there, your only hope of convincing people is to point to them in the formulations that they know. -- Marcus === Subject: Re: Cantor Confusion Lines with finitely many indexes cannot exhaust the column with its > infinitely many An infinite number can. A God can do everything he wants. Infinite numbers are of the same power? Can you apply your reasoning to the following matrix? 1 21 321 4321 ... What ordinal numbers have the following sequences? 3,4,5,...,1,2 3,4,5,...,omega,1,2 === Subject: Re: Cantor Confusion Lines with finitely many indexes cannot exhaust the column with its > infinitely many An infinite number can. A God can do everything he wants. Infinite numbers are of the same > power? > No. God can do anything. Infinite numbers cannot. However, Infinite numbers can do some things. In particular an infinite number of finite lines can exhaust: an infinite number of indexes. > Can you apply your reasoning to the following matrix? 1 > 21 > 321 > 4321 > ... > Reversing the order of the lines does not change thing. There are an infinite number of lines. Each line is finite. There are are omega lines and omega columns. There is no last line. There is no omega th line. There is no inifinite line. > What ordinal numbers have the following sequences? > None. These are not ordinals. An ordinal number must consist of an initial segment of ordinals. However, in a slightly sloppy way they can be said to represent ordinals (note that this representation is no longer uniqe). Note that you can represent any countable ordinal (not just omega) by using just finite integers, you do not need (athough you can use if you want) infinite ordinals. > 3,4,5,...,1,2 A infinite set consisting of finite integers. This represents omega+1 (For n>=3 you have put n in the spot one would usually have n-2, 1 is in the spot where you would usually have omega, and 2 is in the spot where you would usually have omega+1) 3,4,5,...,omega,1,2 > An infinite set consisting of the finite integers and the non-natural number omega. This represents omega+2 (For n>=3 you have put n in the spot one would usually have n-2, 1 is in the spot where you would usually have omega+1, and 2 is in the spot where you would usually have omega+2) What matters is not which ordinals you use to occupy places, but where you put the ... For example. The both sequences 5,4,3,2,1,6,7,8 ... omega, omega*omega, omega+5*omega^3 + 7, 1,2.3, ... represent the ordinal omega. - William Hughes === Subject: Re: Cantor Confusion Lines with finitely many indexes cannot exhaust the column with its > infinitely many An infinite number can. A God can do everything he wants. Infinite numbers are of the same > power? > No. God can do anything. Infinite numbers cannot. However, > Infinite numbers can do some things. > In particular an infinite number of finite lines can > exhaust: an infinite number of indexes. Can you apply your reasoning to the following matrix? 1 > 21 > 321 > 4321 > ... > Reversing the order of the lines does not change thing. It should show you that without an infinite line there is no infinite column. > There > are an infinite number of lines. Each line is finite. There is an infinite number of initial segments of columns. Each one is finite. And nothing more is there. > There are > are omega lines and omega columns. There is no last line. > There is no omega th line. There are no omega lines. > There is no inifinite line. What ordinal numbers have the following sequences? > None. These are not ordinals. > An ordinal number must consist of an > initial segment of ordinals. However, in > a slightly sloppy way they can be said to > represent ordinals (note that this > representation is no longer uniqe). Note that you can > represent any countable ordinal (not just omega) by > using just finite integers, you do not need (athough > you can use if you want) infinite ordinals. 3,4,5,...,1,2 A infinite set consisting of finite integers. > This represents omega+1 No, the sequence above represents omega + 2. > (For n>=3 you have put n in the spot one > would usually have n-2, 1 is in the spot where > you would usually have omega, and 2 is in > the spot where you would usually have omega+1) > 3,4,5,...,omega,1,2 > An infinite set consisting of the finite integers and > the non-natural number omega. This represents > omega+2 No, the sequence above represents omega + 3. But you see the problem. > (For n>=3 you have put n in the spot one > would usually have n-2, 1 is in the spot where > you would usually have omega+1, and 2 is in > the spot where you would usually have omega+2) ? What matters is not which ordinals you use > to occupy places, but where you put the ... > For example. The both sequences 5,4,3,2,1,6,7,8 ... > omega, omega*omega, omega+5*omega^3 + 7, 1,2.3, ... represent the ordinal omega. > The first is omega. The second is not omega. You see it if you replace omega by 1,2,3,... Perhaps you wanted to say that they both represent the *cardinal* omega. That is correct. === Subject: Re: Cantor Confusion > Lines with finitely many indexes cannot exhaust the column with its > infinitely many An infinite number can. A God can do everything he wants. Infinite numbers are of the same > power? > No. God can do anything. Infinite numbers cannot. However, > Infinite numbers can do some things. > In particular an infinite number of finite lines can > exhaust: an infinite number of indexes. Can you apply your reasoning to the following matrix? 1 > 21 > 321 > 4321 > ... > Reversing the order of the lines does not change thing. It should show you that without an infinite line there is no infinite > column. Why should something true show something false? There > are an infinite number of lines. Each line is finite. There is an infinite number of initial segments of columns. Each one is > finite. And nothing more is there. How is it that there are more-than-any-finite-number of lines but not more-than-any-finite-number of first elements of lines? At least according to WM. There are > are omega lines and omega columns. There is no last line. > There is no omega th line. There are no omega lines. There is no omega-th line, but there are omega lines === Subject: Re: Cantor Confusion infinitely many An infinite number can. A God can do everything he wants. Infinite numbers are of the same > power? > No. God can do anything. Infinite numbers cannot. However, > Infinite numbers can do some things. > In particular an infinite number of finite lines can > exhaust: an infinite number of indexes. Can you apply your reasoning to the following matrix? 1 > 21 > 321 > 4321 > ... > Reversing the order of the lines does not change thing. It should show you that without an infinite line there is no infinite > column. > Well, since I already agree there is no infinite column this does not change anything. This is just more of you reading X says there is an infinite number of columns, and concluding therefore X must believe that there is an infinite column. This is where we started Dik believes that 0.111... has an infinite number of places, therefore Dik must believe that there is an infinite place. We have a and b a: Every infinite set contains an infinite element. b: There exists an infinite set that does not contain an infinite element. They cannot both be true. If you want to show that assuming b leads to a contradiction, you cannot first assume b and then say Set A is infinite so it must contain an infinite element. > There > are an infinite number of lines. Each line is finite. There is an infinite number of initial segments of columns. Each one is > finite Which is just what I said (changing names from line to initial segments of columns changes nothing). > And nothing more is there. Nothing more needs to be there. > There are > are omega lines and omega columns. There is no last line. > There is no omega th line. There are no omega lines. The number of lines is greater than any finite natural number. What would you call the number of lines? There is no inifinite line. What ordinal numbers have the following sequences? > None. These are not ordinals. > An ordinal number must consist of an > initial segment of ordinals. However, in > a slightly sloppy way they can be said to > represent ordinals (note that this > representation is no longer uniqe). Note that you can > represent any countable ordinal (not just omega) by > using just finite integers, you do not need (athough > you can use if you want) infinite ordinals. 3,4,5,...,1,2 A infinite set consisting of finite integers. > This represents omega+1 No, the sequence above represents omega + 2. (For n>=3 you have put n in the spot one > would usually have n-2, 1 is in the spot where > you would usually have omega, and 2 is in > the spot where you would usually have omega+1) > 3,4,5,...,omega,1,2 > An infinite set consisting of the finite integers and > the non-natural number omega. This represents > omega+2 No, the sequence above represents omega + 3. But you see the problem. > No. There are two representations of omega + 3. Why is this a problem? - William Hughes === Subject: Re: Cantor Confusion Therefore we can compare the diagonal with every line. We find that the > diagonal is longer than every line. 1 > 1 1 > 1 1 1 > 1 1 1 1 > ... The length of the diagonal is clearly the same as the length of the > first column. And, the first column goes on forever, while each line > does not go on forever. Therefore, for each line L, the length of the > diagonal is longer than the length of line L. Hence, the diagonal is > longer than every line. What is wrong with my reasoning? Nothing. Wrong is only the assumption that goes on forever could be considered a finished infinity, i.e., could be denoted by a fixed cardinal number being larger than any natural number. If you look at the diagonal, you always see that it is only there where lines are. Therefore you will always see that it is of finite length. It goes on forever does not mean actually infinite length. The lengths of the lines also increase from line to line forever. Nevertheless, an infinite length will never be reached. > Each line contains a finite number of 1's, but the first column and the > diagonal contain an infinite number of 1's. This is clear from the > picture. Why should something that is clear from the picture be > impossible? but the first column and the diagonal contain an infinite number of 1's is not clear from the picture. Moreover it is clear from the picture that the length of the diagonal cannot surpass the lengths of all lines. === Subject: Re: Cantor Confusion > What is wrong with my reasoning? Nothing. Wrong is only the assumption that goes on forever could be > considered a finished infinity, i.e., could be denoted by a fixed > cardinal number being larger than any natural number. The set of rationals { (n-1)/n:n in N} goes on forever in the sense that there is no end value IN THE SET beyond which the set does not go , but there is a value not a member of the set that it does not exceed. In the same way, the minimal of sets with members {}, and {{}}, and for each x also x union {x} is ordered by membership, with each being a member of all its successors, and there is not end to these successors. But there can be a set of which they are all members, which will of necessity not be a maximum, but can be a supremum. And in ZF or NBG there /must/ be such a set. === Subject: Re: Cantor Confusion Therefore we can compare the diagonal with every line. We find that the > diagonal is longer than every line. 1 > 1 1 > 1 1 1 > 1 1 1 1 > ... The length of the diagonal is clearly the same as the length of the > first column. And, the first column goes on forever, while each line > does not go on forever. Therefore, for each line L, the length of the > diagonal is longer than the length of line L. Hence, the diagonal is > longer than every line. What is wrong with my reasoning? Nothing. Wrong is only the assumption that goes on forever could be > considered a finished infinity, i.e., could be denoted by a fixed > cardinal number being larger than any natural number. I'm not sure what you mean. In particular, I'm not sure what considered, finished (or finished infinity), denoted, and fixed mean here. > If you look at the diagonal, you always see that it is only there where > lines are. Therefore you will always see that it is of finite length. > It goes on forever does not mean actually infinite length. The > lengths of the lines also increase from line to line forever. > Nevertheless, an infinite length will never be reached. To me, infinite length just means goes on forever. To you, the two phrases have different meanings, it seems. So, what do you mean by the phrase infinite length? -- Marcus === Subject: Re: Cantor Confusion considered a finished infinity, i.e., could be denoted by a fixed > cardinal number being larger than any natural number. I'm not sure what you mean. In particular, I'm not sure what > considered, finished (or finished infinity), denoted, and > fixed mean here. I see. But recently you used the word completed infinity. Be sure that my finished infinity means the same. If you look at the diagonal, you always see that it is only there where > lines are. Therefore you will always see that it is of finite length. > It goes on forever does not mean actually infinite length. The > lengths of the lines also increase from line to line forever. > Nevertheless, an infinite length will never be reached. To me, infinite length just means goes on forever. To you, the two > phrases have different meanings, it seems. So, what do you mean by the > phrase infinite length? Goes on forever is a property of the set of natural numbers, as well of the elements n as of initial the segments 1,2,3,...,n. It does not make you believe that there are infinite natural numbers n. Why does it make you believe that there is an infinite initial segment 1,2,3... ? === Subject: Re: Cantor Confusion Goes on forever is a property of the set of natural numbers, Yes. > as well > of the elements n as of initial the segments 1,2,3,...,n. Yes Let the set of the elements n as of initial the segments 1,2,3,...,n be K. Then K is simply the set of natural numbers. Do you thing the sets are different? >It does not > make you believe that there are infinite natural numbers n. Why does it > make you believe that there is an infinite initial segment 1,2,3... We do not need an infinite natural number ot have an infinite initial segment 1,2,3 ... As you stated Goes on forever is a property of the natural numbers. The set {1,2,3,...} is just the natural numbers. So this set must go on forever. - William Hughes ? > === Subject: Re: Cantor Confusion What is wrong with my reasoning? Nothing. Wrong is only the assumption that goes on forever could be > considered a finished infinity, i.e., could be denoted by a fixed > cardinal number being larger than any natural number. I'm not sure what you mean. In particular, I'm not sure what > considered, finished (or finished infinity), denoted, and > fixed mean here. I see. But recently you used the word completed infinity. I don't think I ever said that. Do you have a quote? > Be sure that my finished infinity means the same. If you look at the diagonal, you always see that it is only there where > lines are. Therefore you will always see that it is of finite length. > It goes on forever does not mean actually infinite length. The > lengths of the lines also increase from line to line forever. > Nevertheless, an infinite length will never be reached. To me, infinite length just means goes on forever. To you, the two > phrases have different meanings, it seems. So, what do you mean by the > phrase infinite length? Goes on forever is a property of the set of natural numbers, as well > of the elements n as of initial the segments 1,2,3,...,n. Yes. > It does not > make you believe that there are infinite natural numbers n. If infinite natural number means a natural number which is larger than every natural number, then I don't believe there are infinite natural numbers. > Why does it > make you believe that there is an infinite initial segment 1,2,3... ? Sorry. I don't understand. 1,2,3,... is the set of natural numbers. forever, and I agreed to it. You seem to be asking me why I believe the set of natural numbers goes on forever. But, we just agreed that was true. So, what are you asking me? -- Marcus === Subject: Re: Cantor Confusion not sufficient. > Why is the supremum not sufficient to define the length of the > diagonal. (Note, there is no line with an infinite index, so > your answer should not depend on a line with an infinite index)? Every column has a fixed number omega of terms. You can add 1 further term and you will get omega + 1 terms in every column. Every line has less than omega terms. If you add 1 term to every line, the number of terms in every line remains less than omega. What about the diagonal? It has to satisfy both conditions, which is impossible. Therefore your conviction, expressed in many postings (it is possible to have an infinite set each of whose elements are finite) is wrong. The length of the diagonal will be longer than every > line if and only if there is no line with maximum length. Therefore also the diagonal cannot have maximum lengh omega > n. As there is no last line, there is no line with maximum length. Exactly. And therefore there is no number omega The length of the set of natural numbers is the supremum of the > lengths of the initial segments. This supremum is omega. However, > this supremum is not a natural number. But it is an ordinal number which can be increased by 1. And if infinity in fact is finished, then it is a maximum. === Subject: Re: Cantor Confusion not sufficient. > Why is the supremum not sufficient to define the length of the > diagonal. (Note, there is no line with an infinite index, so > your answer should not depend on a line with an infinite index)? Every column has a fixed number omega of terms. You can add 1 further > term and you will get omega + 1 terms in every column. Every line has > less than omega terms. If you add 1 term to every line, the number of > terms in every line remains less than omega. What about the diagonal? > It has to satisfy both conditions, which is impossible. No. Adding one element to each line does not change the supremum of the lengths of the lines, so it does not change the length of the diagonal. The number of terms in each line, n, is less than omega. The number of columns is the supremum of the number of terms in each line. The number of columns is omega. The number of lines is omega. The number of columns is equal to the number of lines. Now add one term to each line. The number of terms in each line, n+1, is less than omega. The number of columns is the supremum of the number of terms in each line. The number of columns is omega. The number of lines is omega. The number of columns is equal to the number of lines. Therefore your conviction, expressed in many postings (it is possible > to have an infinite set each of whose elements are finite) is wrong. > The length of the diagonal will be longer than every > line if and only if there is no line with maximum length. Therefore also the diagonal cannot have maximum lengh omega > n. As there is no last line, there is no line with maximum length. Exactly. And therefore there is no number omega The length of the set of natural numbers is the supremum of the > lengths of the initial segments. This supremum is omega. However, > this supremum is not a natural number. But it is an ordinal number which can be increased by 1. And if > infinity in fact is finished, then it is a maximum. No. There is nothing that says that a set must have a maximum to be finished. - William Hughes === Subject: Re: Cantor Confusion > A diagonal consists of certain line elements. Therefore a supremum is > not sufficient. > Why is the supremum not sufficient to define the length of the > diagonal. (Note, there is no line with an infinite index, so > your answer should not depend on a line with an infinite index)? Every column has a fixed number omega of terms. You can add 1 further > term and you will get omega + 1 terms in every column. Every line has > less than omega terms. If you add 1 term to every line, the number of > terms in every line remains less than omega. What about the diagonal? > It has to satisfy both conditions, which is impossible. The whole issue is that according to us the diagonal does NOT have to satisfy the condition of being a line in the listing. So that WM proves our point! Therefore your conviction, expressed in many postings (it is possible > to have an infinite set each of whose elements are finite) is wrong. Has WM just proved 2 + 2 = 1? He has certainly not proved anything else, at least not to any one else's satisfaction. But it is an ordinal number which can be increased by 1. And if > infinity in fact is finished, then it is a maximum. Of what? it is certainly a LUB or supremum in some senses, but how can a anything be the maximum of things of which it is not a member? What WM is claiming is equivalent to saying that 10 is the /maximum/ of one digit decimal integers. What corrupt definition of maximum is WM using? === Subject: Re: Cantor Confusion something that is absurd, this only shows that your mathematical > reasoning leads to absurdities. Everyone who uses mathematical reasoning > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > you will stop proving absurdities. Only then the absurdities would begin. But I would not notice. We need no experts in trees. This tree does nothing else but represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. 0. /a 0 1 /b c / 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge > b. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path which has passed through the first n edges. === Subject: Re: Cantor Confusion If you think that your mathematical proof is valid and you prove > something that is absurd, this only shows that your mathematical > reasoning leads to absurdities. Everyone who uses mathematical reasoning > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > you will stop proving absurdities. Only then the absurdities would begin. But I would not notice. One often does not notice one's own absurdities, so it is quite possible that WM would not notice any such new ones more that he notices his many old ones. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path > which has passed through the first n edges. Then f(n) = n, unless related to has some esoteric meaning as yet unexplained. === Subject: Re: Cantor Confusion > If you think that your mathematical proof is valid and you prove > something that is absurd, this only shows that your mathematical > reasoning leads to absurdities. Everyone who uses mathematical reasoning > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > you will stop proving absurdities. Only then the absurdities would begin. But I would not notice. We need no experts in trees. This tree does nothing else but represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. 0. /a 0 1 /b c / 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge > b. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path > which has passed through the first n edges. Are you saying that f(1) is the number of edges related to the paths that contain edge a? What is the value of f(1)? -- Marcus === Subject: Re: Cantor Confusion something that is absurd, this only shows that your mathematical > reasoning leads to absurdities. Everyone who uses mathematical reasoning > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > you will stop proving absurdities. Only then the absurdities would begin. But I would not notice. We need no experts in trees. This tree does nothing else but represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. 0. /a 0 1 /b c / 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge > b. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path > which has passed through the first n edges. Are you saying that f(1) is the number of edges related to the paths > that contain edge a? What is the value of f(1)? f(1) = 1 is the number of edges related to initial segment (of the path) that contains edge a. We start with the first edge a of a path. Then we see the path splits into two paths. So half of edge a is related to each one and so on. Of course we never get ready, but there is no edge which remains unconsidered. f(n) is a recursion formula like many others. Do you know how to calculate the square root of 2 by x_n+1 = x_n/2 + 1/x_n? You start with x_1, calculate x_2 and so on. Of course we never get ready, but there is no digit o sqrt(2) which remains uncalculated. It is not impossible to achieve more with infinite strings belonging to representations of irrational numbers. === Subject: Re: Cantor Confusion We need no experts in trees. This tree does nothing else but represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. 0. /a 0 1 /b c / 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge > b. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path > which has passed through the first n edges. Are you saying that f(1) is the number of edges related to the paths > that contain edge a? What is the value of f(1)? f(1) = 1 is the number of edges related to initial segment (of the > path) that contains edge a. Still don't get it. Let's try an example. Consider the path where we always go left at each node. This path has initial segments. Each such initial segment contains a. But, what is f(1) supposed to be? > We start with the first edge a of a path. Then we see the path splits > into two paths. So half of edge a is related to each one and so on. Of > course we never get ready, but there is no edge which remains > unconsidered. So, when the height of the tree is 1, there is one path containing edge a. When the height is 2, there are 2 paths containing edge a. When it is 3, there are 4 paths. In general, when the height is n, there are 2^{n- 1} paths containing edge a. If we let the tree go on forever, then there are an infinite number of paths that contain edge a. Is that what you mean? If so, what is the next step in your argument? -- Marcus === Subject: Re: Cantor Confusion > > If you think that your mathematical proof is valid and you prove > something that is absurd, this only shows that your mathematical > reasoning leads to absurdities. Everyone who uses mathematical > reasoning > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > you will stop proving absurdities. Only then the absurdities would begin. But I would not notice. We need no experts in trees. This tree does nothing else but > represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary > strings. > The edges (like a, b, and c below) connect the nodes, i.e., the > binary > digits 0 or 1. 0. /a 0 1 /b c / 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with > 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a > is > inherited by all paths which begin with 0.00, the other half of edge > a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge > b. Suppose we have a finite tree of height n. Then there are 2^n paths. > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > paths. We can define a real-value function on paths by g(e1,e2,...,en) > = > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... Continuing in this > manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 What is f(n)? f(n) is the number of edges related to the initial segment of one path > which has passed through the first n edges. Are you saying that f(1) is the number of edges related to the paths > that contain edge a? What is the value of f(1)? f(1) = 1 is the number of edges related to initial segment (of the > path) that contains edge a. We start with the first edge a of a path. Then we see the path splits > into two paths. So half of edge a is related to each one and so on. But then when those paths split only 1/4 of edge 'a' is related to any of those, and for the next split only 1/8 of 'a' is related, so that for only 1/ (lim-{n -> oo} 2^n) of edge 'a' is related to any infinite path. === Subject: Re: Cantor Confusion <45548d8f$0$97235$892e7fe2@authen.yellow.readfreenews.net > [...] you are working with an infinite triangle [...] ,----[ http://en.wikipedia.org/wiki/Triangle ] > | A triangle is one of the basic shapes of geometry: a polygon with > | three vertices [...] > `---- Are there really three vertices in WM's triangle? If finished infinities like omega do exist, yes. Then you can add 1 term to the first (and any other) column and one term to the diagonal obtaining tree vertices: 1 22 333 ... omega...omega Addition of one term to every line, however, will not yield anything. > Since when do sets have a length? It can be defined: length of a set of natural numbers = cardinal number of that set. === Subject: Re: Cantor Confusion > [...] you are working with an infinite triangle [...] ,----[ http://en.wikipedia.org/wiki/Triangle ] > | A triangle is one of the basic shapes of geometry: a polygon with > | three vertices [...] > `---- Are there really three vertices in WM's triangle? If finished infinities like omega do exist, yes. Then you can add 1 > term to the first (and any other) column and one term to the diagonal > obtaining tree vertices: 1 > 22 > 333 > ... > omega...omega Addition of one term to every line, however, will not yield anything. Since when do sets have a length? It can be defined: length of a set of natural numbers = cardinal number > of that set. > Why is it not the height (or width, or area or volumn, or any other word indicating quantity) of the set rather than its length === Subject: Re: Cantor Confusion [...] >> Are there really three vertices in WM's triangle? If finished infinities [...] Verbiage. F. N. -- xyz === Subject: Re: Cantor Confusion <45548d8f$0$97235$892e7fe2@authen.yellow.readfreenews.net> <4555ddd5$0$97239$892e7fe2@authen.yellow.readfreenews.net > [...] >> Are there really three vertices in WM's triangle? If finished infinities [...] Verbiage. Yes. But, sorry to see, it is the fundament of modern mathematics. === Subject: Re: Cantor Confusion >> [...] > Are there really three vertices in WM's triangle? >> If finished infinities [...] >> Verbiage. Yes. But, sorry to see, it is the fundament of modern mathematics. Finished infinities is your wording. F. N. -- xyz === Subject: Re: Cantor Confusion The diagonal of actually infinite length omega cannot exist without the >>existence of a line of actually infinite length omega. >> Why not? Do you have a proof of this claim? The diagonal of a matrix is defined as consisting of elements of this >matrix. For a diagonal longer than every line (or every column) this >is impossible. Let me restate the situation: We have the 'matrix' 1 > 2 3 > 4 5 6 > 7 8 9 10 > etc and you argue that the length of the diagonal cannot be > infinite. Of course, the first problem is that this is not a square > matrix. If you want to make it into a square matrix, you can > make it into this matrix: 1 0 0 0 0 0 .... > 2 3 0 0 0 0 ... > 4 5 6 0 0 0 ... > .... > .... In which case, it is a square matrix and every line > has infinite length. So the diagonal, which has infinite > length, has the same length as every row and every column. There is no line without trailing zeros. The diagonal has no zeros. An alternative is to look at the incomplete, triangular array > that we started with and notice that in triagular arrays the > length of the diagonal is always at least as large as the > length of each row. In the situation where there are finitely > many rows, the length of the diagonal is the same as the number > of rows. The sticking point is whether the length of the diagonal is > the same as the length of some row even when there are infinitely > many rows. You seem to be claiming the answer is yes. As we have not the least idea of what actual infinity could be, we can do nothing but extrapolate from the finite domain. Would you believe that the diagonal of our infinite triangle can be longer than the first column? > While > it is clear that the length of the diagonal is at least the length of > any row, it seems clear that the length of the diagonal is actually > longer than the length of each row in this case since it is at > least as long as the next row down. Since there is always a > 'next row down', the length of the diagonal is longer than > the length of every row. For every line there is always a next line longer than the first. Nevertheless you do not believe that there are lines with infinite length. === Subject: Re: Cantor Confusion > For every line there is always a next line longer than the first. > Nevertheless you do not believe that there are lines with infinite > length. If every line is no more that one column longer the one before it, And there is always such a line following every line, wherefore does adding one to such a finite make it infinite? === Subject: Re: Cantor Confusion Of course, the first problem is that this is not a square > matrix. If you want to make it into a square matrix, you can > make it into this matrix: 1 0 0 0 0 0 .... > 2 3 0 0 0 0 ... > 4 5 6 0 0 0 ... > .... > .... In which case, it is a square matrix and every line > has infinite length. So the diagonal, which has infinite > length, has the same length as every row and every column. There is no line without trailing zeros. The diagonal has no zeros. So that the diagonal is different from every line, as required. An alternative is to look at the incomplete, triangular array > that we started with and notice that in triagular arrays the > length of the diagonal is always at least as large as the > length of each row. In the situation where there are finitely > many rows, the length of the diagonal is the same as the number > of rows. The sticking point is whether the length of the diagonal is > the same as the length of some row even when there are infinitely > many rows. You seem to be claiming the answer is yes. As we have not the least idea of what actual infinity could be, we > can do nothing but extrapolate from the finite domain. For we read I, as there are lots of people who have quite concrete ideas of what any sort of infinity is. Would you believe that the diagonal of our infinite triangle can be > longer than the first column? Why? While > it is clear that the length of the diagonal is at least the length of > any row, it seems clear that the length of the diagonal is actually > longer than the length of each row in this case since it is at > least as long as the next row down. Since there is always a > 'next row down', the length of the diagonal is longer than > the length of every row. For every line there is always a next line longer than the first. > Nevertheless you do not believe that there are lines with infinite > length. Not unless the construction under consideration provides lines of infinite length === Subject: Re: Cantor Confusion > >>The diagonal of actually infinite length omega cannot exist without the >>existence of a line of actually infinite length omega. >> Why not? Do you have a proof of this claim? The diagonal of a matrix is defined as consisting of elements of this >matrix. For a diagonal longer than every line (or every column) this >is impossible. Let me restate the situation: We have the 'matrix' 1 > 2 3 > 4 5 6 > 7 8 9 10 > etc and you argue that the length of the diagonal cannot be > infinite. Of course, the first problem is that this is not a square > matrix. If you want to make it into a square matrix, you can > make it into this matrix: 1 0 0 0 0 0 .... > 2 3 0 0 0 0 ... > 4 5 6 0 0 0 ... > .... > .... In which case, it is a square matrix and every line > has infinite length. So the diagonal, which has infinite > length, has the same length as every row and every column. There is no line without trailing zeros. The diagonal has no zeros. That is correct. Were you agreeing with Daniel or disagreeing with him? > An alternative is to look at the incomplete, triangular array > that we started with and notice that in triagular arrays the > length of the diagonal is always at least as large as the > length of each row. In the situation where there are finitely > many rows, the length of the diagonal is the same as the number > of rows. The sticking point is whether the length of the diagonal is > the same as the length of some row even when there are infinitely > many rows. You seem to be claiming the answer is yes. As we have not the least idea of what actual infinity could be, we > can do nothing but extrapolate from the finite domain. Using precise definitions and rigorous logic. -- Marcus === Subject: Re: Cantor Confusion [...] > I am quite willing to go to a much more formal set > of terms and definitions, but I try to use the language > preferred by whomoever I am discussing something with > so if WM insists on using informal language that is what > I will use.. The nice thing about informal language is that it allows for talking at cross purposes. > Yes, my switch from length of a line or a diagonal > to length of the natural numbers was an attempt to > move back to slightly more formal language. Whether > this will work I do not know. Note. It has become clear that at the base > we have two statements. a: Any infinite set of numbers must contain > an infinite number b: It is possible to have an infinite set of numbers > that does not contain an infinite number. that cannot both be true. In WM's eyes there are no infinite sets in reality at all. For the sake of continuing the discussion he will not present that argument too early. Otherwise his mission (prevent student from studying set theory) would be in danger ... > WM claims that a is true and that furthermore b leads to a > contradiction. However, all of the > contradictions he has been showing depend on assuming that > a is true. I would like to discuss statements a and b directly, > however, I do not think that WM will cooperate. True. F. N. -- xyz === Subject: Re: Cantor Confusion [...] >> I would prefer discussions either with precisely defined (formalized) >> infinite triangles or better without all these words borrowed from >> Cantor/geometry/physics. In that case, if you discuss anything with WM, you will be > disappointed. Depends on your expectations. You will find a plethora of absurd constraints for thinking. > He has his own defintion for the word definition. He does not at all enact the notion of definition. > By infinite triangle, I meant a function with domain {(n,m)| n,m in > N and m <= n} and range N. OK. So you are writing about abstract entities like functions and domains whereas WM does not. In WM's view a /notation/ like 1 1 2 1 2 3 ... *is* the object under consideration whereas for you it is merely an illustration or reference the abstract entity. What WM calls diagonal means the geometrical object contained in that notation either on screen or written on paper. Mathematically diagonal simply means the sequence of f(i,i) i e N which by definition has cardinality |N|. WM does not belong to the factinista. He is ignorant about the fact that nowadays mathematics is a formal science in the first place. F. N. -- xyz === Subject: Re: Cantor Confusion <45548d8f$0$97235$892e7fe2@authen.yellow.readfreenews.net> <45550ba7$0$97245$892e7fe2@authen.yellow.readfreenews.net> <4555beb3$0$97227$892e7fe2@authen.yellow.readfreenews.net WM does not belong to the factinista. He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works. (Bertrand Russell) Die Mathematik ist wie die Dialektik ein Organ des h?heren Sinns. In der Aus?bung ist sie eine Kunst wie die Beredsamkeit. F?r beide hat nichts Wert als die Form; der Gehalt ist ihnen gleichg?ltig. (Johann Wolfgang von Goethe) Mathematik kann nie durch Logik allein begr?ndet werden. Als Vorbedingung zur Anwendung logischer Schl?sse ist uns bereits immer etwas gegeben: gewisse au?erlogische konkrete Objekte, die anschaulich als unmittelbares Erlebnis vor dem Denken da sind. (Hilbert) Der Lebensnerv der mathematischen Wissenschaft ist bedroht durch die Behauptung, Mathematik sei nichts anderes als ein System von Schl?ssen aus Definitionen und Annahmen, die zwar in sich widerspruchsfrei sein m?ssen, sonst aber von der Willk?r des Mathematikers geschaffen werden. W?re das wahr, dann w?rde die Mathematik keinen intelligenten Menschen anziehen. Sie w?re eine Spielerei mit Definitionen, Regeln und Syllogismen ohne Ziele und Sinn. Die Vorstellung, da? der Verstand sinnvolle Systeme von Postulaten frei erschaffen k?nne, ist eine tr?gerische Halbwahrheit. (R. Courant, H. Robbins) Aber ist denn das eine Wissenschaft, die S?tze beweist, ohne zu wissen, was sie beweist? (G. Frege) Gru?, WM === Subject: Re: Cantor Confusion > WM does not belong to the factinista. He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. The formalists are like a watchmaker who is so absorbed in making his > watches look pretty that he has forgotten their purpose of telling the > time, and has therefore omitted to insert any works. (Bertrand Russell) While WM is so concerned that the works work his way that he has forgot how to tell time. === Subject: Re: Cantor Confusion WM does not belong to the factinista. He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. The formalists are like a watchmaker who is so absorbed in making his > watches look pretty that he has forgotten their purpose of telling the > time, and has therefore omitted to insert any works. (Bertrand Russell) I believe you are confusing the terms formal science and formalists. I don't think Franziska meant them to be the same. They don't mean the same thing to me. -- Marcus === Subject: Re: Cantor Confusion >> I would prefer discussions either with precisely defined (formalized) >> infinite triangles or better without all these words borrowed from >> Cantor/geometry/physics. In that case, if you discuss anything with WM, you will be > disappointed. Depends on your expectations. You will find a plethora of absurd > constraints for thinking. Very true. > He has his own defintion for the word definition. He does not at all enact the notion of definition. By infinite triangle, I meant a function with domain {(n,m)| n,m in > N and m <= n} and range N. OK. So you are writing about abstract entities like functions and domains > whereas WM does not. In WM's view a /notation/ like 1 > 1 2 > 1 2 3 > ... *is* the object under consideration whereas for you it is merely an > illustration or reference the abstract entity. Are you sure? If you ask him if the the above object has less than five lines, I think he will say no. > What WM calls diagonal > means the geometrical object contained in that notation either on > screen or written on paper. Mathematically diagonal simply means the > sequence of f(i,i) i e N which by definition has cardinality |N|. WM does not belong to the factinista. He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. It appears to be willful ignorance. -- Marcus === Subject: Re: Cantor Confusion [...] >> So you are writing about abstract entities like functions and domains >> whereas WM does not. In WM's view a /notation/ like >> >> 1 >> 1 2 >> 1 2 3 >> ... >> >> *is* the object under consideration whereas for you it is merely an >> illustration or reference the abstract entity. Are you sure? If you ask him if the the above object has less than > five lines, I think he will say no. LOL. I am pretty sure that he will not agree that the above object can have aleph_0 lines. F. N. -- xyz === Subject: Re: Cantor Confusion [...] >> So you are writing about abstract entities like functions and domains >> whereas WM does not. In WM's view a /notation/ like >> >> 1 >> 1 2 >> 1 2 3 >> ... >> >> *is* the object under consideration whereas for you it is merely an >> illustration or reference the abstract entity. Are you sure? If you ask him if the the above object has less than > five lines, I think he will say no. LOL. I am pretty sure that he will not agree that the above object can > have aleph_0 lines. I think you are right. So, for WM, it has more than five, but less than aleph_0. Very curious. I think WM would not agree that the object can have any number of lines. -- Marcus === Subject: Re: Cantor Confusion <45548d8f$0$97235$892e7fe2@authen.yellow.readfreenews.net> [...] you are working with an infinite triangle [...] ,----[ http://en.wikipedia.org/wiki/Triangle ] > | A triangle is one of the basic shapes of geometry: a polygon with > | three vertices [...] > `---- Are there really three vertices in WM's triangle? No. But, I don't think an infinite triangle needs to be a triangle. > However, I'm open to suggestions for what to call it. You could call it a two-sided triangle. This might turn out to be useful in a quiz some day. Brian Chandler http://imaginatorium.org === Subject: Re: Cantor Confusion [...] you are working with an infinite triangle [...] ,----[ http://en.wikipedia.org/wiki/Triangle ] > | A triangle is one of the basic shapes of geometry: a polygon with > | three vertices [...] > `---- Are there really three vertices in WM's triangle? No. But, I don't think an infinite triangle needs to be a triangle. > However, I'm open to suggestions for what to call it. You could call it a two-sided triangle. This might turn out to be > useful in a quiz some day. I like that! -- Marcus === Subject: Re: Cantor Confusion On Sat, 11 Nov 2006 12:50:10 -0500, Marcus >> [...] you are working with an infinite triangle [...] >> ,----[ http://en.wikipedia.org/wiki/Triangle ] >> | A triangle is one of the basic shapes of geometry: a polygon with >> | three vertices [...] >> `---- >> Are there really three vertices in WM's triangle? >> No. But, I don't think an infinite triangle needs to be a triangle. >> However, I'm open to suggestions for what to call it. >> >> You could call it a two-sided triangle. This might turn out to be >> useful in a quiz some day. I like that! Next Brian might get to work on a transfinite Zen Abacus. ~v~~ === Subject: Re: Cantor Confusion still all true in modern mathematics. I think that much of his set theory is wrong but more of it is true than of modern set theory. > The first falseness is his assumption that every set can be well-ordered. This assumption is as wrong as the official assumption made by Zermelo. > We all know now that that can not be proven from first principles. According to modern mathematics nothing can be proven from first principles. Cantor considered well-ordering as a first principle, Zermelo introduced it at a first principle = axiom. Cantor was wrong, Zermelo was right? At the most in the ridiculous axiom faith of modern mathematics. > His second falsehood is when he states that a set of first cardinality > (meaning sets of cardinality aleph-0) can only be counted with use of > numbers of the second class (meaning omega and larger). And I think that > especially this quote has lead Wolfgang Mueckenheim astray. Also see > my discussion about this quote with Dave Seaman. The falsity is apparent > if you realise that quote means that every set with cardinality aleph-0 > has an omega-th element. No explicitly stated but implied. Cantor explained it (as I posted recently and repeat it here): Zum Beispiel betrachten wir die Menge aller endlichen positiven ganzen Zahlen, so ist sie in der nat.9frlichen Folge 1, 2, 3, ..., nu... eine wohlgeordnete Menge und hat in dieser Ordnung gedacht die Anzahl: omega. Schreibt man sie aber in der Ordnung (n + 1), (n + 2), ... (n + nu), ....1, 2, 3, ..., n, so hat sie nun die Anzahl omega + n. In der Ordnung 2, 4, 6, ..., 2nu, ..., 1, 3, 5, ..., (2nu + 1), ... hat dieselbe Menge (nu) die Anzahl 2omega u.s.w. u.s.w. Jede Menge von der M.8achtigkeit erster Classe ist abz.8ahlbar durch Zahlen der zweiten Zahlenklasse (II); und zwar l.8a§t sich jede Menge von der M.8achtigkeit erster Classe in solche Succession (als wohlgeordnete Menge) bringen, dass ihre Anzahl mit Bezug auf diese Succession gleich wird einer beliebig vorgeschriebenen Zahl alpha der zweiten Zahlenclasse. (Georg Cantor in a letter to Mittag-Leffler, Dec. 17,1882) There is no element omega. But, of course, if the set is to be counted, then there must be a number omega following after all natural numbers. This is the error. Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen nu zustreben, wenn darunter nichts anderes verstanden wird, als da§ omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt. In the examples above, we have no omega. Introducing it in fact as number which follows on all (even) natural numbers, we get 2, 4, 6, ..., 2nu, ..., omega, 1, 3, 5, ..., (2nu + 1), ..., 2omega That is false. I think this has to do with the problems of the conceptions of potential infinity vs. actual infinity that played an important role in mathematics before the early 1900's. Moreso because there were also religious issues. So Cantor crossed the border (after consultation with his religious leader and in strong opposition from him). The actual infinite was impossible (and unreligious) before that time, but he introduced actual infinity. No. There are hints on the actual infinite in the holy bible and by Saint Augustin. Cantor corresponded with Cardinal Franzelin about that. The Cardinal did not oppose to the idea. He agreed that God will know Cantor's numbers if they are not contradictory. The Cardinal only disagreed with Cantor's proof of the actual infinite. Because Cantor assumed that God had been forced to create it. > And > so, apparently, he thought that if a set was actually infinite, it should > have an actually infinite element, or somesuch. It should have an integer number counting its elements. Of course this number had to appear in the sequence of numbers if it was a number. But where? Obviously after all natural numbers. But then, what is the number of 1,2,3,...,omega ? It cannot be omega + 1, because, according to his equation above, 2,3,4, ....,1 has already the number omega + 1. Therefore omega cannot appear in the sequence. So we have the irrevocable dilemma: 1) omega is the number of countable many numbers like 1,2,3,... or 7,8,9,... and as such a single term in the sequence following the counted terms like 1,2,3,...,omega or 7,8,9,...,omega 2) omega + 1 = 2,3,4,...,1 and therefore omega cannot not be a term in the sequence but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > Indeed. Wolfgang Mueckenheim appears to think that Cantor's writings are > still all true in modern mathematics. > > I think that much of his set theory is wrong but more of it is true > than of modern set theory. Yes, I know you do not like it, but I state something else. You are arguing against modern set theory using statements by Cantor. Even statements by Cantor that are not valid in modern set theory. > The first falseness is his assumption that every set can be well-ordered. > > This assumption is as wrong as the official assumption made by Zermelo. Assumption? > We all know now that that can not be proven from first principles. > > According to modern mathematics nothing can be proven from first > principles. Do you know what that term means in mathematics? It means from a chosen set of axioms using ordinary logic. Within a branch of mathematics the axioms in the first principles are the basic axioms used within that branch. So within geometry the basic axioms might be the postulates by Euclides with the exclusion of the parallel postulate. In that system the parallel postulate can not be proven from first principles. > Cantor considered well-ordering as a first principle, > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > Zermelo was right? Cantor did state it without suggesting either that it was a first principle or something else. He just assumed it. And he was wrong with that assumption. > At the most in the ridiculous axiom faith of modern > mathematics. Oh. Whatever that means. > His second falsehood is when he states that a set of first cardinality > (meaning sets of cardinality aleph-0) can only be counted with use of > numbers of the second class (meaning omega and larger). And I think that > especially this quote has lead Wolfgang Mueckenheim astray. Also see > my discussion about this quote with Dave Seaman. The falsity is apparent > if you realise that quote means that every set with cardinality aleph-0 > has an omega-th element. > > No explicitly stated but implied. Explicitly stated. See my discussion with Dave Seaman where I give the quote, and where there is a long discussion about the meaning. > Cantor explained it (as I posted > recently and repeat it here): For the non-German speaking, Cantor explains that the ordered set has ordinal 1, 2, 3, ... omega n+1, n+2, ..., 1, 2, ... n omega+n 2, 4, 6, ..., 1, 3, 5, ... 2*omega (Note, in current mathematics the last is noted as omega*2, I will use the old notation in the sequel.) He is right and indeed, 1, 2, 4, 8, ..., 3, 6, 12, 24, ..., 5, 10, 20, 40, ... has ordinality omega^2. A set of cardinality aleph-0 can be ordered according to every countable ordinality. > There is no element omega. But, of course, if the set is to be counted, > then there must be a number omega following after all natural numbers. > This is the error. No. The above statement is not implied by the letter of Cantor to Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. To count a set of ordinality omega you do not need omega. In most cases you need omega, but there is one exception. Can you find it? To count a set of ordinality a with ordinals you need only the ordinals smaller than a. (And note that in counting with ordinals you start at 0, because that is the first ordinal.) > In the examples above, we have no omega. Introducing it in fact as > number which follows on all (even) natural numbers, we get > 2, 4, 6, ..., 2nu, ..., omega, 1, 3, 5, ..., (2nu + 1), ..., 2omega > That is false. That ordered set has ordinality 2*omega + 1. What is the problem? And it is easily explained. There are two non-terminating countable subsets followed by a subset of one element. Each of the non-terminating countable subsets gives a term omega, and the subset of one element gives the 1. > I think this has to do with the problems of the conceptions of potential > infinity vs. actual infinity that played an important role in mathematics > before the early 1900's. Moreso because there were also religious issues. > So Cantor crossed the border (after consultation with his religious leader > and in strong opposition from him). The actual infinite was impossible > (and unreligious) before that time, but he introduced actual infinity. > > No. There are hints on the actual infinite in the holy bible and by > Saint Augustin. Cantor corresponded with Cardinal Franzelin about that. > The Cardinal did not oppose to the idea. He agreed that God will know > Cantor's numbers if they are not contradictory. The Cardinal only > disagreed with Cantor's proof of the actual infinite. Because Cantor > assumed that God had been forced to create it. Perhaps. I do not know the bible, neither what Saint Augustin did write. It is irrelevant for mathematics. I just was trying to find the reason. > And so, apparently, he thought that if a set was actually infinite, > it should have an actually infinite element, or somesuch. > > It should have an integer number counting its elements. Of course this > number had to appear in the sequence of numbers if it was a number. But > where? Obviously after all natural numbers. But then, what is the > number of 1,2,3,...,omega ? > It cannot be omega + 1, because, according to his equation above, > 2,3,4, ....,1 has already the number omega + 1. > Therefore omega cannot appear in the sequence. And, I think that Cantor would agree that 1, 3, 4, ..., 2 also has the number omega + 1. > So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega Lack of precision. Aleph-0 is the number of countable many numbers. Omega is the ordinal number of coutably many numbers with a particular ordering. And I fail to see why it should follow. > 2) omega + 1 = 2,3,4,...,1 and therefore omega cannot not be a term in > the sequence but omega is the first part of it, omega = 2,3,4,... Why not? With your notation (which is a bit strange): omega + 1 = 2, 3, 4, ..., omega omega + 1 = omega, 2, 3, 4, ..., 1 omega + 1 = omega, omega + 1, 2, 3, 4, ..., 1 and so on and so forth. This all assuming '=' means 'is the ordinal number of the ordered set that follows'. > This dilemma is what I have been trying to explain for years now. There is no dilemma. Cantor did not yet have the complete idea about ordinals. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion still all true in modern mathematics. I think that much of his set theory is wrong but more of it is true > than of modern set theory. Yes, I know you do not like it, but I state something else. You are > arguing against modern set theory using statements by Cantor. Even > statements by Cantor that are not valid in modern set theory. The first falseness is his assumption that every set can be well-ordered. This assumption is as wrong as the official assumption made by Zermelo. Assumption? AC is taken to be an axiom. That is nothing but an arbitrary assumption. Well-ordering is so easy connected with AC that we can state: Well-ordering has been assumed. We all know now that that can not be proven from first principles. According to modern mathematics nothing can be proven from first > principles. Do you know what that term means in mathematics? It means from a > chosen set of axioms using ordinary logic. The chosing of the set is an arbitrary act. It is not reduced to first principles in modern math. Cantor had first principles! > Within a branch of > mathematics the axioms in the first principles are the basic axioms > used within that branch. So within geometry the basic axioms might > be the postulates by Euclides with the exclusion of the parallel > postulate. In that system the parallel postulate can not be proven > from first principles. Cantor considered well-ordering as a first principle, > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > Zermelo was right? Cantor did state it without suggesting either that it was a first > principle or something else. He just assumed it. And he was wrong > with that assumption. He was as wrong as Zermelo, not a bit more. At the most in the ridiculous axiom faith of modern > mathematics. Oh. Whatever that means. It means you attitude: After a set of axioms has been chosen, we an be sure to make true mathematics. His second falsehood is when he states that a set of first cardinality > (meaning sets of cardinality aleph-0) can only be counted with use of > numbers of the second class (meaning omega and larger). And I think that > especially this quote has lead Wolfgang Mueckenheim astray. Also see > my discussion about this quote with Dave Seaman. The falsity is apparent > if you realise that quote means that every set with cardinality aleph-0 > has an omega-th element. No explicitly stated but implied. Explicitly stated. See my discussion with Dave Seaman where I give the > quote, and where there is a long discussion about the meaning. I hoped to settle this question with my quote. Cantor explained it (as I posted > recently and repeat it here): For the non-German speaking, Cantor explains that > the ordered set has ordinal > 1, 2, 3, ... omega > n+1, n+2, ..., 1, 2, ... n omega+n > 2, 4, 6, ..., 1, 3, 5, ... 2*omega > (Note, in current mathematics the last is noted as omega*2, I will use > the old notation in the sequel.) He is right and indeed, > 1, 2, 4, 8, ..., 3, 6, 12, 24, ..., 5, 10, 20, 40, ... > has ordinality omega^2. A set of cardinality aleph-0 can be ordered > according to every countable ordinality. That did he mean with countable by numbers of class II. You see, he did not use an omegath element. There is no element omega. But, of course, if the set is to be counted, > then there must be a number omega following after all natural numbers. > This is the error. No. The above statement is not implied by the letter of Cantor to > Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. > To count a set of ordinality omega you do not need omega. In most > cases you need omega, but there is one exception. Can you find it? > To count a set of ordinality a with ordinals you need only the > ordinals smaller than a. (And note that in counting with ordinals > you start at 0, because that is the first ordinal.) The latter statement is true in modern mathematics. But nonsense nevertheless.But your assertion To count a set of ordinality omega you do not need omega is just my position. To count the natural numbers, you need not omega, because every set of natural numbers is cunted by natural numbers. Cantor's position, however, was the opposite. In the examples above, we have no omega. Introducing it in fact as > number which follows on all (even) natural numbers, we get > 2, 4, 6, ..., 2nu, ..., omega, 1, 3, 5, ..., (2nu + 1), ..., 2omega > That is false. That ordered set has ordinality 2*omega + 1. What is the problem? That each omega can be substituted by 1,2,3,... or say a,b,c,... yielding 4 omega. === Subject: Re: Cantor Confusion Indeed. Wolfgang Mueckenheim appears to think that Cantor's writings > are > still all true in modern mathematics. I think that much of his set theory is wrong but more of it is true > than of modern set theory. Yes, I know you do not like it, but I state something else. You are > arguing against modern set theory using statements by Cantor. Even > statements by Cantor that are not valid in modern set theory. The first falseness is his assumption that every set can be > well-ordered. This assumption is as wrong as the official assumption made by Zermelo. Assumption? AC is taken to be an axiom. That is nothing but an arbitrary > assumption. Well-ordering is so easy connected with AC that we can > state: Well-ordering has been assumed. It has been shown that ZF with AC will be consistent provided ZF without it is consistent, so that its assumption on top of ZF is reasonable. We all know now that that can not be proven from first principles. According to modern mathematics nothing can be proven from first > principles. Do you know what that term means in mathematics? It means from a > chosen set of axioms using ordinary logic. The chosing of the set is an arbitrary act. It is not reduced to first > principles in modern math. Cantor had first principles! Any first principles over and above pure logic are necessarily axioms ( statements assumed without proof). Within a branch of > mathematics the axioms in the first principles are the basic axioms > used within that branch. So within geometry the basic axioms might > be the postulates by Euclides with the exclusion of the parallel > postulate. In that system the parallel postulate can not be proven > from first principles. Cantor considered well-ordering as a first principle, > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > Zermelo was right? Cantor did state it without suggesting either that it was a first > principle or something else. He just assumed it. And he was wrong > with that assumption. He was as wrong as Zermelo, not a bit more. At the most in the ridiculous axiom faith of modern > mathematics. Oh. Whatever that means. It means you attitude: After a set of axioms has been chosen, we an be > sure to make true mathematics. We can be sure that what is deduced properly from a set of axioms depends only on them. A properly produced theorem in any axiom system can be false only if at least one of the axioms on which it depends is false. That is the great strength of an axiom system. For the non-German speaking, Cantor explains that > the ordered set has ordinal > 1, 2, 3, ... omega > n+1, n+2, ..., 1, 2, ... n omega+n > 2, 4, 6, ..., 1, 3, 5, ... 2*omega > (Note, in current mathematics the last is noted as omega*2, I will use > the old notation in the sequel.) He is right and indeed, > 1, 2, 4, 8, ..., 3, 6, 12, 24, ..., 5, 10, 20, 40, ... > has ordinality omega^2. A set of cardinality aleph-0 can be ordered > according to every countable ordinality. That did he mean with countable by numbers of class II. > You see, he did not use an omegath element. He did not need one to say what he wanted to say. There is no element omega. But, of course, if the set is to be counted, > then there must be a number omega following after all natural numbers. Not in standard mathematics, however screwed up WM's notions may be. The cardinality of {1,2,3,...} union {omega} is the same as that of either{1,2,3,...} or omega. > This is the error. . No. The above statement is not implied by the letter of Cantor to > Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. > To count a set of ordinality omega you do not need omega. In most > cases you need omega, but there is one exception. Can you find it? > To count a set of ordinality a with ordinals you need only the > ordinals smaller than a. (And note that in counting with ordinals > you start at 0, because that is the first ordinal.) The latter statement is true in modern mathematics. But nonsense > nevertheless. > But your assertion To count a set of ordinality omega > you do not need omega is just my position. To count the natural > numbers, you need not omega, because every set of natural numbers is > cunted by natural numbers. Cantor's position, however, was the > opposite. To count a set of ordinality omega one only needs the cardinality of omega. === Subject: Re: Cantor Confusion ... apparently, he thought that if a set was actually infinite, it should > have an actually infinite element, or somesuch. It should have an integer number counting its elements. Of course this > number had to appear in the sequence of numbers if it was a number. But > where? Obviously after all natural numbers. But then, what is the > number of 1,2,3,...,omega ? omega+1 > It cannot be omega + 1 yes it can > because, according to his equation above, > 2,3,4, ....,1 has already the number omega + 1. Yes, there is more that one representation of omega+1. This is like saying 2/4 cannot be one half, because one half already has the representation 1/2. [Note that I cannot tell which ordinal is represented by a set just by looking at the elements of the set. For example, just by using the finite natural numbers I can get: omega 1,2,3,4... omega+1 2,3,4 ...,1 omega+2 3,4,5 ...,1,2 2*omega 1,3,5... 2,4,6... Indeed I can represent any countable ordinal.] [Note further that if we insist that a representation of an ordinal be an initial sequence of ordinals then 2,3,4, ....,1 does not represent an ordinal. If we insist that a representation of an ordinal be an initial segment of ordinals then the representation of an ordinal by a sequence of ordinals is unique.] > Therefore omega cannot appear in the sequence. So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega > 2) omega + 1 = 2,3,4,...,1 If we allow representations like the above, then omega+1 does not have a unique representaion > and therefore omega cannot not be a term in > the sequence It can however be a term in another sequence representing omega+1. There is no single sequence which represents omega+1. > but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. > The explanation is that if we do not insist that an ordinal be represented by an initial seqment of ordinals, then there is no unique representation of an ordinal. - William Hughes === Subject: Re: Cantor Confusion omega+1. There is no single sequence which represents > omega+1. but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. > The explanation is that if we do not insist that an ordinal be > represented by an initial seqment of ordinals, then there > is no unique representation of an ordinal. I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, ... omega + 1 = 1,2,3,..., omega. Cardinal numbers like one, tweo, ... and ordinal numbers like first, second, ... are closely connected. So every iinitial segment of natural numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. Therefore no initial segment of natural numbers can be counted by an unnatural number like omega. This leads to the problem |{1,2,3,...}| = omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = a or a + 1, corresponding to the kind of a. Obviously something has been lost. === Subject: Re: Cantor Confusion > Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Obviously something has > been lost. perhaps it is WM's mind? WM is conflating cardinality with ordinality. The cardinality of a set is the smallest ordinal which can be put in bijection with the set, disregarding any order relation that the set might have defined upon it. The ordinality of a well ordered set (no other kind has an ordinality) is the unique ordinal which is order-isomorphic to the given well ordered set. E.G. the naturals with their usual order have ordinality omega and with any ordering have cardinality equal to that of omega, whereas the rationals with their statndard ordering do not have any ordinality at all but have cardinalty equal to that of omega. It is customary use aleph_0 to denote the cardinality of omega. The cardinality of omega + 1 is the same as the cardinality of omega, as they may be bijected to each other by ignoring their orderings. === Subject: Re: Cantor Confusion omega = 2,4,6,... > omega, omega = 2omega > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 It can however be a term in another sequence representing > omega+1. There is no single sequence which represents > omega+1. but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. > The explanation is that if we do not insist that an ordinal be > represented by an initial seqment of ordinals, then there > is no unique representation of an ordinal. I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, > ... > omega + 1 = 1,2,3,..., omega. Yes. So what? You are not going to do any unjustified formal manipulations are you? Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. Only if we say that {1,2,3,...} is not an initial segment of natural numbers. No set of the form {1,2,3..,n} can be counted by an unnatural number like omega. The set {1,2,3,...} is not a set of the form {1,2,3,...,n} > This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Yes. This is true. It is not however a problem. [Sometimes this is used to argue that starting at 0 is more elegent. Then the ordinal represented by {0,1,2,3,...a} is always a+1 (no matter what type of ordinal a is). While I would agree, I would also say that it is not a major issue.] - William Hughes === Subject: Re: Cantor Confusion omega 1,2,3,4... omega+1 2,3,4 ...,1 omega+2 3,4,5 ...,1,2 2*omega 1,3,5... 2,4,6... omega = 1,3,5,... > omega = 2,4,6,... > omega, omega = 2omega > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 It can however be a term in another sequence representing > omega+1. There is no single sequence which represents > omega+1. but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. > The explanation is that if we do not insist that an ordinal be > represented by an initial seqment of ordinals, then there > is no unique representation of an ordinal. I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, > ... > omega + 1 = 1,2,3,..., omega. Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Obviously something has > been lost. The dots in {1,2,3,...,n} have a different meaning than the dots in {1,2,3,...,omega}. Don't take the notation that literally. The ordered set {1,2,3,...,omega} would perhaps be better written in this context as {1,2,3,...,n,...,omega}. You can't deduce things from notation. -- Marcus === Subject: Re: Cantor Confusion > Therefore omega cannot appear in the sequence. So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega > 2) omega + 1 = 2,3,4,...,1 If we allow representations like the above, then omega+1 > does not have a unique representaion and therefore omega cannot not be a term in > the sequence It can however be a term in another sequence representing > omega+1. There is no single sequence which represents > omega+1. I.e., there are lots of ordered sets that have the same order type. -- Marcus === Subject: Re: Cantor Confusion > And > so, apparently, he thought that if a set was actually infinite, it should > have an actually infinite element, or somesuch. It should have an integer number counting its elements. Please explain what you mean by this. In particular, what does counting its elements mean? > Of course this > number had to appear in the sequence of numbers if it was a number. But > where? Obviously after all natural numbers. But then, what is the > number of 1,2,3,...,omega ? > It cannot be omega + 1, because, according to his equation above, > 2,3,4, ....,1 has already the number omega + 1. > Therefore omega cannot appear in the sequence. So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega > 2) omega + 1 = 2,3,4,...,1 and therefore omega cannot not be a term in > the sequence but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. Oh. -- Marcus === Subject: Re: Cantor Confusion ... > They are. But in mathematics comparing numbers comes in quite late in > the process of definition. > > In mathematics counting and comparing numbers comes before most other > things. Not because I think so, but because mathematics developed this > way. In that case you are talking about a different kind of mathematics than what I am talking about. > Where are they? That there are limit ordinals does not mean there are > limits in set theory. > > Why are they called so? Ask the people who coined those names. > Since Cantor quite a bit has changed. Limits are *not* part of set > theory, they belong to topology and other things build on set theory. > > Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > include limits? Darn. Do read. It does include limits, but not in the branch of set theory. In that branch the term has not been defined, and so is meaningless. If you want to use such a term in set theory, you have to define it in terms of set theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion things. Not because I think so, but because mathematics developed this > way. In that case you are talking about a different kind of mathematics than > what I am talking about. Yes, I am talking about the mathematics which was developed over the last 500 years. Where are they? That there are limit ordinals does not mean there are > limits in set theory. Why are they called so? Ask the people who coined those names. Cantor called them Grenzzahlen and denoted them by Lim n etc. Because by his second creation principle these numbers are created by limit processes. wir nennen sie Zahlen zweiter Art, sind so beschaffen, da§ es f.9fr sie eine n.8achstkleinere?gar nicht gibt; diese gehen aber aus Fundamentalreihen als deren Grenzzahlen hervor Since Cantor quite a bit has changed. Limits are *not* part of set > theory, they belong to topology and other things build on set theory. Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > include limits? Darn. Do read. It does include limits, but not in the branch of set > theory. All modern mathematics is set theory. Set theory is not a branch but the foundation, in fact, it is all. > In that branch the term has not been defined, and so is > meaningless. If you want to use such a term in set theory, you have > to define it in terms of set theory. Cantor did. Should it meanwhile have been forgotten or abolished? a = Lim n (a n) a ist hier die auf s.8amtliche Zahlen a n der Gr.9a§e nach n.8achstfolgende Zahl. === Subject: Re: Cantor Confusion > Since Cantor quite a bit has changed. Limits are *not* part of set > theory, they belong to topology and other things build on set theory. Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > include limits? Darn. Do read. It does include limits, but not in the branch of set > theory. All modern mathematics is set theory. Set theory is not a branch but > the foundation, in fact, it is all. The fact that set theory is the foundation does not mean it is all. That isn't what the words foundation and all mean in English. -- Marcus === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl > > > There is no difference between enumerating all the rational numbers in > the well-known scheme, starting at the corner of an infinite square > matrix > > There is. You can assign numbers to edges that terminate at nodes. And > *that* numbering is different. > > I can assign numbers to edges that start at nodes. I can also assign > numbers to nodes. Yes. In the final tree what edge starts at 1/2? As I stated already some time ago, there are two possible trees to build. One of the trees has edges that terminate at a node, the other tree has edges that start at a node and do not terminate. When completing, neither tree has 1/5 as a node. You should really consult Conway how that is resolved. > It numbers only those edges that are finitely > far away from the root. But that way you do not number all edges in the > infinite tree, because that contains edges that are *not* finitely far > away from the root. > > That is an interesting remark. The paths are isomorphic to the binary > representations of real numbers. So you claim that there are positions > in the binary representation of the real numbers that are infinitely > far from the decimal point. Wrong, again. You are still obsessed with the property that an infinite set of numbers must contain an infinite number. Pray, give a mathematical proof of that property. > There is a difference, see above. You can only number edges that are > finitely far away from the root, but in that way you will only number > edges that terminate at nodes finitely far away from the root. As in > the edges there are nodes infinitely far away from the root you will > never number the edges that terminate there. > > Let us enumerate the nodes. Yes, go ahead. > The situation with the rationals is quite different, because in the > matrix *each* rational is finitely far away from the root. > > Each node is finitely far from the root. (Does Conway really tell what > you reproduce here?) Well, my only advise is, read it. > I have experienced worse opinions. But let us finish with the polemics > (if possible), because we are at the most important point. Please think > over your argument: > 1) Do you say that the nodes cannot be enumerated? Depends on how the tree is built. If it is built with terminating edges the answer is no, if it is built with non-terminating edges the answer is yes. > 2) Do you agree that this implies: There are bit positions infinitely > far from the decimal point (or how this point may be called for binary > numbers). No. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Cantor Confusion the well-known scheme, starting at the corner of an infinite square > matrix There is. You can assign numbers to edges that terminate at nodes. And > *that* numbering is different. I can assign numbers to edges that start at nodes. I can also assign > numbers to nodes. Yes. In the final tree what edge starts at 1/2? None. Every edge starts at a node which has the name 0 or the name 1. If we enumerae the nodes instead of he edges, what would be different according to Conway? > As I stated already > some time ago, there are two possible trees to build. One of the trees > has edges that terminate at a node, the other tree has edges that start > at a node and do not terminate. Look at Cantor's list. Consider its enumeration by natural numbers. There are natural numbers which follow on an even number and there are natural numbers which follow on an odd number. Are there two kinds of lists? It numbers only those edges that are finitely > far away from the root. But that way you do not number all edges in the > infinite tree, because that contains edges that are *not* finitely far > away from the root. That is an interesting remark. The paths are isomorphic to the binary > representations of real numbers. So you claim that there are positions > in the binary representation of the real numbers that are infinitely > far from the decimal point. Wrong, again. You are still obsessed with the property that an infinite > set of numbers must contain an infinite number. Not at all. You said that infinite tree contains edges that are *not* finitely far away from the root. This is a gross mistake. There are no natural numbers which are infinitely far from the first one. But every edge and every node of one single path can be enumerated. > There is a difference, see above. You can only number edges that are > finitely far away from the root, but in that way you will only number > edges that terminate at nodes finitely far away from the root. As in > the edges there are nodes infinitely far away from the root you will > never number the edges that terminate there. Let us enumerate the nodes. Yes, go ahead. > Here are all the nodes of the path 0.000... enumerated by natural numbers: 1,2,3,... No node is infinitely far from the root, as you can easily check. And here are all the nodes of the tree enumerated: 1 2 3 7654 8 ... Also no one is infinitely far from the root. > The situation with the rationals is quite different, because in the > matrix *each* rational is finitely far away from the root. Each node is finitely far from the root. (Does Conway really tell what > you reproduce here?) Well, my only advise is, read it. If he says so, then it wil not be a good idea to waste my time with it. I have experienced worse opinions. But let us finish with the polemics > (if possible), because we are at the most important point. Please think > over your argument: > 1) Do you say that the nodes cannot be enumerated? Depends on how the tree is built. If it is built with terminating edges > the answer is no, if it is built with non-terminating edges the answer > is yes. Let it be built without edges at all. They are only guides for the eye. Here it is 0. 0 1 01 01 ........ 2) Do you agree that this implies: There are bit positions infinitely > far from the decimal point (or how this point may be called for binary > numbers). No. What then do you mean by infinitely far from the root? === Subject: Re: Cantor Confusion Nntp-Posting-Host: apps.cwi.nl ... > I can assign numbers to edges that start at nodes. I can also assign > numbers to nodes. > > Yes. In the final tree what edge starts at 1/2? > > None. Every edge starts at a node which has the name 0 or the name 1. > If we enumerae the nodes instead of he edges, what would be different > according to Conway? Read his book. It is not overly large. > As I stated already > some time ago, there are two possible trees to build. One of the trees > has edges that terminate at a node, the other tree has edges that start > at a node and do not terminate. > > Look at Cantor's list. Consider its enumeration by natural numbers. > There are natural numbers which follow on an even number and there are > natural numbers which follow on an odd number. Are there two kinds of > lists? This makes no sense at all. Moreover, we were talking about a tree, not about a list. In the tree, when constructing from edges that do terminate, for each node there is an edge that terminates at that node, and for each edge there is a node where it terminates. On the other hand, when constructing from edges that do *not* terminate, for each node there are edges that emanate from that node. > Wrong, again. You are still obsessed with the property that an infinite > set of numbers must contain an infinite number. > > Not at all. You said that infinite tree contains edges that are *not* > finitely far away from the root. This is a gross mistake. Why? You are constructing a tree with non-terminating edges. This means that for each node there are edges that emanate from the node. You appear to think that the edges that emanate from the node labelled 1/3 are finitely far away from the root. If there is a node in your tree with that label, that statement is obviously false. I state also that there is in your tree *no* node with that label. > There are no > natural numbers which are infinitely far from the first one. But every > edge and every node of one single path can be enumerated. Yes, so what? > Yes, go ahead. > > Here are all the nodes of the path 0.000... enumerated by natural > numbers: 1,2,3,... > No node is infinitely far from the root, as you can easily check. > > And here are all the nodes of the tree enumerated: > > 1 > 2 3 > 7654 > 8 ... > > Also no one is infinitely far from the root. And so each node has a binary expansion of finitely many binary digits. What is the node number of 1/3? Of 1/5? > The situation with the rationals is quite different, because in the > matrix *each* rational is finitely far away from the root. > > Each node is finitely far from the root. (Does Conway really tell what > you reproduce here?) > > Well, my only advise is, read it. > > If he says so, then it wil not be a good idea to waste my time with it. Do you really think the node 1/3 is finitely far from the root in the tree? > 2) Do you agree that this implies: There are bit positions infinitely > far from the decimal point (or how this point may be called for binary > numbers). > > No. > > What then do you mean by infinitely far from the root? Your node 1/3 is infinitely far from the root because there is no finite (natural) number that can state the distance. BTW, in binary the point is called the binary point. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: p and p^2 + 2 prime => p^3 + 4 prime If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. [One of my profs showed us this and told us to work on it if we're bored with our studies. He said most grad students he asks can't prove this one because it's too easy for their heavy artillery (advanced theorems) to hit. He said it was accessible to us, but might require some creative ideas and a lot of tinkering. ] I don't even know how one would start to prove this one. I tried to do things like, assuming p^3+ 4 was composite and tried to find a contradiction, but got nowhere. === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime > If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. > p /= 2 if p = +-1 (mod 6), then p^2 + 2 = 3 (mod 6) > [One of my profs showed us this and told us to work on it if we're > bored with our studies. He said most grad students he asks can't prove > this one because it's too easy for their heavy artillery (advanced > theorems) to hit. He said it was accessible to us, but might require > some creative ideas and a lot of tinkering. ] > It's a trick question. === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime If p and p^2 + 2 are prime, then p^3 + 4 is prime. > How would I go about proving this? A hint would be appreciated. p /= 2 if p = +-1 (mod 6), then > p^2 + 2 = 3 (mod 6) [One of my profs showed us this and told us to work on it if we're > bored with our studies. He said most grad students he asks can't prove > this one because it's too easy for their heavy artillery (advanced > theorems) to hit. He said it was accessible to us, but might require > some creative ideas and a lot of tinkering. ] It's a trick question. It's also d@mn familiar. ((After searching on Google Groups)) Because a similar question was asked on October 15 of this year (recede, Number Theory: p, p^2+8 are primes, p^3+4 ?): > Give p, where p and p*p+8 are primes. Show that p*p*p+4 is also prime. --- === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime > If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. [One of my profs showed us this and told us to work on it if we're > bored with our studies. Your prof is just fishing for cheaters. He wanted to see which members of his class would post this on sci.math. Now he knows who he can t. I'm tempted to give you the solution just so you can run to him and present it as your own work and try to get your pat on the head, but instead have the rug yanked out from under you....soooo tempting... but I'm just too big a person to put you through that. -- The man without a .sig === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. [One of my profs showed us this and told us to work on it if we're > bored with our studies. Your prof is just fishing for cheaters. He wanted to see > which members of his class would post this on sci.math. > Now he knows who he can t. I'm tempted to give you > the solution just so you can run to him and present it > as your own work and try to get your pat on the head, but > instead have the rug yanked out from under you....soooo > tempting... but I'm just too big a person to put you through > that. p^3 + 4 prime Kenneth Bull a .8ecrit : > If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. [One of my profs showed us this and told us to work on it if we're > bored with our studies. He said most grad students he asks can't prove > this one because it's too easy for their heavy artillery (advanced > theorems) to hit. He said it was accessible to us, but might require > some creative ideas and a lot of tinkering. ] > I don't even know how one would start to prove this one. > I tried to do things like, assuming p^3+ 4 was composite and tried to > find a contradiction, but got nowhere. > Hint: compute the values of p and p^2+2 for small values of p, most of the times p^2 + 2 may be divided by ... Work modulo ... Good luck! Raymond === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime <455666ab$0$5101$ba4acef3@news.orange.fr Kenneth Bull a .8ecrit : > If p and p^2 + 2 are prime, then p^3 + 4 is prime. How would I go about proving this? A hint would be appreciated. [One of my profs showed us this and told us to work on it if we're > bored with our studies. He said most grad students he asks can't prove > this one because it's too easy for their heavy artillery (advanced > theorems) to hit. He said it was accessible to us, but might require > some creative ideas and a lot of tinkering. ] > I don't even know how one would start to prove this one. > I tried to do things like, assuming p^3+ 4 was composite and tried to > find a contradiction, but got nowhere. > Hint: compute the values of p and p^2+2 for small values of p, most > of the times p^2 + 2 may be divided by ... Work modulo ... Good luck! > Raymond Yeah. I only figured out how to prove it after studying for the midterm (ended up learning Fermat's Little theorem). That coupled with your suggestion to figure out values of p^2 + 2 and notice common divisibility, lead me in the right direction.Basically the statement only applies to p=3 because for p <> 3, p^2+2 ends up being divisible by 3 (proved this by Fermat's Little). I must say, this Fermat's Little theorem seems so outrageous, nonintuitive, and powerful. === Subject: Re: p and p^2 + 2 prime => p^3 + 4 prime Kenneth Bull a .8ecrit : >> Kenneth Bull a .8ecrit : > If p and p^2 + 2 are prime, then p^3 + 4 is prime. > ... >> Hint: compute the values of p and p^2+2 for small values of p, most >> of the times p^2 + 2 may be divided by ... Work modulo ... >> Good luck! >> Raymond Yeah. I only figured out how to prove it after studying for the > midterm (ended up learning Fermat's Little theorem). That coupled with > your suggestion to figure out values of p^2 + 2 and notice common > divisibility, lead me in the right direction.Basically the statement > only applies to p=3 because for p <> 3, p^2+2 ends up being divisible > by 3 (proved this by Fermat's Little). > Fine! > I must say, this Fermat's Little theorem seems so outrageous, > nonintuitive, and powerful. In fact you don't need Fermat... if p prime not 3 then p=1 (mod 3) or p=2 (mod 3) in both cases this implies p^2+2=0 (mod 3) Fine continuation! Raymond === Subject: dot product magnitudes I'm having a problem with magnitudes. Can anyone explain how the vectors ... <4,3> dot <3,5> ------------------------ |<4,3>| |<3,5> equals 27 ------------ 5 sq rt (34) It's the magnitude of the denominators I'm having a problem with. === Subject: Re: dot product magnitudes > I'm having a problem with magnitudes. Can anyone explain how the > vectors ... <4,3> dot <3,5 ------------------------ > |<4,3>| |<3,5 > equals 27 > ------------ > 5 sq rt (34) It's the magnitude of the denominators I'm having a problem with. > Use || = sqr(a^2 + b^2). === Subject: Re: dot product magnitudes Trigger a .8ecrit : > I'm having a problem with magnitudes. Can anyone explain how the > vectors ... <4,3> dot <3,5 ------------------------ > |<4,3>| |<3,5>| equals 27 > ------------ > 5 sqrt(34) It's the magnitude of the denominators I'm having a problem with. > Well || = sqrt(x^2+y^2) so ... good luck, Raymond === Subject: Budaya Suap ! Hentikanlah Segera Budaya Suap ! Hentikanlah Segera (www.mediamuslim.info) Sesungguhnya sesuatu yang diharamkan bahkan sangat diharamkan dalam ajaran Islam adalah suap. Suap berarti memberi sejumlah harta benda kepada pihak yang berwenang (pelaku birokrasi) yang mana dengan tanpa pemberian tersebut hal itu memang sudah menjadi kewajibannya yang harus ditunaikan. Hukum suap menjadi sangat diharamkan jika tujuannya adalah memutarbalikkan yang batil menjadi benar atau membenarkan kebatilan atau menganiaya seseorang. Sedang menurut Ibnu Abidin bahwa suap adalah sesuatu yang diberikan seseorang kepada hakim atau lainnya supaya orang itu memutuskan sesuatu hal yang memihak kepadanya atau agar ia memperoleh keinginannya (dengan pemberian tersebut-pent). Sesuatu yang diberikan itu adakalanya berupa harta benda, uang atau apa saja yang bermanfaat bagi si penerima sehingga keinginan penyuap tersebut dapat terwujud. Suap termasuk salah satu dosa besar yang diharamkan Allah Subhannahu wa Ta'ala atas hamba-hamba-Nya, dan Rasulullah Shallallaahu 'alaihi wa Salam pun melaknat pelakunya. Kita wajib menjauhi dan waspada terhadapnya serta memberi peringatan kepada orang-orang yang melakukannya karena suap mengandung kejahatan dan merupakan dosa besar serta berakibat sangat buruk. Allah Subhannahu wa Ta'ala melarang kita untuk bekerjasama dalam dosa dan pelanggaran. Allah Subhannahu wa Ta'ala berfirman: Dan tolong-menolonglah kamu dalam (mengerjakan) kebajikan dan taqwa, dan jangan tolong-menolong dalam berbuat dosa dan pelanggaran. (Al Maidah: 2) Allah Subhannahu wa Ta'ala juga melarang kita memakan harta orang lain dengan cara yang batil, sebagaimana firman-Nya: Dan janganlah sebahagian kamu memakan harta sebahagian yang lain di antara kamu dengan jalan yang batil dan (janganlah) kamu membawa (urusan) harta itu kepada hakim, supaya kamu dapat memakan sebahagian daripada harta benda orang lain itu dengan (jalan berbuat) dosa, padahal kamu menge-tahui. (Al Baqarah: 188) Suap termasuk cara paling buruk dalam memakan harta orang lain dengan jalan batil, karena ia memberi uang kepada orang lain (secara tidak semestinya) dengan maksud untuk menghalangi kebenaran. Pengharaman suap meliputi 3 unsur yaitu: Penyuap, yang disuap dan perantara dari keduanya, sebagai-mana sabda Rasulullah Shallallaahu 'alaihi wa Salam : Allah Subhannahu wa Ta'ala melaknat penyuap, yang disuap dan perantara dari keduanya. (HR. Ahmad dan Thabrani) Laknat Allah Subhannahu wa Ta'ala itu berarti diusir atau dijauhkan dari limpahan rahmat-Nya. (Naudzubillahi min dzalik) dan ini hanya terjadi pada perbuatan dosa besar. Suap merupakan perbuatan buruk dan diharamkan Al Qur'an dan As Sunnah. Dan sungguh Allah Subhannahu wa Ta'ala telah mengancam dan mencela orang-orang Yahudi karena memakan yang haram, sebagaimana firman-Nya: Mereka itu adalah orang-orang yang suka mendengar berita bohong, banyak memakan yang haram. (Al Maidah: 42) Begitu juga firmanNya: Dan kamu akan melihat keba-nyakan dari mereka (orang-orang Yahudi) bersegera membuat dosa, permusuhan dan memakan yang haram. Sesungguhnya amat buruk apa yang mereka telah kerjakan itu. Mengapa orang-orang alim mereka, pendeta-pendeta mereka tidak melarang mereka mengucapkan perkataan bohong dan memakan yang haram Sesungguhnya amat buruk apa yang telah mereka kerjakan itu. (Al Maidah: 62-63) Terdapat banyak hadits yang memberikan peringatan dari perbuatan yang haram ini dan menerangkan akibat buruk bagi pelakunya, di antaranya adalah hadits yang diriwayatkan Ibnu Jarir dari Ibnu Umar Radhiallaahu anhu dari Nabi Shallallaahu 'alaihi wa Salam , beliau bersabda: Setiap daging yang tumbuh dari yang haram maka neraka lebih pantas baginya. Kemudian ditanyakan kepada Nabi Shallallaahu 'alaihi wa Salam : Apakah barang yang haram itu? Nabi Shallallaahu 'alaihi wa Salam menjawab: Suap dalam proses hukum. Diriwayatkan dari Imam Ahmad dari Amr bin Ash Radhiallaahu anhu berkata: Saya men-dengar Rasulullah Shallallaahu 'alaihi wa Salam bersabda: Sesungguhnya Allah Subhannahu wa Ta'ala itu Baik, tidak mau menerima kecuali baik dan sesungguhnya Allah Subhannahu wa Ta'ala menyuruh orang-orang mukmin sebagaimana menyuruh kepada para rasul. Allah Subhannahu wa Ta'ala berfiman: Hai rasul-rasul, makanlah dari makanan yang baik-baik dan kerjakanlah amal yang shalih. (Al Mukminun : 51) Dan Dia berfirman: Hai orang-orang yang beriman, makanlah di antara rezki yang baik-baik yang Kami berikan kepadamu. (Al Baqarah: 172) Kemudian Nabi Shallallaahu 'alaihi wa Salam menuturkan cerita seorang laki-laki yang datang dari tempat yang jauh, rambutnya tidak terurus dan badannya penuh debu sambil menadahkan tangannya ia mengucapkan: Ya Rabbi, Ya Rabbi, sedang makanannya haram, minuman-nya haram, pakaiannya haram dan diberi makan dengan yang haram, maka bagaimana mungkin doanya akan dikabulkan. Wahai kaum muslimin, bertaqwalah kepada Allah Subhannahu wa Ta'ala, jauhilah murka-Nya dan yang menyebabkan kemarahan-Nya. Sesungguhnya Allah Subhannahu wa Ta'ala sangat cemburu jika dilanggar larangan-larangan-Nya. Disebutkan dalam hadits shahih: Tidak ada yang lebih pencemburu selain Allah Subhannahu wa Ta'ala . Kemudian hindarkanlah dirimu dan keluargamu dari harta yang haram dan memakan yang haram, agar kamu dan keluargamu selamat dari api neraka yang dijadikan Allah Subhannahu wa Ta'ala lebih pantas ditempati bagi setiap daging yang tumbuh dari yang haram. Sesungguhnya makanan yang haram menjadi sebab terhalang dan tidak terkabulnya do'a. Sebagaimana hadits yang diriwayatkan Muslim dari Abu Hurairah. Thabrani juga meriwa-yatkan dari Ibnu Abbas Radhiallaahu anhu ia berkata: Dihadapan Rasulullah Shallallaahu 'alaihi wa Salam dibacakan ayat: Hai sekalian manusia makanlah yang halal lagi baik dari apa yang terdapat di bumi. (Al Baqarah: 168) Kemudian Sa'ad bin Abi Waqash berdiri dan berakta: Ya Rasulullah, berdo'alah Anda kepada Allah agar Dia menjadikan aku orang yang selalu dikabulkan bila berdo'a. Lalu Nabi n menjawab: Wahai Sa'ad, bersihkanlah isi perutmu, niscaya engkau menjadi orang yang selalu dikabulkan do'anya, demi jiwa Muhammad yang berada digeng-gamanNya, sesungguhnya seseorang yang menelan sesuap makanan yang haram ke dalam perutnya, maka Allah Subhannahu wa Ta'ala tidak akan menerima ibadahnya selama empat puluh hari. Dan hamba mana saja yang daging (tubuhnya) tumbuh dari yang haram maka neraka lebih pantas baginya. (Dikutip oleh Al Hafizh Ibnu Rajab dalam Kitab Jami'ul Ulum wal Hikam yang diriwayatkan oleh Thabrani). Hadits di atas menerangkan bahwa tidak memilih makanan yang baik dan halal menyebabkan do'a seseorang terhalang, tidak sampai kepada Allah Subhannahu wa Ta'ala, dan cukuplah ia mendapat kesusahan dan kerugian. (Na'udzu billahi min dzalik) Ketahuilah, sesungguhnya Allah Subhannahu wa Ta'ala menyeru agar menjauhkan diri dari neraka dan dari siksa-Nya yang pedih, sebagaimana firman-Nya: Hai orang-orang yang beriman, peliharalah dirimu dan keluargamu dari api neraka yang bahan bakarnya adalah manusia dan batu; penjaganya malaikat-malaikat yang kasar, yang keras, yang tidak mendurhakai Allah terhadap apa yang diperintahkan-Nya kepada mereka dan selalu mengerjakan apa yang diperintahkan. (At Tahrim: 6) Wahai kaum muslimin, sambutlah seruan Allah, taatilah perintah-Nya dan jauhilah larangan-Nya, waspada ter-hadap hal-hal yang menimbulkan murka-Nya, pasti kita semua akan mendapat kebahagiaan di dunia dan di akhirat. Allah Subhannahu wa Ta'ala berfirman: Hai orang-orang beriman, penuhilah seruan Allah dan seruan Rasul apabila Rasul menyeru kamu kepada suatu yang memberi kehidupan kepada kamu, dan ketahuilah bahwa sesungguhnya Allah membatasi antara manusia dan hatinya, dan sesungguhnya kepada-Nyalah kamu akan dikumpulkan. Dan peliharalah dirimu dari pada siksaan yang tidak khusus menimpa orang-orang yang zhalim saja di antara kamu. Dan ketahuilah bahwa Allah amat keras siksaan-Nya. (Al Anfaal: 24-25) Hanya Allah Subhannahu wa Ta'ala lah tempat kita meminta, semoga Allah Subhannahu wa Ta'ala menjadikan kita semua termasuk orang-orang yang mendengarkan firman-Nya, kemudian mengikutinya, dan termasuk orang-orang yang saling tolong menolong dalam kebaikan dan taqwa, senantiasa berpegang teguh dengan Kitabullah dan sunnah Rasul-Nya. Dan semoga Dia melindungi kita dari kejahatan jiwa kita dan keburukan perbuatan kita. Semoga Dia senantiasa menolong agama-Nya dan meninggikan kalimat-Nya, serta memberikan taufiq kepada pemimpin-pemimpin kita yang membawa kebaikan bagi rakyat dan negara. Sesungguhnya Dialah Pelindung dan Yang Maha Kuasa atas segalanya. (Bintu Abiha ) (Dikutip dari: buletin terbitan Daarul Wathan Riyadh judul Ar Risywah, Risalah Terbuka, Syaikh Abdul Aziz bin Abdullah bin Baz.) === Subject: Re: a question about 4 colorings colors? I do not mean the vertices of the graph/topology (whatever > that thing is called where the areas of a map are converted to > vertices), but the natural vertices of the map. -- It's important I > imagine that the coloring of the vertices are independent of the > coloring of the areas. What is the rule for coloring the vertices? They have to be different > colors than all the regions they touch? Or in general, (in non-planar cases), is there a map requiring maximum > colors whos natural vertices also require maximum colors? If there isn't some pigeonhole principle (or something) that excludes > this, how could one construct it? I'm interested in this from an artist's perspective, I'd like to > actually make such an image. -- > Marcus The dual of a map is a map. That is, replace every vertex by a region and shrink every vertex to a point and you will get the dual map. So that will generally need four colors. I have a vague memory that the four color theorem is equivalent to the assertion that the edges of a trivalent map (every vertex has exactly three edges coming into it) can be three colored. That is that you can color the edges in such a way that the three edges coming into each vertex have distinct colors. === Subject: Re: a question about 4 colorings > Pardon if this is some totally dense question! Is there some map that requires 4 colors, who's vertices also require 4 > colors? Yes. A map with 4 regions, where all regions are adjacent to each other. > I do not mean the vertices of the graph/topology (whatever > that thing is called where the areas of a map are converted to > vertices), but the natural vertices of the map. -- It's important I > imagine that the coloring of the vertices are independent of the > coloring of the areas. Or in general, (in non-planar cases), is there a map requiring maximum > colors whos natural vertices also require maximum colors? Almost always; I think there's one exception (the Klein Bottle), where it misses by 1. Otherwise, for any surface S, if k is the largest number of regions in a map on S that can all be adjacent to each other, and c is the coloring number of S (every map on S can be colored with c colors), then k = c. > If there isn't some pigeonhole principle (or something) that excludes > this, how could one construct it? It's a fairly difficult proof; the proof is actually a separate book (not in a journal). I can't use MathSciNet to find it at the moment, but it came out in the 1960s. ((A few minutes later)) Check out http://mathworld.wolfram.com/MapColoring.html . --- > I'm interested in this from an artist's perspective, I'd like to > actually make such an image. === Subject: Re: a question about 4 colorings > Pardon if this is some totally dense question! Is there some map that requires 4 colors, who's vertices also require 4 > colors? Yes. A map with 4 regions, where all regions are adjacent to each > other. Contrary to the rest of my post, this result does NOT hold for higher surfaces. If the regions form a K_n (a clique of size n), then the vertices need not form a K_n. There is a coloring problem called total coloring, where you color all regions, edges, and vertices. --- > I do not mean the vertices of the graph/topology (whatever > that thing is called where the areas of a map are converted to > vertices), but the natural vertices of the map. -- It's important I > imagine that the coloring of the vertices are independent of the > coloring of the areas. Or in general, (in non-planar cases), is there a map requiring maximum > colors whos natural vertices also require maximum colors? Almost always; I think there's one exception (the Klein Bottle), where > it misses by 1. Otherwise, for any surface S, if k is the largest > number of regions in a map on S that can all be adjacent to each other, > and c is the coloring number of S (every map on S can be colored with c > colors), then > k = c. If there isn't some pigeonhole principle (or something) that excludes > this, how could one construct it? It's a fairly difficult proof; the proof is actually a separate book > (not in a journal). I can't use MathSciNet to find it at the moment, > but it came out in the 1960s. ((A few minutes later)) Check out > http://mathworld.wolfram.com/MapColoring.html . --- I'm interested in this from an artist's perspective, I'd like to > actually make such an image. === Subject: Re: Q: solution of a set of exponential equations > Hi - it's driving me nuts; I think, I had it already times > ago, but at the moment I cannot find a path: assume 1,v,w the three complex roots of 1. Then consider the system: a:= a(x) = exp(x) + exp(vx) + exp(wx) > b:= b(x) = exp(x) + v exp(vx) + w exp(wx) > c:= c(x) = exp(x) + w exp(vx) + v exp(wx) Given a, I have 3 eqns in 3 unknowns, and it > should be solvable to find b,c, and x. For that matter, the first equation has only one unknown: x. Solve for it first. David > I think, I even had it using a matrix scheme, > exploiting some reducible terms when squared > or cubed, for instance (a + b + c)^3 = a^3 + b^3 + c^3 + 6abc Also it may involve multiple branching for > negative or complex a, so let's first assume > a is a positive real>0 I'm currently spinning in circles... Some help appreciated - Gottfried Helms === Subject: Re: Q: solution of a set of exponential equations Am 12.11.2006 01:34 schrieb W. Cantrell: >> Hi - >> it's driving me nuts; I think, I had it already times >> ago, but at the moment I cannot find a path: >> assume 1,v,w the three complex roots of 1. >> Then consider the system: >> a:= a(x) = exp(x) + exp(vx) + exp(wx) >> b:= b(x) = exp(x) + v exp(vx) + w exp(wx) >> c:= c(x) = exp(x) + w exp(vx) + v exp(wx) >> Given a, I have 3 eqns in 3 unknowns, and it >> should be solvable to find b,c, and x. For that matter, the first equation has only one unknown: x. > Solve for it first. David - the form of the formula for the first line only is something like a = exp(x)*(1 + 2 * cos( sqrt(3)/4*x) ) I think, this can only be solved numerically; I would like to have it in terms of a single reference to exp(x), sinh(x) or the like first. . What I thought to recall was, that some basic identities (which can be set up because of the fact, that the expressions contain rotations by 1/3 * 2*pi and some cancel out nicely), could be exploited to simplify things much by applying rules of linear algebra. Separating into the above 1,v,w-coefficients into a matrix V and the exp-coefficients into a diagonalmatrix E and (a,b,c) in a matrix A = (( a,b,c),(c,b,a),(b,a,c)) I have 1 1 1 exp(x) . . 1 1 1 a b c 1 v w * . exp(vx) . * 1 v w = b c a 1 w v . . exp(wx) 1 w v c a b V * E * V = A , where V and a are given. Hmm.. Gottfried === Subject: JSH: What are your limits? Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication in a math journal? And even keep manipulating you when the poor journal collapsed and died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? ___JSH === Subject: Re: JSH: What are your limits? > How can they exist? This should be: How can you exist? I.e., your imaginary audience. --- J K Haugland http://home.no.net/zamunda === Subject: Re: JSH: What are your limits? How can they exist? This should be: How can you exist? I.e., your imaginary audience. --- > J K Haugland > http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use pseudonyms to help promote the idea of a LOT of you with a consensus that I'm wrong. So that explains how supposedly publication in a math journal doesn't matter all of a sudden--a bizarre reversal of how the math community supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for Usenet, possibly playing what to you is just a minor game where you feel safe behind the pseudonyms when you use them, never expecting to be held accountable. My guess is that you see this as a small stage with a few players and don't think any of it matters, which reminds me of a while back when there was a strongly held position from sci.math posters replying to me that only a few hundred people read sci.math with mostly a few dozen posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people playing what is to them a minor game. You people think nothing said here matters, so you say anything that suits you. James Harris === Subject: Re: JSH: What are your limits? How can they exist? This should be: How can you exist? I.e., your imaginary audience. --- > J K Haugland > http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use > pseudonyms to help promote the idea of a LOT of you with a consensus > that I'm wrong. So that explains how supposedly publication in a math journal doesn't > matter all of a sudden--a bizarre reversal of how the math community > supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for > Usenet, possibly playing what to you is just a minor game where you > feel safe behind the pseudonyms when you use them, never expecting to > be held accountable. My guess is that you see this as a small stage with a few players and > don't think any of it matters, which reminds me of a while back when > there was a strongly held position from sci.math posters replying to me > that only a few hundred people read sci.math with mostly a few dozen > posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of > thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people > playing what is to them a minor game. You people think nothing said here matters, so you say anything that > suits you. > James Harris I propose a head count. Everyone who is reading James posts should come forward and be counted. I'll start. I'll be number 1. I hope the tens of thousands of people reading this and who believe that James is right should stop being so shy and come forward and support Mr. Harris. === Subject: Re: JSH: What are your limits? How can they exist? This should be: How can you exist? I.e., your imaginary audience. --- > J K Haugland > http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use > pseudonyms to help promote the idea of a LOT of you with a consensus > that I'm wrong. So that explains how supposedly publication in a math journal doesn't > matter all of a sudden--a bizarre reversal of how the math community > supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for > Usenet, possibly playing what to you is just a minor game where you > feel safe behind the pseudonyms when you use them, never expecting to > be held accountable. My guess is that you see this as a small stage with a few players and > don't think any of it matters, which reminds me of a while back when > there was a strongly held position from sci.math posters replying to me > that only a few hundred people read sci.math with mostly a few dozen > posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of > thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people > playing what is to them a minor game. You people think nothing said here matters, so you say anything that > suits you. > James Harris I propose a head count. Everyone who is reading James posts should come > forward and be counted. I'll start. I'll be number 1. I hope the tens of thousands of people reading this and who believe > that James is right should stop being so shy and come forward and > support Mr. Harris. I guess James was right. Three people so far. He commands the attention of the world. === Subject: Re: JSH: What are your limits? > > >How can they exist? >>This should be: How can you exist? I.e., your imaginary audience. >>--- >>J K Haugland >>http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use >pseudonyms to help promote the idea of a LOT of you with a consensus >that I'm wrong. So that explains how supposedly publication in a math journal doesn't >matter all of a sudden--a bizarre reversal of how the math community >supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for >Usenet, possibly playing what to you is just a minor game where you >feel safe behind the pseudonyms when you use them, never expecting to >be held accountable. My guess is that you see this as a small stage with a few players and >don't think any of it matters, which reminds me of a while back when >there was a strongly held position from sci.math posters replying to me >that only a few hundred people read sci.math with mostly a few dozen >posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of >thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people >playing what is to them a minor game. You people think nothing said here matters, so you say anything that >suits you. >James Harris >>I propose a head count. Everyone who is reading James posts should come >>forward and be counted. >>I'll start. I'll be number 1. >>I hope the tens of thousands of people reading this and who believe >>that James is right should stop being so shy and come forward and >>support Mr. Harris. > I guess James was right. Three people so far. He commands the attention > of the world. > And what makes you think that Dr. Doctor, mensantor, and jshucks are three distinct people? Silly fool--they all might be Ullrich's sock puppets. I will admit, though, that this latest foray is bizarre, even for the poster who claims to be James S. Harris. Rick === Subject: Re: JSH: What are your limits? >How can they exist? >>This should be: How can you exist? I.e., your imaginary audience. >>--- >>J K Haugland >>http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use >pseudonyms to help promote the idea of a LOT of you with a consensus >that I'm wrong. So that explains how supposedly publication in a math journal doesn't >matter all of a sudden--a bizarre reversal of how the math community >supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for >Usenet, possibly playing what to you is just a minor game where you >feel safe behind the pseudonyms when you use them, never expecting to >be held accountable. My guess is that you see this as a small stage with a few players and >don't think any of it matters, which reminds me of a while back when >there was a strongly held position from sci.math posters replying to me >that only a few hundred people read sci.math with mostly a few dozen >posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of >thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people >playing what is to them a minor game. You people think nothing said here matters, so you say anything that >suits you. >James Harris >>I propose a head count. Everyone who is reading James posts should come >>forward and be counted. >>I'll start. I'll be number 1. >>I hope the tens of thousands of people reading this and who believe >>that James is right should stop being so shy and come forward and >>support Mr. Harris. > I guess James was right. Three people so far. He commands the attention > of the world. And what makes you think that Dr. Doctor, mensantor, and jshucks are > three distinct people? Silly fool--they all might be Ullrich's > sock puppets. Yes, we have no bananas. We have no bananas today. -- Canaan Banana A genuine sock puppet === Subject: Re: JSH: What are your limits? >How can they exist? >>This should be: How can you exist? I.e., your imaginary audience. >>--- >>J K Haugland >>http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use >pseudonyms to help promote the idea of a LOT of you with a consensus >that I'm wrong. So that explains how supposedly publication in a math journal doesn't >matter all of a sudden--a bizarre reversal of how the math community >supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for >Usenet, possibly playing what to you is just a minor game where you >feel safe behind the pseudonyms when you use them, never expecting to >be held accountable. My guess is that you see this as a small stage with a few players and >don't think any of it matters, which reminds me of a while back when >there was a strongly held position from sci.math posters replying to me >that only a few hundred people read sci.math with mostly a few dozen >posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of >thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people >playing what is to them a minor game. You people think nothing said here matters, so you say anything that >suits you. >James Harris >>I propose a head count. Everyone who is reading James posts should come >>forward and be counted. >>I'll start. I'll be number 1. >>I hope the tens of thousands of people reading this and who believe >>that James is right should stop being so shy and come forward and >>support Mr. Harris. > I guess James was right. Three people so far. He commands the attention > of the world. And what makes you think that Dr. Doctor, mensantor, and jshucks are > three distinct people? Silly fool--they all might be Ullrich's > sock puppets. I will admit, though, that this latest foray is bizarre, even for the > poster who claims to be James S. Harris. How do we know that you aren't a Ullrich sock puppet? And how do you know that I'm not just a Ullrich sock puppet acusing you of being a Ullrich sock puppet? And how do we know that Ullrich is not a JSH sock puppet? After all the only evidence we have that JSH contacted Ullrich's employer comes from JSH or Ullrich. -William (Yeah Right) Hughes === Subject: Re: JSH: What are your limits? [William Hughes, appearing to challenge the Rick Decker poster] > How do we know that you aren't a Ullrich sock puppet? And how do you know that I'm not just a Ullrich sock > puppet acusing you of being a Ullrich sock puppet? And how do we know that Ullrich is not a JSH sock > puppet? After all the only evidence we have that JSH contacted > Ullrich's employer comes from JSH or Ullrich. Not so! I can confirm that -- I'm the guy you get on the phone when you call OSU to complain about Ullrich. In fairness, I haven't admitted that before. Although in brutal fairness, the guy on the phone saying he was James Harris sounded an awful lot like Ullrich, and he is a notorious prankster. > -William (Yeah Right) Hughes No, you really are William Hughes -- or at least that's what your boss tells me when I call her to complain about you. BTW, how's Nora doing? === Subject: Re: JSH: What are your limits? >How can they exist? >>This should be: How can you exist? I.e., your imaginary audience. >>--- >>J K Haugland >>http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use >pseudonyms to help promote the idea of a LOT of you with a consensus >that I'm wrong. So that explains how supposedly publication in a math journal doesn't >matter all of a sudden--a bizarre reversal of how the math community >supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for >Usenet, possibly playing what to you is just a minor game where you >feel safe behind the pseudonyms when you use them, never expecting to >be held accountable. My guess is that you see this as a small stage with a few players and >don't think any of it matters, which reminds me of a while back when >there was a strongly held position from sci.math posters replying to me >that only a few hundred people read sci.math with mostly a few dozen >posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of >thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people >playing what is to them a minor game. You people think nothing said here matters, so you say anything that >suits you. >James Harris >>I propose a head count. Everyone who is reading James posts should come >>forward and be counted. >>I'll start. I'll be number 1. >>I hope the tens of thousands of people reading this and who believe >>that James is right should stop being so shy and come forward and >>support Mr. Harris. > I guess James was right. Three people so far. He commands the attention > of the world. And what makes you think that Dr. Doctor, mensantor, and jshucks are > three distinct people? Silly fool--they all might be Ullrich's > sock puppets. I will admit, though, that this latest foray is bizarre, even for the > poster who claims to be James S. Harris. Rick Shhhhh Rick. Master might get angry :-( === Subject: Re: JSH: What are your limits? How can they exist? This should be: How can you exist? I.e., your imaginary audience. --- > J K Haugland > http://home.no.net/zamunda Well the audience now knows that many of you are fakes who use > pseudonyms to help promote the idea of a LOT of you with a consensus > that I'm wrong. So that explains how supposedly publication in a math journal doesn't > matter all of a sudden--a bizarre reversal of how the math community > supposedly sees publication. It doesn't matter to you people and you're a weird fringe even for > Usenet, possibly playing what to you is just a minor game where you > feel safe behind the pseudonyms when you use them, never expecting to > be held accountable. My guess is that you see this as a small stage with a few players and > don't think any of it matters, which reminds me of a while back when > there was a strongly held position from sci.math posters replying to me > that only a few hundred people read sci.math with mostly a few dozen > posters. That goes back to your assertions about an imaginary audience. While I think there is a worldwide audience in the tens of > thousands--and many of them ted you and people like you. And recent revelations showed they were fools taken in by people > playing what is to them a minor game. You people think nothing said here matters, so you say anything that > suits you. > James Harris I propose a head count. Everyone who is reading James posts should come > forward and be counted. I'll start. I'll be number 1. I hope the tens of thousands of people reading this and who believe > that James is right should stop being so shy and come forward and > support Mr. Harris. Count me also. === Subject: Re: JSH: What are your limits? > How can they exist? >> This should be: How can you exist? I.e., your imaginary audience. >> --- >> J K Haugland >> http://home.no.net/zamunda >> Well the audience now knows that many of you are fakes who use >> pseudonyms to help promote the idea of a LOT of you with a consensus >> that I'm wrong. >> So that explains how supposedly publication in a math journal doesn't >> matter all of a sudden--a bizarre reversal of how the math community >> supposedly sees publication. >> It doesn't matter to you people and you're a weird fringe even for >> Usenet, possibly playing what to you is just a minor game where you >> feel safe behind the pseudonyms when you use them, never expecting to >> be held accountable. >> My guess is that you see this as a small stage with a few players and >> don't think any of it matters, which reminds me of a while back when >> there was a strongly held position from sci.math posters replying to me >> that only a few hundred people read sci.math with mostly a few dozen >> posters. >> That goes back to your assertions about an imaginary audience. >> While I think there is a worldwide audience in the tens of >> thousands--and many of them ted you and people like you. >> And recent revelations showed they were fools taken in by people >> playing what is to them a minor game. >> You people think nothing said here matters, so you say anything that >> suits you. >> James Harris I propose a head count. Everyone who is reading James posts should come > forward and be counted. I'll start. I'll be number 1. I hope the tens of thousands of people reading this and who believe > that James is right should stop being so shy and come forward and > support Mr. Harris. > I'm #2, (I think most JSH info is concentrated in a few areas, that have yet to find applicability, and that are repeated every few weeks or more) === Subject: Re: JSH: What are your limits? Asshole. > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? When will you actually factor a number ? How can they exist? How could such people do what they did? do you mean the Hive? and/or the Group? Who cares, how could they so easily get you all to ignore publication > in a math journa the whole journal was ignored. And even keep manipulating you when the poor journal collapsed and > died? we have always manipulated you, JSH, you are easy. The bigger question now is, are they connected even higher up? Yes. How far up the math chain does this all go? > NSA, the group. The Hive is internet users ___JSH > === Subject: Re: JSH: What are your limits? Asshole. <4556b428$0$97268$892e7fe2@authen.yellow.readfreenews.net [...] > How can they exist? How could such people do what they did? do you mean the Hive? and/or the Group? I think JSH's word-of-the-month is now swarm. Hive and group are so October. --- A. P. Heckman === Subject: Re: JSH: What are your limits? : Yup, many of you were played for fools, but what will you do now? Have a whisky. : What are your limits? With L'hopital's Rule, I can handle many limits. : Will you still just accept the weird group of people who manipulated : and lied to you for years about my math? Nobody's lied to me, I discovered it was wrong all on my own! : Or will you FINALLY go with the math? Been there. : How far up the math chain does this all go? Is it Hammer Time now?! Justin === Subject: Re: JSH: What are your limits? [jstevh@msn.com] > Yup, many of you were played for fools, In truth, only you were -- and it wasn't even intentional! This was most like the time Jim Ferry pretended to be your disciple, and you were the only one who didn't immediately see that it was a put-on. There's a huge difference this time, though: besides that Jim was much wittier, I didn't think for even a nanosecond that anyone would be taken in. Not even you. But then you're endlessly surprising that way. It's like they keep making the Pons Asinorum shorter and shorter, until it's just one atom thick, and you still can't get across it. > but what will you do now? Pretty much the same as yesterday, I imagine. And you? > ... > How far up the math chain does this all go? Consistency, James: remember that I'm not even in the math chain. It seems that you and Wiles are (although only you /should/ be), but I'm not sure you've admitted that any other living being is in it. Hiram Kernoodle Class IV === Subject: Re: JSH: What are your limits? In sci.math, jstevh@msn.com on 11 Nov 2006 17:01:09 -0800 > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? > ___JSH > And the momentous result that we're supposed to accept from you using blind faith is ... ? So far, the only thing I've gotten from your theory is that there is a super-ring of the algebraic integers (the Object Ring) and this Object Ring has exactly two units. Since this is obvious balderdash (a superset of a ring that is also a ring inherits all of the subring's units, and the algebraic integers have an infinite number of units) you're going to have to try a lot harder. -- #191, ewill3@earthlink.net Linux. Because it's not the desktop that's important, it's the ability to DO something with it. -- === Subject: Re: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? > ___JSH > I'm sorry you're not as smart as you think you are and have been told. However, you deserve most the mistreatment you get. I say most and not all because I've seen some remarks made toward you that I felt no one deserves. I'm a mathematician and I don't feel like I'm qualified to comment on topics from Abstract Algebra. What makes you think you are? Dave === Subject: Re: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? You don't have any math. Or will you FINALLY go with the math? You don't know what math is. How can they exist? You don't exist. How could such people do what they did? You've done nothing. Who cares, No one cars how could they so easily get you all to ignore publication > in a math journal? We ignore you. And even keep manipulating you when the poor journal collapsed and > died? Die screaming with a flaming slide rule rammed up your back-side The bigger question now is, are they connected even higher up? Buh? How far up the math chain does this all go? Buh? > ___JSH > Buh? Buh? === Subject: Re: JSH: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? > ___JSH To the top, James. The top. Darth Wiles is just the servant. Who is the Master? You'll never know until it's too late.....mwhahahaah, mwhahahahaha...you poor weak-minded fool...mwhahahaaha === Subject: Re: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore > publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? It doesn't go any higher James. Honestly. Just a bunch of guys larking around. It's no big deal. -- Clive Tooth http://www.shutterstock.com/cat.mhtml?gallery_id=61771 === Subject: Re: What are your limits? > Just a bunch of guys larking around. It's no big deal. I do wish that people replying to jsh would put a [jsh] or [JSH} in the subject line so that we innocent viewers need not be bothered by him. === Subject: Re: JSH: What are your limits? [added JSH: to subject] [Virgil] > I do wish that people replying to jsh would put a [jsh] or [JSH} in the > subject line so that we innocent viewers need not be bothered by him. That may have had more effect had you bothered to do so yourself ;-) Is JSH: not good enough for you? I usually remember to add that marker when it's missing (as I did to this reply), and James often inserts it himself. === Subject: Re: JSH: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? > ___JSH Why would you think that mathematical arguments posted to a newgroups would change with the identity of the author. The problems with your arguments have been spelled out in the open. People read what you posted and then could figure out on their own if you were correct or not, or they could read the replies that others posted. The facts in those posts don't change if the author was using a pseudoname and not By the way, I don't think that mensanator is his real name either. You better get the FBI involved with this travesty of internet injustice. Yours truly, Billy Bob Einstein === Subject: Re: JSH: What are your limits? > Yup, many of you were played for fools, but what will you do now? What are your limits? Will you still just accept the weird group of people who manipulated > and lied to you for years about my math? Or will you FINALLY go with the math? How can they exist? How could such people do what they did? Who cares, how could they so easily get you all to ignore publication > in a math journal? And even keep manipulating you when the poor journal collapsed and > died? The bigger question now is, are they connected even higher up? How far up the math chain does this all go? > ___JSH Why would you think that mathematical arguments posted to a newgroups > would change with the identity of the author. The problems with your > arguments have been spelled out in the open. People read what you > posted and then could figure out on their own if you were correct or > not, or they could read the replies that others posted. The facts in > those posts don't change if the author was using a pseudoname and not By the way, I don't think that mensanator is his real name either. Drat, what tipped you off? But I may as well admit it, it's actually Paul The Berserk. Mensanator is a title - Slayer of the Mensa - in old Norse. > You better get the FBI involved with this travesty of internet > injustice. I hate to pull rank on you, but as the Ace of Clubs, the FBI answers to me. Yours truly, > Billy Bob Einstein === Subject: Re: An infinite debate >> Wonderful, Randy. Now apply that thought process to the Ross-Littlwood >> Paradox, and see if you think there is an answer at noon for that one. > I have, many times. Let me recap: > The experiment does not define the number of balls in > the vase at noon. At all times prior to noon, the number of > balls is finite. > To determine the number of balls at noon, we need to > define what we mean by number of balls at noon, > since it is not defined a priori in the problem. > A reasonable definition is cardinality of the set > of balls which is inserted but not removed. > Since that set is empty, by definition the number of > balls in the vase at noon is 0. > I have said all of this many times before in this NG. > - Randy This is a very nice summary of the problem. In order to answer this question, it has to first be translated into a clear mathematical question. It is not a physical problem, so any appeals to physical possibilities are irrelevant. The balls and vase are just a distraction meant to confuse one's physical intuitions. Stephen === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) > > Undefined. So what? > So, as soon as t=0=1/n, n is infinite. Before that point in time, more naturals remain to be counted. Once the set is complete, it contains an infinite value n, since 1/n=0. === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) > > Undefined. So what? > So, as soon as t=0=1/n, n is infinite. But no one, except those with an excessively tenuous grasp on sanity, ever says t = 0 = 1/n. Before that point in time, more > naturals remain to be counted. Once the set is complete, it contains an > infinite value n, since 1/n=0. === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4555e00f@news2.lightlink.com The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >> Undefined. So what? > So, as soon as t=0=1/n, n is infinite. No. n is undefined does not mean n is infinite. There is no n such that 1/n is 0 (not even if you allow infinite n's) > Before that point in time, more > naturals remain to be counted. Yes > Once the set is complete, it contains all the finite values and nothing else. > an > infinite value n, since 1/n=0. No, nothing is added when t=0, in particular no n that would make 1/n=0 (good thing too, as no such n exists). - William Hughes === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) lim(n->oo: 1/n)<>0? > >> Before that point in time, more >> naturals remain to be counted. Yes > >> Once the set is complete, it contains all the finite values and nothing else. > >> an >> infinite value n, since 1/n=0. No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). And for t<0 the set is not complete. How is it completed when no elements are added? - William Hughes > === Subject: Re: An infinite debate > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). And for t<0 the set is not complete. How is it completed when no > elements are added? It is complete at noon, but as completed implies some action at a time when there is none, it is deliberately misleading. === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) lim(n->oo: 1/n)<>0? William didn't say that. He said that for n a natural number 1/n <> 0. lim_{n->oo} 1/n = 0 doesn't imply that any terms in the sequence have to equal the limit. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). And for t<0 the set is not complete. How is it completed when no > elements are added? when no elements are added. -- Marcus === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? William didn't say that. He said that for n a natural number 1/n <> 0. > lim_{n->oo} 1/n = 0 doesn't imply that any terms in the sequence have to > equal the limit. > Apparently you missed the part in parentheses. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? when no elements are added. > Yes, and that makes no sense. === Subject: Re: An infinite debate > when no elements are added. > Yes, and that makes no sense. The sequence of times 1/n minutes before noon is endless, but is complete at noon, a moment when no more elements are added. === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4555e00f@news2.lightlink.com> <4556175b@news2.lightlink.com> <45565732$1@news2.lightlink.com The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? William didn't say that. He said that for n a natural number 1/n <> 0. > lim_{n->oo} 1/n = 0 doesn't imply that any terms in the sequence have to > equal the limit. > Apparently you missed the part in parentheses. No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? when no elements are added. > Yes, and that makes no sense. Because you are stuck on the idea that complete sequences must have an end. - Randy === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? William didn't say that. He said that for n a natural number 1/n <> 0. > lim_{n->oo} 1/n = 0 doesn't imply that any terms in the sequence have to > equal the limit. Apparently you missed the part in parentheses. Actually, I'm not sure what William meant by that comment, but since the limit as n goes to infinity has nothing to do with infinite n's, I don't > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? when no elements are added. Yes, and that makes no sense. Oh, well. Mathematics is often counter intuitive. That's why we prove things instead of just guessing. Do you disagree that the sup of the set of negative numbers is zero? -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4555e00f@news2.lightlink.com> <4556175b@news2.lightlink.com The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) lim(n->oo: 1/n)<>0? >> Before that point in time, more >> naturals remain to be counted. Yes > Once the set is complete, it contains all the finite values and nothing else. > an >> infinite value n, since 1/n=0. No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). And for t<0 the set is not complete. How is it completed when no > elements are added? There is no time at which the process is completed. The first time the process is complete is when t=0. Repeat 27 time before breakfast: There are processes that do not have a last step. If a process does not have a last step, then it is not completed at the first time for which the process is complete. - William Hughes > - William Hughes > === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? >> Before that point in time, more >> naturals remain to be counted. > Yes > Once the set is complete, it contains > all the finite values and nothing else. > an >> infinite value n, since 1/n=0. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? There is no time at which the process is completed. > The first time the process is complete is when t=0. Right. There is no such time. It would have to be a time greater than all times less than 0, but also less than 0, which is a contradiction. Repeat 27 time before breakfast: There are processes that do not have a last step. Hang it up. If a process does not have a last step, then it is not > completed at the first time for which the process is complete. Did it occur to you yet that this completion of the unending sequence is itself nonsensical and the source of your paradox? > === Subject: Re: An infinite debate >> If a process does not have a last step, then it is not >> completed at the first time for which the process is complete. Did it occur to you yet that this completion of the unending sequence is > itself nonsensical and the source of your paradox? Infinity is paradoxical. Consider a thumb tack that is only moved by natural numbers to new locations. The first move, n=1, the thumb tack moves from 0 to 1/2 and each subsequent natural moves the thumb tack half the distance of the previous move. Then all natural numbers move the thumb tack to the number 1 since 0+1/2+1/4+1/8+1/16+...=1. The nth move occurs at time 1-(1/2)^n and the thumb tack is moved to location 1-(1/2)^n at that time. Every move of the thumb tack moves it to a new location in the set X_n=1-(1/2)^n. So we ask: how is it that the thumb tack, which is restricted to locations X_n for each natural n, can reach the number 1 which is not in the set X_n? The answer is that ALEPH_0 is so huge that it moves the thumb tack to a location that is outside the set. The thumb tack has changed locations ALEPH_0 times putting it past each X_n. R === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another > finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, > after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, >> then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? >> Before that point in time, more >> naturals remain to be counted. > Yes > Once the set is complete, it contains > all the finite values and nothing else. > an >> infinite value n, since 1/n=0. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? There is no time at which the process is completed. > The first time the process is complete is when t=0. Right. There is no such time. It would have to be a time greater than > all times less than 0, but also less than 0, which is a contradiction. Not if such a time need not exist. > Repeat 27 time before breakfast: There are processes that do not have a last step. Hang it up. Then repeat 72 time before breakfast: There are processes that do not have a last step. > Did it occur to you yet that this completion of the unending sequence is > itself nonsensical and the source of your paradox? Except that such unending sequences keep getting completed. Every day the endless sequence of times of 1/n minutes before noon, n in N, local time, gets completed. And this has been going on at least as long as we have had clocks. > === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4555e00f@news2.lightlink.com> <4556175b@news2.lightlink.com> <455656c7@news2.lightlink.com The point here that a finite amount, when incremented by another finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? >> Before that point in time, more >> naturals remain to be counted. > Yes > Once the set is complete, it contains > all the finite values and nothing else. > an >> infinite value n, since 1/n=0. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? There is no time at which the process is completed. > The first time the process is complete is when t=0. Right. There is no such time. It would have to be a time greater than > all times less than 0, but also less than 0, which is a contradiction. > Repeat 27 time before breakfast: There are processes that do not have a last step. Hang it up. > If a process does not have a last step, then it is not > completed at the first time for which the process is complete. Did it occur to you yet that this completion of the unending sequence is > itself nonsensical and the source of your paradox? > It is clearly the point of contention. However, if a process that has no last step can be and is perfomed in a finite time, it can be easily and rigourously shown that there will be a time by which the process will be complete, however, there will be no step of the process at this time. So if you accept a: a process with no last step exists and b: a process with no last step can be done in a finite time. you are stuck with c. if P is a process with no last step that is done in a finite time then there exists a time t_f, which is the first time by which the process is complete, but no step of P is done at time t_f. Your problem is that you think a and b are reasonable so you want to accept them, however you think that c is unreasonable so you do not want to accept it. Rather than attacking the simple argument that a plus b leads to c, you say clearly c is unreasonable so the argument must be wrong. My reply is clearly c is counterintuitive, however the argument is correct. - William Hughes === Subject: Re: An infinite debate > The point here that a finite amount, when incremented by another > finite > amount and infinite number of times, will result in an infinitely > large amount. >> You are still trying to talk about the end of things that don't end. > What is wrong with that? >> Yes I agree there is end... So anyway, after it ends... > Maybe that should be Yes, I agree there is no end... So anyway, > after > it ends.... > - Randy >> Um, what if you start with 0, increment it at 1 minute before noon, >> then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? >> :) >Undefined. So what? > So, as soon as t=0=1/n, n is infinite. > No. n is undefined does not mean n is infinite. There is > no n such that 1/n is 0 (not even if you allow infinite n's) >> lim(n->oo: 1/n)<>0? >> Before that point in time, more >> naturals remain to be counted. > Yes > Once the set is complete, it contains > all the finite values and nothing else. > an >> infinite value n, since 1/n=0. > No, nothing is added when t=0, in particular no > n that would make 1/n=0 (good thing too, as no > such n exists). >> And for t<0 the set is not complete. How is it completed when no >> elements are added? There is no time at which the process is completed. > The first time the process is complete is when t=0. Right. There is no such time. It would have to be a time greater than > all times less than 0, but also less than 0, which is a contradiction. So that it goes from incomplete for every instant before noon to complete at noon without ever having any intervening instant of transition, just as the time goes from before noon to noon without any intervening instant of transition. Does TO argue, like Zeno, that noon can never come because there is no intevening instant before noon when before noon ends? === Subject: Re: An infinite debate Right. There is no such time. It would have to be a time greater than > all times less than 0, but also less than 0, which is a contradiction. So that it goes from incomplete for every instant before noon to > complete at noon without ever having any intervening instant of > transition, just as the time goes from before noon to noon without any > intervening instant of transition. Does TO argue, like Zeno, that noon can never come because there is no > intevening instant before noon when before noon ends? Not only does it never come, but it doesn't exist! That's even better than Zeno. -- Marcus === Subject: Re: An infinite debate > There is no time at which the process is completed. > The first time the process is complete is when t=0. Right. There is no such time. It would have to be a time greater than > all times less than 0, but also less than 0, which is a contradiction. What does it contradict? > If a process does not have a last step, then it is not > completed at the first time for which the process is complete. Did it occur to you yet that this completion of the unending sequence is > itself nonsensical and the source of your paradox? Well, of course that's the source of the paradox. On the other hand, there is nothing nonsensical about it. It is clear that if you do an infinite number of steps in a finite time and there is no last step, then the process will finish at a time that is after each step. The sup of a set need not be an element of the set. You should have learned this as a freshman. -- Marcus === Subject: Re: An infinite debate > Should is a good word to use. When would aleph_0 hash marks be > completed, exactly? At no finite time before noon are all hash marks > completed: more remain to be added in the remaining time. But, AT noon, > no hash marks can be added, or that would imply an n such that 1/n=0, > which cannot be a finite n. If it cannot be completed before noon, and > cannot be completed at noon, how can it be completed by noon? The following are true. 1) At no time before noon are all hash marks completed. 2) At noon, no hash marks are added. Therefore (according to you) 3) The hash marks cannot be complete by noon. Let's try a math problem and use the same logic. For j = 1,2,..., let a_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 2^-j if a_j <= x, 0 if x < a_j. Let g(x) = sum_j f_j(x). Then the following are true. 1a) If t1 < 0, then there is a t2 such that t1 < t2 < 0 and g(t1) < g(t2). 2a) For all j, a_j <> 0. Therefore (using your logic), these should imply the following. 3a) g(0) < 1. However, this is false. In fact, g(0) = 1. -- Marcus === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? The following are true. 1) At no time before noon are all hash marks completed. > 2) At noon, no hash marks are added. Therefore (according to you) 3) The hash marks cannot be complete by noon. Let's try a math problem and use the same logic. For j = 1,2,..., let a_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 2^-j if a_j <= x, > 0 if x < a_j. Let g(x) = sum_j f_j(x). Then the following are true. 1a) If t1 < 0, then there is a t2 such that t1 < t2 < 0 and > g(t1) < g(t2). 2a) For all j, a_j <> 0. Therefore (using your logic), these should imply the following. 3a) g(0) < 1. However, this is false. In fact, g(0) = 1. > lim(x->0: g(x))=1. g(x)<1 for x<0. However, g(x)=1 only at x=0, when the uncountable infinity of infinitesimals are added to complete the sum. Something actually does occur at t=0, which goes beyond natural n. That's the way I see it. === Subject: Re: An infinite debate > That's the way I see it. Get new glasses, TO! === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? The following are true. 1) At no time before noon are all hash marks completed. > 2) At noon, no hash marks are added. Therefore (according to you) 3) The hash marks cannot be complete by noon. Let's try a math problem and use the same logic. For j = 1,2,..., let a_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 2^-j if a_j <= x, > 0 if x < a_j. Let g(x) = sum_j f_j(x). Then the following are true. 1a) If t1 < 0, then there is a t2 such that t1 < t2 < 0 and > g(t1) < g(t2). 2a) For all j, a_j <> 0. Therefore (using your logic), these should imply the following. 3a) g(0) < 1. However, this is false. In fact, g(0) = 1. lim(x->0: g(x))=1. g(x)<1 for x<0. However, g(x)=1 only at x=0, You've got it! Finally. So, you agree that it completes at zero! Excellent. > when > the uncountable infinity of infinitesimals are added to complete the > sum. Something actually does occur at t=0, which goes beyond natural n. > That's the way I see it. Whatever. As long as you agree with the math. -- Marcus === Subject: Re: An infinite debate > Um, what if you start with 0, increment it at 1 minute before noon, then > again at a half minute before noon, then again at a third of a minute > before noon, etc, so that at time noon-1/n the number achieves a value of > n, for n in N? That way we have counted through all of the natural numbers > by noon, correct? What is the value of n at noon? This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If by > `n' we mean a variable name for a number stored in the registry of some kind > of ideal computer then I don't think the value of `n' at noon is defined. > But what if by `n' we mean the number of hash marks that could be made in a > unit interval by noon? That is at 1 minute before noon a hash mark is made > at the location -1 on the real line, at 1/2 minute before noon another hash > mark is made at -1/2 on the real line, at 1/3 of a minute before noon yet > another hash mark is made at -1/3 on the real line, and so on. At noon the > total number of hash marks in the interval [-1,0] should be ALEPH_0. > >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? Indeed. How can Achilles ever reach the tortoise? > At noon, when lim(n->oo: sum(x=1->n: 1/2^n)) reaches 1. === Subject: Re: An infinite debate > Um, what if you start with 0, increment it at 1 minute before noon, then > again at a half minute before noon, then again at a third of a minute > before noon, etc, so that at time noon-1/n the number achieves a value of > n, for n in N? That way we have counted through all of the natural > numbers > by noon, correct? What is the value of n at noon? This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by > `n' we mean a variable name for a number stored in the registry of some > kind > of ideal computer then I don't think the value of `n' at noon is defined. > But what if by `n' we mean the number of hash marks that could be made in > a > unit interval by noon? That is at 1 minute before noon a hash mark is > made > at the location -1 on the real line, at 1/2 minute before noon another > hash > mark is made at -1/2 on the real line, at 1/3 of a minute before noon yet > another hash mark is made at -1/3 on the real line, and so on. At noon > the > total number of hash marks in the interval [-1,0] should be ALEPH_0. > >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? By obeying the rules. === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? Indeed. How can Achilles ever reach the tortoise? At noon, when lim(n->oo: sum(x=1->n: 1/2^n)) reaches 1. How can he reach the tortoise at noon if he doesn't move at noon? -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45555d11@news2.lightlink.com> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? Indeed. How can Achilles ever reach the tortoise? At noon, when lim(n->oo: sum(x=1->n: 1/2^n)) reaches 1. How can he reach the tortoise at noon if he doesn't move at noon? -- > Marcus Infinity. (In the natural numbers.) Ross === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no hash marks can be added, or that would imply an n such that 1/n=0, >> which cannot be a finite n. If it cannot be completed before noon, and >> cannot be completed at noon, how can it be completed by noon? Indeed. How can Achilles ever reach the tortoise? At noon, when lim(n->oo: sum(x=1->n: 1/2^n)) reaches 1. How can he reach the tortoise at noon if he doesn't move at noon? Infinity. (In the natural numbers.) Tell , not me. He's the one saying the hash marks aren't complete at noon. -- Marcus === Subject: Re: An infinite debate > Being wrong is easy, knowing when you're right can be hard, but > actually being right and knowing it, is the hardest thing of all. > -- James S. Harris A trick JSH never mastered. === Subject: Re: An infinite debate Um, what if you start with 0, increment it at 1 minute before noon, then > again at a half minute before noon, then again at a third of a minute > before noon, etc, so that at time noon-1/n the number achieves a value > of n, for n in N? That way we have counted through all of the natural > numbers by noon, correct? What is the value of n at noon? >> This is a great question and I think the answer to it depends on what is >> meant by `increment' and what is meant by the value of `n' at noon. If >> by `n' we mean a variable name for a number stored in the registry of >> some kind of ideal computer then I don't think the value of `n' at noon >> is defined. But what if by `n' we mean the number of hash marks that >> could be made in a unit interval by noon? That is at 1 minute before >> noon a hash mark is made at the location -1 on the real line, at 1/2 >> minute before noon another hash mark is made at -1/2 on the real line, at >> 1/3 of a minute before noon yet another hash mark is made at -1/3 on the >> real line, and so on. At noon the total number of hash marks in the >> interval [-1,0] should be ALEPH_0. >> All the best, >> R Should is a good word to use. When would aleph_0 hash marks be > completed, exactly? At no finite time before noon are all hash marks > completed: more remain to be added in the remaining time. But, AT noon, no > hash marks can be added, or that would imply an n such that 1/n=0, which > cannot be a finite n. If it cannot be completed before noon, and cannot be > completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But that would not change anything since 1+ALEPH_0 equals ALEPH_0. R === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R I'm not sure how that answers the conundrum. It seems you admit it really doesn't. At noon, no element can be added to complete the sequence, yet, at any time before noon, the sequence cannot have been completed, since there are more to be added during that finite interval. So, at what point in time is the set completed? I'm sorry if this wasn't obvious to anyone reading, but it has direct reference to the ball-and-vase problem, the Ross-Littlewood Paradox. :) === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R I'm not sure how that answers the conundrum. It seems you admit it > really doesn't. At noon, no element can be added to complete the > sequence, yet, at any time before noon, the sequence cannot have been > completed, since there are more to be added during that finite interval. > So, at what point in time is the set completed? Noon! If it is not complete before noon but is complete at noon, then the earliest time at which it can be complete is noon. I'm sorry if this wasn't obvious to anyone reading, but it has direct > reference to the ball-and-vase problem, the Ross-Littlewood Paradox. :) === Subject: Re: An infinite debate > >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? Noon! > If it is not complete before noon but is complete at noon, then the > earliest time at which it can be complete is noon. Which elements are added at noon to complete the set, which weren't there before noon? :) >> I'm sorry if this wasn't obvious to anyone reading, but it has direct >> reference to the ball-and-vase problem, the Ross-Littlewood Paradox. :) >> === Subject: Re: An infinite debate > Noon! > If it is not complete before noon but is complete at noon, then the > earliest time at which it can be complete is noon. Which elements are added at noon to complete the set, which weren't > there before noon? :) Some additions or removals take place after every time before noon, so they are not all completed at any time before noon. All additions and removals take place strictly before noon so that by noon they are all completed. === Subject: Re: An infinite debate > >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? Noon! > If it is not complete before noon but is complete at noon, then the > earliest time at which it can be complete is noon. Which elements are added at noon to complete the set, which weren't > there before noon? :) None. Any more irrelevant questions? -- Marcus === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. I'm not sure how that answers the conundrum. It seems you admit it > really doesn't. At noon, no element can be added to complete the > sequence, yet, at any time before noon, the sequence cannot have been > completed, since there are more to be added during that finite interval. > So, at what point in time is the set completed? The set is completed at Noon. > I'm sorry if this wasn't obvious to anyone reading, but it has direct > reference to the ball-and-vase problem, the Ross-Littlewood Paradox. :) Indeed. It shows how simple the paradox is to resolve once you learn some math. -- Marcus === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? The set is completed at Noon. But no elements are added to the set at that point in time. > >> I'm sorry if this wasn't obvious to anyone reading, but it has direct >> reference to the ball-and-vase problem, the Ross-Littlewood Paradox. :) Indeed. It shows how simple the paradox is to resolve once you learn > some math. > Or the definition of a moment, or the significance of the formulaic relations between sets. === Subject: Re: An infinite debate >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? The set is completed at Noon. But no elements are added to the set at that point in time. That is correct. Just keep repeating to yourself, The set is completed at noon even though nothing is added at noon. -- Marcus === Subject: Re: An infinite debate >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > The set is completed at Noon. >> But no elements are added to the set at that point in time. That is correct. Just keep repeating to yourself, The set is completed > at noon even though nothing is added at noon. > Is self-hypnosis a valid mathematical technique? I rather think not. It doesn't occur before t, nor at t, so it doesn't occur by t. The Zeno machine is a deliberate paradox factory. === Subject: Re: An infinite debate > That is correct. Just keep repeating to yourself, The set is completed > at noon even though nothing is added at noon. > Is self-hypnosis a valid mathematical technique? It may be a valid treatment for the mathematically unsane. === Subject: Re: An infinite debate >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > The set is completed at Noon. >> But no elements are added to the set at that point in time. That is correct. Just keep repeating to yourself, The set is completed > at noon even though nothing is added at noon. > Is self-hypnosis a valid mathematical technique? Try it and see. You seem to be impervious to logic. > I rather think not. It > doesn't occur before t, nor at t, so it doesn't occur by t. Ah, well, now you are cheating. You haven't actually shown that it doesn't occur at noon. You have merely shown that no element is added at noon. Conflating two different things is a sure road to paradox. > The Zeno machine is a deliberate paradox factory. paradox 1. something absurd or contradictory: a statement, proposition, or situation that seems to be absurd or contradictory, but in fact is or may be true -- Marcus === Subject: Re: An infinite debate >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. I'm not sure how that answers the conundrum. It seems you admit it > really doesn't. At noon, no element can be added to complete the > sequence, yet, at any time before noon, the sequence cannot have been > completed, since there are more to be added during that finite interval. > So, at what point in time is the set completed? The set is completed at Noon. For some definitions of the word completed. -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> [Quoting two replies to the same question...] >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. I'm not sure how that answers the conundrum. It seems you admit it > really doesn't. At noon, no element can be added to complete the > sequence, yet, at any time before noon, the sequence cannot have been > completed, since there are more to be added during that finite interval. > So, at what point in time is the set completed? William Hughes: > There is no point in time at which the set is completed. Marcus: > The set is completed at Noon. [DM] > For some definitions of the word completed. More precisely, I think, for some definitions of the word at. Consider the two functions which, under the Freedom of Mathematical Notation Act of 1884, I shall refer to as f() and g(): f(x) : 1 if x <= 0, 0 if x > 0 g(x) : 1 if x < 0, 0 if x >= 0 Considering f and g as functions of time: (1) When does f(t) change? (2) When does g(t) change? I suggest that no-one can name a time when either function changes other than noon, I mean t=0, and since both functions are 1 before the change and 0 after, there cannot be any difference between them. Therefore 1 = f(0) = g(0) = 0, and set theory is inconsistent, QED. If you play 's game by attempting to answer his question When does the [vase] become [empty]? (permute arguments at will) without a clear definition of what the question means, you will feed his confusion. (And if you use the i-word with , no good will ever come of it. On a good day he agrees that the set of pofnats goes on and on and never ends; so use his term, which we could abbreviate Goes ON and on And never enDS, for GONADS. Sounds rather neat, actually - the vase and the set of balls that are GONADS.) Brian Chandler http://imaginatorium.org > -- > Marcus === Subject: Re: An infinite debate > [Quoting two replies to the same question...] > William Hughes: > There is no point in time at which the set is completed. Marcus: > The set is completed at Noon. [DM] > For some definitions of the word completed. More precisely, I think, for some definitions of the word at. Probably. > Consider the two functions which, under the Freedom of Mathematical > Notation Act of 1884, I shall refer to as f() and g(): f(x) : 1 if x <= 0, 0 if x > 0 > g(x) : 1 if x < 0, 0 if x >= 0 Considering f and g as functions of time: Under the Act of 1884, don't you have to say, Considering f() and g() as functions of time? > (1) When does f(t) change? > (2) When does g(t) change? And, don't you have to say, When does f() change?? -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> William Hughes: > There is no point in time at which the set is completed. Marcus: > The set is completed at Noon. [DM] > For some definitions of the word completed. More precisely, I think, for some definitions of the word at. Probably. Consider the two functions which, under the Freedom of Mathematical > Notation Act of 1884, I shall refer to as f() and g(): f(x) : 1 if x <= 0, 0 if x > 0 > g(x) : 1 if x < 0, 0 if x >= 0 Considering f and g as functions of time: Under the Act of 1884, don't you have to say, Considering f() and g() > as functions of time? (1) When does f(t) change? > (2) When does g(t) change? And, don't you have to say, When does f() change?? No, you're out of your depth here, I can see. Section 2 of the Act makes it quite clear that the point of all this is to allow your interlocutor to understand your argument, and the freedom to use reasonable alternative terminology, or typographical convention is not constrained by any absolute requirements of petty consistency. Anyway, much of what you've been saying about this makes excellent sense in the context of controlled publishing - the author arranges with the printer to have everything make crystal clear, by use of fancy fonts and whatnot. In a Usenet discussion this isn't realistic, because of the limitations of the format, and also because in responding to someone else, one adapts to their conventions. I know I have argued that this is often a problem, because when a crank says something completely confused and meaningless, typically non-cranks attempt to interpret this as the closest statement that _would_ make sense - with the usual disastrous results. Brian Chandler http://imaginatorium.org -- > Marcus === Subject: Re: An infinite debate > Consider the two functions which, under the Freedom of Mathematical > Notation Act of 1884, I shall refer to as f() and g(): f(x) : 1 if x <= 0, 0 if x > 0 > g(x) : 1 if x < 0, 0 if x >= 0 Considering f and g as functions of time: Under the Act of 1884, don't you have to say, Considering f() and g() > as functions of time? (1) When does f(t) change? > (2) When does g(t) change? And, don't you have to say, When does f() change?? No, you're out of your depth here, I can see. Section 2 of the Act > makes it quite clear that the point of all this is to allow your > interlocutor to understand your argument, and the freedom to use > reasonable alternative terminology, or typographical convention is not > constrained by any absolute requirements of petty consistency. I guess I need to get hold of a copy of the Act. > Anyway, much of what you've been saying about this makes excellent > sense in the context of controlled publishing - the author arranges > with the printer to have everything make crystal clear, by use of fancy > fonts and whatnot. Seems to me that even if you don't have fancy fonts and whatnot, petty consistency is still useful. Besides, what fonts would we need for f, f(), and f(t)? Math italic instead of roman? > In a Usenet discussion this isn't realistic, because > of the limitations of the format, I can write a document in TeX using just ascii. So, the format doesn't seem too limiting. > and also because in responding to > someone else, one adapts to their conventions. I know I have argued > that this is often a problem, because when a crank says something > completely confused and meaningless, typically non-cranks attempt to > interpret this as the closest statement that _would_ make sense - with > the usual disastrous results. -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R I'm not sure how that answers the conundrum. It seems you admit it > really doesn't. At noon, no element can be added to complete the > sequence, yet, at any time before noon, the sequence cannot have been > completed, since there are more to be added during that finite interval. > So, at what point in time is the set completed? There is no point in time at which the set is completed. - William Hughes === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, >> then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what > is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, > at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on > the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, >> no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot >> be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? That is what Zeno kept asking. But it happens anyway. It does not happen > before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? Yes it is wrong that it does not happen by noon. All changes being strictly before noon, it happens by noon. === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> <45555bef@news2.lightlink.com> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen > before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? Yes that is wrong. Let T_c be the set of all times at which an element is added to the sequence. T_c is bounded above and below, so T_c has both an infinmum and a supremum. The infimum is an element of T_c, so T_c has a minimum. The supremum is not an element of T_c so T_c does not have a maximum (This can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. At any time s= t_f the sequence is completed. So there is a time, t_f, such that before t_f the sequence is not complete and by t_f the sequence is complete. In the present case t_f is noon. So before noon the sequence is not complete, and by noon the sequence is complete. - William Hughes === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen >> before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? Yes that is wrong. Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. If that is the LUB of T_c, then there really is no such thing, the way I see it. I know you claim omega to be the smallest infinite ordinal, and some sort of a LUB on N, but I rather see that as antihtetical to the notion that adding any nonzero quantity x, positive or negative, to any quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, violating this principle. If the basics of addition are upheld, then the conclusion that there is no smallest infinity, or LUB on the naturals, is the only conclusion. At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. That implies that the sequence is completed at t_f, except that no elements are added at t_f. That's a contradiction. In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. Which means it's completed at noon, a moment when no elements are added. - William Hughes > === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> <45555bef@news2.lightlink.com> <45560159@news2.lightlink.com> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen >> before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? Yes that is wrong. Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. If that is the LUB of T_c, then there really is no such thing, the way I > see it. T_c is a set of reals bounded above. - William Hughes === Subject: Re: An infinite debate >> Um, what if you start with 0, increment it at 1 minute before noon, >> then >> again at a half minute before noon, then again at a third of a >> minute >> before noon, etc, so that at time noon-1/n the number achieves a >> value >> of n, for n in N? That way we have counted through all of the >> natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what > is > meant by `increment' and what is meant by the value of `n' at noon. > If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at > noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real > line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on > the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT >> noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, >> which >> cannot be a finite n. If it cannot be completed before noon, and >> cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. > But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen >> before noon, nor at noon, so it does not happen by noon. Is that wrong? >> Why? Yes that is wrong. Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. If that is the LUB of T_c, then there really is no such thing, the way I > see it. I know you claim omega to be the smallest infinite ordinal, and > some sort of a LUB on N, but I rather see that as antihtetical to the > notion that adding any nonzero quantity x, positive or negative, to any > quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, > violating this principle. If the basics of addition are upheld, then the > conclusion that there is no smallest infinity, or LUB on the naturals, > is the only conclusion. > At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. That implies that the sequence is completed at t_f, except that no > elements are added at t_f. That's a contradiction. > In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. Which means it's completed at noon, a moment when no elements are added. > It is certainly complete at noon, but since completed implies action at noon, which does not in fact occur, it is, at least, misleading. === Subject: Re: An infinite debate > Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. If that is the LUB of T_c, then there really is no such thing, the way I > see it. I know you claim omega to be the smallest infinite ordinal, and > some sort of a LUB on N, but I rather see that as antihtetical to the > notion that adding any nonzero quantity x, positive or negative, to any > quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, > violating this principle. If the basics of addition are upheld, then the > conclusion that there is no smallest infinity, or LUB on the naturals, > is the only conclusion. T_c is a set of *times*. So, T_c is a set of real numbers between -1 and 0. Are you denying that a set of real numbers between -1 and 0 has a supremum? > At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. That implies that the sequence is completed at t_f, except that no > elements are added at t_f. That's a contradiction. What does it contradict? > In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. Which means it's completed at noon, a moment when no elements are added. Exactly. You've got it! -- Marcus === Subject: Re: An infinite debate > Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. >> If that is the LUB of T_c, then there really is no such thing, the way I >> see it. I know you claim omega to be the smallest infinite ordinal, and >> some sort of a LUB on N, but I rather see that as antihtetical to the >> notion that adding any nonzero quantity x, positive or negative, to any >> quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, >> violating this principle. If the basics of addition are upheld, then the >> conclusion that there is no smallest infinity, or LUB on the naturals, >> is the only conclusion. T_c is a set of *times*. So, T_c is a set of real numbers between -1 and > 0. Are you denying that a set of real numbers between -1 and 0 has a > supremum? > Uh, no, but I am denying that there is a supremum or LUB of N, which is mapped to T_c. > At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. >> That implies that the sequence is completed at t_f, except that no >> elements are added at t_f. That's a contradiction. What does it contradict? The fact that, if t1 So before noon the sequence is not complete, and by noon the > sequence is complete. >> Which means it's completed at noon, a moment when no elements are added. Exactly. You've got it! > Except that in order for the sequence to go from incomplete to complete, elements must be added to it. Do you disagree with that? === Subject: Re: An infinite debate T_c is a set of *times*. So, T_c is a set of real numbers between -1 and > 0. Are you denying that a set of real numbers between -1 and 0 has a > supremum? > Uh, no, but I am denying that there is a supremum or LUB of N, which is > mapped to T_c. So why do you keep harping on what does not have a supremum. when there is something which does? At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. >> That implies that the sequence is completed at t_f, except that no >> elements are added at t_f. That's a contradiction. What does it contradict? The fact that, if t1 is, then there exists a t3 such that t1 elements must be added to it. Do you disagree with that? Yes. What has to be added to before noon to make it become noon? === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> <45555bef@news2.lightlink.com> <45560159@news2.lightlink.com> <4556564c@news2.lightlink.com Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. >> If that is the LUB of T_c, then there really is no such thing, the way I >> see it. I know you claim omega to be the smallest infinite ordinal, and >> some sort of a LUB on N, but I rather see that as antihtetical to the >> notion that adding any nonzero quantity x, positive or negative, to any >> quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, >> violating this principle. If the basics of addition are upheld, then the >> conclusion that there is no smallest infinity, or LUB on the naturals, >> is the only conclusion. T_c is a set of *times*. So, T_c is a set of real numbers between -1 and > 0. Are you denying that a set of real numbers between -1 and 0 has a > supremum? > Uh, no, but I am denying that there is a supremum or LUB of N, which is > mapped to T_c. However, since T_c has a supremum whether of not N has a supremum it is far from clear why you are doing this. At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. >> That implies that the sequence is completed at t_f, except that no >> elements are added at t_f. That's a contradiction. What does it contradict? The fact that, if t1 is, then there exists a t3 such that t1 Well this would be a big problem except for the fact that there exists a t3 such that t1 So before noon the sequence is not complete, and by noon the > sequence is complete. >> Which means it's completed at noon, a moment when no elements are added. Exactly. You've got it! > Except that in order for the sequence to go from incomplete to complete, > elements must be added to it. Do you disagree with that? Yes elements must be added. No, it is not true that there must be a last element added. If there is no last element added then there is no element added at the first time by which the sequence is complete. - William Hughes === Subject: Re: An infinite debate >> That implies that the sequence is completed at t_f, except that no >> elements are added at t_f. That's a contradiction. What does it contradict? The fact that, if t1 is, then there exists a t3 such that t1 In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. >> Which means it's completed at noon, a moment when no elements are added. Exactly. You've got it! > Except that in order for the sequence to go from incomplete to complete, > elements must be added to it. Do you disagree with that? If you are implying that any change must occur in the instant at which it is first complete, we do disagree. By TO's argument no function of a continuous variable can change value, as that change would have to occur at a point, i.e., while the variable is not changing. In mathematics, for a general function of a continuous variable, like time, the value at one argument is totally independent of the values at any others. In order to have values at one argument tied to values at nearby arguments, one must have something like continuity, which is conspicuously missing here. === Subject: Re: An infinite debate > Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. >> If that is the LUB of T_c, then there really is no such thing, the way I >> see it. I know you claim omega to be the smallest infinite ordinal, and >> some sort of a LUB on N, but I rather see that as antihtetical to the >> notion that adding any nonzero quantity x, positive or negative, to any >> quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, >> violating this principle. If the basics of addition are upheld, then the >> conclusion that there is no smallest infinity, or LUB on the naturals, >> is the only conclusion. T_c is a set of *times*. So, T_c is a set of real numbers between -1 and > 0. Are you denying that a set of real numbers between -1 and 0 has a > supremum? Uh, no, but I am denying that there is a supremum or LUB of N, So, you are denying something that William didn't claim. That's nice. > which is mapped to T_c. You will have to explain what you mean by that. > At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. >> That implies that the sequence is completed at t_f, except that no >> elements are added at t_f. That's a contradiction. What does it contradict? The fact that, if t1 is, then there exists a t3 such that t1 In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. >> Which means it's completed at noon, a moment when no elements are added. Exactly. You've got it! Except that in order for the sequence to go from incomplete to complete, > elements must be added to it. Do you disagree with that? No, I don't disagree. The set is complete at/by (depending on your linguistic preference) noon. -- Marcus === Subject: Re: An infinite debate <455388d9@news2.lightlink.com> <4554d4c9@news2.lightlink.com> <45554450@news2.lightlink.com> <45555bef@news2.lightlink.com> <45560159@news2.lightlink.com> Um, what if you start with 0, increment it at 1 minute before noon, then >> again at a half minute before noon, then again at a third of a minute >> before noon, etc, so that at time noon-1/n the number achieves a value >> of n, for n in N? That way we have counted through all of the natural >> numbers by noon, correct? What is the value of n at noon? > This is a great question and I think the answer to it depends on what is > meant by `increment' and what is meant by the value of `n' at noon. If > by `n' we mean a variable name for a number stored in the registry of > some kind of ideal computer then I don't think the value of `n' at noon > is defined. But what if by `n' we mean the number of hash marks that > could be made in a unit interval by noon? That is at 1 minute before > noon a hash mark is made at the location -1 on the real line, at 1/2 > minute before noon another hash mark is made at -1/2 on the real line, at > 1/3 of a minute before noon yet another hash mark is made at -1/3 on the > real line, and so on. At noon the total number of hash marks in the > interval [-1,0] should be ALEPH_0. All the best, > R >> Should is a good word to use. When would aleph_0 hash marks be >> completed, exactly? At no finite time before noon are all hash marks >> completed: more remain to be added in the remaining time. But, AT noon, no >> hash marks can be added, or that would imply an n such that 1/n=0, which >> cannot be a finite n. If it cannot be completed before noon, and cannot be >> completed at noon, how can it be completed by noon? > You could add a hash mark at noon, say at 0, in the interval [-1,0]. But > that would not change anything since 1+ALEPH_0 equals ALEPH_0. R >> I'm not sure how that answers the conundrum. It seems you admit it >> really doesn't. At noon, no element can be added to complete the >> sequence, yet, at any time before noon, the sequence cannot have been >> completed, since there are more to be added during that finite interval. >> So, at what point in time is the set completed? > There is no point in time at which the set is completed. - William Hughes > Then how can you say the event of completion occurs? It does not happen >> before noon, nor at noon, so it does not happen by noon. Is that wrong? Why? Yes that is wrong. Let T_c be the set of all times at which an element is added to the > sequence. T_c is bounded above and below, so T_c has > both an infinmum and a supremum. The infimum is an element > of T_c, so T_c has a minimum. The supremum is not an > element of T_c so T_c does not have a maximum (This > can only occur if T_c has unboundedly many elements). Since T_c does not have a maximum, > there is no time at which the event of completion occurs. Let the supremum of T_c be t_f. If that is the LUB of T_c, then there really is no such thing, the way I > see it. I know you claim omega to be the smallest infinite ordinal, and > some sort of a LUB on N, but I rather see that as antihtetical to the > notion that adding any nonzero quantity x, positive or negative, to any > quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, > violating this principle. If the basics of addition are upheld, then the > conclusion that there is no smallest infinity, or LUB on the naturals, > is the only conclusion. > At any time s At any time t>= t_f the sequence is completed. So there is a time, t_f, such that before t_f > the sequence is not complete and by t_f the sequence is > complete. That implies that the sequence is completed at t_f, except that no > elements are added at t_f. That's a contradiction. > No the fact that a sequence is not compete at any time before t_f, and is complete at t_f does not imply that the sequence is completed at t_f (i.e. there is a last step at t_f). There may be no time at which there is a last step. In the present case t_f is noon. > So before noon the sequence is not complete, and by noon the > sequence is complete. Which means it's completed at noon, a moment when no elements are added. > No. For a process that does not have a last step the first time at which the process is complete is not the same as the time at which the process is completed. Repeat the following sentence 50 times before bed. There are processes that do not have a last step. If a process does not have a last step, then there can be no time *at* which the process is completed. However if we arrange to have all steps happen in a finite time (by having things happen faster and faster) it is possible to have a time after all steps have been done, that is a time *by* which the process is complete. In this case we must have a minimum time, t_f, after all steps have been done. However, nothing happens at this time because t_f is (by definition) a time after all steps have been done. So if we have a process that does not have a last step, but for which all steps happen in a finite time, we must have a first time by which the process is complete, and at this time nothng happens. In our present case we have a process that does not have a last step, but for which all steps happen in a finite time. So the statement it's completed at noon, a moment when no elements are added should read the first time it's complete is noon, a moment when no elements are added exactly what we expect. - William Hughes I === Subject: Google pix amazing Google pix new site click on any image and u will find new software or game FREE for you === Subject: what type of convexity is this? a*f(x) >= f(a*x) ??? what conditions are imposed for a? === Subject: Dumb factoring question Hi all, Struggling with math (as usual) and I can't figure out how to factor this polynomial. I've looked at it for about an hour and can't figure how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: (x+1)(3x-1)(x-4)^2 I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with the two coefficients 3 and 25 Could someone be kind enough to explain how to factor this?? This is Precalc, BTW. pil === Subject: Re: Dumb factoring question > Hi all, Struggling with math (as usual) and I can't figure out how to factor > this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 Well, if your calculator gives the correct answer, what are you worried about? > I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with > the two coefficients 3 and 25 I wish people would stop teaching grouping, because it only works in a few special cases. Most of the time, it's worthless, as you've found out. > Could someone be kind enough to explain how to factor this?? > This is Precalc, BTW. If the other posters haven't helped you, you could always send an e-mail to James Harris, at jstevh@msn.com. He knows a lot about factoring polynomials, and he even published a paper about it. --- === Subject: Re: Dumb factoring question > Hi all, Struggling with math (as usual) and I can't figure out how to factor > this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 Well, if your calculator gives the correct answer, what are you worried > about? I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with > the two coefficients 3 and 25 I wish people would stop teaching grouping, because it only works in a > few special cases. Most of the time, it's worthless, as you've found > out. Well, yes, It's useless because some interfering individual has taken the two terms in x^2 and smushed them together, and the two terms in x and smushed together, so you can't see how the grouping would work. If they'd left the product as 3x^3 - 24x^2 + 48x - x^2 + 8x - 16 then anyone could factor it... :) === Subject: Re: Dumb factoring question > Hi all, >> Struggling with math (as usual) and I can't figure out how to factor >> this polynomial. I've looked at it for about an hour and can't figure >> how to factor it: >> f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) >> My calculator gives the correct answer: >> (x+1)(3x-1)(x-4)^2 Well, if your calculator gives the correct answer, what are you worried > about? > I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with >> the two coefficients 3 and 25 I wish people would stop teaching grouping, because it only works in a > few special cases. Most of the time, it's worthless, as you've found > out. > I presume that you are referring to what is described here. http://personal.jccmi.edu/LairdKristiK/overview6.htm For a quadratic when I was at school (ie high school) for an equation of the form ax^2 + bx + c=0 we used the formula to obtain the roots of the equation: x = -b +- (Sqrt(b^2 - 4ac)) / 2a I know that we were asked to factor the quadratic by grouping. No doubt lower down the school we would be given equations with real and fairly easy solutions - hence the reason for using the grouping method? I imagine that it is horses for courses! Later on, we would have used graphical methods - and numerical analysis at university. Nick === Subject: Re: Dumb factoring question <_LOdneN6csrCssrYnZ2dnUVZ8qadnZ2d@bt.com >> Hi all, >> Struggling with math (as usual) and I can't figure out how to factor >> this polynomial. I've looked at it for about an hour and can't figure >> how to factor it: >> f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) >> My calculator gives the correct answer: >> (x+1)(3x-1)(x-4)^2 Well, if your calculator gives the correct answer, what are you worried > about? > I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with >> the two coefficients 3 and 25 I wish people would stop teaching grouping, because it only works in a > few special cases. Most of the time, it's worthless, as you've found > out. > I presume that you are referring to what is described here. > http://personal.jccmi.edu/LairdKristiK/overview6.htm Not entirely. An example showing when (what I've heard of as) grouping is when you factor the cubic a x^3 + b x^2 + c x + d, where a * d = b * c. For instance, x^3 + 4 x^2 + 3 x + 12 = x^2 (x + 4) + 3 (x + 4) = (x^2 + 3) (x + 4). If you don't have a cubic of this form, then the trick doesn't work. However, the Rational Root Theorem _always_ works and would also find the factorization above. The link you provide shows that you can do this, if you complexify the expression first. (Complexification is the opposite of simplification.) > For a quadratic when I was at school (ie high school) for an equation of the > form ax^2 + bx + c=0 we used the formula to obtain the roots of the > equation: x = -b +- (Sqrt(b^2 - 4ac)) / 2a I know that we were asked to factor the quadratic by grouping. But how do you group a x^2 + b x + c? In some cases, [a x^2 + a x] + [(b - a) x + c] works; in a lot of cases it doesn't. Once again, the quadratic formula or completing the square are guaranteed to give an answer every time. > No doubt > lower down the school we would be given equations with real and fairly easy > solutions - hence the reason for using the grouping method? Yes, but students should not be taught that grouping is a cure-all. And that was my point. --- > I imagine that it is horses for courses! Later on, we would have used graphical methods - and numerical analysis at > university. Nick === Subject: Re: Dumb factoring question Hi all, Struggling with math (as usual) and I can't figure out how to factor > this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 >> Well, if your calculator gives the correct answer, what are you worried >> about? > I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense > with > the two coefficients 3 and 25 >> I wish people would stop teaching grouping, because it only works in a >> few special cases. Most of the time, it's worthless, as you've found >> out. >> I presume that you are referring to what is described here. >> http://personal.jccmi.edu/LairdKristiK/overview6.htm Not entirely. An example showing when (what I've heard of as) > grouping is when you factor the cubic a x^3 + b x^2 + c x + d, where a * d = b * c. For instance, x^3 + 4 x^2 + 3 x + 12 = x^2 (x + 4) + 3 (x + 4) = (x^2 + 3) (x + > 4). If you don't have a cubic of this form, then the trick doesn't work. > However, the Rational Root Theorem _always_ works and would also find > the factorization above. The link you provide shows that you can do this, if you complexify > the expression first. (Complexification is the opposite of > simplification.) > For a quadratic when I was at school (ie high school) for an equation of >> the >> form ax^2 + bx + c=0 we used the formula to obtain the roots of the >> equation: >> x = -b +- (Sqrt(b^2 - 4ac)) / 2a >> I know that we were asked to factor the quadratic by grouping. But how do you group a x^2 + b x + c? In some cases, [a x^2 + a x] + [(b - a) x + c] works; in a lot of cases it doesn't. Once again, the quadratic formula > or completing the square are guaranteed to give an answer every time. > No doubt >> lower down the school we would be given equations with real and fairly >> easy >> solutions - hence the reason for using the grouping method? Yes, but students should not be taught that grouping is a cure-all. And > that was my point. What age are we talking about when we are talking about students? I was using that when I was 14. But then we went on to more complicated methods of solution? It would only be a cure-all if we had been taught it and then stopped there. We weren't. As I haven't been in the (high) school classroom for 30 years having made an abortive attempt at teaching I can't speak for how this is taught these days in the UK. But what I can say is that whilst I started calculus at age 14, on some syllabuses it was started much later. Clearly the way in which it would be approached at the younger age would be different to the older age. Given that I don't think that we experienced imaginary roots to the age of 16 or 17, I doubt at age 14 we would have been presented with quadratics that weren't susceptible to grouping. Those who specialised in maths after age 16 would have had a more sophisticated understanding. Nick === Subject: Re: Dumb factoring question <_LOdneN6csrCssrYnZ2dnUVZ8qadnZ2d@bt.com> <8_mdndgaZvKYt8XYnZ2dnUVZ8qSdnZ2d@bt.com > Hi all, Struggling with math (as usual) and I can't figure out how to factor > this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 >> Well, if your calculator gives the correct answer, what are you worried >> about? > I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with > the two coefficients 3 and 25 >> I wish people would stop teaching grouping, because it only works in a >> few special cases. Most of the time, it's worthless, as you've found out. >> I presume that you are referring to what is described here. >> http://personal.jccmi.edu/LairdKristiK/overview6.htm Not entirely. An example showing when (what I've heard of as) > grouping is when you factor the cubic a x^3 + b x^2 + c x + d, where a * d = b * c. For instance, x^3 + 4 x^2 + 3 x + 12 = x^2 (x + 4) + 3 (x + 4) = (x^2 + 3) (x + 4). If you don't have a cubic of this form, then the trick doesn't work. > However, the Rational Root Theorem _always_ works and would also find > the factorization above. The link you provide shows that you can do this, if you complexify > the expression first. (Complexification is the opposite of > simplification.) > For a quadratic when I was at school (ie high school) for an equation of the >> form ax^2 + bx + c=0 we used the formula to obtain the roots of the equation: >> x = -b +- (Sqrt(b^2 - 4ac)) / 2a >> I know that we were asked to factor the quadratic by grouping. But how do you group a x^2 + b x + c? In some cases, [a x^2 + a x] + [(b - a) x + c] works; in a lot of cases it doesn't. Once again, the quadratic formula > or completing the square are guaranteed to give an answer every time. > No doubt >> lower down the school we would be given equations with real and fairly easy >> solutions - hence the reason for using the grouping method? Yes, but students should not be taught that grouping is a cure-all. And > that was my point. What age are we talking about when we are talking about students? Any age, any student learning how to factor (and solve) polynomials. > I was > using that when I was 14. But then we went on to more complicated methods of > solution? It would only be a cure-all if we had been taught it and then > stopped there. We weren't. You weren't, and I wasn't, but there _are_ college students who don't take anything after college algebra. And _they_ are the ones I'm talking about. --- > As I haven't been in the (high) school classroom for 30 years having made an > abortive attempt at teaching I can't speak for how this is taught these days > in the UK. But what I can say is that whilst I started calculus at age 14, on some > syllabuses it was started much later. Clearly the way in which it would be > approached at the younger age would be different to the older age. Given that I don't think that we experienced imaginary roots to the age of > 16 or 17, I doubt at age 14 we would have been presented with quadratics > that weren't susceptible to grouping. Those who specialised in maths after age 16 would have had a more > sophisticated understanding. Nick === Subject: Re: Dumb factoring question > Struggling with math (as usual) and I can't figure out how to factor > this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with > the two coefficients 3 and 25 Could someone be kind enough to explain how to factor this?? > This is Precalc, BTW. You could guess a root, then divide out the factor: http://www.themathpage.com/aPreCalc/factor-theorem.htm In particular, note the Rational Root Theorem. -- Marcus === Subject: Re: Dumb factoring question >> Struggling with math (as usual) and I can't figure out how to factor >> this polynomial. I've looked at it for about an hour and can't figure >> how to factor it: >> f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) >> My calculator gives the correct answer: >> (x+1)(3x-1)(x-4)^2 >> I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with >> the two coefficients 3 and 25 >> Could someone be kind enough to explain how to factor this?? >> This is Precalc, BTW. You could guess a root, then divide out the factor: http://www.themathpage.com/aPreCalc/factor-theorem.htm In particular, note the Rational Root Theorem. Looks an interesting link. Nick === Subject: Re: Dumb factoring question this polynomial. I've looked at it for about an hour and can't figure > how to factor it: f(x) = (x+1)(3x^3 - 25x^2 + 56x -16) My calculator gives the correct answer: > (x+1)(3x-1)(x-4)^2 I tried grouping 3x^3 - 25x^2 + 56x -16, but it doesn't make sense with > the two coefficients 3 and 25 Could someone be kind enough to explain how to factor this?? > This is Precalc, BTW. If you are in Precalculus, chances are you already learned synthetic division in Algebra 2. Synthetic division works like this: Because (x-4) is a factor, dividing by x as 4 would yield a 0. 4| 3 -25 56 -16 12 -52 16 -- --------------------- 3 -13 4 0 That leaves you with (x+1)(x-4)(3x^2-13x+2) . All that is left is to factor the binomial. === Subject: Re: The determinant of a matrix > Okay, I thought about it and what I get is that your determinant > is, for K > 1: PROD (1 - a^(2n))^(K-n) FOR n = 1 to K-1 > I developed a slightly more general formula, for determinants of kxk matrices whose entries are M(i,j) = a^((i-j+m)^), which I proved inductively by Dodgson's method of condensation: http://en.wikipedia.org/wiki/Dodgson_condensation http://mathworld.wolfram.com/Condensation.html Condensation, as Rev. Dodgson (aka Lewis Carroll) called it, is esp. efficient for Toeplitz matrices such as here. C.L. Dodgson, Condensation of Determinants, Proc. of the Royal Society of London 15(1866), 150-155 The method of condensation neatly generalizes the well- known rule for determinant of a 2x2 matrix as follow: Let M be a (k+1)x(k+1) matrix, and let M1, M2, M3, M4 be four kxk principal submatrices of M, resp. the upper left, upper right, lower left, and lower right, the overlap of which is the (k-1)x(k-1) central submatrix M (omitting first and last rows and columns of M). A determinant identity holds: (det M)(det M) = (det M1)(det M4) - (det M2)(det M3) When det M is nonzero, det M can be calculated from the above, given the values of the smaller determinants. === Subject: Re: The determinant of a matrix > Okay, I thought about it and what I get is that your determinant > is, for K > 1: PROD (1 - a^(2n))^(K-n) FOR n = 1 to K-1 > I developed a slightly more general formula, for determinants > of kxk matrices whose entries are M(i,j) = a^((i-j+m)^), which > I proved inductively by Dodgson's method of condensation: http://en.wikipedia.org/wiki/Dodgson_condensation http://mathworld.wolfram.com/Condensation.html Condensation, as Rev. Dodgson (aka Lewis Carroll) called it, > is esp. efficient for Toeplitz matrices such as here. C.L. Dodgson, Condensation of Determinants, Proc. of the > Royal Society of London 15(1866), 150-155 The method of condensation neatly generalizes the well- > known rule for determinant of a 2x2 matrix as follow: Let M be a (k+1)x(k+1) matrix, and let M1, M2, M3, M4 be > four kxk principal submatrices of M, resp. the upper left, > upper right, lower left, and lower right, the overlap of which > is the (k-1)x(k-1) central submatrix M (omitting first and > last rows and columns of M). A determinant identity holds: (det M)(det M) = (det M1)(det M4) - (det M2)(det M3) When det M is nonzero, det M can be calculated from the > above, given the values of the smaller determinants. To correct a typo in the above and clarify the ultimate result, let me remark that the KxK matrix M whose entries are: M(i,j) = a^((i-j+m)^2) [i'th row, j'th column] has determinant: a^(K*m^2) PROD (1 - a^(2n))^(K-n) FOR n = 1 to K-1 which simplifies for m = 0 to the case Yecloud asks about, setting the factor in front of the product to a^0 = 1. === Subject: Any induction? 2^2 + 3^2 + 4^2 + 14^2 = 15^2 4^2 + 5^2 + 6^2 + 38^2 = 39^2 6^2 + 7^2 + 8^2 + 74^2 = 75^2 8^2 + 10^2 + 12^2 + 122^2 + 123^2 .. .. .. Is there any meaning of this? (without JUST this equation - (2n)^2 + (2n+1)^2 + (2n+2)^2 + (6n^2+6n+3)^2 + (6n^2+6n+4)^2 === Subject: Re: Any induction? > 2^2 + 3^2 + 4^2 + 14^2 = 15^2 > 4^2 + 5^2 + 6^2 + 38^2 = 39^2 > 6^2 + 7^2 + 8^2 + 74^2 = 75^2 > 8^2 + 10^2 + 12^2 + 122^2 + 123^2 .. > .. > .. > Is there any meaning of this? > (without JUST this equation - (2n)^2 + (2n+1)^2 + (2n+2)^2 + > (6n^2+6n+3)^2 + (6n^2+6n+4)^2 > Don't you mean (2n)^2 + (2n+1)^2 + (2n+2)^2 + (6n^2+6n+2)^2 = (6n^2+6n+3)^2 This is the same as As (x+y)(x-y)=x^2-y^2 The last term on the left can be taken away from the term on the right to give (2n)^2 + (2n+1)^2 + (2n+2)^2=12n^2+12n+5 The left side expands to 4n^2+4n^2+4n+1+4n^2+8n+4 which is 12n^2+12n+5 LHS=RHS QED Nick PS The term 8^2 + 10^2 + 12^2 + 122^2 + 123^2 is incorrect and should be: 8^2 + 9^2 + 10^2 + 122^2 = 123^2 === Subject: PhD Fellowships in Combinatorial Scientific Computing Originator: eugene@cse.ucsc.edu Ph.D. Fellowships in Combinatorial Scientific Computing at Old Dominion University We are recruiting two Ph.D. candidates with research interests in Parallel Computing, Algorithms, and/or Scientific Computing, to join the CSCAPES Institute funded by the Department of Energy's Office of Science. The CSCAPES (Combinatorial Scientific Computing and Petascale Simulations) Institute, centered at Old Dominion University, also includes researchers from Sandia and Argonne National Laboratories, and Ohio State and Colorado State Universities. We have received funding for the CSCAPES Institute for five years. The research involves developing algorithms for combinatorial problems arising in scientific computing, such as load-balancing, graph coloring, graph matching, and automatic differentiation, on tera- and peta-scale computers. Students will be guided in their research by Alex Pothen, Assefaw Gebremedhin, and Florin Dobrian. They will have the opportunity to work at Argonne and Sandia Labs during the summers, and will be co-mentored by Lab scientists. Students will benefit from joint seminars with CSCAPES member institutions and partner organizations through the Access Grid, as well as joint workshops with other SciDAC research groups. Further information about the project is available at http://www.cscapes.org. Applicants should hold a Master's degree or equivalent in computer science or computational mathematics. Exceptionally well-qualified students with Bachelor's degrees will also be considered. Excellent academic performance and letters of recommendation attesting to research ability and motivation are required. The research fellowships include a competitive stipend, supplemented by full tuition remission and funds for participating in conferences. Students will be able to pursue their studies in the Computer Science or Mathematics departments depending on educational background and interest. Further information about the departments are available at http://www.cs.odu.edu and http://www.math.odu.edu. The application deadline is January 15, 2007 for students who wish to be considered for the Fall 2007 semester (or Summer 2007). Submitted material should include a completed graduate application, three recommendation letters, a brief research statement, and a CV. See http://admissions.odu.edu/home.php for templates for the application and recommendation letter. Applications could be completed online (preferred) or submitted as hardcopy. For questions about these positions, please contact Alex Pothen (http://www.cs.odu.edu/~pothen, Email: pothen at cs.odu.edu). -- === Subject: Re: Prime numbers, counting tells it all So no Andrew Wiles did not prove Fermat's Last Theorem. But he can > rely on supporters around the world claiming he did no matter how many > ways you prove he failed. Their word against the mathematics. So you've read and understood Andrew Wiles' proof of FLT? If so, and you can > prove that Wiles' proof is in error, you could legitimately earn a place in > Math history. So please, please, *PLEASE* tell us which step in the FLT > proof did Andrew Wiles trip up! Of course, if you are capable of doing this, you should have no difficulty > defining an elliptic curve, and explaining why they are so important to the > study of number theory. We await your reply, but frankly, I'm not holding my breath. Actually, I would like to hear the answer, too. Wiles's proof of FLT seemed to me to rely on three or four unrelated fields of mathematics, and I couldn't follow it when someone presented it at Georgia Tech how ever many years ago, not even an outline of it. (The outline for the 4CT proofs is actually easy: (1) Every planar graph contains a graph in the finite set S; (2) No minimal counterexample (a counterexample with the fewest number of vertices) can contain any graph in S; (3) Since there are no minimal counterexamples, there are no counterexamples, QED.) If JSH has figured out what's going on, then it shouldn't be too difficult for me. --- === Subject: Re: Prime numbers, counting tells it all days. My association with the Department is that of an alumnus. >Actually, I would like to hear the answer, too. Wiles's proof of FLT >seemed to me to rely on three or four unrelated fields of mathematics, >and I couldn't follow it when someone presented it at Georgia Tech how >ever many years ago, not even an outline of it. (The outline for the >4CT proofs is actually easy: (1) Every planar graph contains a graph in >the finite set S; (2) No minimal counterexample (a counterexample with >the fewest number of vertices) can contain any graph in S; (3) Since >there are no minimal counterexamples, there are no counterexamples, >QED.) If you really want to know, I can give you some of the highlights (without me understanding the specifics). Probably a JSH thread is not the best place to do so, though. Let me know. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Prime numbers, counting tells it all >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical argument), every step must either be an axiom, an assumption (later discharged) or follow from previous steps. If a mathematical argument is fallacious, then there must be some step which satisfies none of these conditions. Most probably, this would be a conclusion which does not follow from the premises. So, what is it? Which step? Instead of giving a clear answer, you have always stated that the problem was an informal fallacy about causality. This just is not possible. Mathematics is a deductive science. Mathematical arguments do not involve causal or other inductive reasoning[1]. Your answer is rather like my complaint that the Chicago Bears should have lost last week's game because of the infield fly rule. It just don't apply. But just keep wishing. And in the meantime, figure out which step doesn't follow from the previous ones. Footnotes: [1] Mathematical induction is not inductive in the sense I mean. -- Jesse F. Hughes Mama: I had a very good steak when I was in Bonn. Quincy (Age 4): A stick? I wish you brought it home. Was it very big and did it look like a gun? === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper answer is a surprising one, which is, if you follow through Wiles' entire paper, and assume that you already have a modular form that is not an elliptic curve, you can't find a contradiction with that assumption! It's the null test. I've talked about the null test before and posters dodge it, run away from it and otherwise ignore the remarkable reality that if you just assume the opposition of what Wiles claims he proves and go through his entire paper, you can't find a single thing that contradicts with it. > Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is USUALLY about causality does apply for his highly particular approach to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with four wheels, and trying to come to a conclusion about objects with four wheels, like they have four doors, and then coming upon a child's wagon. Mathematicians found this thing that seemed to connect elliptic curves and modular forms that is about four numbers--the four wheels of my analogy--and Wiles just does a dumb thing of trying to compare infinite sets. It's illogical. > Your answer is rather like my complaint that the Chicago Bears should > have lost last week's game because of the infield fly rule. It just don't apply. > I've explained and explained and explained. People like you think that all you have to do is disagree. And you know that has worked for years on sci.math so I guess that's why you keep doing it, as I connect the dots. Readers can note that I relate my specific objection to the use of the 4 numbers that are used to relate modular forms to elliptic curves and if you do that and really go back and consider what Wiles was doing, it just jumps out at you that it is just so dumb. He tries to COMPARE infinite sets, and it turns out there are technical reasons for why he can maybe have thought he succeeded--that go back to quirks of the ring of algebraic integers which I've shown using non-polynomial factorization--but you don't have to know much math at all to know that logically his approach must fail. And if you do know enough math to try it, doing the null test will just give you that sinking feeling in your gut, so try it!!! Go back through Wiles' work assuming the opposite of what was supposedly proven to be true and try to find a point where that assumption contradicts with ANYTHING in his paper. > But just keep wishing. And in the meantime, figure out which step doesn't follow from the > previous ones. Footnotes: > [1] Mathematical induction is not inductive in the sense I mean. -- > Jesse F. Hughes I've looked over Wiles' work. That's why I like to tell people to do the null test. Try it. Go look at his paper, assume the opposite of what he supposedly proves and try to find a point where that gives a contradiction. Or just keep talking, like you've done for years. It works with these people. So if that is what you think of as an accomplishment, fine. You can fool people too dumb to check you on the mathematical facts. Big deal. So they're too damn stupid to know you're wrong and t you, so what? James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org I've looked over Wiles' work. You don't have the capacity to understand Wiles' work and are not competent to evaluate it. Stop pretending you are. === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why Harris doesn't cite anything specific in those papers or provide a direct quote. This is based solely on Harris' willfully unquestioning acceptance of an extremely stupid secondary source: Wikipedia. Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish a correspondence between semistable elliptic curves over the rational field, and the modular semistable elliptic curves over the rationals, which he did by explicitly showing that there were equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. You cannot prove that a given correspondence between two infinite sets is one-to-one by just showing that the two sets have the same cardinality. But of course this isn't an adequate description of what Wiles did. What Harris has discovered here is that Wikipedia is not a reliable font of all knowledge. Surprise, surprise. Marcus. > It's illogical. Your answer is rather like my complaint that the Chicago Bears should > have lost last week's game because of the infield fly rule. It just don't apply. > I've explained and explained and explained. People like you think that all you have to do is disagree. And you know that has worked for years on sci.math so I guess that's > why you keep doing it, as I connect the dots. Readers can note that I relate my specific objection to the use of the > 4 numbers that are used to relate modular forms to elliptic curves and > if you do that and really go back and consider what Wiles was doing, it > just jumps out at you that it is just so dumb. He tries to COMPARE infinite sets, and it turns out there are technical > reasons for why he can maybe have thought he succeeded--that go back to > quirks of the ring of algebraic integers which I've shown using > non-polynomial factorization--but you don't have to know much math at > all to know that logically his approach must fail. And if you do know enough math to try it, doing the null test will just > give you that sinking feeling in your gut, so try it!!! Go back through Wiles' work assuming the opposite of what was > supposedly proven to be true and try to find a point where that > assumption contradicts with ANYTHING in his paper. But just keep wishing. And in the meantime, figure out which step doesn't follow from the > previous ones. Footnotes: > [1] Mathematical induction is not inductive in the sense I mean. -- > Jesse F. Hughes I've looked over Wiles' work. That's why I like to tell people to do > the null test. Try it. Go look at his paper, assume the opposite of what he > supposedly proves and try to find a point where that gives a > contradiction. Or just keep talking, like you've done for years. It works with these > people. So if that is what you think of as an accomplishment, fine. You can > fool people too dumb to check you on the mathematical facts. Big deal. So they're too damn stupid to know you're wrong and t you, so what? > James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page in a previous discussion on this issue MathWorld had the problem: Here's the messageid: Like I said, you can point out flaws in Wiles' work but it doesn't matter, as you can see posters shifting all over the map and it doesn't matter. The math field in pure math areas is completely corrupted. James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: So? Sure, it's not clear whether your source was MathWorld or Wikipedia. The main point is, you don't cite any quotation whatsoever which it is safe to conclude that you haven't even read the original papers. I say your criticism is based on the stupid summary in Wikipedia (or MathWorld, which may have been the original source). You could easily prove me wrong by pulling out a quotation directly from Wiles. Why don't you do that? > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. > This is bizarre. Who's shifting all over the map? The previous poster on this came up with essentially the same conclusion that I did. Again, the central point: you are claiming to find flaws in Wiles' work, but you are not justifiying those claims with any specific citations whatsoever. Why not? Because you haven't read his papers - all you have read are unreliable secondary sources on the Web. > The math field in pure math areas is completely corrupted. > Looks like to me the corruption is in your own back yard. Marcus. James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld page said something different. > in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so before when I talked about what MathWorld said about Wiles' work the Wiles did, as was just used by a poster attacking what is on the Wikipedia, but he didn't know that I actually talked about MathWorld before or he forgot, so it's a great example to show people how empty you people are of substantive objections. It is just a reality that what Wiles tried to do is logically stupid. Sorry but that's just the logical reality. So when it's explained simply enough what he attempted, I can cite Cum Hoc, Ergo Propter Hoc to show it can't work. So what do you people do? You attack attempts at explaining what he did and try to revert to complicated math-ese where it is more It's so blatant the wonder is that you get away with it, but people want to believe so it's not such a surprise. They don't want to accept that this person they were told is so brilliant failed by a dumb logical error. James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. So when it's explained simply enough what he attempted, I can cite Cum > Hoc, Ergo Propter Hoc to show it can't work. Or, in your case, to quote from Horace: Quodcumque ostendis mihi sic, incredulus odi. Nora B. === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. > poster and I reached the same conclusion: you have never read Wiles' paper, only secondary sources. It's pretty clear that Wikipedia copied MathWorld (which is older), so the fact that the earlier poster found one source and I found another is immaterial. The fact is, that quotation is your only basis for your claim. Other- wise you would do what several people here have asked: cite a direct quote from one or more of Wiles' papers. But you are very conspicuously not doing that. > It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > Sorry, but all you can really claim here is that the quote from Wikipedia/MathWorld is logically stupid. And you are logically stupid and dishonest to boot for claiming that the error was due to Wiles. > So when it's explained simply enough what he attempted, I can cite Cum > Hoc, Ergo Propter Hoc to show it can't work. > As Jesse Hughes has pointed out, that's the wrong logical fallacy anyway. > So what do you people do? You attack attempts at explaining what he > did No. We attack your argument which was based on a secondary source. You are by no means explaining what Wiles did. If you were you would quote him directly. > and try to revert to complicated math-ese where it is more > Where's the complicated math-ese? You say Wiles claimed to show that two sets were the same by showing they have the same number of elements. I say you were relying on a stupid bowdlerized secondary source and that Wiles didn't do that at all. You can prove me wrong by providing a direct quote from Wiles. You could have done that maybe two years ago in reply to the previous poster. But you have never done it. Arrive at the obvious conclusion. You have never done it because you have never read Wiles' paper, only the incompetent popular summaries. Prove me wrong. > It's so blatant the wonder is that you get away with it, but people > want to believe so it's not such a surprise. They don't want to accept > that this person they were told is so brilliant failed by a dumb > logical error. > Prove it - all you have to do is pull out a direct quote from Wiles' work. Simple enough? I'm waiting with bated breath. Marcus. James Harris === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. > poster and I reached the same conclusion: you have never read > Wiles' paper, only secondary sources. It's pretty clear that > Wikipedia copied MathWorld (which is older), so the fact that the > earlier poster found one source and I found another is immaterial. > The fact is, that quotation is your only basis for your claim. Other- > wise you would do what several people here have asked: cite a > direct quote from one or more of Wiles' papers. But you are very > conspicuously not doing that. It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > Sorry, but all you can really claim here is that the quote from > Wikipedia/MathWorld is logically stupid. And you are logically > stupid and dishonest to boot for claiming that the error was due > to Wiles. > But that goes to the delusions of grandeur that you and other Usenet posters continually show. So you attack an electronic math journal reviewed by Mathematical Reviews as a crap journal. And now you assault MathWorld and the Wikipedia. To you people that all makes sense!!! But to most people it shows you are in your own little world and that maybe they've given people like you way to much credit when they listen to you about my research. Quite simply, you people act like there is no authority higher than you, so to you it's a casual thing to claim that MathWorld is stupid as well as say the same thing about the Wikipedia. But, um, who in the hell are you? Last I noticed you were just some Usenet posters. Who appear to shift your identities as well as the latest questions about Greg aka Tim Peters reveal. I guess you feel that you can say anything on Usenet without ever being accountable for it, and don't care what people might be disappointed to learn they have been quite boldly lied to by people who have callously manipulated them. ___JSH === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. > poster and I reached the same conclusion: you have never read > Wiles' paper, only secondary sources. It's pretty clear that > Wikipedia copied MathWorld (which is older), so the fact that the > earlier poster found one source and I found another is immaterial. > The fact is, that quotation is your only basis for your claim. Other- > wise you would do what several people here have asked: cite a > direct quote from one or more of Wiles' papers. But you are very > conspicuously not doing that. It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > Sorry, but all you can really claim here is that the quote from > Wikipedia/MathWorld is logically stupid. And you are logically > stupid and dishonest to boot for claiming that the error was due > to Wiles. > But that goes to the delusions of grandeur that you and other Usenet > posters continually show. So you attack an electronic math journal reviewed by Mathematical > Reviews as a crap journal. And now you assault MathWorld and the Wikipedia. To you people that all makes sense!!! But to most people it shows you are in your own little world and that > maybe they've given people like you way to much credit when they listen > to you about my research. Quite simply, you people act like there is no authority higher than > you, so to you it's a casual thing to claim that MathWorld is stupid as > well as say the same thing about the Wikipedia. But, um, who in the hell are you? Last I noticed you were just some > Usenet posters. Who appear to shift your identities as well as the latest questions > about Greg aka Tim Peters reveal. I guess you feel that you can say anything on Usenet without ever being > accountable for it, and don't care what people might be disappointed to > learn they have been quite boldly lied to by people who have callously > manipulated them. > ___JSH James, you do realize that the info on MathWorld and Wikipedia is not the end all be all on any subject presented on them. They are abridged summations of the subject matter. If they are the only sources you are using for your math education, than you are more of a moron than I ever gave you credit for. The post where Tim Peters signed his name Greg was a joke. Everyone else got it, but of course you didn't. To tell you the truth, who cares if he is using a real name or not. Most people who are on the internet don't use the real names, at least not on public forums. They use screennames. It helps avoid some of the pitfalls of the internet world. It is the quality of his posts that matter, and they seem to hold up on their own. If he was posting things that were incorrect, people would discount him like they discount you. === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. > poster and I reached the same conclusion: you have never read > Wiles' paper, only secondary sources. It's pretty clear that > Wikipedia copied MathWorld (which is older), so the fact that the > earlier poster found one source and I found another is immaterial. > The fact is, that quotation is your only basis for your claim. Other- > wise you would do what several people here have asked: cite a > direct quote from one or more of Wiles' papers. But you are very > conspicuously not doing that. It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > Sorry, but all you can really claim here is that the quote from > Wikipedia/MathWorld is logically stupid. And you are logically > stupid and dishonest to boot for claiming that the error was due > to Wiles. > But that goes to the delusions of grandeur that you and other Usenet > posters continually show. So you attack an electronic math journal reviewed by Mathematical > Reviews as a crap journal. And now you assault MathWorld and the Wikipedia. To you people that all makes sense!!! But to most people it shows you are in your own little world and that > maybe they've given people like you way to much credit when they listen > to you about my research. Quite simply, you people act like there is no authority higher than > you, so to you it's a casual thing to claim that MathWorld is stupid as > well as say the same thing about the Wikipedia. But, um, who in the hell are you? Last I noticed you were just some > Usenet posters. Who appear to shift your identities as well as the latest questions > about Greg aka Tim Peters reveal. I guess you feel that you can say anything on Usenet without ever being > accountable for it, and don't care what people might be disappointed to > learn they have been quite boldly lied to by people who have callously > manipulated them. > ___JSH You're right, but the flaw in Wiles' proof is much more obvious that you realise. I notice that his proof is written in math-ese, therefore false. Evidence enough? If not, then consider this - he never ONCE claims that p mod 3 generates a random sequence. Not only is his proof false, he's clearly unhinged and dangerous. === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. > poster and I reached the same conclusion: you have never read > Wiles' paper, only secondary sources. It's pretty clear that > Wikipedia copied MathWorld (which is older), so the fact that the > earlier poster found one source and I found another is immaterial. > The fact is, that quotation is your only basis for your claim. Other- > wise you would do what several people here have asked: cite a > direct quote from one or more of Wiles' papers. But you are very > conspicuously not doing that. It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > Sorry, but all you can really claim here is that the quote from > Wikipedia/MathWorld is logically stupid. And you are logically > stupid and dishonest to boot for claiming that the error was due > to Wiles. > But that goes to the delusions of grandeur that you and other Usenet > posters continually show. So you attack an electronic math journal reviewed by Mathematical > Reviews as a crap journal. And now you assault MathWorld and the Wikipedia. > Not really. Everyone knows, or should, that Wikipedia is unreliable, and MathWorld is not like a refereed journal. Did you ever notice that they are both almost exclusively descriptive, leaving out proofs for virtually everthing they describe? That's not real math. That's Spectator Math for the masses. The problem is not Wikipedia anyway. It's you, claiming on the basis of an obviously false statement there or in MathWorld that you have found an error in Wiles' work - without ever having read Wiles' work. Again, if you really could prove what you claim, you would be quoting an exact passage from Wiles' papers. I note that in this reply you have yet again passed up the opportunity to do that. Wonder why. > To you people that all makes sense!!! > You think MathWorld and Wikipedia should be accepted as infallible Authorities??? > But to most people it shows you are in your own little world and that > maybe they've given people like you way to much credit when they listen > to you about my research. Quite simply, you people act like there is no authority higher than > you, so to you it's a casual thing to claim that MathWorld is stupid as > well as say the same thing about the Wikipedia. > There's no Authority higher than brute logic and correct mathematics. It's astonishing to hear you suggest that fluff like Wiki and MathWorld are Authorities, when you very blithely claim to refute 150 years of hard, proven and tested mathematics from the likes of Dedekind, Kummer, and Hilbert. Whoyougonna believe? Dedekind or Wikipedia ? Hilbert or Harris? > But, um, who in the hell are you? Last I noticed you were just some > Usenet posters. Who appear to shift your identities as well as the latest questions > about Greg aka Tim Peters reveal. > Your leg has been pulled. Delusional paranoia caused you to fall for it. Wake up and smell the fish. > I guess you feel that you can say anything on Usenet without ever being > accountable for it, and don't care what people might be disappointed to > learn they have been quite boldly lied to by people who have callously > manipulated them. > Everyone's accountable here. You have to prove what you say. If you don't, no one believes you. It's that simple. That's why Tim Peters keeps whipping your ass. In this case you can prove me wrong and tromp me in the dust by quoting the appropriate passage from Wiles. You've had every chance. But you haven't done it. Why not? Marcus. ___JSH === Subject: Re: JSH: Prime numbers, counting tells it all [jstevh@msn.com] > ... > I guess you feel that you can say anything on Usenet without ever > being accountable for it, Nope. And I don't delete my posts either. What does it say about someone who does? I'll tell you what it says to me: that such a person is trying to evade accountability for what they post, by trying to rewrite history. Competing hypotheses are ruled out when someone deletes their posts routinely. > and don't care what people might be disappointed to learn they have > been quite boldly lied to by people who have callously manipulated > them. I'd care about that if it ever happened /in reality/ here. But I try to give enough info, explained clearly enough, so that people /can/ check for themselves. For example, you claim that the sieve form of your prime-counting formula (you haven't been on-topic for a loooong time in this thread, so I will be) is both much more concise than anything that came before, and very fast. I said you were wrong about both. But I didn't /just/ say it, then mindlessly repeat it ad nauseum. I /also/ posted executable code for your sieve form and for the sieve form of Legendre's phi recurrence, backing up my claims with concrete evidence anyone can check. Anyone can count the number of characters in both functions to see that yours is not more concise, and anyone can /run/ both functions to see that yours is actually a bit slower (although this is wholly obvious from inspection, to anyone with programming experience). I also claimed that both were correct implementations of pi(), but did not prove it, because that wasn't at issue. However, if someone sincere asked for a proof, I'd be happy to write one up. I'm actually surprised you didn't, since the code I posted for both didn't bother reducing the second argument so that p_a^2 <= n (I left that out to make the code for both as concise as possible), and it's not obvious why that's /not/ a correctness issue in the code as I gave it. Likewise if someone sincere asked for a proof that your sieve form prime-counting formula is easily derived from the sieve form of the Legendre prime-counting formula, again happy to oblige. That only requires a few lines of mechanical manipulation, but as nobody /has/ asked to see that this time around, I haven't bothered to show it. In fact, the simplest correctness proof (see above) I know of proceeds by proving correctness for Legendre's formula first, and then deriving your formula from that. You want to call this bold lies and callous manipulation? Go right ahead -- compounding false claims erases all doubt. === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org >> ... >> So no Andrew Wiles did not prove Fermat's Last Theorem. But he can >> rely on supporters around the world claiming he did no matter how many >> ways you prove he failed. Their word against the mathematics. >> So you've read and understood Andrew Wiles' proof of FLT? If so, and you can >> prove that Wiles' proof is in error, you could legitimately earn a place in >> Math history. So please, please, *PLEASE* tell us which step in the FLT >> proof did Andrew Wiles trip up! > > His approach fails by the logical fallacy Cum Hoc, Ergo Propter Hoc. You did not answer his question. In deductive proofs (like Wiles's and every other mathematical > argument), every step must either be an axiom, an assumption (later > discharged) or follow from previous steps. If a mathematical argument > is fallacious, then there must be some step which satisfies none of > these conditions. Most probably, this would be a conclusion which > does not follow from the premises. So, what is it? Which step? > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. I've talked about the null test before and posters dodge it, run away > from it and otherwise ignore the remarkable reality that if you just > assume the opposition of what Wiles claims he proves and go through his > entire paper, you can't find a single thing that contradicts with it. Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. It's not even hard. Like I can talk about someone finding objects with > four wheels, and trying to come to a conclusion about objects with four > wheels, like they have four doors, and then coming upon a child's > wagon. Mathematicians found this thing that seemed to connect elliptic curves > and modular forms that is about four numbers--the four wheels of my > analogy--and Wiles just does a dumb thing of trying to compare infinite > sets. > This is not based on a reading of Wiles' papers, which is why > Harris doesn't cite anything specific in those papers or provide a > direct quote. This is based solely on Harris' willfully unquestioning > acceptance of an extremely stupid secondary source: Wikipedia. > Below is the relevant quote from the following site: http://en.wikipedia.org/wiki/Fermat's_last_theorem The main problem that Wiles had to overcome was to establish > a correspondence between semistable elliptic curves over the > rational field, and the modular semistable elliptic curves over > the rationals, which he did by explicitly showing that there were > equal numbers of each. If in fact this is literally all Wiles did, Harris would be right. > You > cannot prove that a given correspondence between two infinite > sets is one-to-one by just showing that the two sets have the same > cardinality. But of course this isn't an adequate description of > what Wiles did. What Harris has discovered here is that Wikipedia > is not a reliable font of all knowledge. Surprise, surprise. Marcus. > Actually it was a MathWorld page And now you are going to explain that the MathWorld > page said something different. in a previous discussion on this issue > MathWorld had the problem: Here's the messageid: > Like I said, you can point out flaws in Wiles' work but it doesn't > matter, as you can see posters shifting all over the map and it doesn't > matter. The math field in pure math areas is completely corrupted. > Or not. I guess it must have said the same thing. - William Hughes My point is that you people make the same excuses no matter what, so > before when I talked about what MathWorld said about Wiles' work the > Wiles did, as was just used by a poster attacking what is on the > Wikipedia, but he didn't know that I actually talked about MathWorld > before or he forgot, so it's a great example to show people how empty > you people are of substantive objections. It is just a reality that what Wiles tried to do is logically stupid. > Sorry but that's just the logical reality. > It is just a reality that Wiles did not try to do what you think he tried to do. It is just a reality that the description that you read was wrong. It does not matter whether you read this on Wikipedia or MathWorld. Sorry but that's just the logical reality. - William Hughes === Subject: Re: Prime numbers, counting tells it all > Actually that's a good thing to make an issue out of, and the proper > answer is a surprising one, which is, if you follow through Wiles' > entire paper, and assume that you already have a modular form that is > not an elliptic curve, you can't find a contradiction with that > assumption! It's the null test. Right. The null test. What an ingenious device that is. But I have never seen you plausibly apply this test with any proof at all. I've talked about the null test before and posters dodge it, run > away from it and otherwise ignore the remarkable reality that if you > just assume the opposition of what Wiles claims he proves and go > through his entire paper, you can't find a single thing that > contradicts with it. Bull, James. Posters don't run away from your null test. They tell you that it is nonsense. They ask for clarifications. They beg you to give an example of a successful application. And you never do. > Instead of giving a clear answer, you have always stated that the >> problem was an informal fallacy about causality. This just is not >> possible. Mathematics is a deductive science. Mathematical arguments >> do not involve causal or other inductive reasoning[1]. > > Actually, no and I've explained how the logical fallacy, which is > USUALLY about causality does apply for his highly particular approach > to a highly particular problem. If it's not about causality, then it is not Cum Hoc. [...] -- Jesse F. Hughes A gorgeous display of homoerotic lust. -- Review blurb found on the back of a Chinese black market Dawn of the Dead DVD === Subject: Re: Prime numbers, counting tells it all days. My association with the Department is that of an alumnus. >Bull, James. Posters don't run away from your null test. They >tell you that it is nonsense. Well, at least once, one person actually answered this challenge in detail. http://tinyurl.com/yjv99b Original in: http://tinyurl.com/yzzmps -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org> <87slgqxdz1.fsf@phiwumbda.org> Bull, James. Posters don't run away from your null test. They >tell you that it is nonsense. Well, at least once, one person actually answered this challenge in > detail. http://tinyurl.com/yjv99b Original in: http://tinyurl.com/yzzmps > Yeah, I kind of think I might have considered that claim. For those who don't want to chase the link I have a quote: The problem then arises on line 3 of pp. 544, i.e., the very last line of the proof of Theorem 5.2. And the post is FROM Arturo Magidin: > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin > magidin-at-member-ams-org To see what he did and consider the problem with his claim you need to know what Theorem 5.2 is, and understand why it is significant that the supposed contradiction occurs at the end. That is, he claims--in his own words--the the problem occurs with the CONCLUSION of what the paper claims is a theorem. Null tests can't end with the conclusion unless the proof is a proof by contradiction. Magidin tries to cleverly change Wiles' words into such. It is an interesting read for those who wish to puzzle through it and see how such people operate. It is a great example of what I call using math-ese. James Harris === Subject: Re: Prime numbers, counting tells it all > It is a great example of what I call using math-ese. In other words, you don't know what they're talking about. Dave === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org> <87slgqxdz1.fsf@phiwumbda.org> A direct reply to Arturo Magidin. This would harm his reputation if such a thing were possible. Bull, James. Posters don't run away from your null test. They >tell you that it is nonsense. Well, at least once, one person actually answered this challenge in > detail. http://tinyurl.com/yjv99b Original in: http://tinyurl.com/yzzmps > Yeah, I kind of think I might have considered that claim. Maybe, but until now you have never replied to it. For those > who don't want to chase the link I have a quote: of the proof of Theorem 5.2. > And the post is FROM Arturo Magidin: It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin > magidin-at-member-ams-org To see what he did and consider the problem with his claim you need to > know what Theorem 5.2 is, and understand why it is significant that > the supposed contradiction occurs at the end. That is, he claims--in his own words--the the problem occurs with the > CONCLUSION of what the paper claims is a theorem. Null tests can't end with the conclusion unless the proof is a proof by > contradiction. > Piffle. Suppose I have a proof that there are an infinite number of primes. Suppose further that I have a direct proof that ends with the statement. Therefore there are an infinite number of primes. The null test would consist of making the assumption There are not an infinite number of primes and looking through the proof for a contradiction. The first such contradiction will end at the conclusion. So it is not true that null tests of a direct proof can't end with the conclusion. - William Hughes === Subject: Re: Prime numbers, counting tells it all days. My association with the Department is that of an alumnus. Sorry for posting this twice. My connection fizzled, and the formatting was a mess. I've cancelled the previous version... A direct reply to Arturo Magidin. This would >harm his reputation if such a thing were possible. Whose reputation? I confess that it reads like you are saying this will harm my reputation, but I don't see why you would say that... [...] >> Null tests can't end with the conclusion unless the proof is a proof by >> contradiction. > >Piffle. Suppose I have a proof that there are an infinite number >of primes. Suppose further that I have a direct proof that >ends with the statement. Therefore there are an infinite number of >primes. The null test would consist of making the assumption There are not >an infinite number of primes and looking through the proof for >a contradiction. The first such contradiction will end at the >conclusion. Here is an easy short direct proof we can try it with. THEOREM. Let a and b be real numbers, a nonzero. If ab=0, then b=0. Proof. Since a is nonzero, we can multiply both sides of ab=0 by 1/a. We get (1/a)(ab) = (1/a)0 [(1/a)a]b = 0 1b = 0 b = 0 Therefore, b=0. QED. Now, oet us try to the null test. This means adding the hypothesis and b is not zero to our hypothesis that a and b are real numbers, a is nonzero, and ab=0. Running through the proof, where do we get a problem? (i) Not at Since a is nonzero. (ii) Not at we can multiply both sides by the same thing. (iii) Not at by 1/a (which we can do by (i) ). (iv) Not at we get (1/a)(ab) = (1/a)0. (v) Not at the set where we use associativity of multiplication. (vi) Not at the step where we use that x0=0 for all x in R. (vii) Not at the step where we use that (1/a)*a = 1. (viii) Not at the step where we use that 1*x=x for all xi n R. (ix) Not at the step where we get b=0. (x) We get our first problem at this point. Therefore, b=0 holds, but b is nonzero is our hypothesis. Contradiction. So we get the first problem at the end. Even though our original proof is a direct proof, not a proof by contradiction. This is the silliest exercise in proof checking. It takes a direct proof, and turns it into a useless proof by contradiction, one in which only the final conclusion is affected by the extra hypothesis. This is what usually happens in all direct proofs that establish the conclusion directly (by construction, say). It is exactly the sort of thing I (at least) warn my students about when they are learning to write proofs: if you are doing what you think is a proof by contradiction, but you do not use the contrary assumption until the end, then what you have can be turned into a direct proof. In which case, you should write it as a direct proof. This is of course not always the case; there are statements that can be proven by contradiction in a manner very different from a direct proof. But in general I would say: if you have a proof by contradiction and attempt the null test, you will not change the proof the at all: you will only be adding a hypothesis that YOU ALREADY HAD ADDED. If, on the other hand, you have a direct proof, then trying this null test will often result in a proof in which the only problem arises at the end. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Prime numbers, counting tells it all Sorry for posting this twice. My connection fizzled, and the formatting > was a mess. I've cancelled the previous version... >A direct reply to Arturo Magidin. This would >harm his reputation if such a thing were possible. Whose reputation? I confess that it reads like you are saying this > will harm my reputation, but I don't see why you would say that... antecedent of his. You made the mistake of reading My appologies. - William (who might acually be Ullrich) Hughes P.S. I just thought of something. If I am Ullrich, I can blame the whole thing on him. This makes me feel better (I think?). === Subject: Re: Prime numbers, counting tells it all <87slgqzp3u.fsf@phiwumbda.org> <87slgqxdz1.fsf@phiwumbda.org> harm his reputation if such a thing were possible. Whose reputation? I confess it seems like you are saying it would harm ->my<- reputation, though I cannot figure out how... > Null tests can't end with the conclusion unless the proof is a proof by > contradiction. > Piffle. Suppose I have a proof that there are an infinite number > of primes. Suppose further that I have a direct proof that > ends with the statement. Therefore there are an infinite number of > primes. The null test would consist of making the assumption There are not > an infinite number of primes and looking through the proof for > a contradiction. The first such contradiction will end at the > conclusion. So it is not true that null tests of a direct proof can't end with > the conclusion. A much simpler example: consider the following statement and proof: Let a and b be real numbers, a different from 0. if ab=0, then b=0. Proof: Consider ab=0. Since a is not zero, we can mutliply both sides by by 1/a. Using the fact that x*0 = 0 for all real numbers zero, that 1*x = x for all real numbers x, and that multiplication is associative we have (1/a)*(ab) = (1/a)*0 [(1/a)*a]*b = 0 1*b = 0 b = 0 Therefore b = 0. QED Now, let us try this null test. We start from the assumptions that a and b are real numbers, that a is different from zero, that ab=0, AND that b is not zero. We run through the proof. The statement since a is not zero... still holds exactly the same as before. So does x*0=0, 1*x = 1, multiplication is associative, and each and every one of the equalities. So does the conclusion Therefore b=0. It is only now that the assumption b is not zero can come into play. How does it come into play? By saying But that contradicts our assumption that b is not zero. No problem until the very last line. As I said 31 months ago: it is the worst kind of proof exercise. Take a direct proof, add the assumption that the conclusion is false. What happens? You have a proof by contradiction where the extra assumption is never used, except to contradict the final conclusion. Exactly the kind of thing I tell students to change into a direct proof. Arturo Magidin, sans .sig === Subject: Re: JSH: Prime numbers, counting tells it all [added JSH: to subject] [jstevh@msn.com] > [... nothing much ...] > It is a great example of what I call using math-ese. And which other people-- the deluded ones --call mathematics? Cool. === Subject: Re: JSH: Prime numbers, counting tells it all [jstevh@msn.com] >> Actually that's a good thing to make an issue out of, and the proper >> answer is a surprising one, which is, if you follow through Wiles' >> entire paper, and assume that you already have a modular form that is >> not an elliptic curve, you can't find a contradiction with that >> assumption! >> It's the null test. [Jesse F. Hughes] > Right. The null test. What an ingenious device that is. But I have never seen you plausibly apply this test with any proof at > all. >> I've talked about the null test before and posters dodge it, run >> away from it and otherwise ignore the remarkable reality that if you >> just assume the opposition of what Wiles claims he proves and go >> through his entire paper, you can't find a single thing that >> contradicts with it. > Bull, James. Posters don't run away from your null test. They > tell you that it is nonsense. They ask for clarifications. They beg > you to give an example of a successful application. And you never do. Jeez. Don't you think it's a bit unseemly for JSH's official archivist to give the appearance of questioning the source of his livelihood? Think of the big picture here. If JSH dismisses you, his wisdom may well die out with him. [also Jesse] > Instead of giving a clear answer, you have always stated that the > problem was an informal fallacy about causality. This just is not > possible. Mathematics is a deductive science. Mathematical arguments > do not involve causal or other inductive reasoning[1]. >> Actually, no and I've explained how the logical fallacy, which is >> USUALLY about causality does apply for his highly particular approach >> to a highly particular problem. > If it's not about causality, then it is not Cum Hoc. And again! James has his own surprising meanings for common English words, like proof, efficient, correct, lie, research, mathematics, ..., on and on. I think that entitles him to play Humpty Dumpty(*) with Latin too. > [...] (*) When /I/ use a word, Humpty Dumpty said, in a rather scornful tone, it means just what I choose it to mean, neither more nor less. The question is, said Alice, whether you can make words mean so many different things. The question is, said Humpty Dumpty, which is to be master - that's all. Lewis Carroll, Alice in Wonderland === Subject: Re: JSH: Prime numbers, counting tells it all days. My association with the Department is that of an alumnus. >(*) When /I/ use a word, Humpty Dumpty said, in a rather scornful > tone, it means just what I choose it to mean, neither more nor > less. The question is, said Alice, whether you can make words mean > so many different things. The question is, said Humpty Dumpty, which is to be master - > that's all. Lewis Carroll, Alice in Wonderland Not at all. Humpty Dumpty does not show up in Alice in Wonderland. He is the subject of Chapter VI of Through the Looking Glass, and what Alice found there. -- The sun was shining on the sea, shining with all his might; He did his very best to make the billows smooth and bright -- And this was odd because it was the middle of the night. The moon was shining sulkily, because she thought the sun Had got no business to be there after the day was done -- 'It's very rude of him', she said, to come and spoil the fun!' -- Lewis Carroll, Through the Looking Glass, and what Alice found there Arturo Magidin magidin-at-member-ams-org === Subject: Re: JSH: Prime numbers, counting tells it all Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Not at all. Humpty Dumpty does not show up in Alice in Wonderland. You're so naive. Do you really think Humpty Dumpty is his real name? -- Nora -- Consideration shall be given to the need for as many as 32 characters in some alphabets - X3.4, 1963. === Subject: Re: JSH: Prime numbers, counting tells it all days. My association with the Department is that of an alumnus. >>Not at all. Humpty Dumpty does not show up in Alice in Wonderland. You're so naive. Do you really think Humpty Dumpty is his real name? Did you hear the NPR piece on two attack ad voice-over artists? They did some spoofs of nursery rhymes, including: Humpty Dumpty sat on a wall. He said he could handle it, but after spending thousands of our tax dollars, not even all the king's horses and all the king's men could put Humpty Dumpty together again. Humpty Dumpty. WRONG on wall sitting. I think Tim got it right. The book's title may be Through the Looking Glass and what Alice found there, and the book's title may be called Through the Looking Glass, but the book is certainly called Alice in Wonderland. And the tune, the tune is of my own invention. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: JSH: Prime numbers, counting tells it all Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >I think Tim got it right. The book's title may be Through the Looking >Glass and what Alice found there, and the book's title may be called >Through the Looking Glass, but the book is certainly called Alice >in Wonderland. But what *is* the book? -- Richard -- Consideration shall be given to the need for as many as 32 characters in some alphabets - X3.4, 1963. === Subject: Re: JSH: Prime numbers, counting tells it all [Tim Peters] >> ... >>(*) When /I/ use a word, Humpty Dumpty said, in a rather scornful >> tone, it means just what I choose it to mean, neither more nor >> less. >> The question is, said Alice, whether you can make words mean >> so many different things. >> The question is, said Humpty Dumpty, which is to be master - >> that's all. >> Lewis Carroll, Alice in Wonderland [Arturo Magidin] > Not at all. Humpty Dumpty does not show up in Alice in Wonderland. > He is the subject of Chapter VI of Through the Looking Glass, and > what Alice found there. Well, perhaps in the way /you/ use words. In /my/ way, the book is /called/ Alice in Wonderland, although its /title/ is Through the Looking Glass, and What Alice Found There, while the /name/ of the book is simply Through the Looking Glass. I suppose tedious pedants will be more inclined to agree with your literal reading, but in a JSH thread I'm sticking to mine ;-) === Subject: Re: JSH: Prime numbers, counting tells it all <87slgqxdz1.fsf@phiwumbda.org> > ... >>(*) When /I/ use a word, Humpty Dumpty said, in a rather scornful >> tone, it means just what I choose it to mean, neither more nor >> less. >> The question is, said Alice, whether you can make words mean >> so many different things. >> The question is, said Humpty Dumpty, which is to be master - >> that's all. >> Lewis Carroll, Alice in Wonderland [Arturo Magidin] > Not at all. Humpty Dumpty does not show up in Alice in Wonderland. > He is the subject of Chapter VI of Through the Looking Glass, and > what Alice found there. Well, perhaps in the way /you/ use words. In /my/ way, the book is /called/ > Alice in Wonderland, although its /title/ is Through the Looking Glass, > and What Alice Found There, while the /name/ of the book is simply Through > the Looking Glass. To be precise, the title is Through the Looking Glass, and what Alice found there. The title is ->called<- Through the looking glass, and the book is called Alice in wonderland. The book itself, of course, is long, but it's very, ->very<- beautiful. Everybody that reads it -- either it brings the ->tears<- into their eyes, or else--- -- or else it doesn't. Arturo Magidin I suppose tedious pedants will be more inclined to > agree with your literal reading, but in a JSH thread I'm sticking to mine > ;-) === Subject: topology with connected component. hello sir~ X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. -------------------------- i think... i don't need topology T generated by B. i can use B directly. so, i know that all components is closed set. so, {1}^c = {2,3,4,5,6,7} {1,2}^c = {3,4,5,6,7} {1,2,3}^c = {4,5,6,7} {1,4,5}^c = {2,3,6,7} {1,2,6,7}^c = {3,4,5} {4,5}^c = {1,2,3,6,7} {6,7}^c = {1,2,3,4,5} empty^c = {1,2,3,4,5,6,7} 1) {2,3,4,5,6,7} is connected 2) {3,4,5,6,7} is connected 3) {4,5,6,7} = {4,5} U {6,7} is not connected 4) {2,3,6,7} is connected 5) {3,4,5} is connected 6) {1,2,3,6,7} = {1,2,3} U {6,7} is not connected 7) {1,2,3,4,5} = {1,2,3} U {4,5} is not connected 8) {1,2,3,4,5,6,7} = {1,2,3} U {4,5,6,7} is not connected. so, {2,3,4,5,6,7} is a component. then, {1} must be a component. but {1} is not a closed set. strange.... how do you think about it ? === Subject: Re: topology with connected component. > hello sir~ X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. -------------------------- > i think... > i don't need topology T generated by B. > i can use B directly. oh...no. sorry. first, i must make topology T generated by B. and since B is finite, components are a open and closed set. so, i must find open and closed set. and then i must investigate with these.(either connected or...) if i don't like to find the open and closed set, i can find directly. so, {1,2,3} is connected. {4,5} is connected. {6,7} is connected. and {1,2,3,4,5} is not connected. {1,2,3,6,7} is not. {1,2,3,4,5,6,7} is not. {4,5,6,7} is not. so, components are {1,2,3}, {4,5}, {6,7}. in fact, this is really open and closed set. === Subject: Re: topology with connected component. hello sir~ X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. -------------------------- > i think... > i don't need topology T generated by B. > i can use B directly. oh...no. sorry. > first, i must make topology T generated by B. > and > since B is finite, components are a open and closed set. > so, > i must find open and closed set. > and then i must investigate with these.(either connected or...) if i don't like to find the open and closed set, > i can find directly. > so, > {1,2,3} is connected. > {4,5} is connected. > {6,7} is connected. > and > {1,2,3,4,5} is not connected. > {1,2,3,6,7} is not. > {1,2,3,4,5,6,7} is not. > {4,5,6,7} is not. sorry. maybe, It's insufficient reason. because, If {1,2,3,4} is possible to be connected, components are possible to be {1,2,3,4}, {5}, {6,7}. etc...possibility. maybe, I would better find the open and closed set. > so, components are {1,2,3}, {4,5}, {6,7}. > in fact, this is really open and closed set. === Subject: Re: topology with connected component. > X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. A connected component is a subset of a clopen subset. The clopen subsets of X are ... > i think... I think you should alway capitalize the pronoun 'I'. === Subject: Re: topology with connected component. > hello sir~ X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. -------------------------- > i think... > i don't need topology T generated by B. > i can use B directly. > so, > i know that all components is closed set. Are you saying that a closed set is connected? That's not true. > so, > {1}^c = {2,3,4,5,6,7} > {1,2}^c = {3,4,5,6,7} > {1,2,3}^c = {4,5,6,7} > {1,4,5}^c = {2,3,6,7} > {1,2,6,7}^c = {3,4,5} > {4,5}^c = {1,2,3,6,7} > {6,7}^c = {1,2,3,4,5} > empty^c = {1,2,3,4,5,6,7} 1) {2,3,4,5,6,7} is connected Why? > 2) {3,4,5,6,7} is connected > 3) {4,5,6,7} = {4,5} U {6,7} is not connected > 4) {2,3,6,7} is connected > 5) {3,4,5} is connected > 6) {1,2,3,6,7} = {1,2,3} U {6,7} is not connected > 7) {1,2,3,4,5} = {1,2,3} U {4,5} is not connected > 8) {1,2,3,4,5,6,7} = {1,2,3} U {4,5,6,7} is not connected. so, {2,3,4,5,6,7} is a component. > then, {1} must be a component. > but {1} is not a closed set. > strange.... how do you think about it ? -- Marcus === Subject: Re: topology with connected component. Marcus ÀÛ.bcº: > hello sir~ X = {1,2,3,4,5,6,7} basis B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}} find all components. -------------------------- > i think... > i don't need topology T generated by B. > i can use B directly. > so, > i know that all components is closed set. Are you saying that a closed set is connected? That's not true. no. i say that connected component => closed. so, i only investigate closed set of X. > so, > {1}^c = {2,3,4,5,6,7} > {1,2}^c = {3,4,5,6,7} > {1,2,3}^c = {4,5,6,7} > {1,4,5}^c = {2,3,6,7} > {1,2,6,7}^c = {3,4,5} > {4,5}^c = {1,2,3,6,7} > {6,7}^c = {1,2,3,4,5} > empty^c = {1,2,3,4,5,6,7} 1) {2,3,4,5,6,7} is connected Why? oh, i see.(may, no) Y={2,3,4,5,6,7} is a subspace of X. so, basis of T Y = {empty, {2}, {2,3}, {4,5}, {2,6,7}, {4,5}, {6,7}} against B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}}. so, Y = {2,3} U {4,5,6,7}. this is not connected. i have thought about only open set of X. i try again. > 2) {3,4,5,6,7} is connected Y={3,4,5,6,7} so, basis of T Y = {emtpy, {3}, {4,5}, {6,7}} against B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}}. so, Y = {3} U {4,5,6,7} this is not connected. > 3) {4,5,6,7} = {4,5} U {6,7} is not connected this is not connected. > 4) {2,3,6,7} is connected Y={2,3,6,7} so, bais of T Y = {{2}, {2,3}, {2,6,7}, {6,7}} against B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}}. so, Y = {2,3} U {6,7} this is not connected. > 5) {3,4,5} is connected Y={3,4,5} so, bais of T Y = {{3},{4,5}} against B = {{1}, {1,2}, {1,2,3}, {1,4,5}, {1,2,6,7}, {4,5}, {6,7}}. so, Y = {3} U {4,5} this is not connected. > 6) {1,2,3,6,7} = {1,2,3} U {6,7} is not connected this is not connected. > 7) {1,2,3,4,5} = {1,2,3} U {4,5} is not connected this is not connected. > 8) {1,2,3,4,5,6,7} = {1,2,3} U {4,5,6,7} is not connected. this is not connected. oh, my god....there exists no connected. what happens me ? === Subject: looking for games using stopping times and coin tossings Hi all, Recently I became interested in card/coin tossing games that needs decision on when to stop optimally or what's the expected return or fair price of the game using the martingale concept and optional sampling and optimal stopping time techniques... i am very interested in these puzzles/games... where can I read more about these games/puzzles and their solutions? Please give me === Subject: Menunda-nunda dan Tergesa-gesa Tergesa-gesa dan Menunda-nunda adalah termasuk pintu masuk setan. Demikian pula berpanjang angan-angan. Sebagian manusia menyebutnya hambatan terbesar. Apa maksudnya? Sebagian orang meletakkan satu perkara yang dianggap harus diprioritaskan sebagai hambatannya, lalu misalnya berkata, 'Kalau aku selesai sekolah, baru --insya Alloh-- aku akan bertobat!' Ini contoh hambatannya berupa sekolah. Tapi setelah selesai sekolahnya, dia berkata, 'Kalau aku sudah mendapat pekerjaan itu, aku bertobat'. Kemudian diperolehnya pekerjaan tadi, namun dia tidak bertobat juga. Demikianlah selanjutnya, dia menyatakan hambatan berikutnya, 'kalau aku berhaji ... kalau aku menikah ...kalau....kalau....' Terus-terusan dia meletakkan satu hambatan di hadapannya, dan menunda-nunda serta hidup dalam berpanjang angan-angan. Akhirnya dia mati tanpa memulai kehidupan hakikinya (dengan beriltizam, memegang teguh dienul Islam)! Sesungguhnya tujuan akhir yang dikehendaki setan darimu adalah menghalangimu agar kamu tidak beramal, atau agar menunda-nundanya, dimana ini merupakan pintu masuk yang membahayakan orang-orang shalih. Setan datang kepadamu dan berkata, 'Kamu belum pantas --sampai sekarang-- untuk mengajari manusia, atau mendakwahi mereka...tunggulah sampai kamu belajar!' Padahal kita diperintahkan untuk menyampaikan apa yang kita punya walau hanya seayat. Maka jika kamu sudah mempelajari sesuatu, ajarkanlah meskipun hanya seayat saja! Imam Ibnul Jauzi dalam buku 'Talbiis Iblis' berkata, 'Betapa banyak orang yang bertekad teguh, dibuat menanti-nanti', yaitu dibuat berkata 'nanti saja' oleh setan. Ibnul Jauzi melanjutkan, 'Betapabanyak pula yang berusaha untuk berbuat baik dipengaruhi setan untuk menunda-nundanya'. Betapa sering seorang alim bertekad untuk mengulang ilmu yang dipelajarinya, dibujuk setan dengan perkataan, 'Istirahatlah sejenak'. Setan terus menerus meniupkan kecintaan pada ekmalasan dan penundaan amal. Bahkan betapa sering setan datang pada ahli ibadah di waktu malam ketika akan shalat malam dengan bujukan, 'Waktu malam kan masih panjang? Tundalah shalatmu!'. Sampai-sampai shubuh datang dan dia tidak shalat malam! (Sumber Rujukan: Madaakhilusy Syaithaan 'ala Ash Shalihin, oleh Dr Abdullah Al Khaathir ) [www.mediamuslim.info] === Subject: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Can p^(p^p) + 8 p^p + 26 be composite? --- === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? > Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Can > p^(p^p) + 8 p^p + 26 be composite? Did you mean p^(p^p) + 8 p^p - 4, as that's obviously prime? Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We T, Inc./. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? <87irhlq867.fsf@nonospaz.fatphil.org Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Is 3^6 + 6008 prime? > Can p^(p^p) + 8 p^p + 26 be composite? > Did you mean p^(p^p) + 8 p^p - 4, as that's obviously prime? > === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? > Can anyone solve this one for me? >> Question: Suppose that p and p^6 + 6008 are both prime. Is 3^6 + 6008 prime? > = 6737 I have checked with every prime, I think, up to 83 (83^2=6589) Unless, I have missed one it is a prime. For p=5 it is not. p^6 + 6008 = 21633 which is not prime (divisible by 3) For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) I assume that in situations where p and p^6 + 6008 are prime. Can there be some clarification? Nick === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? [Proginoskes] > Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. [William Elliot] >> Is 3^6 + 6008 prime? [Nick] > = 6737 I have checked with every prime, I think, up to 83 (83^2=6589) Unless, I have missed one it is a prime. Yes, 6737 is prime. > For p=5 it is not. p^6 + 6008 = 21633 which is not prime > (divisible by 3) For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) I assume that in situations where p and p^6 + 6008 are prime. Can > there be some clarification? p=3 is the only case where both p and p^6 + 6008 are prime. If you ask me (and since I'm answering, I'll assume that you are ;-)), that's the /interesting/ part. It's actually very easy to show this, if approached in the right way (you're moving toward it empirically: can you show that p^6 + 6008 is /always/ divisible by 3 when p is a prime larger than 3? hint: all such p are congruent to +1 or -1 modulo 3). The less interesting (and by far, if you ask me) part is then proving that 7625597485199 (3^(3^3) + 8*3^3 - 4) is prime -- unless Christopher has an elegant way of knowing that I'm not seeing. Of course it's dead easy to prove that 7625597485199 is prime by using appropriate software. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? > [Proginoskes] >> Can anyone solve this one for me? >> Question: Suppose that p and p^6 + 6008 are both prime. [William Elliot] > Is 3^6 + 6008 prime? [Nick] >> = 6737 >> I have checked with every prime, I think, up to 83 (83^2=6589) >> Unless, I have missed one it is a prime. Yes, 6737 is prime. > For p=5 it is not. p^6 + 6008 = 21633 which is not prime >> (divisible by 3) >> For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) >> I assume that in situations where p and p^6 + 6008 are prime. Can >> there be some clarification? p=3 is the only case where both p and p^6 + 6008 are prime. If you ask me > (and since I'm answering, I'll assume that you are ;-)), that's the > /interesting/ part. It's actually very easy to show this, if approached > in the right way (you're moving toward it empirically: can you show that > p^6 + 6008 is /always/ divisible by 3 when p is a prime larger than 3? > hint: all such p are congruent to +1 or -1 modulo 3). I did this stuff over 30 years ago, in my maths degree - I haven't studied maths at university level for nearly 30 years - so that you presume too much (intransitive or otherwise). I covered it in my first year and the terms mean something - some of the things that I have already done here I haven't done in a long time. I am interested - otherwise I wouldn't be participating. But I am getting beyond my depth. My question is therefore was the question a trick one ie did the OP know that it only occurred for p=3. And if it is the case for only p=3, why wasn't p=3 substituted for p in the question? Nick === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? [Proginoskes] > Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. [William Elliot] >> Is 3^6 + 6008 prime? [Nick] > = 6737 I have checked with every prime, I think, up to 83 (83^2=6589) Unless, I have missed one it is a prime. [Tim Peters] >> Yes, 6737 is prime. [Nick] > For p=5 it is not. p^6 + 6008 = 21633 which is not prime > (divisible by 3) For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) I assume that in situations where p and p^6 + 6008 are prime. Can > there be some clarification? >> p=3 is the only case where both p and p^6 + 6008 are prime. If you >> ask me (and since I'm answering, I'll assume that you are ;-)), >> that's the /interesting/ part. It's actually very easy to show this, >> if approached in the right way (you're moving toward it empirically: >> can you show that p^6 + 6008 is /always/ divisible by 3 when p is a >> prime larger than 3? hint: all such p are congruent to +1 or -1 >> modulo 3). [Nick] > I did this stuff over 30 years ago, in my maths degree - I haven't > studied maths at university level for nearly 30 years - so that you > presume too much (intransitive or otherwise). I don't think so -- I think you underestimate yourself here. Take it one step at a time. Suppose p leaves a remainder of 1 when divided by 3. Can you figure out what remainder p^6 leaves when divided by 3? If you ever studied any number theory, it's immediate that a = b (mod c) implies a^i = b^i (mod c) for every i >= 0. Or you could do it the hard way: if p leaves a remainder of 1 when divided by 3, then p = 3*j+1 for some integer j, so p^6 = (3*j+1)^6 = 729*j^6 + 1458*j^5 + 1215*j^4 + 540*j^3 + 135*j^2 + 18*j + 1 and each piece of that is divisible by 3 except for the trailing +1. So p^6 leaves a remainder of 1 when divided by 3, so p^6+6008 is exactly divisible by 3. You can do the same kind of thing to show that when p leaves a remainder of 2 when divided by 3, p^6+6008 must also be divisible by 3. If you give yourself the briefest of refresher courses in how congruences work, the easy way is just this simple: p prime > 3 implies p leaves a remainder of 1 or 2 when divided by 3 implies p = +/- 1 (mod 3) implies p^6 = (+/- 1)^6 (mod 3) implies p^6 = 1 (mod 3) implies p^6 + 6008 = 1 + 6008 = 1 + 2 = 3 = 0 (mod 3) implies p^6 + 6008 is composite (as it's divisible by 3) > I covered it in my first year and the terms mean something - some of > the things that I have already done here I haven't done in a long time. I am interested - otherwise I wouldn't be participating. But I am getting beyond my depth. I hope the above was enough to jog your memory. There isn't anything hard here, /except/ for seeing the right approach to take. This was very much a trick question. > My question is therefore was the question a trick one ie did the OP > know that it only occurred for p=3. Oh yes, he certainly did. In fact, it's really an obscured variation of a simpler trick question posted a few weeks ago. > And if it is the case for only p=3, why wasn't p=3 substituted for p > in the question? Because that would have given away the trick :-) The /real/ point of this question is to try to fool people who know a lot about primes and congruences, not those who know a little. People who know a lot have a decent chance of getting frated by trying to apply high-powered results that are wholly unnecessary (and pretty much wholly useless ;-)) here. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? <87irhlq867.fsf@nonospaz.fatphil.org> Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. [William Elliot] >> Is 3^6 + 6008 prime? [Nick] > = 6737 I have checked with every prime, I think, up to 83 (83^2=6589) Unless, I have missed one it is a prime. [Tim Peters] >> Yes, 6737 is prime. [Nick] > For p=5 it is not. p^6 + 6008 = 21633 which is not prime > (divisible by 3) For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) I assume that in situations where p and p^6 + 6008 are prime. Can > there be some clarification? > p=3 is the only case where both p and p^6 + 6008 are prime. If you >> ask me (and since I'm answering, I'll assume that you are ;-)), >> that's the /interesting/ part. It's actually very easy to show this, >> if approached in the right way (you're moving toward it empirically: >> can you show that p^6 + 6008 is /always/ divisible by 3 when p is a >> prime larger than 3? hint: all such p are congruent to +1 or -1 >> modulo 3). [Nick] > I did this stuff over 30 years ago, in my maths degree - I haven't > studied maths at university level for nearly 30 years - so that you > presume too much (intransitive or otherwise). I don't think so -- I think you underestimate yourself here. Take it one > step at a time. Suppose p leaves a remainder of 1 when divided by 3. Can > you figure out what remainder p^6 leaves when divided by 3? If you ever > studied any number theory, it's immediate that a = b (mod c) implies a^i = b^i (mod c) for every i >= 0. Or you could do it the hard way: if p leaves a > remainder of 1 when divided by 3, then p = 3*j+1 for some integer j, so p^6 = > (3*j+1)^6 = > 729*j^6 + 1458*j^5 + 1215*j^4 + 540*j^3 + 135*j^2 + 18*j + 1 and each piece of that is divisible by 3 except for the trailing +1. So > p^6 leaves a remainder of 1 when divided by 3, so p^6+6008 is exactly > divisible by 3. You can do the same kind of thing to show that when p leaves a remainder of > 2 when divided by 3, p^6+6008 must also be divisible by 3. If you give yourself the briefest of refresher courses in how congruences > work, the easy way is just this simple: p prime > 3 > implies > p leaves a remainder of 1 or 2 when divided by 3 > implies > p = +/- 1 (mod 3) > implies > p^6 = (+/- 1)^6 (mod 3) > implies > p^6 = 1 (mod 3) > implies > p^6 + 6008 = 1 + 6008 = 1 + 2 = 3 = 0 (mod 3) > implies > p^6 + 6008 is composite (as it's divisible by 3) I covered it in my first year and the terms mean something - some of > the things that I have already done here I haven't done in a long time. I am interested - otherwise I wouldn't be participating. But I am getting beyond my depth. I hope the above was enough to jog your memory. There isn't anything hard > here, /except/ for seeing the right approach to take. This was very much a > trick question. And it was in the manner of a similar trick question posted a few days ago, and another one a month ago. > My question is therefore was the question a trick one ie did the OP > know that it only occurred for p=3. Oh yes, he certainly did. In fact, it's really an obscured variation of a > simpler trick question posted a few weeks ago. and 3 or 4 weeks before that one. I seemed to be the only one who thought it was familiar, and who took the trouble to search the archives, so I decided to see if anyone was still paying attention. Conclusion: Outlook not so good. > And if it is the case for only p=3, why wasn't p=3 substituted for p > in the question? Because that would have given away the trick :-) The /real/ point of this question is to try to fool people who know a lot > about primes and congruences, not those who know a little. People who > know a lot have a decent chance of getting frated by trying to apply > high-powered results that are wholly unnecessary (and pretty much wholly > useless ;-)) here. --- P.S. I used Maple for prime verification. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? <87irhlq867.fsf@nonospaz.fatphil.org> Question: Suppose that p and p^6 + 6008 are both prime. Is 3^6 + 6008 prime? > = 6737 I have checked with every prime, I think, up to 83 (83^2=6589) > Unless, I have missed one it is a prime. For p=5 it is not. p^6 + 6008 = 21633 which is not prime (divisible by 3) > For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) I assume that in situations where p and p^6 + 6008 are prime. You assume what? That we're psychic? > Can there be some clarification? > p = 3. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? Question: Suppose that p and p^6 + 6008 are both prime. >> Is 3^6 + 6008 prime? >> = 6737 >> I have checked with every prime, I think, up to 83 (83^2=6589) >> Unless, I have missed one it is a prime. >> For p=5 it is not. p^6 + 6008 = 21633 which is not prime (divisible by >> 3) >> For p=7 p^6 + 6008 = 123657 is not prime (divisible by 3) >> I assume that in situations where p and p^6 + 6008 are prime. You assume what? That we're psychic? > Can there be some clarification? > p = 3. I myself am asking for clarification of the problem. I obtained my degree 30 years ago. I am trying to understand the problem. I am trying to understand the question. When I said I assume I didn't mean that I understood the solution - rather what the OP (Proginoskes) meant by the question. Hence my request for clarification. I haven't dealt with such maths for nearly 30 years. Nick === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? <87irhlq867.fsf@nonospaz.fatphil.org> > I assume that in situations where p and p^6 + 6008 are prime. > You assume what? That we're psychic? > Can there be some clarification? > p = 3. I myself am asking for clarification of the problem. I obtained my degree 30 > years ago. I am trying to understand the problem. I am trying to understand the question. When I said I assume I didn't mean > that I understood the solution - rather what the OP (Proginoskes) meant by > the question. 'Assume' isn't an intransitive verb. It's a transitive verb requiring an object. > Hence my request for clarification. I haven't dealt with such maths for > nearly 30 years. > The problem reads, Question: Suppose that p and p^6 + 6008 are both prime. and then he made another statement, which you in your forsight clipped, about something being composite. He's asking if p and p^6 + 6008 are both prime, then is that something composite? Since p = 3, that question about something becomes 'is that something evaluated at p = 3, composite?' === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? > Question: Suppose that p and p^6 + 6008 are both prime. I assume that in situations where p and p^6 + 6008 are prime. >> You assume what? That we're psychic? > Can there be some clarification? > p = 3. >> I myself am asking for clarification of the problem. I obtained my degree >> 30 >> years ago. I am trying to understand the problem. >> I am trying to understand the question. When I said I assume I didn't >> mean >> that I understood the solution - rather what the OP (Proginoskes) meant >> by >> the question. 'Assume' isn't an intransitive verb. It's a transitive verb requiring an > object. > I was abbreviating that I assume that this refers to situations where p and p^6 + 6008 are prime. I was querying the meaning of the statement (in proper English) of Suppose that p and p^6 + 6008 are both prime. Clearly being patronising is your forte as evidenced in http://schools.mylounge.com/t34060-binomial-theorem----negative-exponents.ht ml. Nick === Subject: [] p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? <87irhlq867.fsf@nonospaz.fatphil.org> <_omdnaenybxGz8rYnZ2dnUVZ8smdnZ2d@bt.com > You're welcome. The correct spelling of ;!) is ;( . === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Is 3^6 + 6008 prime? In what way is it easier to ask that on a medium like usenet than to simply ask the computer that you are no doubt sitting at? $ echo $((3**6+6008)) | factor factor> this number is prime! factor> exit $ echo $((3**6+6008)) | /usr/bin/factor 6737: 6737 $ echo isprime($((3**6+6008))) | gp -q 1 $ calc isprime($((3**6+6008))) 1 $ pfgw -t -q'3^6+6008' 3^6+6008 is prime! (0.0001s+0.0165s) $ perl -e 'print((xx(3**6+6008))=~/^(..+)1+$/?compositen:primen)' prime Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We T, Inc./. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? [Proginoskes] > Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Can > p^(p^p) + 8 p^p + 26 be composite? Do you just want the answer, or do you also want a proof? A complete proof would be a lot simpler if you grant Tim's Third Lemma: every 13-digit integer starting with 7 that isn't obviously composite is prime ;-) === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Can > p^(p^p) + 8 p^p + 26 be composite? Do you just want the answer, or do you also want a proof? A complete proof > would be a lot simpler if you grant Tim's Third Lemma: every 13-digit > integer starting with 7 that isn't obviously composite is prime ;-) You're not Fermat, so I'll insist on a proof of your lemma. 8-) --- === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? On 12 Nov 2006 00:38:37 -0800, Proginoskes Can anyone solve this one for me? Question: Suppose that p and p^6 + 6008 are both prime. Can >p^(p^p) + 8 p^p + 26 be composite? > No. It is guaranteed to be a prime! A new way to find large primes! Call the NSA, FBI, CIA! I'm off to buy a year's worth of canned food and bottled water and more shotgun ammo. -- Wim Benthem === Subject: Planes of constant sums Great obstacles must be overcome, when we try to imagine a five dimensional plane simplex. Its envelope composes from five four dimensional planes, tetrahedrons having one coordinate zero: (a, b, c, d, 0) (a, b, c, 0, e) (a, b, 0, d, e) (a, 0, c, d, e) (0, b, c, d, e) In 3-dimensional space, the tetrahedron sides interfere. We can simply take two tetrahedrons, and glue their triangle sides. We have the simplex as a trigonal bipyramide. We see in one moment two tetrahedrons, say abcd and abce, having the common side abc as the base of two trigonal pyramides, tetrahedrons, and in other moment three tetrahedrons having a common edge de, which goes through the bipyramide. But these are only sides of the simplex and its inside lies between these five tetrahedrons. We must move them aside before we come inside. Or the tetrahedron can be flattened by force to its base. We get it as a triangle with axes leading to the top vertex. Over this base, we place the fifth vertex and connect it with the previous four vertices. We get a new tetrahedron. Inside it, we can find the four remaining tetrahedons, superposed in two pairs. They cover the pyramid twice, once as two tetrahedrons abce and acde, once as two tetrahedrons abde and bcde. The third possibility, how to vizualize a five dimensional plane simplex, is to rotate the tetrahedron into the plane. This operation can not be made without deformations. We use them, and deform the tetrahedron into a square with both diagonals. Again, we place over this base the fifth vertex and connect it with the previous four vertices. The obtained pyramid contains the four remaining tetrahedons, superposed in pairs, again. Into 2~dimensional plane, the 5~dimensional plane simplex is projected as the pentagram. In all cases the plane simplex is distorted by squeezing it into the lower dimensional space. In the ideal state, all edges should have equal length. The 5~dimensional plane simplexes of the 6~dimensional plane simplex cover their 3~dimensional projection trice. The projection in the form of~the~tetragonal bipyramide can be divided into two pyramides having the common side abcd as the base, and then into four simplexes along the axis ef as before at the 5~dimensional simplex. Or one 5-dimensional plane simplex can be flattened into the regular pentagon, and over this base, five 5-dimensional plane simplexes have place which cover the base of the pentagonal pyramid 3 times, the corners of the pentagram 4 times, its center 5 times. This makes the analysis of the 7-dimensional plane simplex difficult, since the pentagonal bipyramide is its simplest model. For studying higher dimensional planes, we must therefore use another techniques. kunzmilan === Subject: Re: Math as Religion Who has said that mathematics is THE truth? The mainstream > folks disagreeing with Timothy Golden have made no claims > of truth. Timothy Golden is the one who seems to think that > it is THE truth that magnitude is more fundamental than the reals. > If he wishes to define magnitude rigourously and then define > the reals based on that, he is free to. As has been pointed out on numerous times, this is in fact an old idea, > going back to the Greeks. Landau, for a modern example, develops > positive reals from second order arithmetic of > positive integers, and goes on from there. This has some advantages, in > particular that the positive rationals, as the ratios of positive > integers, may be constructed without worrying about division by zero, > and then the positive reals (or magnitude) can be constructed next. One can use polysigned numbers, if one so chooses, for constructions. > But Tim seems unable to say why we should. Gene, like Dik, represents the establishment. Oh. This is Gene on a prior thread: > I've pointed out several times that you do not have such a > construction. I'll repeat it: you have NOT constructed the reals. > This is because your definition requires that the reals have > already been constructed. And Gene was right. The important distinction that allows this conflict is in how we > dissect the number system. No. The important distinction is that you use terms that you define > in existing terms (the reals). So you assume the existence of the > reals. Otherwise the definition is void. In the quote on top Gene *did* show how you could define magnitude > without any reference to the reals. But for some reason you do not > want to follow that road. Because the polysign construction imposes the identity law > Sum for s = 1 to n ( s x ) = 0 Again that basic notational flaw. Proper notation would be > sum for i = 1 to n ( s_i x) = 0 This is a primitive and productive > construction that poses and answers many questions: Are the field criteria accurate? Eh? What can be inaccurate in criteria? Must a linear system obey the magnitudinal law > | A B | = | A | | B | ? To me this makes no sense. In a linear system we have a linear operator > that transforms input to output. I think that with A and B you mean the > linear operators. But in that case you have to define the meaning of |.|. > In linear algebra, when we define the norm of a vector as the Euclidean > norm, and the norm of a matrix as > sup |A.x|/|x| > the above certainly does not hold. In that case we can only show: > | A B | <= | A | | B |. Does time correspond to P1? > Do improper transformations model electron spin? > Do n-poles exist? > Why spacetime? These are not mathematical problems. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ I am willing to be flexible with notation and I don't believe that there is a singular way. Above in this thread I have criticized your underscore notation as lacking generality and I would suggest on that issue if you want to take it up to take it up there: Your refutation of the polysign construction as relying on the reals is exactly what this thread is trying to address. I am looking squarely at this from one side and you from the other. So this is fine. This is the debate that I wish to have. And so this debate leads to the question of whether a fine mathematician such as yourself is capable of breaking such a rule in the interest of improving mathematics. Hindsight tells me that breaking this rule is not a conflict ad so this rule is no law. It is merely a tradition which has cost mathematics the deficit of the polysign numbers. Your ability to go here is the difference between scientific mathematics and religious mathematics. Currently you are practicing religious mathematics. Breaking rules is a lot of fun; especially false rules, for these shroud the truth and should be broken. It is a scientific mathematicians duty to tread here. -Tim === Subject: Re: Math as Religion > I am willing to be flexible with notation and I don't believe that > there is a singular way. Above in this thread I have criticized your > underscore notation as lacking generality and I would suggest on that > issue if you want to take it up to take it up there: Your refutation of the polysign construction as relying on the reals is > exactly what this thread is trying to address. I am looking squarely at > this from one side and you from the other. > So this is fine. This is the debate that I wish to have. And so this > debate leads to the question of whether a fine mathematician such as > yourself is capable of breaking such a rule in the interest of > improving mathematics. Hindsight tells me that breaking this rule is > not a conflict ad so this rule is no law. It is merely a tradition > which has cost mathematics the deficit of the polysign numbers. Your ability to go here is the difference between scientific > mathematics and religious mathematics. Currently you are practicing > religious mathematics. Breaking rules is a lot of fun; especially false > rules, for these shroud the truth and should be broken. It is a > scientific mathematicians duty to tread here. Which rule do you want to break? -- Marcus === Subject: Re: Math as Religion there is a singular way. Above in this thread I have criticized your > underscore notation as lacking generality and I would suggest on that > issue if you want to take it up to take it up there: Your refutation of the polysign construction as relying on the reals is > exactly what this thread is trying to address. I am looking squarely at > this from one side and you from the other. > So this is fine. This is the debate that I wish to have. And so this > debate leads to the question of whether a fine mathematician such as > yourself is capable of breaking such a rule in the interest of > improving mathematics. Hindsight tells me that breaking this rule is > not a conflict ad so this rule is no law. It is merely a tradition > which has cost mathematics the deficit of the polysign numbers. Your ability to go here is the difference between scientific > mathematics and religious mathematics. Currently you are practicing > religious mathematics. Breaking rules is a lot of fun; especially false > rules, for these shroud the truth and should be broken. It is a > scientific mathematicians duty to tread here. Which rule do you want to break? -- > Marcus Formal mathematics seems to deny magnitude as having fundamental status. With the polysign numbers the identity law: Sum for s = 1 to n ( s x ) = 0 takes care of all sign information. So this information need not be defined within either of the components s or x. This leaves x a raw magnitude. Traditionally magnitude seems to be defined as a real number and so people gripe. I don't personally care but if the polysign construction is to be palatable to others this small conflict needs to be adressed. Otherwise the strict mathematician will claim a flaw and the system is refuted. The language of mathematics leaves no room for conflict whatsoever. There is another benefit in allowing magnitude to be fundamental: the unity problem. Traditionally when mathematicians define the real line they do so via the natural numbers. This description is inherently carried about when we study Cartesian spaces such as 3D physical space. Where is unity? It is an arbitrary decision, not a fixed choice. In other words the correspondence of Cartesian coordinates to 3D space is imperfect. By granting magnitude as fundamental the unit value can take on a nondefinite position 'u' and yet the natural numbers will now be recoverable under magnitude via u, u+u, u+u+u, ... which is symbolically identical to a, a+a, a+a+a. Because the natural numbers are so easily recovered from magnitude and this definition does not suffer the unity problem I see this as another reason to grant magnitude as fundamental. These natural numbers are somewhat more like scalars but that is fine; when we talk about having three apples and two oranges the scalar product is apparent. The most convincing argument I call the gorilla conjecture: magnitude is a primitive feature of existence that we needn't even be tought to understand. A gorilla can be tought to sort sticks by length. All of mathematics as a human construct is a priori subject to this feature. Is this minor? In my opinion it is, but the consequences are actually quite large if magnitude is granted as a primitive. Existing mathematics will be vastly simplified, particularly the definition(s) of real numbers. -Tim === Subject: Re: Math as Religion > Which rule do you want to break? Formal mathematics seems to deny magnitude as having fundamental > status. That's not a rule. If anything, it's a comment on the standard way things are done. If you want to do things differently, go ahead. > With the polysign numbers the identity law: > Sum for s = 1 to n ( s x ) = 0 > takes care of all sign information. As I said before, you need to be precise about what you are doing so people will have a hope of understanding you. It seems you are defining some sort of algebraic structure. If so, you should study how this is normally done so you can get the hang of it. > So this information need not be > defined within either of the components s or x. This leaves x a raw > magnitude. Traditionally magnitude seems to be defined as a real number > and so people gripe. I don't personally care but if the polysign > construction is to be palatable to others this small conflict needs to > be adressed. Otherwise the strict mathematician will claim a flaw and > the system is refuted. The language of mathematics leaves no room for > conflict whatsoever. There is another benefit in allowing magnitude to be fundamental: the > unity problem. Traditionally when mathematicians define the real line > they do so via the natural numbers. This description is inherently > carried about when we study Cartesian spaces such as 3D physical space. > Where is unity? It is an arbitrary decision, not a fixed choice. Don't know what this means. > In > other words the correspondence of Cartesian coordinates to 3D space is > imperfect. By granting magnitude as fundamental the unit value can take > on a nondefinite position 'u' and yet the natural numbers will now be > recoverable under magnitude via u, u+u, u+u+u, ... which is > symbolically identical to a, a+a, a+a+a. Because the natural numbers > are so easily recovered from magnitude and this definition does not > suffer the unity problem I see this as another reason to grant > magnitude as fundamental. These natural numbers are somewhat more like > scalars but that is fine; when we talk about having three apples and > two oranges the scalar product is apparent. The most convincing argument I call the gorilla conjecture: magnitude > is a primitive feature of existence that we needn't even be tought to > understand. A gorilla can be tought to sort sticks by length. All of > mathematics as a human construct is a priori subject to this feature. Is this minor? In my opinion it is, but the consequences are actually > quite large if magnitude is granted as a primitive. Existing > mathematics will be vastly simplified, particularly the definition(s) > of real numbers. If you wish us to believe this, first tell us one of the standard definitions of the real numbers. Then tell us yours. -- Marcus === Subject: Re: Math as Religion status. That's not a rule. If anything, it's a comment on the standard way > things are done. If you want to do things differently, go ahead. With the polysign numbers the identity law: > Sum for s = 1 to n ( s x ) = 0 > takes care of all sign information. As I said before, you need to be precise about what you are doing so > people will have a hope of understanding you. It seems you are defining > some sort of algebraic structure. If so, you should study how this is > normally done so you can get the hang of it. So this information need not be > defined within either of the components s or x. This leaves x a raw > magnitude. Traditionally magnitude seems to be defined as a real number > and so people gripe. I don't personally care but if the polysign > construction is to be palatable to others this small conflict needs to > be adressed. Otherwise the strict mathematician will claim a flaw and > the system is refuted. The language of mathematics leaves no room for > conflict whatsoever. There is another benefit in allowing magnitude to be fundamental: the > unity problem. Traditionally when mathematicians define the real line > they do so via the natural numbers. This description is inherently > carried about when we study Cartesian spaces such as 3D physical space. > Where is unity? It is an arbitrary decision, not a fixed choice. Don't know what this means. In > other words the correspondence of Cartesian coordinates to 3D space is > imperfect. By granting magnitude as fundamental the unit value can take > on a nondefinite position 'u' and yet the natural numbers will now be > recoverable under magnitude via u, u+u, u+u+u, ... which is > symbolically identical to a, a+a, a+a+a. Because the natural numbers > are so easily recovered from magnitude and this definition does not > suffer the unity problem I see this as another reason to grant > magnitude as fundamental. These natural numbers are somewhat more like > scalars but that is fine; when we talk about having three apples and > two oranges the scalar product is apparent. The most convincing argument I call the gorilla conjecture: magnitude > is a primitive feature of existence that we needn't even be tought to > understand. A gorilla can be tought to sort sticks by length. All of > mathematics as a human construct is a priori subject to this feature. Is this minor? In my opinion it is, but the consequences are actually > quite large if magnitude is granted as a primitive. Existing > mathematics will be vastly simplified, particularly the definition(s) > of real numbers. If you wish us to believe this, first tell us one of the standard > definitions of the real numbers. Then tell us yours. -- > Marcus I've gone over the entire construction with you. Can't you see that all sign operations are handled above the magnitude layer? P2 are the real numbers. All that needs to be granted is magnitude. I am aware that the standard definition of the real numbers is covered many different ways and that one of these descriptions effectively forms an entire book by the time the construction is completed. For this reason I can't honor your request to build the real numbers. The contortions that are gone through to get the continuum are obnoxiously complex given that an ape already gets it. The physical correspondence does not seem to be of value for mathematicians. Is unity a specific place on the real line? Are Cartesian coordinates apropriate representations of the space we inhabit? Which is more fundamental, magnitude or the real number? Which has less structure? Why are the symbols seperate discrete things( a sign and a number)? When we build things we start with the simpler things and yield more complicated things, not the other way around. The standard approach is backwards. This backward method has hidden a portion of mathematics. There is a small hole in this method that I am proposing. It is the magnitudinal product. As you can see unity does still take a special behavior under product. The product and the integral are closely related and an answer may lie there. There is a distance systems that won't encounter the unity problem with good physical correspondence: y = 1 / ( x + 1 ) . This transform allows classical forces to obey F = q1 q2 without the inverse square relation. Here the magnitude of q is its transformed distance and its charge the discrete sign(inherently quantized). I have yet to settle this. So this product concern is still open, though its no worse than what already exists leaving unity fixed. I suppose my real problem is that I am concerned with nature more than mathematics. Mathematicians are divorced from nature. Otherwise they'd be physicists. Is it accurate to say that natural correspondence has no meaning to a mathematician? -Tim === Subject: Re: Math as Religion > I've gone over the entire construction with you. Really? I even looked at your website, but I don't see a construction. > Can't you see that all > sign operations are handled above the magnitude layer? P2 are the real > numbers. All that needs to be granted is magnitude. By magnitude do you mean the nonnegative real numbers? > I am aware that the standard definition of the real numbers is covered > many different ways and that one of these descriptions effectively > forms an entire book by the time the construction is completed. No idea what you mean. The standard constructions aren't that long. It can take some work to prove that they satisfy the properties of a complete ordered field, but the constructions themselves only take a few sentences. Instead of constructing the real numbers, another approach to defining them is to specify their properties. There are just thirteen axioms for a complete ordered field. These axioms characterize the real numbers. The reason to do a construction is to prove that a complete ordered field exists. > Is unity a specific place on the real line? Don't know what that question means. > Are Cartesian coordinates apropriate representations of the space we > inhabit? That's a physics question. > Which is more fundamental, magnitude or the real number? Don't know what magnitude means. > Which has less structure? > Why are the symbols seperate discrete things( a sign and a number)? We use notations that are convenient. Sometimes the notation is telling us something important, sometimes it isn't. > When we build things we start with the simpler things and yield more > complicated things, not the other way around. The standard approach is > backwards. This backward method has hidden a portion of mathematics. What standard approach are you referring to? > I suppose my real problem is that I am concerned with nature more than > mathematics. Mathematicians are divorced from nature. Otherwise they'd > be physicists. > Is it accurate to say that natural correspondence has no meaning to a > mathematician? Of course not. However, mathematicians are not only interested in mathematics that can be applied. And, often mathematics that didn't appear to have any application when it was developed, does turn out to have application. I would say your problem is you keep telling us what is wrong with mathematics instead of telling us what physical problem you would like to attack that you don't know how to attack with the mathematics that you know. -- Marcus === Subject: Re: Math as Religion sign operations are handled above the magnitude layer? P2 are the real > numbers. All that needs to be granted is magnitude. By magnitude do you mean the nonnegative real numbers? I am aware that the standard definition of the real numbers is covered > many different ways and that one of these descriptions effectively > forms an entire book by the time the construction is completed. No idea what you mean. The standard constructions aren't that long. It > can take some work to prove that they satisfy the properties of a > complete ordered field, but the constructions themselves only take a few > sentences. Instead of constructing the real numbers, another approach to > defining them is to specify their properties. There are just thirteen > axioms for a complete ordered field. These axioms characterize the real > numbers. The reason to do a construction is to prove that a complete > ordered field exists. Is unity a specific place on the real line? Don't know what that question means. Are Cartesian coordinates apropriate representations of the space we > inhabit? That's a physics question. Which is more fundamental, magnitude or the real number? Don't know what magnitude means. Which has less structure? > Why are the symbols seperate discrete things( a sign and a number)? We use notations that are convenient. Sometimes the notation is telling > us something important, sometimes it isn't. When we build things we start with the simpler things and yield more > complicated things, not the other way around. The standard approach is > backwards. This backward method has hidden a portion of mathematics. What standard approach are you referring to? I suppose my real problem is that I am concerned with nature more than > mathematics. Mathematicians are divorced from nature. Otherwise they'd > be physicists. > Is it accurate to say that natural correspondence has no meaning to a > mathematician? Of course not. However, mathematicians are not only interested in > mathematics that can be applied. And, often mathematics that didn't > appear to have any application when it was developed, does turn out to > have application. I would say your problem is you keep telling us what is wrong with > mathematics instead of telling us what physical problem you would like > to attack that you don't know how to attack with the mathematics that > you know. -- > Marcus Alright. I guess we'd better just leave it at that. -Tim === Subject: Re: Math as Religion there is a singular way. Above in this thread I have criticized your > underscore notation as lacking generality and I would suggest on that > issue if you want to take it up to take it up there: Your refutation of the polysign construction as relying on the reals is > exactly what this thread is trying to address. I am looking squarely at > this from one side and you from the other. > So this is fine. This is the debate that I wish to have. And so this > debate leads to the question of whether a fine mathematician such as > yourself is capable of breaking such a rule in the interest of > improving mathematics. Hindsight tells me that breaking this rule is > not a conflict ad so this rule is no law. It is merely a tradition > which has cost mathematics the deficit of the polysign numbers. Your ability to go here is the difference between scientific > mathematics and religious mathematics. Currently you are practicing > religious mathematics. Breaking rules is a lot of fun; especially false > rules, for these shroud the truth and should be broken. It is a > scientific mathematicians duty to tread here. Which rule do you want to break? -- > Marcus Formal mathematics seems to deny magnitude as having fundamental status. With the polysign numbers the identity law: Sum for s = 1 to n ( s x ) = 0 takes care of all sign information. So this information need not be defined within either of the components s or x. This leaves x a raw magnitude. Traditionally magnitude seems to be defined as a real number and so people gripe. I don't personally care but if the polysign construction is to be palatable to others this small conflict needs to be adressed. Otherwise the strict mathematician will claim a flaw and the system is refuted. The language of mathematics leaves no room for conflict whatsoever. There is another benefit in allowing magnitude to be fundamental: the unity problem. Traditionally when mathematicians define the real line they do so via the natural numbers. This description is inherently carried about when we study Cartesian spaces such as 3D physical space. Where is unity? It is an arbitrary decision, not a fixed choice. In other words the correspondence of Cartesian coordinates to 3D space is imperfect. By granting magnitude as fundamental the unit value can take on a nondefinite position 'u' and yet the natural numbers will now be recoverable under magnitude via u, u+u, u+u+u, ... which is symbolically identical to a, a+a, a+a+a. Because the natural numbers are so easily recovered from magnitude and this definition does not suffer the unity problem I see this as another reason to grant magnitude as fundamental. These natural numbers are somewhat more like scalars but that is fine; when we talk about having three apples and two oranges the scalar product is apparent. The most convincing argument I call the gorilla conjecture: magnitude is a primitive feature of existence that we needn't even be tought to understand. A gorilla can be tought to sort sticks by length. All of mathematics as a human construct is a priori subject to this feature. Is this minor? In my opinion it is, but the consequences are actually quite large if magnitude is granted as a primitive. Existing mathematics will be vastly simplified, particularly the definition(s) of real numbers. -Tim === Subject: Re: Math as Religion The fundamental law which I have applied is > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is > magnitude) I also have to accuse you of insincerity. Where? You stated a formula above. You state for s = 1 to n. The only > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else. But the cheap revisions that he poses here are weak and I have seen > this behavior before. Why cheap? I indicate to you basic mathematical problems with your > model. It is cheap because you ignored the specification that s is sign. I see in some other post you suggest putting an underscore. The point is that you choose to ignore parts of the communication. I can go back to past threads and find the same poor style of debate. It only weakens your position. For you to criticize my communication by this method is paradoxical. I concede that I can do a better job of communicating the properties of polysign numbers. In particular there are places where the notation is seriously conflicted due to the usage in traditional mathematics of the '+' sign as summation whereas under the polysign approach this does not hold true. So to do algebra with a '+' sign as summation has to be explicity stated and cannot mix with concrete instances. For instance in my original communication there > is an explicit declaration that s is representing sign. In that case you should not write for s = 1 to n. The elemental > s x > representation does need to be explicitly introduced to allow > adaptation to the new system, But in mathematics it is essential that you *define* your terms. but let's not forget that as children we > were taught the real numbers without higher level mathematics. The same > can be done for polysign numbers. Right. When teaching children you skip a lot. But *not* higher level > mathematics, but lower level mathematics. When children grow in their > mathematics education they learn more and more about those lower basics. I prefer this approach since it does > not require much knowledge and so is more accessible to a less educated > person. You may prefer that, but your preferences do not make it a mathematical > model. But you present it as such. None the less, here is a perfectly capable mathematician who > is incapable of accepting a new construction. I do not accept a new construction if the basic definitions are missing. > You state that I must forget about the reals and only consider magnitude. > But when I consider magnitude I can not ignore the reals, because > magnitude is defined in relation to the reals. So unless you provide > a proper definition of magnitude, a am at a loss to proceed in a > mathematical way. This > construction diminishes the real numbers and favors magnitude as > fundamental. Perhaps. But that is not the case when you do not define magnitude as > something independent from the reals. As long as magnitude is defined > in terms of the reals, there is a problem. However, you ignore completely the way I *did* solve some of your > division problems. That was by looking at algebras over the reals, > and regarding your polysign numbers as equivalence classes within > those algebras. So when solving the division problems the reals > were much more fundamental. They allowed to calculate solutions. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ You are a wonderful antagonist Dik. Rather than chase about in a circle it would be preferable to simply acknowledge the conflict with existing mathematics. If there is something wrong with the polysign construction then it should be pointed out. What consequence makes it fail? We do have the division problem in P4+, but without it there would be no claim for spacetime support. I believe you are disputing my form of communication rather than the polysign numbers. You have not granted that some new room must be made if we are to generalize sign. To generalize sign is a new concept. New concepts need room for representation. Most of the old ways can be applied but the meaning is not the same. We can use ordered coordinates: ( 3.4, 1.2, 5.6, 9.0 ) to represent a value in P4. Alternatively we can use symbolic signs: # 3.4 - 1.2 + 5.6 * 9.0 . We could even use unit vectors: 3.4 + 1.2 i + 5.6 j + 9.0 k . All of these work. Their meaning is the same. There is no one right way. Beneath all of these ways is a new concept that is sign in its general form. The generalized sign is inherently a natural number. Each one of these instances above does this implicitly. In order to generate these forms compactly we have: Sum from s = 1 to n ( s x_s ) where s is sign and x_s is the sth magnitudinal component. Do you believe that I have generalized sign? -Tim === Subject: Re: Math as Religion > The fundamental law which I have applied is > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is > magnitude) I also have to accuse you of insincerity. Where? You stated a formula above. You state for s = 1 to n. The only > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else. But the cheap revisions that he poses here are weak and I have seen > this behavior before. Why cheap? I indicate to you basic mathematical problems with your > model. It is cheap because you ignored the specification that s is sign. > I see in some other post you suggest putting an underscore. > The point is that you choose to ignore parts of the communication. > I can go back to past threads and find the same poor style of debate. > It only weakens your position. > For you to criticize my communication by this method is paradoxical. If you write sum s=1 to n, then it means s is an integer. If that's not what you mean, then don't write that. You can't say, s is a sign and then say s=1 to n. The reason is that 1 is not a sign. You can have a first sign, but then you need to give it a name, e.g., s_1 (where the underscore means subscript). If you say that s_i is the i-th sign, then you can write sum_{i=1}^n s_i x. > I concede that I can do a better job of communicating the properties of > polysign numbers. > In particular there are places where the notation is seriously > conflicted due to the usage in traditional mathematics of the '+' sign > as summation whereas under the polysign approach this does not hold > true. So to do algebra with a '+' sign as summation has to be explicity > stated and cannot mix with concrete instances. If you are defining a new operation, then state that + means your new operation, not the usual addition. If you need to use both operations, then make up a new symbol for your new operation. -- Marcus === Subject: Re: Math as Religion The fundamental law which I have applied is > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is > magnitude) I also have to accuse you of insincerity. Where? You stated a formula above. You state for s = 1 to n. The only > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else. But the cheap revisions that he poses here are weak and I have seen > this behavior before. Why cheap? I indicate to you basic mathematical problems with your > model. It is cheap because you ignored the specification that s is sign. > I see in some other post you suggest putting an underscore. > The point is that you choose to ignore parts of the communication. > I can go back to past threads and find the same poor style of debate. > It only weakens your position. > For you to criticize my communication by this method is paradoxical. If you write sum s=1 to n, then it means s is an integer. If that's > not what you mean, then don't write that. You can't say, s is a sign > and then say s=1 to n. The reason is that 1 is not a sign. You can > have a first sign, but then you need to give it a name, e.g., s_1 (where > the underscore means subscript). If you say that s_i is the i-th sign, > then you can write sum_{i=1}^n s_i x. I concede that I can do a better job of communicating the properties of > polysign numbers. > In particular there are places where the notation is seriously > conflicted due to the usage in traditional mathematics of the '+' sign > as summation whereas under the polysign approach this does not hold > true. So to do algebra with a '+' sign as summation has to be explicity > stated and cannot mix with concrete instances. If you are defining a new operation, then state that + means your new > operation, not the usual addition. If you need to use both operations, > then make up a new symbol for your new operation. -- > Marcus The simplest and most intuitive sign symbology uses - + * # ... Here the first symbol in one line, the second symbol two lines, etc. These symbols represent natural numbers that will be married to a magnitude to achieve -1.2 , +35.6 , *5 , etc. if the sign in these elemental values is regarded as a natural number then the generic form is s x where s is sign and x is magnitude. These are two different data types; they combine but they do not evaluate to a singular type. They retain structure. That s is a natural number does not deny it this possibility. In particular operations on the sx form go like: ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 s1 x1 + s1 x2 = s1( x1 + x2 ) These form the basic operations of polysign numbers, the first being product and the second being superpostion or summation where the '+' operator means superposition. The product causes a sign sum s1+s2 which is actually a mod n type of sum. Really these rules are trying to explain something that is more practically learned from my website using the second grader approach. But since mathematicians like things like this sx terminology there it is. Really there is just one more statement and the polysign construction is completed: Sum for s = 1 to n ( s x ) = 0 where s is sign and x is magnitude. Because noone has gneralized sign before this notation may be uncomfortable. Does the s_i symbol mean anything different? I suppose it is like a unit vector approach where we might write 1.2 + 3.4 i + 4.5 j . The most compact form and the least cumbersome is the - 1 + 3 # 2.5 type of notation which for the sx notation turns into s1 x1 + s2 x2 + s3 x3 where s1=1,x1=1,s2=2,x2=3,s3=4,x3=2.5 and likewise in your notation s_1 x1 + s_2 x2 + s_4 x3 where the sign is carried around inherently so we only need specify the x components. However these signs are not generic and so when we go over to the generic form: s_i x1 + s_j x2 + s_k x3 we have now lost the correspondence of the numerical indices. Based on this analysis I would like to remain with the sx form. If you see something wrong with this analysis I hope you will share it. Your notation works, but it may cause some difficulty. Some room must be granted for the generalization of sign. It does not fit into existing mathematics. -Tim === Subject: Re: Math as Religion > The fundamental law which I have applied is > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is > magnitude) I also have to accuse you of insincerity. Where? You stated a formula above. You state for s = 1 to n. The only > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else. But the cheap revisions that he poses here are weak and I have seen > this behavior before. Why cheap? I indicate to you basic mathematical problems with your > model. It is cheap because you ignored the specification that s is sign. > I see in some other post you suggest putting an underscore. > The point is that you choose to ignore parts of the communication. > I can go back to past threads and find the same poor style of debate. > It only weakens your position. > For you to criticize my communication by this method is paradoxical. If you write sum s=1 to n, then it means s is an integer. If that's > not what you mean, then don't write that. You can't say, s is a sign > and then say s=1 to n. The reason is that 1 is not a sign. You can > have a first sign, but then you need to give it a name, e.g., s_1 (where > the underscore means subscript). If you say that s_i is the i-th sign, > then you can write sum_{i=1}^n s_i x. I concede that I can do a better job of communicating the properties of > polysign numbers. > In particular there are places where the notation is seriously > conflicted due to the usage in traditional mathematics of the '+' sign > as summation whereas under the polysign approach this does not hold > true. So to do algebra with a '+' sign as summation has to be explicity > stated and cannot mix with concrete instances. If you are defining a new operation, then state that + means your new > operation, not the usual addition. If you need to use both operations, > then make up a new symbol for your new operation. -- > Marcus The simplest and most intuitive sign symbology uses > - + * # ... You are free to use whatever symbols you wish, as long as you state clearly what you are doing. > Here the first symbol in one line, the second symbol two lines, etc. > These symbols represent natural numbers that will be married to a > magnitude to achieve > -1.2 , +35.6 , *5 , etc. > if the sign in these elemental values is regarded as a natural number > then the generic form is > s x > where s is sign and x is magnitude. I don't know what these symbols represent natural numbers or is regarded as a natural number mean. It looks like you've got a sequence of signs. The normal notation for a sequence is something like s_i, where s_1 is the first element in the sequence, s_2 is the second, etc. So, you could define s_1 to be -, s_2 to be +, and, in general, s_i to be your i-th sign. This would let you refer to the i-th sign as s_i. It is just another name for the same thing. > These are two different data types; they combine but they do not > evaluate to a singular type. So, you have orderd pairs. s x means the ordered pair (s,x) where s is a sign and x is a natural number. > They retain structure. > That s is a natural number does not deny it this possibility. > In particular operations on the sx form go like: > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 > s1 x1 + s1 x2 = s1( x1 + x2 ) > These form the basic operations of polysign numbers, the first being > product and the second being superpostion or summation where the '+' > operator means superposition. Hold on. You are using + as one of your signs. You shouldn't now use it for something else. If you do, then you'll get formulas like ( + 5 )( + 6 ) = ( + + + ) 5 6 What is that supposed to mean? It is very confusing. > The product causes a sign sum s1+s2 which is actually a mod n type of > sum. > Really these rules are trying to explain something that is more > practically learned from my website using the second grader approach. > But since mathematicians like things like this sx terminology there it > is. Really there is just one more statement and the polysign > construction is completed: > Sum for s = 1 to n ( s x ) = 0 > where s is sign and x is magnitude. Because noone has gneralized sign before this notation may be > uncomfortable. > Does the s_i symbol mean anything different? If we define what s_i is, then it is just another name for the thing. If I say x_22 = 55, then x_22 and 55 are both names for the number 55 (note the use of double quotes to refer to the string of characters). -- Marcus === Subject: Re: Math as Religion Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is > magnitude) I also have to accuse you of insincerity. Where? You stated a formula above. You state for s = 1 to n. The only > way I see that can be read is for s = 1, 2, 3, ..., n, not anything else. But the cheap revisions that he poses here are weak and I have seen > this behavior before. Why cheap? I indicate to you basic mathematical problems with your > model. It is cheap because you ignored the specification that s is sign. > I see in some other post you suggest putting an underscore. > The point is that you choose to ignore parts of the communication. > I can go back to past threads and find the same poor style of debate. > It only weakens your position. > For you to criticize my communication by this method is paradoxical. If you write sum s=1 to n, then it means s is an integer. If that's > not what you mean, then don't write that. You can't say, s is a sign > and then say s=1 to n. The reason is that 1 is not a sign. You can > have a first sign, but then you need to give it a name, e.g., s_1 (where > the underscore means subscript). If you say that s_i is the i-th sign, > then you can write sum_{i=1}^n s_i x. I concede that I can do a better job of communicating the properties of > polysign numbers. > In particular there are places where the notation is seriously > conflicted due to the usage in traditional mathematics of the '+' sign > as summation whereas under the polysign approach this does not hold > true. So to do algebra with a '+' sign as summation has to be explicity > stated and cannot mix with concrete instances. If you are defining a new operation, then state that + means your new > operation, not the usual addition. If you need to use both operations, > then make up a new symbol for your new operation. -- > Marcus The simplest and most intuitive sign symbology uses > - + * # ... You are free to use whatever symbols you wish, as long as you state > clearly what you are doing. Here the first symbol in one line, the second symbol two lines, etc. > These symbols represent natural numbers that will be married to a > magnitude to achieve > -1.2 , +35.6 , *5 , etc. > if the sign in these elemental values is regarded as a natural number > then the generic form is > s x > where s is sign and x is magnitude. I don't know what these symbols represent natural numbers or is > regarded as a natural number mean. It looks like you've got a sequence > of signs. The normal notation for a sequence is something like s_i, > where s_1 is the first element in the sequence, s_2 is the second, etc. > So, you could define s_1 to be -, s_2 to be +, and, in general, s_i to > be your i-th sign. This would let you refer to the i-th sign as s_i. It > is just another name for the same thing. These are two different data types; they combine but they do not > evaluate to a singular type. So, you have orderd pairs. s x means the ordered pair (s,x) where s is a > sign and x is a natural number. They retain structure. > That s is a natural number does not deny it this possibility. > In particular operations on the sx form go like: > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 > s1 x1 + s1 x2 = s1( x1 + x2 ) > These form the basic operations of polysign numbers, the first being > product and the second being superpostion or summation where the '+' > operator means superposition. Hold on. You are using + as one of your signs. You shouldn't now use > it for something else. If you do, then you'll get formulas like ( + 5 )( + 6 ) = ( + + + ) 5 6 What is that supposed to mean? It is very confusing. The product causes a sign sum s1+s2 which is actually a mod n type of > sum. > Really these rules are trying to explain something that is more > practically learned from my website using the second grader approach. > But since mathematicians like things like this sx terminology there it > is. Really there is just one more statement and the polysign > construction is completed: > Sum for s = 1 to n ( s x ) = 0 > where s is sign and x is magnitude. Because noone has gneralized sign before this notation may be > uncomfortable. > Does the s_i symbol mean anything different? If we define what s_i is, then it is just another name for the thing. If > I say x_22 = 55, then x_22 and 55 are both names for the number 55 > (note the use of double quotes to refer to the string of characters). -- > Marcus construction that is new. I have written out the description many times and will do so again here for you. The proper tutorial is my website: http://www.bandtechnology.com/PolySigned but let's just start from scratch. It really won't take that long to describe the entire system. Just suppose that there is a new domain of numbers called three-signed numbers. Now we need a third sign. The existing signs are '-' and '+' and so we choose '*' for the third sign since when we write it down on a piece of paper it has three lines. So these symboles are numerical mnemonics. Now, we hunt for a symmetrical property in the reals that can extend to this three-signed system and find that - x + x = 0 can be expanded to - x + x * x = 0 (the identity law). Under the reals when we do superposition if the values have the same sign we simply add the values and preserve the sign. The same will happen here so that: - 1.2 - 2.3 = - 3.5 . * 5.6 * 1.1 = * 6.7 . The way that the identity law gets applied is when we have a value like - 2.3 + 4.5 * 1.1 This value can be broken down to - 1.2 + 3.4 - 1.1 + 1.1 * 1.1 where the last three values are equivalent to zero. So really -2.3 + 4.5 * 1.1 = - 1.2 + 3.4 . Now you will be capable of performing any concrete summation in three-signed math. The product has a rotational character that matches the real numbers. Let's just look at the sign rules of the real product as if a plus sign jumps twice and a minus sign jumps once. In effect this is just a count that keeps wrapping at two where the amount to count is represented by the sign mnemonic. This sign product rule extends to three-signed numbers so that: - - = + - + = * - * = - + * = + + + = - * * = * . If you want to insert ones to make these concrete products that is fine. Really you won't need the table. its just addition. So for example the third line of this table can be used to do: ( - 3 )( * 4 ) = - 12 . This is no different than what we might write in the reals: ( - 2 )( - 3 ) = + 6 . The distributive, commutative, and associative properties work so that ( - 2 + 3 )( - 1 * 2 ) = + 2 - 4 * 3 + 6 = - 4 + 8 * 3 = - 1 + 5 . The last line is the reduced form but it really doesn't matter if you reduce. Perhaps you already see that these three signed numbers are two-dimensional. Upon graphing a value the reduction takes place automatically. Please see my website for a drawing. It turns out that they are the complex numbers in a natural form that extends directly from the real numbers by generalizing sign. The proof is on my website. The rules you have hopefully just learned are extensible and allow algebraic geometry in any dimension. In order to discuss the entire family in general we need a generic representation. It has taken me a long time to arrive at the sx notation. I am still integrating it into my website. The use of modulo sum is clunky until the zero sign is introduced and I don't bother with that yet since it adds another element of confusion. Your concern over the '+' symbol is exposed here in three-sign (P3) math. In the reals (P2) we can write: + 5 - ( - 1 ) = + 6 . and we are used to + preserving sign since it jumps two places: - 4 + ( - 3 ) = - 7 . above I could have done something more like: ( - a + b )( - c * d ) = (-a)(-c) * (-a)(*d) * (+b)(-c) * (+b)(*d) where all letters are magnitudes. So usually I am careful to say for example In P9 ( z1 + z2 ) z3 = z1 z3 + z2 z3 = z3 z1 + z2 z3 where '+' means superposition. This is also true for any sign level Pn. Otherwise someone might substitute in values and forget to change the plus's to s_9's (your underscore notation). This is a slightly different description than I usually write; I try to shake it up to see how it works so I would appreciate your feedback on what is confusing and of course feel free to ask questions. Dik and I have already been through this level months ago. His claim of misunderstanding the sign representation is not a real position. He completely understands the construction. He's just giving me a hard time about notation. His suggestion I guess is what you are pushing but I'll stand by my earlier critique. The underscore notation will not be as clean. Anyhow notational variation is allowed. If you want to use the underscore notation I can follow it while communicating with you. It does not alter the underlying construction. -Tim === Subject: Re: Math as Religion construction that is new. I have written out the description many > times and will do so again here for you. The proper tutorial is my > website: > http://www.bandtechnology.com/PolySigned > but let's just start from scratch. It really won't take that long to > describe the entire system. > Just suppose that there is a new domain of numbers called three-signed > numbers. > Now we need a third sign. > The existing signs are '-' and '+' and so we choose '*' for the third > sign since when we write it down on a piece of paper it has three > lines. So these symboles are numerical mnemonics. > Now, we hunt for a symmetrical property in the reals that can extend to > this three-signed system and find that > - x + x = 0 > can be expanded to > - x + x * x = 0 (the identity law). Are you defining a new type of number or are you defining a new binary operation on an existing type of number, both, or neither? -- Marcus === Subject: Re: Math as Religion operation on an existing type of number, both, or neither? He's defining real commutative algebras, I think. === Subject: Re: Math as Religion Are you defining a new type of number or are you defining a new binary > operation on an existing type of number, both, or neither? He's defining real commutative algebras, I think. -- Marcus === Subject: Re: Math as Religion construction that is new. I have written out the description many > times and will do so again here for you. The proper tutorial is my > website: > http://www.bandtechnology.com/PolySigned > but let's just start from scratch. It really won't take that long to > describe the entire system. > Just suppose that there is a new domain of numbers called three-signed > numbers. > Now we need a third sign. > The existing signs are '-' and '+' and so we choose '*' for the third > sign since when we write it down on a piece of paper it has three > lines. So these symboles are numerical mnemonics. > Now, we hunt for a symmetrical property in the reals that can extend to > this three-signed system and find that > - x + x = 0 > can be expanded to > - x + x * x = 0 (the identity law). Are you defining a new type of number or are you defining a new binary > operation on an existing type of number, both, or neither? -- > Marcus The construction defines a family of number systems: P1, P2, P3, P4, ... where P1 is one-signed numbers, P2 are two-signed numbers, etc. P2 are consistent with the real numbers by design. P3 are equivalent to the complex numbers. Pn are n-1 dimensional spaces that obey the geometry of a simplex as their coordinate system structure. They have a product and sum defined and they obey the algebraic principles just as the real and complex numbers do with the exception that P4 and above fail the field division criteria under special circumstances. Still the higher sign systems are algebraically well behaved under product and sum. P1 has exact time correspondence and due to the break at P4 the system supports spacetime. -Tim === Subject: Re: Math as Religion > construction that is new. I have written out the description many > times and will do so again here for you. The proper tutorial is my > website: > http://www.bandtechnology.com/PolySigned > but let's just start from scratch. It really won't take that long to > describe the entire system. > Just suppose that there is a new domain of numbers called three-signed > numbers. > Now we need a third sign. > The existing signs are '-' and '+' and so we choose '*' for the third > sign since when we write it down on a piece of paper it has three > lines. So these symboles are numerical mnemonics. > Now, we hunt for a symmetrical property in the reals that can extend to > this three-signed system and find that > - x + x = 0 > can be expanded to > - x + x * x = 0 (the identity law). Are you defining a new type of number or are you defining a new binary > operation on an existing type of number, both, or neither? The construction defines a family of number systems: > P1, P2, P3, P4, ... > where P1 is one-signed numbers, P2 are two-signed numbers, etc. > P2 are consistent with the real numbers by design. > P3 are equivalent to the complex numbers. > Pn are n-1 dimensional spaces that obey the geometry of a simplex as > their coordinate system structure. They have a product and sum defined > and they obey the algebraic principles just as the real and complex > numbers do with the exception that P4 and above fail the field division > criteria under special circumstances. Still the higher sign systems are > algebraically well behaved under product and sum. P1 has exact time > correspondence and due to the break at P4 the system supports > spacetime. I was asking what the lines above were doing, not what is done later. Would you first state just the definitions without including the discussion/motivation? -- Marcus === Subject: Re: Math as Religion Sorry. The identity law... The dimensionality of the system can be broached by this law. The operation of rendering or graphing a value invokes this law implicitly. Otherwise its usage is completely optional to the arithmetic system. Because it can be applied to a value I do believe that it deserves operator status. Perhaps it deserves a bit more than that. Your question is challenging. -Tim === Subject: Re: Math as Religion But you asked for no text. Here is the ultra-compact form: Sum for s = 1 to n ( s x ) = 0 . s1 x1 + s1 x2 = s1( x1 + x2 ) . ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. -Tim === Subject: Re: Math as Religion > But you asked for no text. No, I asked for no discussion/remarks. Words are fine (and a good idea). > Here is the ultra-compact form: Sum for s = 1 to n ( s x ) = 0 . > s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you are defining. You can either construct the object or tell us the properties that characterize it. For example, we can give a construction of the real numbers via Cauchy sequences or Dedekind cuts or we can write down the properties that characterize a complete ordered field and prove that there is only one object that satisfies all the properties. -- Marcus === Subject: Re: Math as Religion s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you > are defining. You can either construct the object or tell us the > properties that characterize it. I think it might help to take s to be a primitive nth root of unity, in the sense that s^n=1, but s^i is not 1 for 0 s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you > are defining. You can either construct the object or tell us the > properties that characterize it. I think it might help to take s to be a primitive nth root of unity, in > the sense that s^n=1, but > s^i is not 1 for 0 assuming s is a complex number. Then x1, x2, etc are nonnegative real > numbers. So you have things like x1 s + x2 s^2 + ... + xn s^n. Now the > question is modulo what--when are two such expressions equated? So in Pn you want to know when z1=z2 ? Your construction looks fine. There is the hair-splitting issue of constructing it out of the reals but this will work. The sign vectors are nonorthogonal according to the identity law. In running the numbers through the identity law they will be reduced and so if all of their reduced components are equal they will be equal. You say 'modulo what' but I don't think you need to worry about any modulo math to test for equality. The modulo math is just for sign products. With your construction s^n = s^0 which is good. If you were to multiply two values z1 and z2 you'd distribute the terms out and the signs would combine by addition modulo n, and since you've implemented the zero sign you'll be fine. Is this a branch of existing mathematics? -Tim === Subject: Re: Math as Religion s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you > are defining. You can either construct the object or tell us the > properties that characterize it. For example, we can give a construction > of the real numbers via Cauchy sequences or Dedekind cuts or we can > write down the properties that characterize a complete ordered field and > prove that there is only one object that satisfies all the properties. -- > Marcus I don't really know how to answer you. If you want to know what area of mathematics it is this then I would have to say that it fits beneath the real numbers if we were to view mathematics as a branching tree. It derives dimensionality without useage of a Cartesian product. There is implicit geometry. I think you could tell me the answer better than I can tell you. I am not versed in mathematics at the level that you are trying to discuss this at. It's polysign numbers. Sorry that this is such poor communication. Do you understand the polysign numbers? -Tim === Subject: Re: Math as Religion > But you asked for no text. No, I asked for no discussion/remarks. Words are fine (and a good idea). Here is the ultra-compact form: Sum for s = 1 to n ( s x ) = 0 . > s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you > are defining. You can either construct the object or tell us the > properties that characterize it. For example, we can give a construction > of the real numbers via Cauchy sequences or Dedekind cuts or we can > write down the properties that characterize a complete ordered field and > prove that there is only one object that satisfies all the properties. -- > Marcus I don't really know how to answer you. If you want to know what area of mathematics it is this then I would > have to say that it fits beneath the real numbers if we were to view > mathematics as a branching tree. No, that wasn't what I asked. > It derives dimensionality without useage of a Cartesian product. > There is implicit geometry. I think you could tell me the answer better than I can tell you. Perhaps. > I am not versed in mathematics at the level that you are trying to > discuss this at. How much mathematics have you had? > It's polysign numbers. Sorry that this is such poor communication. > Do you understand the polysign numbers? Nope. -- Marcus === Subject: Re: Math as Religion s1 x1 + s1 x2 = s1( x1 + x2 ) . > ( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2 . The sum (s1 + s2) is a modulo n sum as described already. I asked for a definition of the object (commutative algebra?) that you > are defining. You can either construct the object or tell us the > properties that characterize it. For example, we can give a construction > of the real numbers via Cauchy sequences or Dedekind cuts or we can > write down the properties that characterize a complete ordered field and > prove that there is only one object that satisfies all the properties. -- > Marcus I don't really know how to answer you. If you want to know what area of mathematics it is this then I would > have to say that it fits beneath the real numbers if we were to view > mathematics as a branching tree. No, that wasn't what I asked. It derives dimensionality without useage of a Cartesian product. > There is implicit geometry. I think you could tell me the answer better than I can tell you. Perhaps. I am not versed in mathematics at the level that you are trying to > discuss this at. How much mathematics have you had? I've taken a topology course though I didn't appreciate it. Real analysis, complex analysis, and engineering math beneath that. You should probably just spend some time on my website to get familiar with the polysign construction. I've found some neat stuff. The lattice is pretty interesting and the P4 product also. There is quite a lot of work still to do. I mostly look for physics models. Since they are capable of generating spacetime they could make a very tight and natural physical model. Do you see a conflict with taking magnitude as fundamental? This has been a sticky point for mathematicians since they build the reals then get magnitude. This system takes magnitude and builds the reals. -Tim It's polysign numbers. Sorry that this is such poor communication. > Do you understand the polysign numbers? Nope. -- > Marcus === Subject: Re: Math as Religion I've taken a topology course though I didn't appreciate it. > Real analysis, complex analysis, and engineering math beneath that. > You should probably just spend some time on my website to get familiar > with the polysign construction. I've found some neat stuff. The lattice > is pretty interesting and the P4 product also. There is quite a lot of > work still to do. I mostly look for physics models. Since they are > capable of generating spacetime they could make a very tight and > natural physical model. Do you see a conflict with taking magnitude as fundamental? This has > been a sticky point for mathematicians since they build the reals then > get magnitude. This system takes magnitude and builds the reals. If you can really do it, then there is no conflict. I don't think you understand enough mathematics to even know if you are really doing it. I think what other people have been trying to tell you is that your construction doesn't do quite what you think it does. Although, if you are mainly interested in applications, I'm not sure why you would care whether you use the reals to build what you want or build it from other things. -- Marcus === Subject: Re: Math as Religion > But the only relevant issue with respect to Math as Religion is > whether it is true in universal terms. Isn't the only relevant issue with religion whether it is true? -- Marcus === Subject: Re: Math as Religion On Fri, 10 Nov 2006 23:09:31 -0500, Marcus >> But the only relevant issue with respect to Math as Religion is >> whether it is true in universal terms. Isn't the only relevant issue with religion whether it is true? Of course. I took it for granted that was understood. Mathematics is in a different category because most assume it's true but can't prove it. In fact apart from certain categorical doctrines I as well assume it's true. The problem is demonstrating that truth. Otherwise you're reduced to name calling and ad hominem nonsense such as we've seen recently. The search for truth in terms of syllogistic inference is Aristotle's intellectual legacy both to religion and science. Religion just uses different axioms which are certainly less likely true than mathematical axioms and science in general. But religious arguments based on their axiomatic assumptions can be quite rigorous although obviously they're often meretricious. The difficulty is that we can't judge these issues critically without a rigorous definition for truth in universal terms. (Actually looking back at your question I notice the qualification only so I suppose I should qualify my comments because religion can have other relevant issues than truth alone in terms of empirical social, emotional, and personal utility which I don't see applying to mathematics in general. Either mathematics is demonstrably true or it becomes a more reasonable variant of belief systems in general.) ~v~~ === Subject: Re: Math as Religion > Sure. Any who wish to mold mathematics to suit their > al) to Constructivists (E. Bishop, et al) to Formalists > (Russell, Frege, Hilbert, et al)to Platonists (Godel, > Penrose, et al). Penrose is a Platonist? I think he's a physicist, not a mathematician. -- Marcus === Subject: Re: Math as Religion On Fri, 10 Nov 2006 23:08:01 -0500, Marcus >> Sure. Any who wish to mold mathematics to suit their >> al) to Constructivists (E. Bishop, et al) to Formalists >> (Russell, Frege, Hilbert, et al)to Platonists (Godel, >> Penrose, et al). Penrose is a Platonist? I think he's a physicist, not a mathematician. I take platonist in general to mean mystic. And there are certainly plenty of those among professionals in both camps. ~v~~ === Subject: Re: Math as Religion <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> geometry. In math among other things he came up with Penrose tilings > and the Moore-Penrose pseudoinverse of a matrix. I knew about the tilings, but didn't realize he was the Penrose of the pseduoinverse. Regardless, he clearly doesn't understand mathematical logic. In fact, if he was posting to sci.math instead of writing books, we'd call him a crank. I think we should give him to physics. -- Marcus === Subject: Re: Math as Religion <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> pseduoinverse. Regardless, he clearly doesn't understand mathematical logic. Happily, mathematicians are not required to have a high-level understanding of mathematical logic to count as real mathematicians. > In fact, if he was posting to sci.math instead of writing books, we'd call him a > crank. I think we should give him to physics. Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at best a dodgy claim; I think some people would call *you* a crank for making it. Is there a word for notions such as that the nature of consciousness probably depends on unknown features of gravitation? How about goofball? === Subject: Re: Math as Religion In fact, if he was posting to sci.math instead of writing books, we'd call him a > crank. I think we should give him to physics. Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him > a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at > best a dodgy claim; I think some people would call *you* a crank for > making it. Is there a word for notions such as that the nature of > consciousness probably depends on unknown features of gravitation? How > about goofball? Goofball certainly applies. However, I think the fact that his argument is based on a mathematical error, and he denies that it is an error, qualifies him as a mathematical crank. He is using an incorrect mathematical argument to support his other goofball notions. Here is Penrose's books have often been reviewed in mathematical journals, e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point out that Penrose makes a mathematical error. Specifically, Penrose misunderstands G.9adel's Incompleteness Theorem, and so erroneously concludes that human mathematicians can do something that a machine cannot. Chapter 6, Section 2, of the recent book [Franz.8en] also provides a clear discussion of Penrose's error. Unfortunately, Penrose's misunderstanding of G.9adel's work is the primary basis for his argument that new physics is needed to explain consciousness. Once we correct the mathematical error, little remains. As [Barr] says in the final sentence of his review, The only real criticism [of the book] is that there is no valid evidence brought to bear on its main thesis. Penrose's misunderstanding is on a point that is often confusing to students of mathematical logic. However, anyone who has taken a course in mathematical logic should be able to find the error in Penrose's argument. As such, it is rather surprising that Penrose should write two books based on this error. As [Putnam] says of Shadows of the Mind: our current intellectual life. Roger Penrose is the Rouse Ball Professor of Mathematics at Oxford University and has shared the prestigious Wolf Price in physics with Stephen Hawking, but he is convinced by--and has produced this book as well as the earlier The emperor's new mind to defend--an argument that all experts in mathematical logic have long rejected as fallacious. The fact that the experts all reject Lucas's infamous argument counts for nothing in Penrose's eyes. He mistakenly believes that he has a philosophical disagreement with the logical community, when in fact this is a straightforward case of a mathematical fallacy. Penrose's books are of course also reviewed and discussed in the non-mathematical literature, e.g., [Dewdney], [Landauer], [Smith], [Zurek]. Such reviewers are typically unaware of the mathematical error that invalidates Penrose's argument. Despite this, they often disagree with Penrose's conclusions. It is ironic that while Penrose sees G.9adel's work as demonstrating the need for new physics to understand consciousness, Douglas Hofstadter (in the Pulitzer-Prize winning [Hofstadter]) argued that by studying the self-reference displayed in the works of G.9adel, Escher, and Bach, we could better understand the analogous self-reference displayed in consciousness. Barr, Michael, review of The Emperor's New Mind, American Mathematical Monthly, Vol. 97, No. 12, Dec. 1990, pp. 938-942. Dewdney, A.K., A Pandora's Box of Minds, Machines and Metaphysics, Computer Recreations, Scientific American, Dec. 1989, pp. 140-142. Faris, William, review of Shadows of the Mind, Notices of the American Mathematical Society, Vol. 43, No. 2, Feb. 1996, pp. 203-208. Franz.8en, Torkey, G.9adel's Theorem, an Incomplete Guide to Its Use and Abuse, A.K. Peters, Ltd., Wellesley, Massachusetts, 2005. Hofstadter, Douglas R., G.9adel, Escher, Bach: An Eternal Golden Braid, Basic Books, 20th Anniversary Ed., Jan. 1999. Landauer, Rolf, Is the Mind More than an Analytical Engine?, review of The Emperor's New Mind, Physics Today, June 1990, pp. 73-75. McCarthy, John, review of The Emperor's New Mind, Bulletin of the American Mathematical Society, Vol. 23, No. 2, Oct. 1990, pp. 606-616. Putnam, Hilary, review of Shadows of the Mind, Bulletin of the American Mathematical Society, Vol. 32, No. 3, July 1995, pp. 370-373. Smith, John Maynard, What Can't the Computer Do?, review of The Emperor's New Mind, The New York Review, March 15, 1990, pp. 21-25. Zurek, Wojciech H., Physics, Mathematics, and Minds, review of The Emperor's New Mind, Science, Vol. 248, May 18, 1990, pp. 880-881. -- Marcus === Subject: Re: Math as Religion <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> argument is based on a mathematical error, and he denies that it is an > error, qualifies him as a mathematical crank. No, because it isn't mathematics, it's philosophy. Calling it a mathematical error is a mischaractization. > Penrose's books have often been reviewed in mathematical journals, > e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point > out that Penrose makes a mathematical error. Citations saying specificially that he makes a *mathematical* error? > Specifically, Penrose > misunderstands G.9adel's Incompleteness Theorem, and so erroneously > concludes that human mathematicians can do something that a machine > cannot. Arguing on the basis of G.9adel that humans can do things machines cannot is not mathematics. > He mistakenly believes that he has a > philosophical disagreement with the logical community, when in fact > this is a straightforward case of a mathematical fallacy. OK, here is a claim that Penrose commits a mathematical fallacy. How is it even *possible* to commit a mathematical fallacy on the question of what humans can or cannot do? This makes no sense to me. === Subject: Re: Math as Religion Goofball certainly applies. However, I think the fact that his > argument is based on a mathematical error, and he denies that it is an > error, qualifies him as a mathematical crank. No, because it isn't mathematics, it's philosophy. Calling it a > mathematical error is a mischaractization. If I say that I am basing my philosophy on the fact that 1 = 0, that I am doing this because I can prove that 1 = 0 is true, and you tell me why my proof is wrong, but I insist that it is correct, am I making a mathematical error or are we arguing philosophy? > Penrose's books have often been reviewed in mathematical journals, > e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point > out that Penrose makes a mathematical error. Citations saying specificially that he makes a *mathematical* error? They all say that Penrose thinks the proof of Godel's Theorem proves something that it doesn't prove. I would call that a mathematical error. I don't have Penrose's book anymore, but when I read it, I agreed that Penrose says he proves something that he doesn't prove. (See below for more on this.) > Specifically, Penrose > misunderstands G=F6del's Incompleteness Theorem, and so erroneously > concludes that human mathematicians can do something that a machine > cannot. Arguing on the basis of G=F6del that humans can do things machines > cannot is not mathematics. If you base your physics/biology/whatever on incorrect mathematics, aren't you still making a mathematical error? > He mistakenly believes that he has a > philosophical disagreement with the logical community, when in fact > this is a straightforward case of a mathematical fallacy. OK, here is a claim that Penrose commits a mathematical fallacy. How > is it even *possible* to commit a mathematical fallacy on the question > of what humans can or cannot do? This makes no sense to me. Penrose says that we (mathematicians/logicians) prove that the Godel sentence (say for ZFC) is true. He agrees that the proof shows that ZFC can't prove the Godel sentence, but insists that the proof is a valid mathematical proof showing that the Godel sentence is true. He concludes that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say that we don't prove the Godel sentence is true. All we do is prove that if ZFC is consistent, the Godel sentence is true. Which is precisely what ZFC proves. Admittedly, as the logicians kept trying to tell Penrose that he was wrong, his counter arguments got more involved, but I think the preceding is a fair description of his basic error. Is that a mathematical error or a philosophical error? -- Marcus === Subject: Re: Math as Religion >> OK, here is a claim that Penrose commits a mathematical fallacy. How >> is it even *possible* to commit a mathematical fallacy on the question >> of what humans can or cannot do? This makes no sense to me. Penrose says that we (mathematicians/logicians) prove that the Godel > sentence (say for ZFC) is true. He agrees that the proof shows that ZFC > can't prove the Godel sentence, but insists that the proof is a valid > mathematical proof showing that the Godel sentence is true. He concludes > that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say > that we don't prove the Godel sentence is true. All we do is prove that > if ZFC is consistent, the Godel sentence is true. Which is precisely > what ZFC proves. Admittedly, as the logicians kept trying to tell Penrose that he was > wrong, his counter arguments got more involved, but I think the > preceding is a fair description of his basic error. Is that a mathematical error or a philosophical error? It seems to me extremely improbable that Penrose has misunderstood anything about Godel's Theorem, which is not very difficult to comprehend, as he is something of an expert in this area, as in several others. Perhaps you could give an exact citation of this supposed mathematical error? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Math as Religion > It seems to me extremely improbable that Penrose has misunderstood anything > about Godel's Theorem, which is not very difficult to comprehend, > as he is something of an expert in this area, What makes you say he is an expert in this area? > as in several others. Perhaps you could give an exact citation > of this supposed mathematical error? Unfortunately, I don't have a copy of any of Penrose's books. My memory is the error is quite easy to spot. It is near the beginning of his first book. I do have copies of the book reviews I mentioned. They give some quotes from Penrose's books and point out his errors. If you don't have access to the journals they appeared in, feel free to email me, and I'll scan them and email you copies. Torkel Franz.8en's book has a couple of sections on Penrose. Torkel frequently posted to usenet before his recent death. -- Marcus === Subject: Re: Math as Religion >> It seems to me extremely improbable that Penrose has misunderstood >> anything about Godel's Theorem, which is not very difficult to >> comprehend, as he is something of an expert in this area, What makes you say he is an expert in this area? He made some contributions on accessible ordinals. >> Perhaps you could give an exact citation >> of this supposed mathematical error? Unfortunately, I don't have a copy of any of Penrose's books. My memory > is the error is quite easy to spot. Well, I have copies of all of them, and I haven't spotted any error. I find his views on micro-tubules rather speculative, but then so does he. I think if you say an eminent mathematician has made an error, it is up to you to produce evidence of this, and at the very least to say precisely where the error was made. Perhaps you could go to your local library, and find the exact statement of Penrose in question. If it is easy to spot, as you say, this cannot be a very difficult task. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Math as Religion > It seems to me extremely improbable that Penrose has misunderstood > anything about Godel's Theorem, which is not very difficult to > comprehend, as he is something of an expert in this area, >> >> What makes you say he is an expert in this area? He made some contributions on accessible ordinals. Perhaps you could give an exact citation of this supposed > mathematical error? >> >> Unfortunately, I don't have a copy of any of Penrose's books. My >> memory is the error is quite easy to spot. Well, I have copies of all of them, and I haven't spotted any > error. Get a copy of Michael Barr's book review (reference below). He goes through what Penrose says in some detail and points out the errors. Barr's summary of one of Penrose's arguments is as follows. G.9adel showed that for every set of axioms which is sufficiently strong for arithmetic there is what is now called a G.9adel statement: a statement that is true in ordinary arithmetic but that cannot be proven from those axioms. Moreover, it is even possible to give an algorithm to construct such a sentence, given a set of axioms, much as T_k is constructed to confound the machine H. But now, Penrose and Lucas argue, a person can readily see that this proposition is true since it was constructed to be true but unprovable in this system. Therefore there can be no axiomatization of the human thought process. As Barr then explains in detail, the flaw is that all we know is that if the system is consistent, then the sentence is true. And, we don't know that the system is consistent. Barr then says: On page 116, Penrose dismisses the undecidability of consistency, which he describes as, 'historically perhaps the most important part of his [G.9adel's] argument', as irrelevant to the purposes of his book. To the contrary, this undecidability is the key to the unraveling of his argument. On page 110, there is a curious statement, 'The insight whereby we concluded that the G.9adel proposition P_k(k) is actually a true statement of arithmetic is an example of a general type of procedure known to logicians as a *reflection principle*: thus by 'reflecting' upon the *meaning* of the axiom system and rules of procedure, and convincing oneself that these indeed provide valid ways of arriving at mathematical truths, one may be able to code this insight into further true mathematical statements that were not deducible from these very axioms and rules.' This is nothing less than a claim that logicians use their reflections and insights, rather than formal reasoning to prove their theorems. This is an unbelievable howler of an error that is a libel on the world of logicians. There are subjective aspects in the choices of which sets of axioms you accept and the reflection principle (which is, in fact, a rather esoteric axiom of set theory that says that certain properties of very large infinite cardinal numbers have 'reflections' in the properties of more ordinary cardinals) is just as objective as any other axiom of logic. One of the striking things about all these arguments, including Searle's, is that they are completely formal. It is possible to imagine a planet full of robots that we have programmed to study philosophy and logic, using some heuristic procedures that are even now being worked on, to have discovered all these arguments and to be drawing exactly the same conclusions. Except that if their programming were actually consistent, they would see the flaws in the arguments and would not thereby conclude that their own thought processes could not be modeled by a computer. > I find his views on micro-tubules rather speculative, but then so > does he. I think if you say an eminent mathematician has made an error, As I said, I think he is a physicist. Let's see him publish his putative proof that the G.9adel sentence is true in a logic journal. > it is up to you to produce evidence of this, and at the very least > to say precisely where the error was made. It ain't just me! The following reviews and books say the same. Barr, Michael, review of The Emperor's New Mind, American Mathematical Monthly, Vol. 97, No. 12, Dec. 1990, pp. 938-942. Faris, William, review of Shadows of the Mind, Notices of the American Mathematical Society, Vol. 43, No. 2, Feb. 1996, pp. 203-208. Franz.8en, Torkel, G.9adel's Theorem, an Incomplete Guide to Its Use and Abuse, A.K. Peters, Ltd., Wellesley, Massachusetts, 2005. McCarthy, John, review of The Emperor's New Mind, Bulletin of the American Mathematical Society, Vol. 23, No. 2, Oct. 1990, pp. 606-616. Putnam, Hilary, review of Shadows of the Mind, Bulletin of the American Mathematical Society, Vol. 32, No. 3, July 1995, pp. 370-373. > Perhaps you could go to your local library, and find the exact > statement of Penrose in question. If it is easy to spot, as you > say, this cannot be a very difficult task. Have you read the reviews and books I mentioned above? If you've read them and still can't find the error, I'll go find a copy of Penrose's book. -- Marcus === Subject: Re: Math as Religion > Have you read the reviews and books I mentioned above? If you've read > them and still can't find the error, I'll go find a copy of Penrose's > book. YOU claim that YOU have seen an obvious error in a book by Penrose. It is up to YOU to cite the error, not up to me to read some comment by someone else on the book. If as you claim it is easy to spot the error it shouldn't take you long to find it. Penrose upset a lot of AI people like John McCarthy - I recall going to hear a panel discuss the future of AI, and McCarthy refused to appear if Penrose was on the panel. I thought at the time this seemed a bit over the top, but it was suggested to me that a lot of grant money might be lost if Penrose's negative views on AI got too much publicity. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Math as Religion Have you read the reviews and books I mentioned above? If you've read > them and still can't find the error, I'll go find a copy of Penrose's > book. YOU claim that YOU have seen an obvious error in a book by Penrose. > It is up to YOU to cite the error, > not up to me to read some comment by someone else on the book. I guess that means that you haven't read them. Penrose says that mathematicians know the Godel sentence is true. This is around page 100 (give or take 10-20 pages) of the Emperor's New Mind. Clearly, this is false. Are you saying that Penrose doesn't actually say this or that you don't agree it is false? > If as you claim it is easy to spot the error > it shouldn't take you long to find it. If I had the book, I could find it. However, I don't. Feel free to send me your copy. If you don't wish to believe Penrose is in error, that's your prerogative. It doesn't change the facts. Anyone who has taken a course in mathematical logic can find at least some of the errors. It isn't like there are any logicians standing up to say Penrose is right. > Penrose upset a lot of AI people like John McCarthy - > I recall going to hear a panel discuss the future of AI, > and McCarthy refused to appear if Penrose was on the panel. I thought at the time this seemed a bit over the top, > but it was suggested to me that a lot of grant money might be lost > if Penrose's negative views on AI got too much publicity. Plus, Penrose is a crank, so non-cranks could reasonably be expected to be reluctant to appear on a panel with him. It would be like inviting a proponent of intelligent design to be on a panel on the future of biology. -- Marcus === Subject: Re: Math as Religion I find it incredibly pretentious and totally disingenuous that you would use the term true several times in the context of this post without being able to define the term while demanding I define it for you in other posts. Is this an example of the exact and precise meanings you claim mathematicians are trained to employ? On Sat, 11 Nov 2006 19:05:10 -0500, Marcus >> >> Goofball certainly applies. However, I think the fact that his >> argument is based on a mathematical error, and he denies that it is an >> error, qualifies him as a mathematical crank. >> >> No, because it isn't mathematics, it's philosophy. Calling it a >> mathematical error is a mischaractization. If I say that I am basing my philosophy on the fact that 1 = 0, that I >am doing this because I can prove that 1 = 0 is true, and you tell me >why my proof is wrong, but I insist that it is correct, am I making a >mathematical error or are we arguing philosophy? > Penrose's books have often been reviewed in mathematical journals, >> e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point >> out that Penrose makes a mathematical error. >> >> Citations saying specificially that he makes a *mathematical* error? They all say that Penrose thinks the proof of Godel's Theorem proves >something that it doesn't prove. I would call that a mathematical error. >I don't have Penrose's book anymore, but when I read it, I agreed that >Penrose says he proves something that he doesn't prove. (See below for >more on this.) > Specifically, Penrose >> misunderstands G=F6del's Incompleteness Theorem, and so erroneously >> concludes that human mathematicians can do something that a machine >> cannot. >> >> Arguing on the basis of G=F6del that humans can do things machines >> cannot is not mathematics. If you base your physics/biology/whatever on incorrect mathematics, >aren't you still making a mathematical error? > He mistakenly believes that he has a >> philosophical disagreement with the logical community, when in fact >> this is a straightforward case of a mathematical fallacy. >> >> OK, here is a claim that Penrose commits a mathematical fallacy. How >> is it even *possible* to commit a mathematical fallacy on the question >> of what humans can or cannot do? This makes no sense to me. Penrose says that we (mathematicians/logicians) prove that the Godel >sentence (say for ZFC) is true. He agrees that the proof shows that ZFC >can't prove the Godel sentence, but insists that the proof is a valid >mathematical proof showing that the Godel sentence is true. He concludes >that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say >that we don't prove the Godel sentence is true. All we do is prove that >if ZFC is consistent, the Godel sentence is true. Which is precisely >what ZFC proves. Admittedly, as the logicians kept trying to tell Penrose that he was >wrong, his counter arguments got more involved, but I think the >preceding is a fair description of his basic error. Is that a mathematical error or a philosophical error? ~v~~ === Subject: Re: Math as Religion > I find it incredibly pretentious and totally disingenuous that you > would use the term true several times in the context of this post > without being able to define the term while demanding I define it for > you in other posts. Is this an example of the exact and precise > meanings you claim mathematicians are trained to employ? Let's look at where I used the word true: > If I say that I am basing my philosophy on the fact that 1 = 0, that I > am doing this because I can prove that 1 = 0 is true, and you tell me > why my proof is wrong, but I insist that it is correct, am I making a > mathematical error or are we arguing philosophy? In this case, the word is redundant with prove. Feel free to delete it. > Penrose says that we (mathematicians/logicians) prove that the Godel > sentence (say for ZFC) is true. He agrees that the proof shows that ZFC > can't prove the Godel sentence, but insists that the proof is a valid > mathematical proof showing that the Godel sentence is true. He > concludes that we have proven something that ZFC can't prove. In this case, I am saying what Penrose says. Unfortunately, Penrose is very confused about the meanings of true and prove. > On the other hand, all the book reviews written by mathematicians say > that we don't prove the Godel sentence is true. In this case, the word is again redundant with prove. Feel free to delete it. Does that answer your questions? > On Sat, 11 Nov 2006 19:05:10 -0500, Marcus >> >> Goofball certainly applies. However, I think the fact that his >> argument is based on a mathematical error, and he denies that it is an >> error, qualifies him as a mathematical crank. >> >> No, because it isn't mathematics, it's philosophy. Calling it a >> mathematical error is a mischaractization. If I say that I am basing my philosophy on the fact that 1 = 0, that I >am doing this because I can prove that 1 = 0 is true, and you tell me >why my proof is wrong, but I insist that it is correct, am I making a >mathematical error or are we arguing philosophy? > Penrose's books have often been reviewed in mathematical journals, >> e.g., [Barr], [Faris], [McCarthy], [Putnam]. All such reviews point >> out that Penrose makes a mathematical error. >> >> Citations saying specificially that he makes a *mathematical* error? They all say that Penrose thinks the proof of Godel's Theorem proves >something that it doesn't prove. I would call that a mathematical error. >I don't have Penrose's book anymore, but when I read it, I agreed that >Penrose says he proves something that he doesn't prove. (See below for >more on this.) > Specifically, Penrose >> misunderstands G=F6del's Incompleteness Theorem, and so erroneously >> concludes that human mathematicians can do something that a machine >> cannot. >> >> Arguing on the basis of G=F6del that humans can do things machines >> cannot is not mathematics. If you base your physics/biology/whatever on incorrect mathematics, >aren't you still making a mathematical error? > He mistakenly believes that he has a >> philosophical disagreement with the logical community, when in fact >> this is a straightforward case of a mathematical fallacy. >> >> OK, here is a claim that Penrose commits a mathematical fallacy. How >> is it even *possible* to commit a mathematical fallacy on the question >> of what humans can or cannot do? This makes no sense to me. Penrose says that we (mathematicians/logicians) prove that the Godel >sentence (say for ZFC) is true. He agrees that the proof shows that ZFC >can't prove the Godel sentence, but insists that the proof is a valid >mathematical proof showing that the Godel sentence is true. He concludes >that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say >that we don't prove the Godel sentence is true. All we do is prove that >if ZFC is consistent, the Godel sentence is true. Which is precisely >what ZFC proves. Admittedly, as the logicians kept trying to tell Penrose that he was >wrong, his counter arguments got more involved, but I think the >preceding is a fair description of his basic error. Is that a mathematical error or a philosophical error? -- Marcus === Subject: Re: Math as Religion > In this case, I am saying what Penrose says. Unfortunately, Penrose is > very confused about the meanings of true and prove. It seems to me very much more likely that you are confused about what Penrose is saying. The words true and prove are used in different senses by different people, or even by the same person at different times. Some people, for example, do not accept proofs using the law of the excluded middle, while other people do. All you are saying, as far as I can see, is that you use these words differently to Penrose. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Math as Religion In this case, I am saying what Penrose says. Unfortunately, Penrose is > very confused about the meanings of true and prove. It seems to me very much more likely that you are confused > about what Penrose is saying. The words true and prove are used in different senses > by different people, or even by the same person at different times. > Some people, for example, do not accept proofs > using the law of the excluded middle, while other people do. All you are saying, as far as I can see, > is that you use these words differently to Penrose. A Google search for Penrose Godel will turn up lots of interesting reading. -- Marcus === Subject: Re: Math as Religion In this case, I am saying what Penrose says. Unfortunately, Penrose is > very confused about the meanings of true and prove. It seems to me very much more likely that you are confused > about what Penrose is saying. Want to bet on that? > The words true and prove are used in different senses > by different people, or even by the same person at different times. > Some people, for example, do not accept proofs > using the law of the excluded middle, while other people do. If Penrose meant something like that, I think he could have said it. > All you are saying, as far as I can see, > is that you use these words differently to Penrose. Well, that's now what I'm saying. -- Marcus === Subject: Re: Math as Religion <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> cannot is not mathematics. If you base your physics/biology/whatever on incorrect mathematics, > aren't you still making a mathematical error? But it isn't incorrect as mathematics. > Penrose says that we (mathematicians/logicians) prove that the Godel > sentence (say for ZFC) is true. He agrees that the proof shows that ZFC > can't prove the Godel sentence, but insists that the proof is a valid > mathematical proof showing that the Godel sentence is true. He concludes > that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say > that we don't prove the Godel sentence is true. All we do is prove that > if ZFC is consistent, the Godel sentence is true. Which is precisely > what ZFC proves. You are simply ignoring his point here, which is that proving something on the assumption that ZFC is consistent is regarded in the wider world of mathematics as a proof of its truth. Now, I think Penrose is trying to milk this for a conclusion he can't get, but that's a philosophy argument. There's nothing wrong mathematically with making a philosophical mountain out of a mathematical molehill, which I think is the deal here. > Is that a mathematical error or a philosophical error? It is entirely a philosophical argument. === Subject: Re: Math as Religion Arguing on the basis of G=F6del that humans can do things machines > cannot is not mathematics. If you base your physics/biology/whatever on incorrect mathematics, > aren't you still making a mathematical error? But it isn't incorrect as mathematics. If you mean that physics/biology/whatever isn't mathematics, then I agree. But, if a person insists that incorrect mathematics is correct mathematics, isn't that incorrect as mathematics? > Penrose says that we (mathematicians/logicians) prove that the Godel > sentence (say for ZFC) is true. He agrees that the proof shows that ZFC > can't prove the Godel sentence, but insists that the proof is a valid > mathematical proof showing that the Godel sentence is true. He concludes > that we have proven something that ZFC can't prove. On the other hand, all the book reviews written by mathematicians say > that we don't prove the Godel sentence is true. All we do is prove that > if ZFC is consistent, the Godel sentence is true. Which is precisely > what ZFC proves. You are simply ignoring his point here, which is that proving something > on the assumption that ZFC is consistent is regarded in the wider world > of mathematics as a proof of its truth. That's not what he says. And, I don't think he means that. If he really does mean that, he has a very strange way of saying it. And, if he did say/mean that, I don't think we would have logicians complaining in math journals that Penrose just doesn't get it. I think you are giving Penrose too much credit. > Now, I think Penrose is trying to milk this for a conclusion he can't > get, but that's a philosophy argument. There's nothing wrong > mathematically with making a philosophical mountain out of a > mathematical molehill, which I think is the deal here. Is that a mathematical error or a philosophical error? It is entirely a philosophical argument. If you are right that he is merely saying that mathematicians believe that ZFC is consistent and so believe the Godel sentence is true, then OK--anything from then on is not mathematics. But, suppose that is not what he is saying. Would it still be just a philosophical argument? And, if a person claims that something is proven mathematically when it isn't and uses this to argue that this mathematical fact supports their physics/biology views in writings that are read by people in those fields who do not have the mathematical training to detect the mathematical error, what is that person? Have you read the relevant sections in The Emperor's New Mind? If not, I suggest you do. -- Marcus === Subject: Re: Math as Religion On Sat, 11 Nov 2006 19:39:52 -0500, Marcus >> >> Arguing on the basis of G=F6del that humans can do things machines >> cannot is not mathematics. >> If you base your physics/biology/whatever on incorrect mathematics, >> aren't you still making a mathematical error? >> >> But it isn't incorrect as mathematics. If you mean that physics/biology/whatever isn't mathematics, then I >agree. But, if a person insists that incorrect mathematics is correct >mathematics, isn't that incorrect as mathematics? And perhaps you'd be so good as to define correct and incorrect mathematics for the heathen? Or even just the term mathematics might do for a starter. I mean purely in the interests of clear and precise meanings modern mathematicians are trained to employ. >> Penrose says that we (mathematicians/logicians) prove that the Godel >> sentence (say for ZFC) is true. He agrees that the proof shows that ZFC >> can't prove the Godel sentence, but insists that the proof is a valid >> mathematical proof showing that the Godel sentence is true. He concludes >> that we have proven something that ZFC can't prove. >> On the other hand, all the book reviews written by mathematicians say >> that we don't prove the Godel sentence is true. All we do is prove that >> if ZFC is consistent, the Godel sentence is true. Which is precisely >> what ZFC proves. >> >> You are simply ignoring his point here, which is that proving something >> on the assumption that ZFC is consistent is regarded in the wider world >> of mathematics as a proof of its truth. That's not what he says. And, I don't think he means that. If he really >does mean that, he has a very strange way of saying it. And, if he did >say/mean that, I don't think we would have logicians complaining in math >journals that Penrose just doesn't get it. I think you are giving >Penrose too much credit. > Now, I think Penrose is trying to milk this for a conclusion he can't >> get, but that's a philosophy argument. There's nothing wrong >> mathematically with making a philosophical mountain out of a >> mathematical molehill, which I think is the deal here. >> >> Is that a mathematical error or a philosophical error? >> >> It is entirely a philosophical argument. If you are right that he is merely saying that mathematicians believe >that ZFC is consistent and so believe the Godel sentence is true, then >OK--anything from then on is not mathematics. But, suppose that is not >what he is saying. Would it still be just a philosophical argument? And, if a person claims that something is proven mathematically when it >isn't and uses this to argue that this mathematical fact supports their >physics/biology views in writings that are read by people in those >fields who do not have the mathematical training to detect the >mathematical error, what is that person? Have you read the relevant sections in The Emperor's New Mind? If not, I >suggest you do. ~v~~ === Subject: Re: Math as Religion > On Sat, 11 Nov 2006 19:39:52 -0500, Marcus > >> >> Arguing on the basis of G=F6del that humans can do things machines >> cannot is not mathematics. >> If you base your physics/biology/whatever on incorrect mathematics, >> aren't you still making a mathematical error? >> >> But it isn't incorrect as mathematics. If you mean that physics/biology/whatever isn't mathematics, then I >agree. But, if a person insists that incorrect mathematics is correct >mathematics, isn't that incorrect as mathematics? And perhaps you'd be so good as to define correct and incorrect > mathematics for the heathen? Correct means the definitions and theorems are clearly stated and the proofs are all valid with each step following from previous steps or axioms. Incorrect means not correct. > Or even just the term mathematics might > do for a starter. That's a hard word to define. However, if you use definitions, theorems, and proofs, you are probably doing mathematics. -- Marcus === Subject: Re: Math as Religion <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> that ZFC is consistent and so believe the Godel sentence is true, then > OK--anything from then on is not mathematics. I think he is saying we know it in our inner gizzard, whereas a robot doesn't have an inner gizzard and just has to follow the rules. > Have you read the relevant sections in The Emperor's New Mind? If not, I > suggest you do. Some time ago. What you see me say is what I thought he was saying. === Subject: Re: Math as Religion If you are right that he is merely saying that mathematicians believe > that ZFC is consistent and so believe the Godel sentence is true, then > OK--anything from then on is not mathematics. I think he is saying we know it in our inner gizzard, whereas a robot > doesn't have an inner gizzard and just has to follow the rules. If that's what he's saying, then it is just philosophy. But, then why give the Godel sentence as an example of this? I would think there are lots simpler examples of things our inner gizzard says are true that we can't prove. For that matter, one of my logic professors said that the more you work with the ZFC axioms, the less you feel you understand them. Having taken several logic courses in graduate school, I feel the same way. I wouldn't place a bet that ZFC is consistent. I might be willing to bet that people will be able to fix any problems that are found, but that's a rather different bet. > Have you read the relevant sections in The Emperor's New Mind? If not, I > suggest you do. Some time ago. What you see me say is what I thought he was saying. OK. Does my thinking Penrose is a crank still make me a crank? -- Marcus === Subject: Re: Math as Religion On Sat, 11 Nov 2006 20:36:02 -0500, Marcus >> >> If you are right that he is merely saying that mathematicians believe >> that ZFC is consistent and so believe the Godel sentence is true, then >> OK--anything from then on is not mathematics. >> >> I think he is saying we know it in our inner gizzard, whereas a robot >> doesn't have an inner gizzard and just has to follow the rules. If that's what he's saying, then it is just philosophy. But, then why >give the Godel sentence as an example of this? I would think there are >lots simpler examples of things our inner gizzard says are true that we >can't prove. For that matter, one of my logic professors said that the more you work >with the ZFC axioms, the less you feel you understand them. Having taken >several logic courses in graduate school, I feel the same way. I >wouldn't place a bet that ZFC is consistent. I might be willing to bet >that people will be able to fix any problems that are found, but that's >a rather different bet. > Have you read the relevant sections in The Emperor's New Mind? If not, I >> suggest you do. >> >> Some time ago. What you see me say is what I thought he was saying. OK. Does my thinking Penrose is a crank still make me a crank? No but your thinking you know what you're talking about does. ~v~~ === Subject: ($$ logic wars $$(a disciple speaks(re: math as religion))) <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> I knew about the tilings, but didn't realize he was the Penrose of the > pseduoinverse. Regardless, he clearly doesn't understand mathematical logic. Happily, mathematicians are not required to have a high-level > understanding of mathematical logic to count as real mathematicians. so what is required for proof? a fuzzy feeling? a social contract? you do realise this one quotable quote underscores quite clearly where the feeling of religious application in mathematics comes from... > In fact, if he was posting to sci.math instead of writing books, we'd call him a > crank. I think we should give him to physics. Baloney. He wouldn't be a *mathematical* crank, so you'd be calling him > a philosophical crank, like Ayn Rand or Alfred Korzybski. That is at > best a dodgy claim; I think some people would call *you* a crank for > making it. Is there a word for notions such as that the nature of > consciousness probably depends on unknown features of gravitation? How > about goofball? plus the ability to disparage others without need for explanation _or_proof_ just one's word... here is an axiom for living: BEWARE THE MATHEMATICIAN WHO DISTINGUISHES HERSELF FROM PHILOSOPHY FOR THEY ARE RUNNING AWAY FROM SOMETHING IMPORTANT -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: ($$ logic wars $$(a disciple speaks(re: math as religion))) <15525287.1163183684607.JavaMail.jakarta@nitrogen.mathforum.org> a social contract? (1) Find a math journal (3) Read The sort of thing you see is what is required for a proof. === Subject: Re: Math as Religion On Sat, 11 Nov 2006 00:31:35 -0500, Marcus >> >> Penrose is a Platonist? I think he's a physicist, not a mathematician. >> >> His PhD from Cambridge is in math, on the use of tensors in algebraic >> geometry. In math among other things he came up with Penrose tilings >> and the Moore-Penrose pseudoinverse of a matrix. I knew about the tilings, but didn't realize he was the Penrose of the >pseduoinverse. Regardless, he clearly doesn't understand mathematical logic. In fact, >if he was posting to sci.math instead of writing books, we'd call him a >crank. I think we should give him to physics. Hey the lot of you belong in faith based pseudologic. ~v~~ === Subject: Re: number of cycles in digraphs with maximum density Complete graphs K(n) have (n choose 2) edges, but twice of this number of arcs. This fact is changing your calculations. kunzmilan === Subject: Re: number of cycles in digraphs with maximum density > Complete graphs K(n) have (n choose 2) edges, but twice of this number > of arcs. This fact is changing your calculations. But the OP (which you thoughtlessly did not quote) was only considering acyclic digraphs. This means, in particular, that you cannot have (a,b) and (b,a) as arcs, since a-b-a would then be a directed cycle. Thus, there can only be (n choose 2) directed edges in such a digraph, not twice this number. Usenet Rule #37 (Faisal Nameer Jawdat): Read the thread from the beginning, or else. --- === Subject: Re: Something for sci.math's amateur mathematicians? (Part 2) This was my opportunity. A table starts usually at upper left, a graph from bottom left. This is a graph: 1 4 10 20 1 3 6 10 1 2 3 4 1 1 1 1 It shows the number of ways, how the corresponding points of the square can be reached by unit steps, a nad b, from the beginning point 1. The table was obtained by truncation of the full table of binomial coefficients. This technique can be used for triangular numbers as well as for any dimensional positive quadrants. Cubes are obtained by truncation of plane complexes. Only difficulty is, that we can not see them, and we must rely on calculations. We start with generating functions and arrange results into tables of combinatorial functions. Here we reduce too rich space of strings at first to get equivalent points of the lattice. We can generate n dimensional cubes directly but then we miss some functions, this space is not complete. kunzmilan === Subject: How to declare integer in mathematica ? format=flowed; reply-type=original So the engine could calculate i.e.: Cos[k*Pi]=(-1)^k ?? Tom === Subject: Re: How to declare integer in mathematica ? > So the engine could calculate i.e.: Cos[k*Pi]=(-1)^k ?? Simplify[Cos[k Pi], Element[k, Integers]] yields (-1)^k. David === Subject: Re: How to declare integer in mathematica ? format=flowed; reply-type=original thx U[YAcute]ytkownik W. Cantrell napisa¸ w wiadomoæci >> So the engine could calculate i.e.: Cos[k*Pi]=(-1)^k >> ?? Simplify[Cos[k Pi], Element[k, Integers]] yields (-1)^k. === Subject: Re: what are all the integer numbers n so that 500|7-11n ? To read this post you need LaTeX and amsfonts installed. After studying congruences and modulo arithmetic, I came up with a solution to the problem: First, $500|7-11n Rightarrow 11n cong 7 mod 500$. But we know that 11 is reversible (mod 500), because gcd(500,11)=1. So the problem can be solved easily, if we find an x, so $11x cong 1 mod 500$. Using the euclidean algorithm we find the x to be equal to 91 (Hint: We know that for a,b integers, there are $x,y in mathbb{Z}$ such as gcd(a,b)=ax+by) Indeed 91*11=1 mod 500. We conclude that $n cong 91 7 (mod 500) cong 637 (mod 500) cong 137 (mod 500)$. All the integers n are $k 500 + 137, k in mathbb{Z}$ > I need some help on this. I'd like some advice on how to think to solve > it > === Subject: Magic Square Given a square with 3x3 dimensions. In the first row there are written the numbers a, b, c; In the second row there are written the numbers d, e, f; In the third row there are written the numbers g, h, i; So we have a square like this: a b c d e f g h i We know that a+b+c = d+e+f = g+h+i = a+e+i = g+e+c = a+d+g = b+e+h = c+f+i i.e. we have a Magic Square(sum of all columns, rows and diagonals are equale to each other) Proove, that 2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 === Subject: Re: Magic Square > Given a square with 3x3 dimensions. > In the first row there are written the numbers a, b, c; > In the second row there are written the numbers d, e, f; > In the third row there are written the numbers g, h, i; So we have a square like this: > a b c > d e f > g h i > We know that > a+b+c = d+e+f = g+h+i = a+e+i = g+e+c = a+d+g = b+e+h = c+f+i > i.e. we have a Magic Square(sum of all columns, rows and diagonals are > equale to each other) > Proove, that > 2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 A Maple solution: > S:={a+b+c=d+e+f,a+b+c=g+h+i,a+b+c=a+e+i,a+b+c=g+e+c,a+b+c=a+d+g,a+b+c=b+e+h, a+b+c = c+f+i}: > A:=solve(S,{a,b,c,d,e,f}): A := {a = 2/3*g+2/3*h-1/3*i, d = -2/3*g+1/3*h+4/3*i, e = 1/3*g+1/3*h+1/3*i, f = 4/3*g+1/3*h-2/3*i, c = -1/3*g+2/3*h+2/3*i, b = 2/3*g-1/3*h+2/3*i} > simplify(subs(A,-2*(a^3+c^3+g^3+i^3)+b^3+d^3+f^3+h^3+4*e^3)); 0 Mate === Subject: Re: Magic Square days. My association with the Department is that of an alumnus. >Given a square with 3x3 dimensions. >In the first row there are written the numbers a, b, c; >In the second row there are written the numbers d, e, f; >In the third row there are written the numbers g, h, i; So we have a square like this: >a b c >d e f >g h i >We know that >a+b+c = d+e+f = g+h+i = a+e+i = g+e+c = a+d+g = b+e+h = c+f+i >i.e. we have a Magic Square(sum of all columns, rows and diagonals are >equale to each other) >Proove, that >2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 It is not hard to verify that you must have e=5, that the corner entries a, c, g, and i must be the numbers 2, 4, 6, and 8 in some order, and that the entries b, d, f, and h must be the remaining way to do with without proving that there is one and only one (up to rotation and reflection) 3 x 3 magic square, but that's certainly one way to go. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: Magic Square >Given a square with 3x3 dimensions. >In the first row there are written the numbers a, b, c; >In the second row there are written the numbers d, e, f; >In the third row there are written the numbers g, h, i; So we have a square like this: >a b c >d e f >g h i >We know that >a+b+c = d+e+f = g+h+i = a+e+i = g+e+c = a+d+g = b+e+h = c+f+i >i.e. we have a Magic Square(sum of all columns, rows and diagonals are >equale to each other) >Proove, that >2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 It is not hard to verify that you must have e=5, that the corner > entries a, c, g, and i must be the numbers 2, 4, 6, and 8 in some > order, and that the entries b, d, f, and h must be the remaining > way to do with without proving that there is one and only one (up to > rotation and reflection) 3 x 3 magic square, but that's certainly one > way to go. I think you're making the unjustified assumption that OP wants a, b, ..., i to be 1, 2, ..., 9 in some order. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Magic Square >Given a square with 3x3 dimensions. >>In the first row there are written the numbers a, b, c; >>In the second row there are written the numbers d, e, f; >>In the third row there are written the numbers g, h, i; >>So we have a square like this: >>a b c >>d e f >>g h i >>We know that >>a+b+c = d+e+f = g+h+i = a+e+i = g+e+c = a+d+g = b+e+h = c+f+i >>i.e. we have a Magic Square(sum of all columns, rows and diagonals are >>equale to each other) >>Proove, that >>2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 >> >> It is not hard to verify that you must have e=5, that the corner >> entries a, c, g, and i must be the numbers 2, 4, 6, and 8 in some >> order, and that the entries b, d, f, and h must be the remaining >> way to do with without proving that there is one and only one (up to >> rotation and reflection) 3 x 3 magic square, but that's certainly one >> way to go. I think you're making the unjustified assumption that >OP wants a, b, ..., i to be 1, 2, ..., 9 in some order. Ah, perhaps. You tell me Magic Square, and I always assume it contains the numbers 1 through n^2, unless you say otherwise explicitly. You're probably right. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Urgent question: MCMC in a sparse, large Dirichlet space In a n-Dim (n>100) Dirichlet distribution Dir(a1,...,an) with parameters ai>=0, using random walk techniques (Markov Chain Monte Carlo,etc) to move from an origin sample to the target distribution point. How to decide methods to update ai? This is a rather urgent problem for me... Rebecca === Subject: Liouvilles formula Hi We were discussing the proof of Liouville's formula in Shaums outlines Differential Geometry. The teacher claims it is wrong. It says: P=partial derivative .... dg1/ds=Pg1/Pu*du/ds+Pg1/Pv*dv/ds=dg1/ds1*ds1/du*du/ds+dg1/ds2*ds2/dv*dv/ds where s1 is arclength along the u-parameter curves and s2 is arclength along the v-parameter curves. .... what bothers the teacher is that s1 depends both on u and v but here they claim it depends only on u. (g1=(Px/Pu)/|Px/Pu| is a unit vector in the direction of the parameter curves of the surface x(u,v), and the parameter curves are orthogonal.) === Subject: Re: Liouvilles formula > Hi > We were discussing the proof of Liouville's formula in Shaums outlines > Differential Geometry. The teacher claims it is wrong. > It says: P=partial derivative > .... > dg1/ds=Pg1/Pu*du/ds+Pg1/Pv*dv/ds=dg1/ds1*ds1/du*du/ds+dg1/ds2*ds2/dv*dv/ds > where s1 is arclength along the u-parameter curves and s2 is arclength along > the v-parameter curves. > .... > what bothers the teacher is that s1 depends both on u and v but here they > claim it depends only on u. > (g1=(Px/Pu)/|Px/Pu| is a unit vector in the direction of the parameter > curves of the surface x(u,v), and the parameter curves are orthogonal.) The arc length s1 of a stand-alone curve in space on parameter u will depend on u alone.For example,the arc length of a circle comes from its u - parameterization only,it is the same whether it is sitting on a sphere,a cone or a hyperboloid and its v - parameterization can be anything.However, geodesic curvature which is to be found in Liouville's formula depends on u and v. Narasimham === Subject: Re: Liouvilles formula >> Hi >> We were discussing the proof of Liouville's formula in Shaums outlines >> Differential Geometry. The teacher claims it is wrong. >> It says: P=partial derivative >> .... >> dg1/ds=Pg1/Pu*du/ds+Pg1/Pv*dv/ds=dg1/ds1*ds1/du*du/ds+dg1/ds2*ds2/dv*dv/ds >> where s1 is arclength along the u-parameter curves and s2 is arclength >> along >> the v-parameter curves. >> .... >> what bothers the teacher is that s1 depends both on u and v but here they >> claim it depends only on u. >> (g1=(Px/Pu)/|Px/Pu| is a unit vector in the direction of the parameter >> curves of the surface x(u,v), and the parameter curves are orthogonal.) The arc length s1 of a stand-alone curve in space on parameter u will > depend on u alone.For example,the arc length of a circle comes from its > u - parameterization only,it is the same whether it is sitting on a > sphere,a cone or a hyperboloid and its v - parameterization can be > anything.However, geodesic curvature which is to be found in > Liouville's formula depends on u and v. How would you then write a formula for the arc length s1? I told the teacher that s1(u)=integral(norm(x'_u(u,v)),from u_0 to u) and g1(s1(u),s2(v)) but he says it's wrong because s1 seems to depend on v as well. but to me the above formula indicates that. What is the correct notation? === Subject: Re: JSH: Some prime counting facts, facing the swarm > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has signed his name Clive something or other for years, and now you deduce that his real name is Clive. Very impressive. >Why try to block here? Jesus. A hint: When people are trying to hide things they don't usually post them on usenet. >So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting to me anyway, may as well come clean: _I_ am sci.math, also alt.math.undergrad, and every other usenet group containing math in the name. All the posts you see in any of those groups come from me - none of the other supposed posters actually exist. I know I shouldn't say that - when the Mathematical Community gave me this job I took an oath never to reveal the secret. But to hell with the consequences, I can't stand it anymore, pretending to be thousands of different people just for the sake of creating a facade to insulate the rest of the Mathematical Community from JSH. It's going to be interesting to see what happens now that he's on to the whole thing. >By the way, how's the weather in New Zealand? >___JSH ************************ === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math Aha. If you are sci.math then you must also be JSH, just as I have always suspected. But, wait a minute. If you are sci.math then you must also be me, and my schizophrenia must be getting really bad. The whole thing is making my head ache. You had better go lie down. -William (in reality Ullrich) Hughes === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are all in working together, just further revealing the cooperative nature of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake is made. The simple reality of posters lying, playing some weird game to manipulate the discussions, and believing they are immune from consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are completely stupid? I guess you do. You've lied about the mathematics for years now, so you probably DO consider the readers of these newsgroups to be completely stupid. ___JSH === Subject: Re: JSH: Some prime counting facts, facing the swarm [jstevh@msn.com] > The attempts at damage control by your group show how dedicated you > are all in working together, just further revealing the cooperative > nature of the group effort to control the discussion. No, but it does confirm how delusional you've become. Or that you're a troll with an admirably self-deprecating sense of humor. > You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? There is one person here behaving so stupidly that adequate words fail all observers. > I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. If that's true, all evidence so far appears to support the idea that they are all that stupid: you're the only one who's noticed the HUGE deception, despite you ranting about it endlessly. Do note that, unlike you, I haven't deleted any of my posts from the archives. If you think you found a smoking gun, it's still there, and I'll never remove it. You would do new readers a favor by posting a link to it, so they can see for themselves the new evidence. LOL -- of course you won't do that -- but why not? === Subject: Re: JSH: Some prime counting facts, facing the swarm > [jstevh@msn.com] >> The attempts at damage control by your group show how dedicated you >> are all in working together, just further revealing the cooperative >> nature of the group effort to control the discussion. No, but it does confirm how delusional you've become. Or that > you're a troll with an admirably self-deprecating sense of humor. > You protect each other--or try very hard to do so, when a major >> mistake >> is made. >> The simple reality of posters lying, playing some weird game to >> manipulate the discussions, and believing they are immune from >> consequences isn't so easy to hide. >> So you people think you can cover now as if these newsgroups are >> completely stupid? There is one person here behaving so stupidly that adequate words > fail all observers. > I guess you do. You've lied about the mathematics for years now, >> so >> you probably DO consider the readers of these newsgroups to be >> completely stupid. If that's true, all evidence so far appears to support the idea that > they are all that stupid: you're the only one who's noticed the > HUGE deception, despite you ranting about it endlessly. Do note that, unlike you, I haven't deleted any of my posts from the > archives. If you think you found a smoking gun, it's still there, > and I'll never remove it. You would do new readers a favor by > posting a link to it, so they can see for themselves the new > evidence. LOL -- of course you won't do that -- but why not? Nice work, Greg. I am sure we are well on the way to re-establishing the status quo. I have already thanked Fedor for his outstanding work on the noise-bots; they are almost human :) He says that JWMS is nearly ready, that should be quite something! Looking back on it, calling *all* of the version 2.1 script engines something was a bit silly. Oh well, onwards and upwards. As usual, our friends over at Google are co-operating fully on Operation Sardine. Gerald. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com> <5K6dnUzV4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > The attempts at damage control by your group show how dedicated you > So you people think you can cover now as if these newsgroups are > completely stupid? There is one person here behaving so stupidly that adequate words fail all > observers. > Oh come on Tim (or Greg or whatever your name is). That one was just too easy. I managed to resist and I'm only a sock puppet. You are responsible for starting this. You and your I'll make this post so ridiculous that even JSH will see the joke. When will you learn not to underestimate the depth of JSH's stupidity. - William (paragon of self restraint) Hughes === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any value in your work, so we must be doing something right. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I appreciate you bragging about it at this point--again you possibly think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The mathematical results don't just disappear. I think that's where people like you are lost, as you think there's some way to win completely. Like if you can just convince people long enough that it's over, but then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced to keep at it, ever more frated that you can't just smear me and my research completely and walk away satisfied that the dirty deed is done. James Harris === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris Did I mention that Darth Wiles is your father? And you have a sister, but we've kept her hidden from you to protect you both from the Dark Side. All will be revealed in time. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris People don't need to smear your work. People can read it for themselves and see if it is correct. That has been true all along. No one has ever deleted any of your posts. Those that you didn't delete are still there for anyone to read. So are the arguments that contradict your theories. People are able to make up their own minds about who is right and who is wrong. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris People don't need to smear your work. People can read it for themselves > and see if it is correct. That has been true all along. No one has ever > deleted any of your posts. Those that you didn't delete are still there > for anyone to read. So are the arguments that contradict your > theories. People are able to make up their own minds about who is right > and who is wrong. something was wrong with a paper they had sent through the formal peer review process and published? Smear campaigns don't exist because someone is wrong and not a threat because they are wrong. Smear campaigns are about convincing people when the truth does not help you. So with my prime counting function, I can go through detail after detail showing how what I have is different from what was known, but people like you just lie about the mathematics. And you get a group together to lie to help be convincing, and then come back to lie about lying. ALL of that is about working to help make up people's minds for them. Why else would one person--me--require dedicated posting efforts and flame websites from dozens of other individuals? If I'm wrong, why does it take a village to work convincing the world that I am? Where's the beef in what you're saying? It MAKES NO SENSE that so many people around the world go to so much effort and work to convince others about mathematical research that is supposedly wrong. It just makes no sense. What DOES make sense is a lot of effort by some threatened people to smear work that is absolutely right, and incredibly important. James Harris === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris People don't need to smear your work. People can read it for themselves > and see if it is correct. That has been true all along. No one has ever > deleted any of your posts. Those that you didn't delete are still there > for anyone to read. So are the arguments that contradict your > theories. People are able to make up their own minds about who is right > and who is wrong. something was wrong with a paper they had sent through the formal peer > review process and published? Smear campaigns don't exist because someone is wrong and not a threat > because they are wrong. Smear campaigns are about convincing people when the truth does not > help you. So with my prime counting function, I can go through detail after > detail showing how what I have is different from what was known, but > people like you just lie about the mathematics. And you get a group together to lie to help be convincing, and then > come back to lie about lying. ALL of that is about working to help make up people's minds for them. Why else would one person--me--require dedicated posting efforts and > flame websites from dozens of other individuals? If I'm wrong, why does it take a village to work convincing the world > that I am? Where's the beef in what you're saying? It MAKES NO SENSE that so many people around the world go to so much > effort and work to convince others about mathematical research that is > supposedly wrong. It just makes no sense. What DOES make sense is a lot of effort by some threatened people to > smear work that is absolutely right, and incredibly important. > James Harris You have no proof that more than one person ever e-mailed the journal. Also, you have stated that the people on sci.math are on the fringes of math. Why would a respected journal cave to a few fringe people? I thought that you said that it was only a few dedicated people against you on the math newsgroups. How does that amount to , It MAKES NO SENSE that so many people around the world go to so much effort and work to convince others about mathematical research that is supposedly wrong. Which is it James, are there alot of people on usenet against you or just a few fringe mathematicians? What doesn't make sense is somebody posting the same crap over and over again for years when it is so obviously wrong. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com [long discussion deleted] > [...] > You have no proof that more than one person ever e-mailed the journal. > Also, you have stated that the people on sci.math are on the fringes of > math. Why would a respected journal cave to a few fringe people? Because they caved into JSH by publishing his paper? --- === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com [long discussion deleted] > [...] > You have no proof that more than one person ever e-mailed the journal. > Also, you have stated that the people on sci.math are on the fringes of > math. Why would a respected journal cave to a few fringe people? Because they caved into JSH by publishing his paper? --- I think it was more an act of desperation. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris People don't need to smear your work. People can read it for themselves > and see if it is correct. That has been true all along. No one has ever > deleted any of your posts. Those that you didn't delete are still there > for anyone to read. So are the arguments that contradict your > theories. People are able to make up their own minds about who is right > and who is wrong. something was wrong with a paper they had sent through the formal peer > review process and published? Smear campaigns don't exist because someone is wrong and not a threat > because they are wrong. Smear campaigns are about convincing people when the truth does not > help you. So with my prime counting function, I can go through detail after > detail showing how what I have is different from what was known, but > people like you just lie about the mathematics. And you get a group together to lie to help be convincing, and then > come back to lie about lying. ALL of that is about working to help make up people's minds for them. Why else would one person--me--require dedicated posting efforts and > flame websites from dozens of other individuals? If I'm wrong, why does it take a village to work convincing the world > that I am? Where's the beef in what you're saying? It MAKES NO SENSE that so many people around the world go to so much > effort and work to convince others about mathematical research that is > supposedly wrong. It just makes no sense. What DOES make sense is a lot of effort by some threatened people to > smear work that is absolutely right, and incredibly important. > James Harris You have no proof that more than one person ever e-mailed the journal. > Also, you have stated that the people on sci.math are on the fringes of > math. Why would a respected journal cave to a few fringe people? > The journal was brave enough to post a controversial paper from an admitted amateur. As for the numbers my understanding is that it was more than one sci.math'er as several talked about emailing the journal, working out a strategy in posts to the newsgroup. > I thought that you said that it was only a few dedicated people against > you on the math newsgroups. How does that amount to , It MAKES NO > SENSE that so many people around the world go to so much effort and > work to convince others about mathematical research that is supposedly > wrong. > You are a few dedicated people from around the world, like Clive Tooth is in New Zealand, and Ullrich is in Oklahoma, while there is some guy in France, Denis is it? > Which is it James, are there alot of people on usenet against you or > just a few fringe mathematicians? > It takes a village is a good way to put it. There is like a global village of you people working together from around the world. > What doesn't make sense is somebody posting the same crap over and over > again for years when it is so obviously wrong. If it were obviously wrong it wouldn't take a group of dedicated posters who continually engage in a smear campaign INCLUDING going after a paper of mine at the journal with emails. The reality is that to even barely hold the line against my research it takes a dedicated effort from people around the world, who use email, Usenet and web pages. It takes that much effort, and you can't stop even when I point that out. So many of you dedicated to try and block my research. It dominates your lives but you deny the importance of research you spend so much effort fighting. James Harris === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH Well let's face it, after 11 years of trying, NOBODY recognises any > value in your work, so we must be doing something right. Well, you have done a very effective job, I'll give you that, and I > appreciate you bragging about it at this point--again you possibly > think anonymity gives you safety. But you forget--the math doesn't change. So what if you can make fools of much of the public for years? The > mathematical results don't just disappear. I think that's where people like you are lost, as you think there's > some way to win completely. Like if you can just convince people long enough that it's over, but > then it's not, as you can never accomplish total victory. You cannot be certain that the lies will hold, so you are stuck, forced > to keep at it, ever more frated that you can't just smear me and my > research completely and walk away satisfied that the dirty deed is > done. > James Harris People don't need to smear your work. People can read it for themselves > and see if it is correct. That has been true all along. No one has ever > deleted any of your posts. Those that you didn't delete are still there > for anyone to read. So are the arguments that contradict your > theories. People are able to make up their own minds about who is right > and who is wrong. something was wrong with a paper they had sent through the formal peer > review process and published? Smear campaigns don't exist because someone is wrong and not a threat > because they are wrong. Smear campaigns are about convincing people when the truth does not > help you. So with my prime counting function, I can go through detail after > detail showing how what I have is different from what was known, but > people like you just lie about the mathematics. And you get a group together to lie to help be convincing, and then > come back to lie about lying. ALL of that is about working to help make up people's minds for them. Why else would one person--me--require dedicated posting efforts and > flame websites from dozens of other individuals? If I'm wrong, why does it take a village to work convincing the world > that I am? Where's the beef in what you're saying? It MAKES NO SENSE that so many people around the world go to so much > effort and work to convince others about mathematical research that is > supposedly wrong. It just makes no sense. What DOES make sense is a lot of effort by some threatened people to > smear work that is absolutely right, and incredibly important. > James Harris You have no proof that more than one person ever e-mailed the journal. > Also, you have stated that the people on sci.math are on the fringes of > math. Why would a respected journal cave to a few fringe people? > The journal was brave enough to post a controversial paper from an > admitted amateur. As for the numbers my understanding is that it was more than one > sci.math'er as several talked about emailing the journal, working out a > strategy in posts to the newsgroup. I thought that you said that it was only a few dedicated people against > you on the math newsgroups. How does that amount to , It MAKES NO > SENSE that so many people around the world go to so much effort and > work to convince others about mathematical research that is supposedly > wrong. > You are a few dedicated people from around the world, like Clive Tooth > is in New Zealand, and Ullrich is in Oklahoma, while there is > some guy in France, Denis is it? Which is it James, are there alot of people on usenet against you or > just a few fringe mathematicians? > It takes a village is a good way to put it. There is like a global village of you people working together from > around the world. > What doesn't make sense is somebody posting the same crap over and over > again for years when it is so obviously wrong. If it were obviously wrong it wouldn't take a group of dedicated > posters who continually engage in a smear campaign INCLUDING going > after a paper of mine at the journal with emails. The reality is that to even barely hold the line against my research it > takes a dedicated effort from people around the world, who use email, > Usenet and web pages. It takes that much effort, and you can't stop even when I point that > out. So many of you dedicated to try and block my research. It dominates > your lives but you deny the importance of research you spend so much > effort fighting. > James Harris Well, you named 3 people, not exactly a global village. People have a right to post to public newsgroups. Most of the people who reply to your posts also make other posts to the newsgroups. So it doesn't exactly take a huge effort to reply to your posts and try to make sense of your math so that the casual observer doesn't get sucked into your delusions. A rational person would have quit making posts like your along time ago. You continue to post your mathese, so others continue to make sure that those without the background to see your crap as crap, don't get the wool pulled over there eyes. I have made maybe 1/100th the number of posts that you have on usenet. I only see one person who's life is dominated by the crap you post, and you see him every morning in the mirror. You have made thousands of posts over the years, and 99.9% of them have been about you and your work. Sounds a little self-obsessed to me, wouldn't you agree. Let's see the progress you have made. Oh, that's right, you have made zero progress. Way to make your life fulfilling there James. === Subject: Re: JSH: Some prime counting facts, facing the swarm <4rn7ttFsau0fU1@mid.individual.net> <1ebel2190te51af5rvipau2gc68psgnikr@4ax.com > [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? >> [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the >> Nora Baron character for replies like this, or maybe even assign >> it to Rupert (his reasonable script isn't generating much JSH >> hatred -- time to ratchet him up a notch?). /Some/ smart kid is >> going to dig back and notice that Ullrich is in fact one of the >> over and over again names Harris is urging them to notice. Fedor >> isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg >> Greg, you freakin' maniac!! >> You are supposed to sign them Tim. >> I'll send you an email about this. >> We may need to revive Libert until this blows over. >> Gerald. Nice try. You're Clive something or other who lives in New Zealand. Wow. Now _that's_ an impressive bit of detective work! The guy has > signed his name Clive something or other for years, and now you > deduce that his real name is Clive. Very impressive. Why try to block here? Jesus. A hint: When people are trying to hide things they > don't usually post them on usenet. So Tim Peters is just another freaking pseudonym and people reading >and ting that they were getting a real name were just pulled in and >fooled--yet again--by people who think they're immune to consequences. Ok, looks like I'm busted. The strain of living a lie has been getting > to me anyway, may as well come clean: _I_ am sci.math > Aha. If you are sci.math then you must also be JSH, just > as I have always suspected. But, wait a minute. If you > are sci.math then you must also be me, and my > schizophrenia must be getting really bad. The whole thing is making my head ache. You > had better go lie down. -William (in reality Ullrich) Hughes The attempts at damage control by your group show how dedicated you are > all in working together, just further revealing the cooperative nature > of the group effort to control the discussion. You protect each other--or try very hard to do so, when a major mistake > is made. The simple reality of posters lying, playing some weird game to > manipulate the discussions, and believing they are immune from > consequences isn't so easy to hide. So you people think you can cover now as if these newsgroups are > completely stupid? I guess you do. You've lied about the mathematics for years now, so > you probably DO consider the readers of these newsgroups to be > completely stupid. ___JSH What damage? Lets say for the sake of argument that most of the posters to the math newsgroups have been using fake names. So what? Does that change the substance of their posts? Have they prevented anyone from reading your posts? It seems like you are the only one who regularly deletes his posts. Anyone visiting these newsgroups can read your posts and those that reply. Are you trying to say that the people who read your posts aren't able to reply in support of you if they feel like it? If so, how is anyone stopping them? If there are thousands of people reading your posts like you say, are they all intimidated by the couple of people who post using fake names on the math newsgroups? Where are all of the threats and warnings not to endorse your theories? Where are the posts of protest against your unfair treatment? Shouldn't the newsgroups be bombarded by the thousands of people from around the world who agree with your brilliant ideas. === Subject: Re: JSH: Some prime counting facts, facing the swarm > _I_ am sci.math, also alt.math.undergrad, and every other usenet > group containing math in the name. All the posts you see in any > of those groups come from me - none of the other supposed posters > actually exist. How about stuff crossposted to both sci.logic and sci.math? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: JSH: Some prime counting facts, facing the swarm >> [jstevh@msn.com] >[...] Notice how posters swarm over these threads as I create them, babbling >nonsense to block out the facts? Read through those threads and see the same names over and over again, >and consider, how many of those people are actually acting alone? Top mathematicians can ignore me, but how can they be sure that maybe >some of you might not pick up on these ideas and start asking >uncomfortable questions? So why wouldn't they have an attack squad out here on Usenet to protect >them? >> [] >> I bet I can answer this one. Gimme a minute... no, that's not it, wait >> a second... right. >> I think I got it. They don't have an attack squad here on Usenet >> because they're not total friggin lunatics. >> Is that it? I can hardly wait to find out if I got it right. >> Terry, Fedor suggests that maybe it would be better to revive the Nora >> Baron character for replies like this, or maybe even assign it to Rupert >> (his reasonable script isn't generating much JSH hatred -- time to ratchet >> him up a notch?). /Some/ smart kid is going to dig back and notice that >> Ullrich is in fact one of the over and over again names Harris is urging >> them to notice. Fedor isn't worried yet, but better safe than in jail, eh? >> Love to Clarissa and the kids, >> Greg Greg? So the Tim Peters is a pseudonym? And Nora Baron was that a previous one for you? Are you insinuating >that Rupert is one as well? I had guessed that Nora Baron was Ullrich, who has usually >refrained from talking actual math in his posts, so a pseudonym would >give him a chance to try to do so. Maybe now some mysteries will get cleared up. I assume Clarissa is >Ullrich's wife and have no interest in that area or in the kids. It IS of interest that you reveal a personal relationship with Ullrich >at this time. Kind of odd timing in fact... My god you can be stupid. Tim's post was a silly joke. The joke being to play on your paranoia about who's who. See, he even put my name in quotes: , as though there was some question about whether that was my real name - when he put my name in quotes like that he was poking fun at _you_, the way you sometimes refer to a 'Joe Blow' or whatever it is instead of just Joe Blow No, I'm not Nora Baron. Yes, Nora Baron was a fake name. (Yes, I know who NB was, or I did once - I'm not going to tell you.) Yes, Tim knew my name before he arrived on sci.math/sci.logic (or at least before I knew of his presence here) from my stupid questions and stupid comments on comp.lang.python. Whether that counts as a personal relationship we leave to your tortured imagination. No, Clarissa is not my wife's name. He just made that up as part of the joke. On the other hand, the fact that you think it would be of interest if he _had_ revealed a personal relationship between the two of us is fascinating, at least to people who are fascinated by abnormal psychology. >James Harris ************************ === Subject: Re: JSH: Some prime counting facts, facing the swarm [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting >>alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? [] > I bet I can answer this one. Gimme a minute... no, that's not > it, wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. Terry, Fedor suggests that maybe it would be better to revive the > Nora > Baron character for replies like this, or maybe even assign it to > Rupert > (his reasonable script isn't generating much JSH hatred -- time > to ratchet > him up a notch?). /Some/ smart kid is going to dig back and > notice that > Ullrich is in fact one of the over and over again names Harris > is urging > them to notice. Fedor isn't worried yet, but better safe than in > jail, eh? Love to Clarissa and the kids, > Greg >>Greg? So the Tim Peters is a pseudonym? >>And Nora Baron was that a previous one for you? Are you >>insinuating >>that Rupert is one as well? >>I had guessed that Nora Baron was Ullrich, who has usually >>refrained from talking actual math in his posts, so a pseudonym >>would >>give him a chance to try to do so. >>Maybe now some mysteries will get cleared up. I assume Clarissa is >>Ullrich's wife and have no interest in that area or in the kids. >>It IS of interest that you reveal a personal relationship with >>Ullrich >>at this time. Kind of odd timing in fact... My god you can be stupid. Tim's post was a silly joke. The joke being to play on your > paranoia about who's who. See, he even put my name in quotes: > , as though there was some question about > whether that was my real name - when he put my name in quotes > like that he was poking fun at _you_, the way you sometimes > refer to a 'Joe Blow' or whatever it is instead of just Joe Blow No, I'm not Nora Baron. Yes, Nora Baron was a fake name. > (Yes, I know who NB was, or I did once - I'm not going to > tell you.) Yes, Tim knew my name before he arrived on sci.math/sci.logic > (or at least before I knew of his presence here) from my > stupid questions and stupid comments on comp.lang.python. > Whether that counts as a personal relationship we leave > to your tortured imagination. No, Clarissa is not my wife's name. He just made that up > as part of the joke. On the other hand, the fact that you think it would be > of interest if he _had_ revealed a personal relationship > between the two of us is fascinating, at least to people > who are fascinated by abnormal psychology. James, all that says is totally true. It was all just a silly joke. Really. Please believe me. Gerald. === Subject: Re: JSH: Some prime counting facts, facing the swarm <07ael29bvp5n960fdss9guitjupbsr7si0@4ax.com> <4rovhbFrvrmoU1@mid.individual.net [jstevh@msn.com] >>[...] >>Notice how posters swarm over these threads as I create them, >>babbling >>nonsense to block out the facts? >>Read through those threads and see the same names over and over >>again, >>and consider, how many of those people are actually acting >>alone? >>Top mathematicians can ignore me, but how can they be sure that >>maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >>So why wouldn't they have an attack squad out here on Usenet to >>protect >>them? [] > I bet I can answer this one. Gimme a minute... no, that's not > it, wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. Terry, Fedor suggests that maybe it would be better to revive the > Nora > Baron character for replies like this, or maybe even assign it to > Rupert > (his reasonable script isn't generating much JSH hatred -- time > to ratchet > him up a notch?). /Some/ smart kid is going to dig back and > notice that > Ullrich is in fact one of the over and over again names Harris > is urging > them to notice. Fedor isn't worried yet, but better safe than in > jail, eh? Love to Clarissa and the kids, > Greg >>Greg? So the Tim Peters is a pseudonym? >>And Nora Baron was that a previous one for you? Are you >>insinuating >>that Rupert is one as well? >>I had guessed that Nora Baron was Ullrich, who has usually >>refrained from talking actual math in his posts, so a pseudonym >>would >>give him a chance to try to do so. >>Maybe now some mysteries will get cleared up. I assume Clarissa is >>Ullrich's wife and have no interest in that area or in the kids. >>It IS of interest that you reveal a personal relationship with >>Ullrich >>at this time. Kind of odd timing in fact... My god you can be stupid. Tim's post was a silly joke. The joke being to play on your > paranoia about who's who. See, he even put my name in quotes: > , as though there was some question about > whether that was my real name - when he put my name in quotes > like that he was poking fun at _you_, the way you sometimes > refer to a 'Joe Blow' or whatever it is instead of just Joe Blow No, I'm not Nora Baron. Yes, Nora Baron was a fake name. > (Yes, I know who NB was, or I did once - I'm not going to > tell you.) Yes, Tim knew my name before he arrived on sci.math/sci.logic > (or at least before I knew of his presence here) from my > stupid questions and stupid comments on comp.lang.python. > Whether that counts as a personal relationship we leave > to your tortured imagination. No, Clarissa is not my wife's name. He just made that up > as part of the joke. On the other hand, the fact that you think it would be > of interest if he _had_ revealed a personal relationship > between the two of us is fascinating, at least to people > who are fascinated by abnormal psychology. James, all that says is totally true. > It was all just a silly joke. > Really. Please believe me. Gerald. Damage control continues... === Subject: Re: JSH: Some prime counting facts, facing the swarm [, explains the obvious] >> ... [jstevh@msn.com] > Damage control continues... James, you believe you know that is real. You missed that, as said, See, he even put my name in quotes: , as though there was some question about whether that was my real name ... Did you also miss that I started by calling him Terry (without quotes)? Terry, Fedor suggests that maybe it would be better ... God knows I don't insist that you make sense, but how do you account for this? Is really a fake name too, and Terry his real name? Do we have real names, fake names we use on Usenet, and secret code names inside the conspiracy we only use with each other (so like David's real name is David, but Terry is his secret code name)? Did I screw up /so/ badly that I not only let the conspiratorial cat out of the bag, but also forgot that David's name is David, right after laboriously inserting quotes by hand around David's name? Or what? Just curious. /My/ loony story is that it was just a silly put-on, and deliberately included many obvious signs (like calling Terry indeed) of that. What's your loony story? Fedor isn't worried yet, but better safe than in jail, eh? Oops. I suppose I just repeated that by accident too. === Subject: Re: JSH: Some prime counting facts, facing the swarm >[...] >> Notice how posters swarm over these threads as I create them, >> babbling nonsense to block out the facts? >> Read through those threads and see the same names over and over >> again, and consider, how many of those people are actually acting >> alone? >> Top mathematicians can ignore me, but how can they be sure that >> maybe some of you might not pick up on these ideas and start asking >> uncomfortable questions? >> So why wouldn't they have an attack squad out here on Usenet to >> protect them? [] > I bet I can answer this one. Gimme a minute... no, that's not it, > wait a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. [someone claiming to be Tim Peters ] >> Terry, Fedor suggests that maybe it would be better to revive >> the Nora Baron character for replies like this, or maybe even >> assign it to Rupert (his reasonable script isn't generating >> much JSH hatred -- time to ratchet him up a notch?). /Some/ >> smart kid is going to dig back and notice that Ullrich is in >> fact one of the over and over again names Harris is urging >> them to notice. Fedor isn't worried yet, but better safe than in >> jail, eh? >> Love to Clarissa and the kids, >> Greg > [jstevh@msn.com] > Greg? So the Tim Peters is a pseudonym? Check the headers for a forgery. That happens to you a lot more than to me, > but I'm certainly not immune. As I told Proginoskes recently, my real full name is Dick Wang Peters. I > was putting /him/ on, though (he's a little gullible -- sometimes I just > can't resist). I didn't fall for that. Besides, there is no such word as 'gullible'; check your dictionary if you don't believe me. But you need to know that I really am Archimedes Plutonium's alter ego. Someone _almost_ figured it out earlier this year; the rest of you are pea-brains. Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies www.archimedesplutonium.com www.iw.net/~a_plutonium > In any case, no, my name isn't, wasn't, and probably never > will be, Greg. And Nora Baron was that a previous one for you? Are you insinuating > that Rupert is one as well? I'm not insinuating anything. I don't believe your conspiracy theories, so > my best guess is that someone was trying to make it /look/ like I really > was part of a conspiracy, and accidentally hit the reply to group > instead of the reply to sender button when I was replying to the Ullrich > poster. OTOH, you'll have a different take on it, and I really can't help you with > that. Or it /could/ just be that someone is trying to create trouble between you > and me. Who knows? I had guessed that Nora Baron was Ullrich, who has usually > refrained from talking actual math in his posts, so a pseudonym would > give him a chance to try to do so. Don't know; can't say. Maybe now some mysteries will get cleared up. I assume Clarissa is > Ullrich's wife and have no interest in that area or in the kids. Ditto. It IS of interest that you reveal a personal relationship with Ullrich > at this time. Kind of odd timing in fact... /Too/ odd, don't you think? I think it's a put-on -- but I suppose you'd > expect me to say that regardless. So let's suppose you're really right about this conspiracy. Then why on > Earth would they admit to it like this? Maybe you said something that > spooked them, and they got a bit careless? Or it's deliberate > misinformation, revealing just a bit of the truth hoping people will vaguely > remember it, and knee-jerk dismiss stronger evidence later with oh, I > already saw that? Or ...? No offense intended, but those all sound crazy to me, in the technical sense > of paranoia. === Subject: Re: JSH: Some prime counting facts, facing the swarm ... [Tim Peters] >> ... >> As I told Proginoskes recently, my real full name is Dick Wang >> Peters. I was putting /him/ on, though (he's a little gullible -- >> sometimes I just can't resist). [Proginoskes] > I didn't fall for that. Well, duh ;-) In context, it was a just a transparent attempt to distract James from that I was putting /him/ on by admitting to trying to fool you. I don't know whether he fell for it, but falling for a he's a little gullible line applied to someone else is the kind of rich outcome worth taking risks to achieve :-) If you feel used and exploited, great, two birds with one stone ;-) > Besides, there is no such word as 'gullible'; check your > dictionary if you don't believe me. Oh no, I knew that. I didn't believe it the first time I heard it, but that was ... about 30 years ago. Very surprising! > But you need to know that I really am Archimedes Plutonium's alter ego. > Someone _almost_ figured it out earlier this year; the rest of you are > pea-brains. Who's the potential genius? I'm betting it was one of my dozen pseudonyms -- or one of yours? > Archimedes Plutonium > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies > www.archimedesplutonium.com > www.iw.net/~a_plutonium Have you checked the first of those links recently? Honest! Why in the name of sacred Plutonium would ehow.com buy ArchiePu's domain name?! === Subject: Re: JSH: Some prime counting facts, facing the swarm On Sun, 12 Nov 2006 02:20:37 -0500, Tim Peters ... > As I told Proginoskes recently, my real full name is Dick Wang > Peters. I was putting /him/ on, though (he's a little gullible -- > sometimes I just can't resist). [Proginoskes] >> I didn't fall for that. Well, duh ;-) In context, it was a just a transparent attempt to distract >James from that I was putting /him/ on by admitting to trying to fool you. Ah, now it makes sense. See, at first I thought you were trying to confuse _me_. That didn't make any sense, because I know you know I'm much too smart to be confused by a silly little usenet post (for example I almost always get my wife's name right when I think about it). You were actually trying to confuse _him_. That makes more sense - >I don't know whether he fell for it, but falling for a he's a little >gullible line applied to someone else is the kind of rich outcome worth >taking risks to achieve :-) If you feel used and exploited, great, two birds with one stone ;-) > Besides, there is no such word as 'gullible'; check your >> dictionary if you don't believe me. Oh no, I knew that. I didn't believe it the first time I heard it, but that >was ... about 30 years ago. Very surprising! > But you need to know that I really am Archimedes Plutonium's alter ego. >> Someone _almost_ figured it out earlier this year; the rest of you are >> pea-brains. Who's the potential genius? I'm betting it was one of my dozen >pseudonyms -- or one of yours? > Archimedes Plutonium >> whole entire Universe is just one big atom where dots >> of the electron-dot-cloud are galaxies >> www.archimedesplutonium.com >> www.iw.net/~a_plutonium Have you checked the first of those links recently? Honest! Why in the >name of sacred Plutonium would ehow.com buy ArchiePu's domain name?! > ************************ === Subject: Re: JSH: Some prime counting facts, facing the swarm [] > Ah, now it makes sense. See, at first I thought you were trying to confuse _me_. That didn't > make any sense, because I know you know I'm much too smart to be > confused by a silly little usenet post (for example I almost always > get my wife's name right when I think about it). You were actually trying to confuse _him_. That makes more sense - Glad you're straightened out! It remains very puzzling to me, but only for the obvious reasons. You'd think that after years of me never misspelling your surname, /someone/ would have figured out we're related -- or even the same person. By the way, if you see Occam tonight, remind him to return my razor. === Subject: Re: JSH: Some prime counting facts, facing the swarm > Greg? So the Tim Peters is a pseudonym? And Nora Baron was that a previous one for you? Are you insinuating > that Rupert is one as well? I had guessed that Nora Baron was Ullrich, who has usually > refrained from talking actual math in his posts, so a pseudonym would > give him a chance to try to do so. Man. That's just sad. -- Jesse F. Hughes Please. I was a philosophy major. Nobody can 'know' anything. And I DO know. -- George Greene embarrasses philosophy majors everywhere. === Subject: Re: JSH: Some prime counting facts, facing the swarm >[...] >Notice how posters swarm over these threads as I create them, babbling >>nonsense to block out the facts? >Read through those threads and see the same names over and over again, >>and consider, how many of those people are actually acting alone? >Top mathematicians can ignore me, but how can they be sure that maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >So why wouldn't they have an attack squad out here on Usenet to protect >>them? [] > I bet I can answer this one. Gimme a minute... no, that's not it, wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. Terry, Fedor suggests that maybe it would be better to revive the Nora > Baron character for replies like this, or maybe even assign it to Rupert > (his reasonable script isn't generating much JSH hatred -- time to ratchet > him up a notch?). /Some/ smart kid is going to dig back and notice that > Ullrich is in fact one of the over and over again names Harris is urging > them to notice. Fedor isn't worried yet, but better safe than in jail, eh? Love to Clarissa and the kids, > GregGreg? So the Tim Peters is a pseudonym? And Nora Baron was that a previous one for you? Are you insinuating > that Rupert is one as well? I had guessed that Nora Baron was Ullrich, who has usually > refrained from talking actual math in his posts, so a pseudonym would > give him a chance to try to do so. Maybe now some mysteries will get cleared up. I assume Clarissa is > Ullrich's wife and have no interest in that area or in the kids. It IS of interest that you reveal a personal relationship with Ullrich > at this time. Kind of odd timing in fact... Well gang, the cat seems to be out of the bag now. Looks like we need to think up some new nyms. Nora B. === Subject: Re: JSH: Some prime counting facts, facing the swarm >[...] >Notice how posters swarm over these threads as I create them, babbling >>nonsense to block out the facts? >Read through those threads and see the same names over and over again, >>and consider, how many of those people are actually acting alone? >Top mathematicians can ignore me, but how can they be sure that maybe >>some of you might not pick up on these ideas and start asking >>uncomfortable questions? >So why wouldn't they have an attack squad out here on Usenet to protect >>them? [] > I bet I can answer this one. Gimme a minute... no, that's not it, wait > a second... right. I think I got it. They don't have an attack squad here on Usenet > because they're not total friggin lunatics. Is that it? I can hardly wait to find out if I got it right. Terry, Fedor suggests that maybe it would be better to revive the Nora > Baron character for replies like this, or maybe even assign it to Rupert > (his reasonable script isn't generating much JSH hatred -- time to ratchet > him up a notch?). /Some/ smart kid is going to dig back and notice that > Ullrich is in fact one of the over and over again names Harris is urging > them to notice. Fedor isn't worried yet, but better safe than in jail, eh? Love to Clarissa and the kids, > GregGreg? So the Tim Peters is a pseudonym? And Nora Baron was that a previous one for you? Are you insinuating > that Rupert is one as well? I had guessed that Nora Baron was Ullrich, who has usually > refrained from talking actual math in his posts, so a pseudonym would > give him a chance to try to do so. Maybe now some mysteries will get cleared up. I assume Clarissa is > Ullrich's wife and have no interest in that area or in the kids. It IS of interest that you reveal a personal relationship with Ullrich > at this time. Kind of odd timing in fact... Well gang, the cat seems to be out of the bag now. > Looks like we need to think up some new nyms. Nora B. And I think back to all those posts I'd read from people who believed in Nora Baron who'd accuse me of ignoring cogent arguments when I'd shred them, only to have the poster using the pseudonym ignore them. READERS wake up. This is Usenet. These people lie about my mathematical research because they think they are immune from consequences because they DO NOT TELL YOU WHO THEY REALLY ARE. I, on the other hand, am James Harris. === Subject: Re: JSH: Some prime counting facts, facing the swarm in Nora Baron who'd accuse me of ignoring cogent arguments when I'd > shred them, only to have the poster using the pseudonym ignore them. READERS wake up. This is Usenet. These people lie about my mathematical research because they think they > are immune from consequences because they DO NOT TELL YOU WHO THEY > REALLY ARE. Golly, do I ever feel a cunt. You know I actually believed that your mathematical claims were wrong when Nora Baron posted explicit counterexamples to them. Now that I know she was using a pseudonym I realise that you were right all along about everything. > I, on the other hand, am James Harris. Knowing your real name adds so much weight to you argument. -Sally Tyllas === Subject: Re: JSH: Some prime counting facts, facing the swarm > However, when it was pointed out that the solutions to a partial > difference equation might have nothing to do with the solutions to a > similar looking partial differential equation you dropped this > claim. By dropped, dear brother, you mean continued to claim the same damn thing even though you should have known better. -- Sorry, wakeup to the real world. You're on your own dependent on me as your guide. Luckily for you, I'm self-correcting to a large extent, so if the proof were wrong, I'd tell you. It's not wrong. --- James Harris confirms that his proof is correct. === Subject: Re: JSH: Some prime counting facts, facing the swarm <871woazmn7.fsf@phiwumbda.org > However, when it was pointed out that the solutions to a partial > difference equation might have nothing to do with the solutions to a > similar looking partial differential equation you dropped this > claim. By dropped, dear brother, you mean continued to claim the same damn > thing even though you should have known better. > Strangely in this case no. While James continued to claim that the partial difference equation was important because it led to a partial differential equation, he stopped claiming (at least explicitely and in the posts I read in sci.math) that at solution ot the partial difference equation was a solution to the partial differential equation. - William Hughes === Subject: Re: Some prime counting facts, facing the swarm > Clive Tooth >> http://www.shutterstock.com/cat.mhtml?gallery_id=61771 >> Gerald, your pictures are awesome. Would you mind if I posted the above >> link to a forum I frequent? It's a psychedelic music forum and I think >> the guys there will like it. >> -Sarah Sharas Damage control continues. These posters have no shame, and clearly believe Usenet is just >completely stupid. Nope. Posting different names to try and cover for Tim Peters aka >Greg won't work, I hope. The reality of the conspiracy is too much to hide now. How many of you are there? Four? Or three? I know there is Clive, and Ullrich is a real person at Oklahoma State >University, as I talked to his freaking boss on the phone a few years >back, so no one is acting like him without his knowledge. You don't know any such thing. You dialed a number that you thought was OSU, you talked to someone who claimed to be head of the math department. How do you know that that's who he really was? >That's kind of sad, but also kind of amazing---maybe three or four >individuals with very low morals who manipulated and used an entire >math newsgroup, convincing otherwise intelligent people that some >incredible mathematics was wrong. Yup. The _hundreds_ of people who've been laughing at you here all these years, it's really just two or three. Good to know that, because now we know that the ten or twenty people who've agreed with you are actually the majority - the apparent vast majority who think you're an idiot are just two or three people. No wait. It's not ten or twenty people who've agreed with you, it's actually _nobody_ who's agreed with you in all these years. Sorry I had that figure wrong a second ago. >That is an accomplishment. Indeed. Not only have we three or four people succeeded in looking like hundreds of people for years, we've _somehow_ managed to _suppress_ the post from all the people who otherwise _would_ have been making posts agreeing with you! How do you imagine we did that, btw? Just curious. >And I am the real James Harris. >James Harris ************************ === Subject: =?iso-8859-1?q?Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_Than _15=B0F=3F=3F=3F=3F?= : Scales of measurement that have both equal intervals and absolute zero points are spoken of as ratio scales, for the simple reason that they permit the meaningful calculation of ratios. If you find, for example, that object A is 5 inches wide and object B is 15 inches wide, it is legitimate and meaningful to conclude that object B is three times as wide as object A, or alternatively, that object A is only one-third as wide as object B. Similarly, it makes sense to say that 15 students are three times as many as 5 students, and that 5 students are only one-third as many as 15 students. If the high temperatures on two successive winter days are 5¡F and 15¡F, on the other hand, it makes no sense at all to conclude that the second day is three times as warm as the first-because the zero point from which 5¡F and 15¡F are starting out is only an arbitrary marker on a scale that potentially extends all the way down to about -460¡F. In order to make such ratio judgments concerning temperatures we would have to use a scale, such as the Kelvin scale, whose zero point does mark an absolute zero level of temperature. === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? Because the lower the number the colder it is. Warmer is higher , colder is lower. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= Warmer is higher , colder is lower. That's what I'm thinking. Apparently, that's incorrect. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= Warmer is higher , colder is lower. That's what I'm thinking. Apparently, that's incorrect. === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? > Because the lower the number the colder it is. >> Warmer is higher , colder is lower. > That's what I'm thinking. Apparently, that's incorrect. > I think that you need to do some work on different types of scales as is normally covered in statistics. See There are 4 types of scales of measurement: 1 Nominal or categorical data Where data is simply categorised (and clearly doesn't apply to maths) 2 Ordinal data Where there is a ranking or ordering - one thing is greater or less than another 3 Data on an interval scale In the link the example given is of exam marks where there is a zero - but a better example would be temperature where the zero is arbitrary and depends on the scale (unless it is Absolute Zero). 4 Ratio Scales If data have a natural zero then they can be assigned to a ratio scale. The link below gives the permitted arithmetic operations on each. Scale type Permitted operation Nominal Counting Ordinal Greater than or less than operations Interval Addition and subtraction of scale values Ratio Multiplication and division of scale values Note: As one goes down the table the operations above can be carried out on the scales below eg Ratio scales also permit greater than or less than operations. *** Examples of each scale *** ** Nominal ** Gender. Ethnicity. Marital Status. Region numeric ID Census Tract Numbers ** Ordinal ** Cities to live in: best, good, average, poor U.S.D.A. quality of beef ratings (good, choice, prime). The rank order of anything. ** Interval ** Temperature in degrees of Fahrenheit or Centigrade Curved grades. Any consistent bias in measurement. Suppose the clock started 3 minutes after the race began. ** Ratio ** Degrees in Kelvin units (natural zero ie Absolute Zero) Annual income in dollars. Geographic distances NB These examples are not mine - they are for geography students. Nick (Only posted to sci.math) === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= It's the language. First, you have 5 and 15 reversed. Second, 15 is three times as warm, but two times warmer. OK? Dr. Michael W. Ecker Associate Professor of Mathematics Pennsylvania State University Wilkes-Barre Campus Lehman, PA 18627 > : Scales of measurement that have both equal intervals and absolute zero > points are spoken of as ratio scales, for the simple reason that they > permit the meaningful calculation of ratios. If you find, for example, > that object A is 5 inches wide and object B is 15 inches wide, it is > legitimate and meaningful to conclude that object B is three times as > wide as object A, or alternatively, that object A is only one-third as > wide as object B. Similarly, it makes sense to say that 15 students are > three times as many as 5 students, and that 5 students are only > one-third as many as 15 students. If the high temperatures on two > successive winter days are 5¡F and 15¡F, on the other hand, it makes > no sense at all to conclude that the second day is three times as warm > as the first-because the zero point from which 5¡F and 15¡F are > starting out is only an arbitrary marker on a scale that potentially > extends all the way down to about -460¡F. In order to make such > ratio judgments concerning temperatures we would have to use a scale, > such as the Kelvin scale, whose zero point does mark an absolute zero > level of temperature. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= > It's the language. First, you have 5 and 15 reversed. Second, 15 is three times as warm, but two times warmer. OK? Dr. Michael W. Ecker > Associate Professor of Mathematics > Pennsylvania State University > Wilkes-Barre Campus > Lehman, PA 18627 Hmm! Three times *as* warm, but two times warm-er...but the Professor at Vassar explicitly states that it makes no sense at all to conclude that the second day is three times as warm as the first-because the zero point from which 5¡F and 15¡F are starting out is only an arbitrary marker on a scale that potentially extends all the way down to about -460¡F. In order to make such ratio judgments concerning temperatures we would have to use a scale, such as the Kelvin scale, whose zero point does mark an absolute zero level of temperature. It is indeed the language (hence the x-post to alt.english.usage), and another thing I don't understand is what a ratio judgment is...how is the difference between 5F and 15F a ratio judgment??? === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? On 12 Nov 2006 11:56:17 -0800, Prisoner at War > It's the language. >> First, you have 5 and 15 reversed. >> Second, 15 is three times as warm, but two times warmer. >> OK? >> Dr. Michael W. Ecker >> Associate Professor of Mathematics >> Pennsylvania State University >> Wilkes-Barre Campus >> Lehman, PA 18627 >Hmm! Three times *as* warm, but two times warm-er...but the Professor >at Vassar explicitly states that it makes no sense at all to conclude >that the second day is three times as warm as the first-because the >zero point from which 5¡F and 15¡F are starting out is only an >arbitrary marker on a scale that potentially extends all the way down >to about -460¡F. In order to make such ratio judgments concerning >temperatures we would have to use a scale, such as the Kelvin scale, >whose zero point does mark an absolute zero level of temperature. It is indeed the language (hence the x-post to alt.english.usage), and >another thing I don't understand is what a ratio judgment is...how is >the difference between 5F and 15F a ratio judgment??? Perhaps Dr. Ecker replied too quickly. Some of us sometimes do that. :-) I have another thought to add to your confusion. Suppose you have an iron brick. In Physics you talk about the relation between its temperature and the amount of heat it contains. It's been eons since I sat in a Physics class, but with appropriate units, the amout of heat in, for example, calories = (specific heat of iron) times (temperature). Of course, for there to be no heat in the brick, its temperature must be 0 degrees Kelvin. Now suppose the brick is at 2 degrees F. (It has added a lot of heat because its temperature has been raised from -460F). Does it seem to you that raising it from 2 to 4 degrees F should double its heat? Hopefully not. The first change of temperature was 462 degrees F and the second 2 degrees F. No comparison in the change it its amount of heat. -- === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= :-) Yes -- that's how I'd originally reversed 15F and 5F in the subject line! > I have another thought to add to your confusion. Suppose you have an > iron brick. In Physics you talk about the relation between its > temperature and the amount of heat it contains. It's been eons since I > sat in a Physics class, but with appropriate units, the amout of heat > in, for example, calories = (specific heat of iron) times > (temperature). Whoa! Wait a minute, that's right...calories...that's a measure of heat, too! Wait, so what's temperature measuring, exactly? They both measure heat? Why two measures of heat for the one object? Or is calories, more strictly speaking, the measure of the expenditure of heat -- the measure of heat loss, as opposed to a steady state of heat, like temperature? Anyway, I digress.... > Of course, for there to be no heat in the brick, its > temperature must be 0 degrees Kelvin. Now suppose the brick is at 2 > degrees F. (It has added a lot of heat because its temperature has > been raised from -460F). Does it seem to you that raising it from 2 to > 4 degrees F should double its heat? Hopefully not. The first change of > temperature was 462 degrees F and the second 2 degrees F. No > comparison in the change it its amount of heat. Ah, yes, this is Brett Magill's apples analogy above (in the Google thread tree). By Odin's lost eye, I think I understand where y'all are coming from! And it proves, once again, the semanticists' contention that all confusion is over semantics! Indeed, there is a great deal of difference between 0K and 2F, which is not the same as between 2F and 4F. But, still, y'all keep referring to some absolute scale (K), though the issue is on the F scale. Pray tell, then, whether it would be correct to say, at least, that 15 degrees Fahrenheit is three times warmer, in degrees Fahrenheit, than 5 degrees Fahrenheit?? If it feels cold outside (5F) but not as cold in a simple wooden shed (15F), I'd say it was warmer in the shed. If I check my thermometer, I'd say it was three times warmer. Why is that wrong? What do I care about absolute zero? > -- === Subject: Re: Please Explain: How Is 5?F ***NOT*** Three Times Warmer Than 15?F???? On 12 Nov 2006 18:33:17 -0800, Prisoner at War Pray tell, then, whether it would be correct to say, at least, that > 15 degrees Fahrenheit is three times warmer, in degrees Fahrenheit, > than 5 degrees Fahrenheit?? A statement that narrow? Yes, I would say that you could say it. The problem is, Nobody is apt to care, and they may think you are strange for saying it that way. If it feels cold outside (5F) but not as cold in a simple wooden shed > (15F), I'd say it was warmer in the shed. If I check my thermometer, > I'd say it was three times warmer. Why is that wrong? What do I > care about absolute zero? That statement is not so narrow. This time, Nobody knows what you are talking about, especially if *they* are well-informed. That's not a desirable model for good communications. If you are communicating to your soul-buddy, say what you want! Should your personal friend, who knows your idiosyncratic communications, now ask you, Is it 3 degrees compared to 1 degree, or exactly what are you saying? So you haven't said much to him, either. One thing that has not been mentioned is that a zero point and a ratio-scale *can* be fixed by your present need, and not by someone else's scale. Forty degrees F. is twice as far from the freezing point of water than 36 degrees F. is. -- That is a legitimate ratio, if someone has the need for it (it can be measured in degrees C), for numbers above the zero. -- Rich Ulrich, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= :-) Yes -- that's how I'd originally reversed 15F and 5F in the subject > line! I have another thought to add to your confusion. Suppose you have an > iron brick. In Physics you talk about the relation between its > temperature and the amount of heat it contains. It's been eons since I > sat in a Physics class, but with appropriate units, the amout of heat > in, for example, calories = (specific heat of iron) times > (temperature). Whoa! Wait a minute, that's right...calories...that's a measure of > heat, too! Wait, so what's temperature measuring, exactly? They both > measure heat? No: temperature does not measure heat (at least, not in physics). I can have two identical bricks at the same temperature. Taken together, they contain twice as much heat as one brick, but their temperature does not change. Temperature measures an abstract concept called temperature! Back in the days when the laws governing heat, etc., were being discovered, people found it necessary to have a quantity they subsequently called temperature. It does correspond roughly to your everyday concept of 'hot' and 'cold', but it is nevertheless a different concept from 'heat'. We now know that it essentially measures the object you are speaking of; however, the originators of the subject did not know that. The connection between temperature and average 19th century. I suggest that you go to the library and take out a book on thermodynamics and statistical mechanics for a more complete story. > Why two measures of heat for the one object? Or is > calories, more strictly speaking, the measure of the expenditure of > heat -- the measure of heat loss, as opposed to a steady state of > heat, like temperature? Anyway, I digress.... Of course, for there to be no heat in the brick, its > temperature must be 0 degrees Kelvin. Now suppose the brick is at 2 > degrees F. (It has added a lot of heat because its temperature has > been raised from -460F). Does it seem to you that raising it from 2 to > 4 degrees F should double its heat? Hopefully not. The first change of > temperature was 462 degrees F and the second 2 degrees F. No > comparison in the change it its amount of heat. Ah, yes, this is Brett Magill's apples analogy above (in the Google > thread tree). By Odin's lost eye, I think I understand where y'all are coming from! > And it proves, once again, the semanticists' contention that all > confusion is over semantics! Indeed, there is a great deal of difference between 0K and 2F, which is > not the same as between 2F and 4F. But, still, y'all keep referring to > some absolute scale (K), though the issue is on the F scale. Pray tell, then, whether it would be correct to say, at least, that > 15 degrees Fahrenheit is three times warmer, in degrees Fahrenheit, > than 5 degrees Fahrenheit?? No, and this has been explained to you over and over again--it's just that you don't seem to listen. There is, perhaps a limited (and useless) sense in which 15 F is 3 times 5F: suppose we start with an object at 0 degrees F and heat it up by pumping energy into it. Then (provided we are careful and arrange the experiment in an appropriate manner) we will find that we need to expend three times as much energy to raise the object to 15 degrees F as 5 degrees F (starting from 0 F in both cases). However, this statement is essentially useless. What happens if we start, instead, with an object at 25 degrees below zero F and raise it to 5 F or to 15 F? We will now find that we only need to pump in 33% more energy to raise it to 15, so now the ratio of 3 no longer applies. If we started from 50 below the ratio would be different again. I think part of you problem is word overloading, inasmuch as you want to keep the meanings of words from everyday conversation in technical discussions. If it feels cold outside (5F) but not as cold in a simple wooden shed > (15F), I'd say it was warmer in the shed. Yes, that is correct. The Farenheit (or Celsius) scales are so-called interval scales, so object B is hotter than object A if you increase (or add to) the temperature of A to get B. Note: this says you ADD; it says nothing at all about MULTIPLYING. An interval scale allows meaningful addition and subtraction, but not multiplication or division. This distinction is more than mere semantics because it makes a genuine difference to the types of valid statistical tests that are available to analyze data. > If I check my thermometer, > I'd say it was three times warmer. Why is that wrong? Because you would be talking nonsense. The statement has no meaning. It is like saying that this object is three times as yellow as that object. You can say it, but it means nothing and communicates nothing. This has already been explained to you several times, but you are offering no evidence that you have even read, let alone thought about the answers you have been given. > What do I > care about absolute zero? Many people don't care at all, and they manage to live perfectly well even so. However, if they want to do chemistry, chemical engineering, physics, etc., then they DO need to care. R.G. Vickson -- === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= have two identical bricks at the same temperature. Taken together, they > contain twice as much heat as one brick, but their temperature does not > change. Temperature measures an abstract concept called temperature! > Back in the days when the laws governing heat, etc., were being > discovered, people found it necessary to have a quantity they > subsequently called temperature. It does correspond roughly to your > everyday concept of 'hot' and 'cold', but it is nevertheless a > different concept from 'heat'. We now know that it essentially measures > the object you are speaking of; however, the originators of the subject > did not know that. The connection between temperature and average > 19th century. I suggest that you go to the library and take out a book > on thermodynamics and statistical mechanics for a more complete story. I think I will. Folks have noted that temperature is more complex than I'd thought. > No, and this has been explained to you over and over again--it's just > that you don't seem to listen. I don't understand! How's that not listening??? > There is, perhaps a limited (and > useless) sense in which 15 F is 3 times 5F: suppose we start with an > object at 0 degrees F and heat it up by pumping energy into it. Then > (provided we are careful and arrange the experiment in an appropriate > manner) we will find that we need to expend three times as much energy > to raise the object to 15 degrees F as 5 degrees F (starting from 0 F > in both cases). However, this statement is essentially useless. Why is it useless? I suppose synonyms are useless, too, then -- why computer laptop and computer notebook?? > What > happens if we start, instead, with an object at 25 degrees below zero F > and raise it to 5 F or to 15 F? We will now find that we only need to > pump in 33% more energy to raise it to 15, so now the ratio of 3 no > longer applies. If we started from 50 below the ratio would be > different again. Sorry, how did you arrive at the 33% more figure? 33% more than what?? Between -25F and 15F there's, what, 40 degrees' difference.... > I think part of you problem is word overloading, inasmuch as you want > to keep the meanings of words from everyday conversation in technical > discussions. Yes, I do stumble over semantics (wording, meaning). And that motivates part of my complaint: we should have different words in technical discussions; technical discussions should not employ everyday words. Why should the eskimos have fifty different words for snow, and not one single word for snow-in-general, yet our civilization uses the minus-sign in three different ways??? > Yes, that is correct. The Farenheit (or Celsius) scales are so-called > interval scales, so object B is hotter than object A if you increase > (or add to) the temperature of A to get B. Note: this says you ADD; it > says nothing at all about MULTIPLYING. An interval scale allows > meaningful addition and subtraction, but not multiplication or > division. Whoa -- for me, addition is first cousins with multiplication! How is it possible to add but not multiply?? I've always thought of multiplication as a kind of patterned aggregate addition.... > This distinction is more than mere semantics because it makes > a genuine difference to the types of valid statistical tests that are > available to analyze data. Indeed -- but unfortunately I still fail to see why. Not to worry: I expect a bit of wrestling with some word-problems to help clear up the matter...in time.... > Because you would be talking nonsense. The statement has no meaning. So how about all them public television science shows that sprout factoids like a million times hotter than the sun and so forth?? > It > is like saying that this object is three times as yellow as that > object. Now that's a most interesting analogy, and most convenient: bear with me and you'll see where I'm coming from with all this.... Obviously, it's possible to say 3x more yellow meaningfully when speaking of color values in something like Photoshop, say. Similarly, I'm wondering what why 15F isn't 3X warmer than 5F, even within the context of the Fahrenheit scale. > You can say it, but it means nothing and communicates nothing. > This has already been explained to you several times, but you are > offering no evidence that you have even read, let alone thought about > the answers you have been given. Can you try and put yourself in my shoes before claiming that these shoes you so kindly cobbled for me fit me perfectly fine? I find everyone's responses most interesting, and if I can integrate them all I'm sure I'll have the answer (but that takes some time). However, you are all answering the question from your own already-enlightened perspectives, so of course your answers make perfect sense -- to you all. But to help me most effectively, you ought to get at what's causing confusion on my end -- of which factor I myself have no real notion, though folks' mentioning everything from over-wording to 'temperature' as an abstraction different from 'heat' and 'warmth' seems to be about right, in the aggregate -- and, falling short of diagnosing my illness, as it were, it does no good for you as the doctor to say, up, man! Walk! > Many people don't care at all, and they manage to live perfectly well > even so. However, if they want to do chemistry, chemical engineering, > physics, etc., then they DO need to care. Not sure why. Water boils when it boils; why is it necessary for the scale to refer to an absolute point? Some set of ticks/numbers on either side of some arbitrary point: as long as I know which one water boils at, freezes at, etc, why does it matter? > R.G. Vickson === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? On 12 Nov 2006 06:52:20 -0800, Prisoner at War : Scales of measurement that have both equal intervals and absolute zero >points are spoken of as ratio scales, for the simple reason that they >permit the meaningful calculation of ratios. If you find, for example, >that object A is 5 inches wide and object B is 15 inches wide, it is >legitimate and meaningful to conclude that object B is three times as >wide as object A, or alternatively, that object A is only one-third as >wide as object B. Similarly, it makes sense to say that 15 students are >three times as many as 5 students, and that 5 students are only >one-third as many as 15 students. If the high temperatures on two >successive winter days are 5¡F and 15¡F, on the other hand, it makes >no sense at all to conclude that the second day is three times as warm >as the first-because the zero point from which 5¡F and 15¡F are >starting out is only an arbitrary marker on a scale that potentially >extends all the way down to about -460¡F. In order to make such >ratio judgments concerning temperatures we would have to use a scale, >such as the Kelvin scale, whose zero point does mark an absolute zero >level of temperature. That is well stated! In addition to Jose's reply, think about what happens with -5 deg F and -15 deg F. -15 deg F is three times the first temperature -- but it is colder. Or what if we had -5 and +5? Or -5 and zero? These are not explanations, but are examples to make it clearer that you cannot take ratios on the F (or C) scales -- because they do not have a proper zero, meaning nothing. This is an important issue in chemistry or physics, where one really does want meaningful ratios of temperatures. To do that, the Kelvin (absolute) scale is used. The key point of that scale is that 0 K means nothing. bob === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= That is well stated! In addition to Jose's reply, think about what happens with -5 deg F > and -15 deg F. -15 deg F is three times the first temperature -- but > it is colder. I'm still leery of arithmetic operations on negative numbers myself, precisely due to things like that. Too weird when you translate it into everyday language. > Or what if we had -5 and +5? Or -5 and zero? These are > not explanations, but are examples to make it clearer that you > cannot take ratios on the F (or C) scales -- because they do not have > a proper zero, meaning nothing. And that's another psychological stumbling block for me -- if zero is nothing, then how can there be negative numbers at all? What sense is there in speaking of something that is less than nothing (seems a bit like claiming that infinity is smaller than infinity-plus-one!). Yes yes I understand negative balances and deficits, sure, but in terms of so-called real numbers, why, negative numbers seem as made up to me as the imaginary number i (square root of negative two). > This is an important issue in chemistry or physics, where one really > does want meaningful ratios of temperatures. To do that, the Kelvin > (absolute) scale is used. The key point of that scale is that 0 K > means nothing. Hmm! Now that's interesting...how did scientists come up with such scales as F and C in the first place, then? Whatever is the point, then, if they're so, well, useless.... > bob === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? On 12 Nov 2006 09:57:45 -0800, Prisoner at War >> This is an important issue in chemistry or physics, where one really >> does want meaningful ratios of temperatures. To do that, the Kelvin >> (absolute) scale is used. The key point of that scale is that 0 K >> means nothing. Hmm! Now that's interesting...how did scientists come up with such >scales as F and C in the first place, then? Whatever is the point, >then, if they're so, well, useless.... > In the case of Mr F, he set zero as the lowest T he could achieve. I am not sure if he knew that lower T were possible. Mr C set zero as the freezing point of water, for convenience. I donât think that either of them had any understanding of what a true zero for T might be. That idea came relatively late. That is, T scales were developed for practical use before T was well understood. When the idea of a true T zero was understood, indeed a scale based on it was developed (Kelvin). But in common usage, it really doesnât matter which scale we use, so the world in general does not use K. I do understand the colloquial usage of 3 times hotter. It has no real significance, but at least we all understand what one is saying. But even that only works if both values are positive. Although you posted your Q to a math group, I donât think the math is really the key issue here. It is the understanding of T, which comes from physics. T is simply a more complex property than is, say, length (or bank balance). I know that the nature of T has been discussed in other parts of this thread. bob === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= am not sure if he knew that lower T were possible. Mr C set zero as > the freezing point of water, for convenience. I don't think that either of them had any understanding of what a true > zero for T might be. That idea came relatively late. That is, T scales > were developed for practical use before T was well understood. When the idea of a true T zero was understood, indeed a scale based on > it was developed (Kelvin). But in common usage, it really doesn't > matter which scale we use, so the world in general does not use K. I > do understand the colloquial usage of 3 times hotter. It has no > real significance, but at least we all understand what one is > saying. But even that only works if both values are positive. > Although you posted your Q to a math group, I don't think the math is > really the key issue here. It is the understanding of T, which comes > from physics. T is simply a more complex property than is, say, length > (or bank balance). I know that the nature of T has been discussed in > other parts of this thread. > bob That's very interesting. If nothing else, I've learned from all this that's a big part of the problem here: folks are using a scientific notion of temperature to answer my question which is based on a lay understanding of temperature. Semantics, as usual! === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= nothing, then how can there be negative numbers at all? What sense is > there in speaking of something that is less than nothing (seems a bit > like claiming that infinity is smaller than infinity-plus-one!). Yes > yes I understand negative balances and deficits, sure, but in terms of > so-called real numbers, why, negative numbers seem as made up to me > as the imaginary number i (square root of negative two). And another thing: how is zero a number at all?? It's just nothing. How the heck is nothing a number?? Artists say that black isn't, technically speaking, a color, but that it is the complete absence of color. That's how I think of zero. And partly why so-called negative numbers seem incredibly arbitrary to me. Which of course it all is, right, since these are just ideas humanity has made up. I really do hope it gets easier with practice. Yes yes, I've been through kindergarten and primary school. Believe it or not, I was typically in the top 25 percentile when it came to math and arithmetic, even in college. But I have no idea what the heck I'm doing. I'm just memorizing algorithms to apply in recognizable situations. But I've never really had much of an idea why. And simple things like this stuff that we talk about still trip me up if I think about it. === Subject: Re: Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? Q) Please Explain: How Is 5¡F ***NOT*** Three Times Warmer Than 15¡F???? === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= More accurately, A) You're are a newsgroup troll. Barry, we're not distant cousins, are we? You are the first Grouper I've seen in newsgroups besides my alias-groupers. I see that Prisoner has successfully reeled in many species of catch from her troll, from those who nitpicked warmer and colder to those who relied on actual examples converted to F or C to show that differences do not convert to ratio without a meaningful scale relative some an absolute scale -- all failing to break the troll line. My answer to the question on the SUBJECT line is this (making use of the explanation Prisoner quoted from her Vasser prof: VP> the Kelvin scale, whose zero point does mark an absolute zero VP> level of temperature. Since Kelvin has an absolute zero, relative to that RATIO scale, 15 degrees K is 3 times as warm as 5 degrees K, and 2 times warmer, as one of the trolled victims bought back. BUT, 5 degrees F is INFINITELY (on a ratio scale) warmer than 0 degree Kelvin! So is 15 degrees F. Therefore, you have two different temperatures that are both infinitely warmer than O degree K, one cannot be three times warmer or three times colder than the other! infinity IS infinity, in mathematics, the omnipotent God of mathematics: You add 5 to infinity, you get infinity. You subtract 15 from infinity, you get infinity. You add one googol (not Google), which is 1 followed by 100 zeros, to infinity, you still have only infinity. Add one googolplex to infinity, you get infinity right back. So, dear trolled beloved and friends and victimes of the Prisoner at War, that is the TRUE reason that NOBODY can tell whether one temperature is colder or warmer than another, because whether you measure it in Fahrenheit or Centigrade, or in BTU, each corresponds to SOME temperature in Kelvin, but all of them are infinitely warmer than absolute zero! Q.E.D. Quack Endeth Duck. -- Reef Fish Bob. === Subject: Re: Please Explain: How Is 5?F ***NOT*** Three Times Warmer Than 15?F???? On 12 Nov 2006 15:15:03 -0800, Reef Fish degree Kelvin! Really? 5 divided by 0 is not infinity these days. It is undefined. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= BUT, 5 degrees F is INFINITELY (on a ratio scale) warmer than 0 >degree Kelvin! Really? 5 divided by 0 is not infinity these days. It is undefined. What IS infinity these days, Paul? 0 divided by 0 is undefined, 5/0 is still plus infinity, in my book. -- Reef Fish Bob. === Subject: Re: Please Explain: How Is 5?F ***NOT*** Three Times Warmer Than 15?F???? On 12 Nov 2006 16:50:36 -0800, Reef Fish > On 12 Nov 2006 15:15:03 -0800, Reef Fish BUT, 5 degrees F is INFINITELY (on a ratio scale) warmer than 0 >>degree Kelvin! >> Really? 5 divided by 0 is not infinity these days. It is undefined. What IS infinity these days, Paul? 0 divided by 0 is undefined, No. It's indeterminate. >5/0 is still plus infinity, in my book. Then your book is out of date. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= On 12 Nov 2006 15:15:03 -0800, Reef Fish BUT, 5 degrees F is INFINITELY (on a ratio scale) warmer than 0 >>degree Kelvin! >> Really? 5 divided by 0 is not infinity these days. It is undefined. What IS infinity these days, Paul? 0 divided by 0 is undefined, No. It's indeterminate. If an infinite sequence oscillates between +1 and -1, then the limit of the sequence is indeterminate. +inifnitity is the limit of 5/t as t ---> zero. 0/ t as t ---> zero is undefined. >5/0 is still plus infinity, in my book. Then your book is out of date. It may be out of print, but is dated, so how can it be out of date? :-) Now Prisoner at War may come back and argued that dated means 'out of date', and that's why the English language is a perfect language for trolling. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= :-) Now Prisoner at War may come back and argued that dated means > 'out of date', and that's why the English language is a perfect > language > for trolling. What's with the accusation of trolling? You almost sound like you're jealous; this is the second time you've mentioned it. If my problems don't interest you, please start your own thread. I don't see why you should hijack this one and yet accuse me of trolling -- and by implication accuse the others of being suckers. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= :-) Now Prisoner at War may come back and argued that dated means > 'out of date', and that's why the English language is a perfect > language for trolling. > What's with the accusation of trolling? Because it was so obvious, so blatent. An undergrad at Vasser displays the mentality of Pow-Wow junior community college is bad enough with the initial troll; but to continue the troll with each lucid exposition by the trollee, your continued troll just became every clearer with each post. Elementary, Prisoner. > You almost sound like you're > jealous; this is the second time you've mentioned it. Why should I be jealous of a troll who is so blatently obvious? A seasoned troll would be much more subtle with the trolls. > If my problems > don't interest you, please start your own thread. I don't see why you > should hijack this one and yet accuse me of trolling -- and by > implication accuse the others of being suckers. Your confession of a troll is manifested by your use of hijacked when I responded to the absurdity in your troll with something even more absurd, and you are devastated that your troll had been exposed and accused me of hijacking it. You are presumptious in thinking the others are all suckers. Some of them may be. Others are just stringing YOU along with your troll line and making fun of YOU. That's the worst fate for a troll -- worse than death -- to turn from a troll to be the trollee. As Paul says, Perhaps we are enjoying ourselves? I certainly am, watching the poor Troll jumping from one trollee to the next, in 6 different direction, each time trying to play Dumb and Dumber, not realizing some of the lines were trolling back at the Prisoner, like Jaw's revenge. :) === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= Dude, you need to learn how to read. Hint: I'm not a Vassar undergrad. Glad you enjoy this. But if you think this is fun, you might try going to the movies once in a while. With someone. Good luck. Good-bye. Good riddance! > Because it was so obvious, so blatent. An undergrad at Vasser > displays the mentality of Pow-Wow junior community college is > bad enough with the initial troll; but to continue the troll with each > lucid exposition by the trollee, your continued troll just became > every clearer with each post. Elementary, Prisoner. Why should I be jealous of a troll who is so blatently obvious? > A seasoned troll would be much more subtle with the trolls. Your confession of a troll is manifested by your use of hijacked > when I responded to the absurdity in your troll with something > even more absurd, and you are devastated that your troll had been > exposed and accused me of hijacking it. You are presumptious in thinking the others are all suckers. > Some of them may be. Others are just stringing YOU along with > your troll line and making fun of YOU. That's the worst fate for a troll -- worse than death -- to turn from > a troll to be the trollee. As Paul says, Perhaps we are enjoying ourselves? I certainly am, watching the poor Troll jumping from one trollee > to the next, in 6 different direction, each time trying to play > Dumb and Dumber, not realizing some of the lines were > trolling back at the Prisoner, like Jaw's revenge. :) === Subject: Re: Please Explain: How Is 5?F ***NOT*** Three Times Warmer Than 15?F???? On 12 Nov 2006 17:35:11 -0800, Prisoner at War > It may be out of print, but is dated, so how can it be out of date? >> :-) >> Now Prisoner at War may come back and argued that dated means >> 'out of date', and that's why the English language is a perfect >> language >> for trolling. >What's with the accusation of trolling? You almost sound like you're >jealous; this is the second time you've mentioned it. If my problems >don't interest you, please start your own thread. I don't see why you >should hijack this one and yet accuse me of trolling -- and by >implication accuse the others of being suckers. Perhaps we are enjoying ourselves? === Subject: Re: Please Explain: How Is 5?F ***NOT*** Three Times Warmer Than 15?F???? On 12 Nov 2006 17:20:49 -0800, Reef Fish > On 12 Nov 2006 16:50:36 -0800, Reef Fish BUT, 5 degrees F is INFINITELY (on a ratio scale) warmer than 0 >degree Kelvin! Really? 5 divided by 0 is not infinity these days. It is undefined. >>What IS infinity these days, Paul? >>0 divided by 0 is undefined, >> No. It's indeterminate. If an infinite sequence oscillates between +1 and -1, then the >limit of the sequence is indeterminate. +inifnitity is the limit of 5/t as t ---> zero. 0/ t as t ---> zero is undefined. What is the value of 0/0? (Is it really undefined or are there an infinite number of values?) There's a special word for stuff like this, where you could conceivably give it any number of values. That word is indeterminate. It's not the same as undefined. It essentially means that if it pops up somewhere, you don't know what its value will be in your case. For instance, if you have the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1 in the first case and 7 in the second case. Indeterminate. http://mathforum.org/dr.math/faq/faq.divideby0.html >5/0 is still plus infinity, in my book. >> Then your book is out of date. It may be out of print, but is dated, so how can it be out of date? >:-) Now Prisoner at War may come back and argued that dated means >'out of date', and that's why the English language is a perfect >language >for trolling. Plato would have managed just fine in Greek. === Subject: =?iso-8859-1?q?Re:_Please_Explain:_How_Is_5=B0F_***NOT***_Three_Times_Warmer_ Than_15=B0F=3F=3F=3F=3F?= LOL Actually, the correct answer may be that I have brain damage. Um, long story, but as capable as I am in other areas, when it comes to math, especially involving fractions and negative numbers, it's like my mind just...I don't know...stops. Like, it just stops. Not stop working, but simply stops. Sigh. I hope it's simply a matter of practice. === Subject: Re: Please Explain: How Is =?ISO-8859-1?Q?5=B0F_***NOT***_Th?= =?ISO-8859-1?Q?ree_Times_Warmer_Than_15=B0F=3F=3F=3F=3F?= > : Scales of measurement that have both equal intervals and absolute zero > points are spoken of as ratio scales, for the simple reason that they > permit the meaningful calculation of ratios. If you find, for example, > that object A is 5 inches wide and object B is 15 inches wide, it is > legitimate and meaningful to conclude that object B is three times as > wide as object A, or alternatively, that object A is only one-third as > wide as object B. Similarly, it makes sense to say that 15 students are > three times as many as 5 students, and that 5 students are only > one-third as many as 15 students. If the high temperatures on two > successive winter days are 5¡F and 15¡F, on the other hand, it makes > no sense at all to conclude that the second day is three times as warm > as the first-because the zero point from which 5¡F and 15¡F are > starting out is only an arbitrary marker on a scale that potentially > extends all the way down to about -460¡F. In order to make such > ratio judgments concerning temperatures we would have to use a scale, > such as the Kelvin scale, whose zero point does mark an absolute zero > level of temperature. Because if the assertion I am twice as tall as my son is true for some scale, then it remains true for any other scale that I use to measure both me and my son. If, say, I am 186 centimeters tall and my son is 93 centimeters tall, then the number of my height in centimeters is twice the number of his height in centimeters and if I now decide that we shall be measured in, say, inches, then my height measured in inches will be twice his height measured in inches. This is *not* true for scales of temperature in which the 0 point is arbitrary; just convert 5¡F and 15¡F to Celsius degrees to see what I mean. Besides, the question at the subject of your post should have the numbers 5 and 15 exchanged. === Subject: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net Because if the assertion I am twice as tall as my son is true for some > scale, then it remains true for any other scale that I use to measure > both me and my son. If, say, I am 186 centimeters tall and my son is > 93 centimeters tall, then the number of my height in centimeters is > twice the number of his height in centimeters and if I now decide that > we shall be measured in, say, inches, then my height measured in inches > will be twice his height measured in inches. This is *not* true for > scales of temperature in which the 0 point is arbitrary; just convert > 5¡F and 15¡F to Celsius degrees to see what I mean. Now this seems just the kind of semantic stumbling block I often have with mathematics: half the time I'm battling definitions! By simple arithmetic 15¡F is indeed 3X warmer than 5¡F -- I don't see why other scales and their intervals have to do with it. Sure, on other scales the same increase in temperature would post bigger or smaller numbers, but no matter how it is measured, the increase is that much greater! It just so happens that in Farenheit, it's 3X greater and since we're talking Farenheit here, 15¡F is 3X warmer than 5¡F! The author should have let alone the colloquial, quotidian way of parsing things, but note that on other scales, with their different intervals, blah blah blah -- instead of saying no it's not warmer, which sounds like an ontological statement about the physical condition instead of the technical hair-splitting semantic observation that it is. I guess I just take math much too literally. I'm rather attuned to the nuances of words and their context-dependent definitions and connotations, but it's probably an even bigger semantic mess in mathematics for me -- until recently, I never understood how there could be a negative exponent, philosophically speaking: what does it *mean* when an exponent means to increase, but negatively??? I followed the rules for calculating such things, of course, but I never understood how a quantity can be increased negatively (have a negative exponent). Well, it seems that the minus sign for the exponent isn't so much an ontological statement (positive or negative) as simply an instruction or direction as to what to do with that number: a negative exponent isn't a statement about the quantity, but an instruction on what to do with that quantity. Okay, but then why don't we use some other symbol? This is like the ol' Abott and Costello Who's on first, Watts on second routine! Mathematical homonyms??? > Besides, the question at the subject of your post should have the > numbers 5 and 15 exchanged. Indeed! This is a personality quirk with me: when I get confused over one little thing, I start babbling and wind up totally confused over everything (like the ol' Blue Screen of Death in Windows, as you see). === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > Because if the assertion I am twice as tall as my son is true for some > scale, then it remains true for any other scale that I use to measure > both me and my son. If, say, I am 186 centimeters tall and my son is > 93 centimeters tall, then the number of my height in centimeters is > twice the number of his height in centimeters and if I now decide that > we shall be measured in, say, inches, then my height measured in inches > will be twice his height measured in inches. This is *not* true for > scales of temperature in which the 0 point is arbitrary; just convert > 5¡F and 15¡F to Celsius degrees to see what I mean. Now this seems just the kind of semantic stumbling block I often have > with mathematics: half the time I'm battling definitions! By simple arithmetic 15¡F is indeed 3X warmer than 5¡F -- I don't see > why other scales and their intervals have to do with it. That would be like measuring height as the number of cm taller than, say, 20 cm. The amount of heat in a physical object is proportional to its absolute temperature, in, say, degrees Rankin (in which 1¡R = 1¡F). A difference in temperature in ¡R is the same as the difference in temps in ¡F, but they have different 0 points. 0 ¡R = -459.67 ¡F, or 459.67 ¡R = 0 ¡F 15 ¡F = 474.67 ¡R 5 ¡F = 464.67 ¡R so 15¡F is only about 2% warmer than 5 ¡F in absolute terms. > Sure, on > other scales the same increase in temperature would post bigger or > smaller numbers, but no matter how it is measured, the increase is > that much greater! It just so happens that in Farenheit, it's 3X > greater and since we're talking Farenheit here, 15¡F is 3X warmer > than 5¡F! The author should have let alone the colloquial, quotidian > way of parsing things, but note that on other scales, with their > different intervals, blah blah blah -- instead of saying no it's not > warmer, which sounds like an ontological statement about the physical > condition instead of the technical hair-splitting semantic observation > that it is. I guess I just take math much too literally. I'm rather attuned to the > nuances of words and their context-dependent definitions and > connotations, but it's probably an even bigger semantic mess in > mathematics for me -- until recently, I never understood how there > could be a negative exponent, philosophically speaking: what does it > *mean* when an exponent means to increase, but negatively??? I > followed the rules for calculating such things, of course, but I never > understood how a quantity can be increased negatively (have a negative > exponent). Well, it seems that the minus sign for the exponent isn't > so much an ontological statement (positive or negative) as simply an > instruction or direction as to what to do with that number: a negative > exponent isn't a statement about the quantity, but an instruction on > what to do with that quantity. Okay, but then why don't we use some > other symbol? This is like the ol' Abott and Costello Who's on first, > Watts on second routine! Mathematical homonyms??? Besides, the question at the subject of your post should have the > numbers 5 and 15 exchanged. Indeed! This is a personality quirk with me: when I get confused over > one little thing, I start babbling and wind up totally confused over > everything (like the ol' Blue Screen of Death in Windows, as you see). === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> say, 20 cm. Hmm! I'm sure there's a 3-blind-men-and-an-elephant thing going on here.... Okay, so, just as I was beginning to suspect, y'all (or at least you) are coming from an absolutist perspective (which I'd called Platonic elsewhere in this thread). Your height analogy proves that: you're thinking of heat in terms of absolute cold (the way you think of height in terms of a lack of height, which is obviously correct), whereas I'm merely thinking of heat in terms of whatever scale I'm using to measure heat. So can I at least say, then, that on the Fahrenheit scale, 15F is warmer, in degrees Fahrenheit, than 5F?? > The amount of heat in a physical object is proportional to its > absolute temperature, in, say, degrees Rankin (in which 1¡R = 1¡F). What?!?! There's a Rankin scale, too??? Now if this doesn't prove the theory of parallel universes, I don't know what will! How in heck is there so many scales to measure the one thing, heat??? I mean, okay, historically (as Morrison has so kindly pointed out elsewhere in this thread), various scales were used...but still, this is crazy, like if folks used different time clocks (carcadian, atomic, lunar, solar, whatever) to measure the same thing, time.... > A difference in temperature in ¡R is the same as the difference in temps > in ¡F, but they have different 0 points. Hmmm...! > 0 ¡R = -459.67 ¡F, > or 459.67 ¡R = 0 ¡F 15 ¡F = 474.67 ¡R > 5 ¡F = 464.67 ¡R > so 15¡F is only about 2% warmer than 5 ¡F in absolute terms. Aha! So you are an absolutist! Semantics indeed! No, I'm not trying to be cute here, I'm serious: so you guys have been critiquing 15F is 3X warmer than 5F all along from some absolutist scale like Kelvin or, evidently, Rankin! But, then, within Fahrenheit, wouldn't it make sense that 15F is 3X warmer than 5F in the sense of, more strictly speaking, 15F is 3X warmer in Fahrenheit degrees than 5F??? === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > That would be like measuring height as the number of cm taller than, > say, 20 cm. Hmm! I'm sure there's a 3-blind-men-and-an-elephant thing going on here.... Okay, so, just as I was beginning to suspect, y'all (or at least you) > are coming from an absolutist perspective (which I'd called > Platonic elsewhere in this thread). Your height analogy proves that: > you're thinking of heat in terms of absolute cold (the way you think of > height in terms of a lack of height, which is obviously correct), > whereas I'm merely thinking of heat in terms of whatever scale I'm > using to measure heat. To physicists ( of which I am not one) the heat of an object is the amount of a certain type of energy in that object, and the zero amount of that kind of energy coincides with the absolute zero of temperature scales. I tend to follow their thought processes imagining objects as containers of various amounts of heat energy proportional to, among other things, the temperatures of those objects. So can I at least say, then, that on the Fahrenheit scale, 15F is > warmer, in degrees Fahrenheit, than 5F?? The amount of heat in a physical object is proportional to its > absolute temperature, in, say, degrees Rankin (in which 1¡R = 1¡F). What?!?! There's a Rankin scale, too??? There are two degree sizes, that of Celsius and that of Fahrenheit. with 9/5 of one degree F equalling one degree C. The corresponding scales in which 0 means no heat energy at all are Kevlin and Rankin, with Kelvin degree size equal to Celsius degree size and Rankin degree size equal to fahrenheit degree size. Now if this doesn't prove the theory of parallel universes, I don't > know what will! How in heck is there so many scales to measure the one thing, heat??? > I mean, okay, historically (as Morrison has so kindly pointed out > elsewhere in this thread), various scales were used...but still, this > is crazy, like if folks used different time clocks (carcadian, atomic, > lunar, solar, whatever) to measure the same thing, time.... Why does the USA still cling to feet and pounds and acres and miles when the rest of the world has all gone metric? A difference in temperature in ¡R is the same as the difference in temps > in ¡F, but they have different 0 points. Hmmm...! 0 ¡R = -459.67 ¡F, > or 459.67 ¡R = 0 ¡F 15 ¡F = 474.67 ¡R > 5 ¡F = 464.67 ¡R > so 15¡F is only about 2% warmer than 5 ¡F in absolute terms. Aha! So you are an absolutist! In matters of relative temperatures, at. least, yes! Semantics indeed! No, I'm not trying to be cute here, I'm serious: so you guys have been > critiquing 15F is 3X warmer than 5F all along from some absolutist > scale like Kelvin or, evidently, Rankin! But, then, within Fahrenheit, wouldn't it make sense that 15F is 3X > warmer than 5F in the sense of, more strictly speaking, 15F is 3X > warmer in Fahrenheit degrees than 5F??? How many times warmer is 15¡F that -15¡F ? === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > By simple arithmetic 15¡F is indeed 3X warmer than 5¡F -- I don't see > why other scales and their intervals have to do with it. Sure, on > other scales the same increase in temperature would post bigger or > smaller numbers, but no matter how it is measured, the increase is > that much greater! It just so happens that in Farenheit, it's 3X > greater and since we're talking Farenheit here, 15¡F is 3X warmer > than 5¡F! The author should have let alone the colloquial, quotidian > way of parsing things, but note that on other scales, with their > different intervals, blah blah blah -- instead of saying no it's not > warmer, which sounds like an ontological statement about the physical > condition instead of the technical hair-splitting semantic observation > that it is. Nope, it's not hair-splitting. 5 degrees Farenheit doesn't mean 5 more degrees worth of heat than no heat at all; it means 5 more degrees of heat than are necessary to make ice melt. It takes 459 degrees worth of heat to make ice melt, so 5 degrees F is 464 degrees higher than no heat at all, and 15 degrees F is 474 degrees higher than no heat at all. 474 is 1.02 times 464, not three times. Essentially, the problem is that when you take ratios of quantities that have arbitrary zero points, the ratio no longer refers to the original units the quantities were expressed in. In particular, the magnitude of the ratio is entirely dependent on where you set the arbitrary zero point. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) >> By simple arithmetic 15¡F is indeed 3X warmer than 5¡F -- I don't see >> why other scales and their intervals have to do with it. Sure, on >> other scales the same increase in temperature would post bigger or >> smaller numbers, but no matter how it is measured, the increase is >> that much greater! It just so happens that in Farenheit, it's 3X >> greater and since we're talking Farenheit here, 15¡F is 3X warmer >> than 5¡F! The author should have let alone the colloquial, quotidian >> way of parsing things, but note that on other scales, with their >> different intervals, blah blah blah -- instead of saying no it's not >> warmer, which sounds like an ontological statement about the physical >> condition instead of the technical hair-splitting semantic observation >> that it is. Nope, it's not hair-splitting. 5 degrees Farenheit doesn't mean 5 more > degrees worth of heat than no heat at all; it means 5 more degrees of > heat than are necessary to make ice melt. It takes 459 degrees worth of > heat to make ice melt, so 5 degrees F is 464 degrees higher than no heat > at all, and 15 degrees F is 474 degrees higher than no heat at all. 474 > is 1.02 times 464, not three times. Essentially, the problem is that when you take ratios of quantities that > have arbitrary zero points, the ratio no longer refers to the original > units the quantities were expressed in. In particular, the magnitude of > the ratio is entirely dependent on where you set the arbitrary zero > point. Well, actually I thought that the latent heat of fusion of ice referred to the amount of heat to convert a unit of ice to water (334 kJ per kg at 0 deg C). http://www.physchem.co.za/Heat/Latent.htm Nick (only posting this to sci.math) === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> degrees worth of heat than no heat at all; it means 5 more degrees of > heat than are necessary to make ice melt. It takes 459 degrees worth of > heat to make ice melt, so 5 degrees F is 464 degrees higher than no heat > at all, and 15 degrees F is 474 degrees higher than no heat at all. 474 > is 1.02 times 464, not three times. What scale are you using? In any case, I do believe I understand the issue of different 'numerical widths' given different scales, and different starting points and different referents for those starting points (one scale uses zero for the absolute lack of heat/motion [so does this mean time stops at this point??], another the freezing point of water, etc.). BTW, I didn't mean hair-splitting in a perjorative way. Indeed, I'm hair-splitting myself here, insofar as I'm now requesting that quotidian terms like warmer be used more carefully by mathematicians (see my comment below). > Essentially, the problem is that when you take ratios of quantities that > have arbitrary zero points, the ratio no longer refers to the original > units the quantities were expressed in. In particular, the magnitude of > the ratio is entirely dependent on where you set the arbitrary zero > point. I think I understand that -- it's just that, as far as the original statement went, to my way of thinking, 5 Fahrenheit is simply one-third the heat, as measured in Fahrenheit, of 15 Fahrenheit. But to casually say that 15F is not 3X warmer than 5F is confusing if what you really mean is that 15F to 5F isn't the same 'numerical width' as its equalvent scores in Celsius of -15 and -9.4, respectively. That seems to have been what the Professor meant by his statements. It was just confusingly put. It reminds me of Ogden Nash's poem Proessor Twist. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> why other scales and their intervals have to do with it. Sure, on > other scales the same increase in temperature would post bigger or > smaller numbers, but no matter how it is measured, the increase is > that much greater! It just so happens that in Farenheit, it's 3X > greater and since we're talking Farenheit here, 15¡F is 3X warmer > than 5¡F! The author should have let alone the colloquial, quotidian > way of parsing things, but note that on other scales, with their > different intervals, blah blah blah -- instead of saying no it's not > warmer, which sounds like an ontological statement about the physical > condition instead of the technical hair-splitting semantic observation > that it is. Nope, it's not hair-splitting. 5 degrees Farenheit doesn't mean 5 more > degrees worth of heat than no heat at all; it means 5 more degrees of > heat than are necessary to make ice melt. It takes 459 degrees worth of > heat to make ice melt, so 5 degrees F is 464 degrees higher than no heat > at all, and 15 degrees F is 474 degrees higher than no heat at all. 474 > is 1.02 times 464, not three times. Ice doesn't melt at 0 degres F. Essentially, the problem is that when you take ratios of quantities that > have arbitrary zero points, the ratio no longer refers to the original > units the quantities were expressed in. In particular, the magnitude of > the ratio is entirely dependent on where you set the arbitrary zero > point. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) On 12 Nov 2006 08:35:27 -0800, Prisoner at War >By simple arithmetic 15¡F is indeed 3X warmer than 5¡F No it isn't. By simple arithmetic it is 3x the number of degrees Fahrenheit. Not the same thing at all. I stand on top of the 5 ft pillar which is on top of a 3000 ft mountain. Then I stand on the next pillar a few feet away which is 15 ft tall. Am I now three times as high as I was before? Hint: 1st height is 3005 ft. 2nd height is 3015 ft. It's the same with degrees fahrenheit - 0 degrees isn't at the bottom, but is a long way up the scale! === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> Fahrenheit. Not the same thing at all. Well, that's why I had noted the issue of semantics: to say 3X warmer, one has to know what it means > I stand on top of the 5 ft pillar which is on top of a 3000 ft mountain. > Then I stand on the next pillar a few feet away which is 15 ft tall. Am I now three times as high as I was before? Hint: 1st height is 3005 ft. 2nd height is 3015 ft. It's the same with degrees fahrenheit - 0 degrees isn't at the bottom, but > is a long way up the scale! match here: your referent is the base of the mountain, but colloquially, if you'd claimed to be 3X taller, folks would have understood you to mean 3X taller vis-.88-vis your first position on the peak of the mountain. mine now is that I wish mathematicians would be as sensitive to language as the rest of us are supposed to be strictly correct in our calculations! It doesn't help to say 15 Fahrenheit isn't 3X warmer than 5 Fahrenheit when of course that's precisely how we speak when talking about the weather. No, he should have said, instead, something like but notice that the same warmth [the degree of atomic movement] as measured in Celsisus yields barely over five degrees in difference (-15 Celsius to -9.4 Celsius)...to simply say 15F isn't 3X (warmer than) 5F is confusing! Seriously, I support universal numeracy -- I'm a struggling adult learner -- but I say that mathematicians ought to improve their literacy as well! It's amazing how simple math can be once I understand what the heck is really being said!! === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > No it isn't. By simple arithmetic it is 3x the number of degrees > Fahrenheit. Not the same thing at all. Well, that's why I had noted the issue of semantics: to say 3X warmer, > one has to know what it means I stand on top of the 5 ft pillar which is on top of a 3000 ft mountain. > Then I stand on the next pillar a few feet away which is 15 ft tall. Am I now three times as high as I was before? Hint: 1st height is 3005 ft. 2nd height is 3015 ft. It's the same with degrees fahrenheit - 0 degrees isn't at the bottom, but > is a long way up the scale! match here: your referent is the base of the mountain, but > colloquially, if you'd claimed to be 3X taller, folks would have > understood you to mean 3X taller vis-.88-vis your first position on the > peak of the mountain. mine now is that I wish mathematicians would be as sensitive to > language as the rest of us are supposed to be strictly correct in our > calculations! It doesn't help to say 15 Fahrenheit isn't 3X warmer > than 5 Fahrenheit when of course that's precisely how we speak when > talking about the weather. No, he should have said, instead, something > like but notice that the same warmth [the degree of atomic movement] > as measured in Celsisus yields barely over five degrees in difference > (-15 Celsius to -9.4 Celsius)...to simply say 15F isn't 3X (warmer > than) 5F is confusing! And how does one deal with comparing -5 ¡F with +15 ¡F ? One should obviously not say that 15¡ is -3 times warmer than -5¡. Seriously, I support universal numeracy -- I'm a struggling adult > learner -- but I say that mathematicians ought to improve their > literacy as well! It's amazing how simple math can be once I > understand what the heck is really being said!! === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> One should obviously not say that 15¡ is -3 times warmer than -5¡. Huh? Isn't that simply 15¡ is twenty times warmer than -5¡?? I know now that it's not -- but I don't know why it's not! There's something I'm missing in all this...(LOL, no, not my left hemisphere, I'm sure). === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > And how does one deal with comparing -5 ¡F with +15 ¡F ? > One should obviously not say that 15¡ is -3 times warmer than -5¡. > Huh? Isn't that simply 15¡ is twenty times warmer than -5¡?? It is 20 degrees warmer, but that is 15 - (-5), not 15 / (-5). I know now that it's not -- but I don't know why it's not! There's something I'm missing in all this...(LOL, no, not my left > hemisphere, I'm sure). === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> >>And how does one deal with comparing -5 ¡F with +15 ¡F ? >>One should obviously not say that 15¡ is -3 times warmer than -5¡. Huh? Isn't that simply 15¡ is twenty times warmer than -5¡?? I know now that it's not -- but I don't know why it's not! There's something I'm missing in all this...(LOL, no, not my left > hemisphere, I'm sure). > === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> <4557E08B.7070804@verizon.net temperature is a way of measuring heat. > Heat is a way of saying amount of intramolecular motion. > Zero degrees Kelvin is when there is NO intramolecular motion. > Zero degrees centigrade/Celsius is the freezing point of water. There are different meanings to zero. It can refer to an abstract notion such as the number line. It can also refer to counts. It can also refer to an arbitrary reference point such as temperature, > longitude, latitude, any Z-score, etc. Other arbitrary reference > points are 100 for IQ, 500 for GRE, SAT, and 50 for T-scores used in > psychometrics. > Any of the examples after Z-scores above use the mean as a reference point. One way to distinguish ratio level of measurement from interval level of > measurement is to ask whether the zero point has any relation to > nothingness or nonbeing in some way. but insofar as it works, it works. I guess I'm like Samuel Morse who knew how to harness some little aspect of electricity but had no idea what it really is. Thus I'm a good math student but no mathematician at all. > Art Kendall > Social Research Consultants === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) >> And how does one deal with comparing -5 ¡F with +15 ¡F ? >> One should obviously not say that 15¡ is -3 times warmer than -5¡. > Huh? Isn't that simply 15¡ is twenty times warmer than -5¡?? I know now that it's not -- but I don't know why it's not! Er, because -5 * 20 is not 15? -- Customers have come to SCO asking what they can do to respect and help protect the rights of the SCO intellectual property in Linux. SCO has created the Intellectual Property License for Linux in response to these customers needs. -- SCO responds to needs. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> Fahrenheit. Not the same thing at all. Well, that's why I had noted the issue of semantics: to say 3X warmer, > one has to know what it means I stand on top of the 5 ft pillar which is on top of a 3000 ft mountain. > Then I stand on the next pillar a few feet away which is 15 ft tall. Am I now three times as high as I was before? Hint: 1st height is 3005 ft. 2nd height is 3015 ft. It's the same with degrees fahrenheit - 0 degrees isn't at the bottom, but > is a long way up the scale! match here: your referent is the base of the mountain, but > colloquially, if you'd claimed to be 3X taller, folks would have > understood you to mean 3X taller vis-.88-vis your first position on the > peak of the mountain. mine now is that I wish mathematicians would be as sensitive to > language as the rest of us are supposed to be strictly correct in our > calculations! In a technical subject (including mathematics) we sometimes use words a bit differently than in casual, across-the-back-fence conversation. For example, in physics, the word momentum has a definite meaning that is not the same as its usage in ordinary conversation. In mathematics, the word group means something different than it does in the coffee shop or in Shakespeare. The passage you were complaining about was attempting to get the reader to drop the ingrained habit of always assuming that 15 something is three time 5 something, or, at least, to stop thinking it must always mean anything. It is not that mathematicians are insensitive to language (although some may be); rather, it is that they want to speak with more than normal precision, at least when they are talking about their own subject. If you are worrying about these things for hours on end, you are spending far too much time on inessentials and far too little time actually learning concepts and methods. R.G. Vickson > It doesn't help to say 15 Fahrenheit isn't 3X warmer > than 5 Fahrenheit when of course that's precisely how we speak when > talking about the weather. No, he should have said, instead, something > like but notice that the same warmth [the degree of atomic movement] > as measured in Celsisus yields barely over five degrees in difference > (-15 Celsius to -9.4 Celsius)...to simply say 15F isn't 3X (warmer > than) 5F is confusing! Seriously, I support universal numeracy -- I'm a struggling adult > learner -- but I say that mathematicians ought to improve their > literacy as well! It's amazing how simple math can be once I > understand what the heck is really being said!! === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> bit differently than in casual, across-the-back-fence conversation. For > example, in physics, the word momentum has a definite meaning that is > not the same as its usage in ordinary conversation. In mathematics, the > word group means something different than it does in the coffee shop > or in Shakespeare. The passage you were complaining about was > attempting to get the reader to drop the ingrained habit of always > assuming that 15 something is three time 5 something, or, at least, to > stop thinking it must always mean anything. It is not that > mathematicians are insensitive to language (although some may be); > rather, it is that they want to speak with more than normal precision, > at least when they are talking about their own subject. If you are > worrying about these things for hours on end, you are spending far too > much time on inessentials and far too little time actually learning > concepts and methods. R.G. Vickson Believe it or not, I used to be a star at a local community (junior) college math lab with trig and algebra. I was an unpaid tutor, able to help folks! I was rarely stumped. In college (senior, four-year college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: I've been applying methods successfully, but with no idea or feeling for the concepts at all! That's why I seem to be belaboring this small point. As well, given that it is raised at all by the author of that text, right in the beginning within the context of a pedagogy (so stated in his introduction) of focusing on fundamentals first, it doesn't seem that small a point. It is, indeed, what's been eating me all my mathematical life -- I follow orders, but I have no idea what they mean. But I suspect you're right: more practice. That's why I'm about to go for a second (undergraduate) degree, in mathematics, just for the heck of it. I feel terribly embarrassed not knowing what a fraction really is (I know about pizza slices and all that, yeah, but I can't begin to explain [essentially, visualize] what it means to take two-thirds of a pie and subtract from that a hundred-fortieth of a pie, and then multiply the result by six-fifths of a pie). I do hope it all gets better with practice. Simply learning enough required math for the required courses for the time required to get a course grade just isn't doing it. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> *snip* > Believe it or not, I used to be a star at a local community (junior) > college math lab with trig and algebra. I was an unpaid tutor, able to > help folks! I was rarely stumped. In college (senior, four-year > college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: > I've been applying methods successfully, but with no idea or feeling > for the concepts at all! That's why I seem to be belaboring this small > point. As well, given that it is raised at all by the author of that > text, right in the beginning within the context of a pedagogy (so > stated in his introduction) of focusing on fundamentals first, it > doesn't seem that small a point. It is, indeed, what's been eating me > all my mathematical life -- I follow orders, but I have no idea what > they mean. *snip* I don't know if this is relevant to your difficulty, but my experience as a tutor (not math) has taught me that when a technical point seems obscure or confusing, there is usually a _historical_ explanation. People mess things up all the time, usually with good intentions, and then the resulting mess has enough inertia that it can't be supplanted by a more logical solution. I don't claim to be an expert, but perhaps a little story about temperature measurement will make things clearer to you. It may not; it will only explain why temperature measurement is messed up, not what to do about it. The Fahrenheit scale was proposed by Gabriel Fahrenheit in 1724, who (according to Wikipedia) was annoyed that Ole Roemer's temperature scale produced negative numbers as measurements in everyday situations. Fahrenheit wanted to make a new scale which produced 'friendlier' numbers. Hooray! Good intentions! Back in those days the thing to do was to pick two useful-seeming temperatures, call one of them 0 and the other 100. Fahrenheit did this, and it was called the Fahrenheit scale. The story I heard was that Fahrenheit picked the freezing point of sea water (useful for sailors, I guess) and body temperature. Wikipedia has more variations on the story if you're interested. In 1742 Anders Celsius did the same kind of thing, with the freezing point and boiling point of pure water. It was called the Celsius scale. about the lack of a temperature scale rooted at 'absolute zero', the and so forth), which is about -273 Celsius. His scale became known as the kelvin scale. The kelvin scale is still a bit arbitrary; 0 kelvin is good because you can't get any colder than no heat at all, but the magnitude of a degree kelvin is still related to the properties of water. So, now we have degrees kelvin which starts at absolute zero instead of the freezing point of water. This kind of scale is much more useful for physicists because 15 kelvin is three times as hot as 5 kelvin in a meaningful sense, because 0 kelvin _actually means_ no heat, unlike 0 Fahrenheit or 0 Celsius which just mean 'pretty cold'. But the kelvin scale never caught on in popular use, perhaps because weather reporters feel silly saying it's going to be a chilly 280 degrees today. Fahrenheit, Celsius and kelvin all measure the same property (temperature), but they use different scales to do it. They are all an abstraction of the same thing, and it's the abstraction process which introduces mathematical problems. Another way of looking at it is that Fahrenheit is a bad abstraction of the real world, precisely _because_ 15F != 5F. This unfortunate fact is because of the way the abstraction was made. I won't try to talk about things like whether particular abstractions obey associativity rules -- I think that would be a job for someone in sci.logic, and I know just enough to get myself into trouble. I apologise if that wasn't helpful for you, or you already knew all about it (I see some sci.* newsgroups in the reply-to field, I probably offended someone there with a dangerously over-simplified science story). The point I'm trying to get at is that 15F is not three times 5F because of a quirk of history, not because of an obscure mathematical principle. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> reminds me of how algebraic notation came about -- I understand that historically, it was even more confusing before Descartes or somebody started using letters and symbols instead of writing the whole thing out in everyday language! So yes, I can better appreciate my confusion now given the historical development you cite. Unfortunately, for some reason I just don't get it. Maybe I should be posting to a philosophy or even psychology newsgroup as well. I just don't understand why not having a meaningful zero (in the cases of F and C) has anything to do with why 15F is not 3X warmer than 5F. Would it be correct to simply say 15F is 10 degrees warmer than 5F, at least? So why not 15F is three times warmer than 5F? It seems that everyone is saying that without an absolute zero, the adjective warmer, even when used numerically like in 3X warmer, has no meaning...but for me, the meaning is simply arithmetic -- 5F three times gives 15F! So how about 3X more than?? Would it be correct to say 15F is 3X more than 5F? Seems like the comparative warmer is the key issue here...folks want to say that warmer must refer to some actual scale of temperature that's rather Platonic (in the sense of ideal Platonic Forms)...it seems that I'm just letting the comparative mean, in effect, 3X more arithmetically -- which certainly feels warmer to me, I'm sure -- while others insist that it must mean truly warmer vis-?-vis some Platonic Form of Temperature, some absolute notion of Temperature.... > I don't know if this is relevant to your difficulty, but my experience > as a tutor (not math) has taught me that when a technical point seems > obscure or confusing, there is usually a historical explanation. > People mess things up all the time, usually with good intentions, and > then the resulting mess has enough inertia that it can't be supplanted > by a more logical solution. I don't claim to be an expert, but perhaps > a little story about temperature measurement will make things clearer > to you. It may not; it will only explain why temperature measurement is > messed up, not what to do about it. The Fahrenheit scale was proposed by Gabriel Fahrenheit in 1724, who > (according to Wikipedia) was annoyed that Ole Roemer's temperature > scale produced negative numbers as measurements in everyday situations. > Fahrenheit wanted to make a new scale which produced 'friendlier' > numbers. Hooray! Good intentions! Back in those days the thing to do was to pick two useful-seeming > temperatures, call one of them 0 and the other 100. Fahrenheit did > this, and it was called the Fahrenheit scale. The story I heard was > that Fahrenheit picked the freezing point of sea water (useful for > sailors, I guess) and body temperature. Wikipedia has more variations > on the story if you're interested. In 1742 Anders Celsius did the same kind of thing, with the freezing > point and boiling point of pure water. It was called the Celsius scale. > about the lack of a temperature scale rooted at 'absolute zero', the > and so forth), which is about -273 Celsius. His scale became known as > the kelvin scale. The kelvin scale is still a bit arbitrary; 0 kelvin > is good because you can't get any colder than no heat at all, but the > magnitude of a degree kelvin is still related to the properties of > water. So, now we have degrees kelvin which starts at absolute zero instead of > the freezing point of water. This kind of scale is much more useful for > physicists because 15 kelvin is three times as hot as 5 kelvin in a > meaningful sense, because 0 kelvin actually means no heat, unlike 0 > Fahrenheit or 0 Celsius which just mean 'pretty cold'. But the kelvin > scale never caught on in popular use, perhaps because weather reporters > feel silly saying it's going to be a chilly 280 degrees today. Fahrenheit, Celsius and kelvin all measure the same property > (temperature), but they use different scales to do it. They are all an > abstraction of the same thing, and it's the abstraction process which > introduces mathematical problems. Another way of looking at it is that Fahrenheit is a bad abstraction of > the real world, precisely because 15F != 5F. This unfortunate fact is > because of the way the abstraction was made. I won't try to talk about > things like whether particular abstractions obey associativity rules -- > I think that would be a job for someone in sci.logic, and I know just > enough to get myself into trouble. > I apologise if that wasn't helpful for you, or you already knew all > about it (I see some sci.* newsgroups in the reply-to field, I probably > offended someone there with a dangerously over-simplified science > story). The point I'm trying to get at is that 15F is not three times > 5F because of a quirk of history, not because of an obscure > mathematical principle. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) reminds me of how algebraic notation came about -- I understand that > historically, it was even more confusing before Descartes or somebody > started using letters and symbols instead of writing the whole thing > out in everyday language! So yes, I can better appreciate my > confusion now given the historical development you cite. Unfortunately, for some reason I just don't get it. Maybe I should be > posting to a philosophy or even psychology newsgroup as well. I just > don't understand why not having a meaningful zero (in the cases of F > and C) has anything to do with why 15F is not 3X warmer than 5F. Would > it be correct to simply say 15F is 10 degrees warmer than 5F, at > least? Of course, 10? warmer is correct But if you insist that 15?F should be 3 times a warm as 5 ?F, then it would also have to be -3 times as warm as -5 ?F, which is a good deal less acceptable. So for any scale allowing both positive and negative values, it is not too sensible to allude to one measurement representing a certain number of times another. It is only when Zero is an absolute minimum (or maximum) that one measurement in that scale can be be meaningfully a number of times another. > So why not 15F is three times warmer than 5F? It seems that > everyone is saying that without an absolute zero, the adjective > warmer, even when used numerically like in 3X warmer, has no > meaning...but for me, the meaning is simply arithmetic -- 5F three > times gives 15F! But if zero is not absolute, how do you deal with -5 compared to +15? So how about 3X more than?? Would it be correct to say 15F is 3X > more than 5F? Ask yourself how it would work for 15 ?F and -5?F. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) On 12 Nov 2006 16:41:37 -0800, Prisoner at War >Platonic Form of Temperature, some absolute notion of >Temperature.... It's not Platonic - it's real warmth we're discussing. And 15F is nowhere near three times as 5F in terms of amount of heat present. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> near three times as 5F in terms of amount of heat present. I didn't mean Platonic in a dismissive way -- to Platonists, the Platonic is the only real thing around! I used that word in its absolutist sense, which seems to be the sense in which folks are thinking of three times warmer (three times the amount of heat)...if the Fahrenheit scale is not measuring heat, then what's it measuring? And if it's measuring heat (temperature -- these are synonyms here, right?), then how is 15F not 3X more (warm, 3X more warm, 3X warmer) than 5F?? Sorry, I'm sure everyone's annoyed with me now, but believe me, it's not easy being me.... === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > It's not Platonic - it's real warmth we're discussing. And 15F is nowhere > near three times as 5F in terms of amount of heat present. > I didn't mean Platonic in a dismissive way -- to Platonists, the > Platonic is the only real thing around! I used that word in its absolutist sense, which seems to be the sense > in which folks are thinking of three times warmer (three times the > amount of heat)...if the Fahrenheit scale is not measuring heat, then > what's it measuring? And if it's measuring heat (temperature -- these > are synonyms here, right?), then how is 15F not 3X more (warm, 3X more > warm, 3X warmer) than 5F?? Consider the inconsistencies that arise if you carry your method further. For example, would you say that 15F is infinitely warmer than 0F ? For another example, if you accept that 5F is the same temperature as -15C, and that 15F is about the same temperature as -9.4C, then by referential transparency would you say that -15C is about three times warmer than -9.4C ? Note that given ANY two different temperatures, one could define a temperature scale such that the second was 3 times warmer than the first by your method (even if the second were actually colder!) > Sorry, I'm sure everyone's annoyed with me now, but believe me, it's > not easy being me.... -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> further. For example, would you say that 15F is infinitely warmer than 0F ? No, unless we're doing some kind of Zeno's Paradox trick.... > For another example, if you accept that 5F is the same temperature as > -15C, and that 15F is about the same temperature as -9.4C, then by > referential transparency would you say that -15C is about three times > warmer than -9.4C ? Indeed not! But I'd say that on the Fahrenheit scale, 15F is 3x 5F, and insofar as the scale measures temperature and temperature is heat, 15F is 3x warmer than 5F! I suppose that I'm being too literal-minded, in a way, about all this.... > Note that given ANY two different temperatures, one could define a > temperature scale such that the second was 3 times warmer than the > first by your method (even if the second were actually colder!) Now see, this is just the thing -- I'm taking the scale to be the final arbiter of heat, whereas y'all seem to keep referring to some outside absolute point in determining heat. Kinda like I want to say something's legal and you folks continue to ask whether it's moral.... > -- > --------------------------- > | BBB b Barbara at LivingHistory stop co stop uk > | B B aa rrr b | > | BBB a a r bbb | Quidquid latine dictum sit, > | B B a a r b b | altum viditur. > | BBB aa a r bbb | > ----------------------------- === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > Sorry, I'm sure everyone's annoyed with me now, but believe me, it's > not easy being me.... It should be at least 3 times easier that being someone else. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> not easy being me.... It should be at least 3 times easier that being someone else. LOL Actually, I love that line from Europa Europa where the Jewish kid is asking the German soldier, who was an actor before the War, whether it's hard to be someone else. The answer was that, no, it's easier than being yourself. I'm sure psychologists would all agree with that one. Believe me, people think I'm some kind of a math nerd. I was even approached in the NYC subway once, totally out of the blue, ? propos nothing at all (I mean, it wasn't like I had a pocket protector with a million pens or something like that), by a woman who wanted help with her algebra homework! True story; scout's honor. As well, I got top grades in college maths (algebra, trig, stats) and Symbolic Logic I and II. Not that I really understood anything: I just followed the rules. As a matter of fact, I graduated with a degree in English! But I've always wondered why I could do so well, apparently, in math and yet have no idea whatsoever what I'm doing. Can you imagine acing exams on Shakespearean sonnets simply by regurgitating Cliff Notes? === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) On 12 Nov 2006 17:01:13 -0800, Prisoner at War >.if the Fahrenheit scale is not measuring heat, then >what's it measuring? Arbitrary increments of the amount of heat present above and below the freezing point of alcohol, I believe. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> bit differently than in casual, across-the-back-fence conversation. For > example, in physics, the word momentum has a definite meaning that is > not the same as its usage in ordinary conversation. In mathematics, the > word group means something different than it does in the coffee shop > or in Shakespeare. The passage you were complaining about was > attempting to get the reader to drop the ingrained habit of always > assuming that 15 something is three time 5 something, or, at least, to > stop thinking it must always mean anything. It is not that > mathematicians are insensitive to language (although some may be); > rather, it is that they want to speak with more than normal precision, > at least when they are talking about their own subject. If you are > worrying about these things for hours on end, you are spending far too > much time on inessentials and far too little time actually learning > concepts and methods. R.G. Vickson > Believe it or not, I used to be a star at a local community (junior) > college math lab with trig and algebra. I was an unpaid tutor, able to > help folks! I was rarely stumped. In college (senior, four-year > college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: > I've been applying methods successfully, but with no idea or feeling > for the concepts at all! That's why I seem to be belaboring this small > point. As well, given that it is raised at all by the author of that > text, right in the beginning within the context of a pedagogy (so > stated in his introduction) of focusing on fundamentals first, it > doesn't seem that small a point. I think I know why the author did that: he/she knows it is an issue that will bother students over and over again, so wants to deal with it right away. Before retiring, I taught Operations Research models and methods for more than 30 years, and every year, year in and year out, the same problem always arose: how to get students to stop thinking of a number of objects as necessarily being an integer. For example, a model might involve production quantities, and the variables in the model might be R = number of red widgets, B =number of blue widgets to produce per week. So, there would be constraints on manpower, money, productive capacity, raw materials, etc., and the final model might be a linear programming problem in variables R and B. Many students would insist on adding the constrtaints that R and B must be whole numbers; after all, these quantities are defined as *numbers* of objects per week. In fact, in a _linear_ model, we let R and B be fractional, and later deal with the issue of a fractional solution if it arises. The students were trying to put the cart before the horse, or to run before they could walk. Now, you seem to be bothered by several concepts of number; you mentioned zero, negative numbers, fractions, etc. Although you didn't do so, you could have also mentioned irrational numbers such as pi or the square root of 2. Where do these things come from? Well, it all has to do with extension. We start off with whole numbers, then generalize to negative integers, then generalize to fractions, then to real numbers and finally to complex numbers. Why do we do this? If we assume that many of the concepts and methods arise from practical problems, it must be the case that we use fractions because we need them, and use negative numbers because we need them. Of course, to speak of -2 apples seems nonsensical, but that might be just because we should, perhaps, have invented a different term than number. Or, it may be, as in many business and industrial situations, that an inventory of -2 apples means we have 2 apples on backorder. If my bank balance is -$1000 it means that I owe the bank $1000. Accounting statements indicate this by putting brackets around the number ($1000) instead of a minus sign, but it really amounts to the same thing. As for complex numbers: while the man on the street might find complex and/or imaginary numbers to be nonsensical, to the electical engineer a complex-valued voltage is no more mysterious than a dozen eggs and to the physicist a complex-valued quantum-mechanical wave function is no less real than a lottery ticket. To return to your original example: to speak of 15 degrees F as being 3 times as warm as 5 degrees F is wrong, since we first need a warmness measure that makes physical sense. The one that does have physical significance is the absolute temperature scale, and in those terms, 5 and 15 degrees F are only very slightly different. In casual conversation we might think about these matters differently, but the author was, after all, discussing a technical subject, in which words might be used a bit differently than in their everyday occurrences. Finally, I am not sure who is to blame, but I am not convinced that the types of scale material that is bothering you can be laid at the feet of mathematicians at all. Many of these things seem to arise in psychology or sociology and the like. If you Google the term interval scale, you will be led to several web pages that discuss scales of all types (ordinal, nominal, interval and ratio). Different scales are used in different contexts and for different reasons. The point being made by the author of the passage bothering you is that certain quantities are measured in a ratio scale, but others are not. Standard temperature measures used in meteorology are examples of quantities measured on an interval, but not ratio, scale. On the other hand, a chemical engineer would typically use temperature as measured on a ratio scale by going to degrees Kelvin or their Farenheit equivalent. Different tools for different purposes! R.G. Vickson Adjunct Professor, University of Waterloo > It is, indeed, what's been eating me > all my mathematical life -- I follow orders, but I have no idea what > they mean. But I suspect you're right: more practice. That's why I'm about to go > for a second (undergraduate) degree, in mathematics, just for the heck > of it. I feel terribly embarrassed not knowing what a fraction really > is (I know about pizza slices and all that, yeah, but I can't begin to > explain [essentially, visualize] what it means to take two-thirds of a > pie and subtract from that a hundred-fortieth of a pie, and then > multiply the result by six-fifths of a pie). I do hope it all gets > better with practice. Simply learning enough required math for the > required courses for the time required to get a course grade just isn't > doing it. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) > In a technical subject (including mathematics) we sometimes use words a > bit differently than in casual, across-the-back-fence conversation. For > example, in physics, the word momentum has a definite meaning that is > not the same as its usage in ordinary conversation. In mathematics, the > word group means something different than it does in the coffee shop > or in Shakespeare. The passage you were complaining about was > attempting to get the reader to drop the ingrained habit of always > assuming that 15 something is three time 5 something, or, at least, to > stop thinking it must always mean anything. It is not that > mathematicians are insensitive to language (although some may be); > rather, it is that they want to speak with more than normal precision, > at least when they are talking about their own subject. If you are > worrying about these things for hours on end, you are spending far too > much time on inessentials and far too little time actually learning > concepts and methods. R.G. Vickson > Believe it or not, I used to be a star at a local community (junior) > college math lab with trig and algebra. I was an unpaid tutor, able to > help folks! I was rarely stumped. In college (senior, four-year > college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: > I've been applying methods successfully, but with no idea or feeling > for the concepts at all! That's why I seem to be belaboring this small > point. As well, given that it is raised at all by the author of that > text, right in the beginning within the context of a pedagogy (so > stated in his introduction) of focusing on fundamentals first, it > doesn't seem that small a point. It is, indeed, what's been eating me > all my mathematical life -- I follow orders, but I have no idea what > they mean. But I suspect you're right: more practice. That's why I'm about to go > for a second (undergraduate) degree, in mathematics, just for the heck > of it. I feel terribly embarrassed not knowing what a fraction really > is (I know about pizza slices and all that, yeah, but I can't begin to > explain [essentially, visualize] what it means to take two-thirds of a > pie and subtract from that a hundred-fortieth of a pie, and then > multiply the result by six-fifths of a pie). I do hope it all gets > better with practice. Simply learning enough required math for the > required courses for the time required to get a course grade just isn't > doing it. Part of the problem is that many people think that mathematics is more like music than medicine. One can listen to and appreciate music quite easily and without much mental effort, though it does require effort to produce it, so that one gets the notion that one knows a lot more about it than is actually the case. Most of us, without having gone though the intensive study of a medical degree of some sort are quite happy to acknowledge our lack of any deep understanding of most things medical. In that sense math is more like medical studies than musical ones. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> What an interesting idea! Actually, I think that before the new math and math anxiety revolutions, most folks did assume mathematics to be more like medicine than music, something requiring not only expertise but a rare kind of talent. I am an adult learner who was a pretty successful (~87% A's and B's) math parrot all my life but now I want to really understand this stuff inside-out, and I'm making such an attempt on the assumption that math is indeed more like music than medicine (though I choke on it the way I choke on medicine!). What's perhaps even more puzzling to me, now, is why I'm so befuddled by what appears to be such a simple petty matter. There is a crucial assumption that's hidden from me which prevents me from my eureka! moment here...I sense what folks are saying about scales and arbitrary zeros versus meaningful zeros but it's not exactly gelling together just yet.... > Part of the problem is that many people think that mathematics is more > like music than medicine. One can listen to and appreciate music quite easily and without much > mental effort, though it does require effort to produce it, so that one > gets the notion that one knows a lot more about it than is actually the > case. Most of us, without having gone though the intensive study of a medical > degree of some sort are quite happy to acknowledge our lack of any deep > understanding of most things medical. In that sense math is more like medical studies than musical ones. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) On 12 Nov 2006 16:49:47 -0800, Prisoner at War What an interesting idea! Actually, I think that before the new math and math anxiety >revolutions, most folks did assume mathematics to be more like >medicine than music, something requiring not only expertise but a rare >kind of talent. I am an adult learner who was a pretty successful (~87% A's and B's) >math parrot all my life but now I want to really understand this >stuff inside-out, and I'm making such an attempt on the assumption that >math is indeed more like music than medicine (though I choke on it the >way I choke on medicine!). What's perhaps even more puzzling to me, now, is why I'm so befuddled >by what appears to be such a simple petty matter. There is a crucial >assumption that's hidden from me which prevents me from my eureka! >moment here...I sense what folks are saying about scales and arbitrary >zeros versus meaningful zeros but it's not exactly gelling together >just yet.... Just accept that you're thick. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> Just accept that you're thick. You'll never believe what I do for a living. I wonder as to the nature of confusion. It seems that, mental infirmaties aside, confusion exists simply because all the pieces of the puzzle aren't present. Confounding the confusion is not knowing what's not known (? la Rumsfeld's known unknowns and unknown unknowns). Think about it: why have some folks developed so much math and others have not? Something like zero seems so simple, and yet it was tens of thousands of years, apparently, before anyone came up with the idea! Was the human race, even Pythagoras himself, thick? In any case, I must press on, press my case to the annoyance of all. I've got to know, and know why I don't know! I take comfort in the fact that, just as a teenager knows as much as Pythagoras ever did (if that teen studies, anyway), one day a typical teen will know as much as, say, Ed Witten does (and just how's he determined to be Einstein's true successor, anyway???). === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> bit differently than in casual, across-the-back-fence conversation. For > example, in physics, the word momentum has a definite meaning that is > not the same as its usage in ordinary conversation. In mathematics, the > word group means something different than it does in the coffee shop > or in Shakespeare. The passage you were complaining about was > attempting to get the reader to drop the ingrained habit of always > assuming that 15 something is three time 5 something, or, at least, to > stop thinking it must always mean anything. It is not that > mathematicians are insensitive to language (although some may be); > rather, it is that they want to speak with more than normal precision, > at least when they are talking about their own subject. If you are > worrying about these things for hours on end, you are spending far too > much time on inessentials and far too little time actually learning > concepts and methods. R.G. Vickson > Believe it or not, I used to be a star at a local community (junior) > college math lab with trig and algebra. I was an unpaid tutor, able to > help folks! I was rarely stumped. In college (senior, four-year > college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: > I've been applying methods successfully, but with no idea or feeling > for the concepts at all! That's why I seem to be belaboring this small > point. As well, given that it is raised at all by the author of that > text, right in the beginning within the context of a pedagogy (so > stated in his introduction) of focusing on fundamentals first, it > doesn't seem that small a point. I think I know why the author did that: he/she knows it is an issue that will bother students over and over again, so wants to deal with it right away. Before retiring, I taught Operations Research models and methods for more than 30 years, and every year, year in and year out, the same problem always arose: how to get students to stop thinking of a number of objects as necessarily being an integer. For example, a model might involve production quantities, and the variables in the model might be R = number of red widgets, B =number of blue widgets to produce per week. So, there would be constraints on manpower, money, productive capacity, raw materials, etc., and the final model might be a linear programming problem in variables R and B. Many students would insist on adding the constrtaints that R and B must be whole numbers; after all, these quantities are defined as *numbers* of objects per week. In fact, in a _linear_ model, we let R and B be fractional, and later deal with the issue of a fractional solution if it arises. The students were trying to put the cart before the horse, or to run before they could walk. Now, you seem to be bothered by several concepts of number; you mentioned zero, negative numbers, fractions, etc. Although you didn't do so, you could have also mentioned irrational numbers such as pi or the square root of 2. How did all these thing come to be? Well, it all has to do with extension. We start off with whole numbers, then generalize to negative integers, then generalize to fractions, then to real numbers and finally to complex numbers. Why do we do this? If we assume that many of the concepts and methods arise from practical problems, it must be the case that we use fractions because we need them, and use negative numbers because we need them. Ditto for complex numbers: to an electical engineer, a complex voltage is no more mysterious than a dozen eggs, and to a physicist, a complex-valued quantum-mechanical wave function is just as meaningful as a lottery ticket. Of course, to speak of -2 apples seems nonsensical, but that might be just because we should, perhaps, have invented a different term than number. Or, it may be, as in many business and industrial situations, that an inventory of -2 apples means we have 2 apples on backorder. If my bank balance is -$1000 it means that I owe the bank $1000. Accounting statements indicate this by putting brackets around the number ($1000) instead of a minus sign, but it really amounts to the same thing. To return to your original example: to speak of 15 degrees F as being 3 times as warm as 5 degrees F is wrong, since we first need a warmness measure that makes physical sense. The one that does have physical significance is the absolute temperature scale, and in those terms, 5 and 15 degrees F are only very slightly different. In casual conversation we might think about these matters differently, but the author was, after all, discussing a technical subject, in which words might be used a bit differently than in their everyday occurrences. R.G. Vickson Adjunct Professor, University of Waterloo > It is, indeed, what's been eating me > all my mathematical life -- I follow orders, but I have no idea what > they mean. But I suspect you're right: more practice. That's why I'm about to go > for a second (undergraduate) degree, in mathematics, just for the heck > of it. I feel terribly embarrassed not knowing what a fraction really > is (I know about pizza slices and all that, yeah, but I can't begin to > explain [essentially, visualize] what it means to take two-thirds of a > pie and subtract from that a hundred-fortieth of a pie, and then > multiply the result by six-fifths of a pie). I do hope it all gets > better with practice. Simply learning enough required math for the > required courses for the time required to get a course grade just isn't > doing it. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> bit differently than in casual, across-the-back-fence conversation. For > example, in physics, the word momentum has a definite meaning that is > not the same as its usage in ordinary conversation. In mathematics, the > word group means something different than it does in the coffee shop > or in Shakespeare. The passage you were complaining about was > attempting to get the reader to drop the ingrained habit of always > assuming that 15 something is three time 5 something, or, at least, to > stop thinking it must always mean anything. It is not that > mathematicians are insensitive to language (although some may be); > rather, it is that they want to speak with more than normal precision, > at least when they are talking about their own subject. If you are > worrying about these things for hours on end, you are spending far too > much time on inessentials and far too little time actually learning > concepts and methods. R.G. Vickson > Believe it or not, I used to be a star at a local community (junior) > college math lab with trig and algebra. I was an unpaid tutor, able to > help folks! I was rarely stumped. In college (senior, four-year > college), I posted top grades in math, much as I've done all my life. But you see, that's just the problem that's been nagging me all along: > I've been applying methods successfully, but with no idea or feeling > for the concepts at all! That's why I seem to be belaboring this small > point. As well, given that it is raised at all by the author of that > text, right in the beginning within the context of a pedagogy (so > stated in his introduction) of focusing on fundamentals first, it > doesn't seem that small a point. I think I know why the author did that: he/she knows it is an issue that will bother students over and over again, so wants to deal with it right away. Before retiring, I taught Operations Research models and methods for more than 30 years, and every year, year in and year out, the same problem always arose: how to get students to stop thinking of a number of objects as necessarily being an integer. For example, a model might involve production quantities, and the variables in the model might be R = number of red widgets, B =number of blue widgets to produce per week. So, there would be constraints on manpower, money, productive capacity, raw materials, etc., and the final model might be a linear programming problem in variables R and B. Many students would insist on adding the constrtaints that R and B must be whole numbers; after all, these quantities are defined as *numbers* of objects per week. In fact, in a _linear_ model, we let R and B be fractional, and later deal with the issue of a fractional solution if it arises. The students were trying to put the cart before the horse, or to run before they could walk. Now, you seem to be bothered by several concepts of number; you mentioned zero, negative numbers, etc. Although you didn't do so, you could have also mentioned fractions (what does it mean to have 2/7 of an apple pie?), and irrational numbers such as pi or the square root of 2. Well, it all has to do with extension. We start off with whole numbers, then generalize to negative integers, then generalize to fractions, then to real numbers and finally to complex numbers. Why do we do this? If we assume that many of the concepts and methods arose from practical problems, it must be the case that we use fractions because we need them, and use negative numbers because we need them. Of course, to speak of -2 apples seems nonsensical, but that might be just because we should, perhaps, have invented a different term than number. Or, it may be, as in many business and industrial situations, that an inventory of -2 apples means we have 2 apples on backorder. If my bank balance is -$1000 it means that I owe the bank $1000. Accounting statements indicate this by putting brackets around the number ($1000) instead of a minus sign, but it really amounts to the same thing. To return to your original example: to speak of 15 degrees F as being 3 times as warm as 5 degrees F is wrong, since we first need a warmness measure that makes physical sense. The one that does have physical significance is the absolute temperature scale, and in those terms, 5 and 15 degrees F are only very slightly different. In casual conversation we might think about these matters differently, but the author was, after all, discussing a technical subject, in which words might be used a bit differently than in their everyday occurrences. R.G. Vickson Adjunct Professor, University of Waterloo > It is, indeed, what's been eating me > all my mathematical life -- I follow orders, but I have no idea what > they mean. But I suspect you're right: more practice. That's why I'm about to go > for a second (undergraduate) degree, in mathematics, just for the heck > of it. I feel terribly embarrassed not knowing what a fraction really > is (I know about pizza slices and all that, yeah, but I can't begin to > explain [essentially, visualize] what it means to take two-thirds of a > pie and subtract from that a hundred-fortieth of a pie, and then > multiply the result by six-fifths of a pie). I do hope it all gets > better with practice. Simply learning enough required math for the > required courses for the time required to get a course grade just isn't > doing it. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> Here is a simple example. Suppose I have 20 apples, but redefine the scale such that 10 is zero. On my redefined scale, where zero is arbitrary and meaningless, the count is 10. Now I add 10 apples to my original number. On my new scale, my count is 20, numerically 2X as much. But that does not reflect the real change in the number of apples, that is from 20 to 30. I can say that I have exactly 10 more apples, but not twice as many. That is the scale is interval, not ratio. The issue is the relationship with some real and meaningful zero point. Scales like F for temperature set an arbitrary zero, that is not related to anything meaningful. This does not mean they are not useful. === Subject: Re: Arrgghh -- Semantics and Mathematical Homonyms (Watts on second?) <4rotagFsbp26U1@mid.individual.net> On my redefined scale, where zero is arbitrary and meaningless, the > count is 10. Now I add 10 apples to my original number. On my new > scale, my count is 20, numerically 2X as much. Um, don't you mean 40 on this redefined scale?? Or are you starting with no apples (instead of the 20 you seem to say you originally had)??? > But that does not > reflect the real change in the number of apples, that is from 20 to 30. > I can say that I have exactly 10 more apples, but not twice as many. Ah, okay, I see, kind of like z-scores in Statistics, converting one scale into another...except with z-scores, the intervals have a one-to-one correspondance across scales...?? > That is the scale is interval, not ratio. HMMM!!! Interval...that's width between one tick and the next (on a ruler, say)...but what's a ratio here, then?? Hmm, it's my old problem with fractions again -- no wonder I'm confused.... I sense what you're saying, though, even if I can't quite put my finger around this (amazing I can get A's in math for years and not have any idea what I'm doing).... Ratio...interval...ratio is numerically meaningful width between two points...interval is any arbitrary width between two points...right?? > The issue is the relationship with some real and meaningful zero point. > Scales like F for temperature set an arbitrary zero, that is not > related to anything meaningful. This does not mean they are not useful. Why was it set to something arbitrary when it seems so much more logical that if they're measures of heat, their original referent ought to be the absolute absence of heat??? Kinda like how the original referent of height is the total lack of height.... === Subject: How can I solve negative factorial that now i am sending with this title? (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? === Subject: Re: How can I solve negative factorial that now i am sending with this title? >(-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? > > As others have pointed out, a! is not defined when a is a negative integer. (One could argue that a! is defined *only* for a a natural number, but a natural extension is a! = Gamma(a+1).) However, I have seen the binomial coeffcient C(a,n) defined for any a, by the formula a (a-1) (a-2) ... (a-n+1) / n! For example, this comes up in the binomial series expansion of f(x) = (b + x)^a when a is not a natural number. (Would that make a an *unnatural* number?) I may have seen this come up in probability as well, but I don't remember exactly. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: How can I solve negative factorial that now i am sending with this title? > (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? > Binomial coefficient (-2k-1) choose n cannot be defined by division of factorials. In general, define binomial coefficient t choose n by: t*(t-1)*(t-1) ... [n factors] / n! For example: -3 choose 2 = (-3)*(-4)/2! = 6, no need for negative factorials at all. === Subject: Re: How can I solve negative factorial that now i am sending with this title? > (-2k-1)! / ( (n!) * (-2k-1-n)! ) As others have pointed out, the factorial is undefined for negative integer values. However, if we proceed formally, we can recognize that your desired quantity is the coefficient of x^n in the binomial expansion of (1+x)^(-2k-1). Therefore, if it has any sensible meaning at all, this would likely be (-2k-1)*(-2k-2)*...*(-2k-1-(n-1))/n! = (-2k-1)*...*(-2k-n)/n!. R.G. Vickson can anyone help me? === Subject: Re: How can I solve negative factorial that now i am sending with this title? >> (-2k-1)! / ( (n!) * (-2k-1-n)! ) As others have pointed out, the factorial is undefined for negative > integer values. However, if we proceed formally, we can recognize > that your desired quantity is the coefficient of x^n in the binomial > expansion of (1+x)^(-2k-1). Therefore, if it has any sensible meaning > at all, this would likely be (-2k-1)*(-2k-2)*...*(-2k-1-(n-1))/n! = > (-2k-1)*...*(-2k-n)/n!. Has anyone heard of Smarandacheials? http://www.gallup.unm.edu/~smarandache/Smarandacheials.htm Smarandache defines a k-factorial as n(n-k)(n-2k)., terminating when n-xk is positive and n-(x+1)k is 0 or negative. Smarandacheials extend this definition into the negative numbers such that the factorial terminates when |n-xk| is less than or equal to n and |n-(x+1)k| is greater than n. http://www.users.globalnet.co.uk/~perry/maths/kfactorials/kfactorials.htm Nick === Subject: Re: How can I solve negative factorial that now i am sending with this title? > (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? No. The factorial function is not defined for numbers less than 0. Even the extension of it (the gamma function) is not defined at negative integers. coefficient (the number of ways to choose n objects from -2k-1 objects, were that possible), so perhaps you made an error when doing the work leading up to this expression. === Subject: Re: How can I solve negative factorial that now i am sending with this title? > (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? No. The factorial function is not defined for numbers less than 0. Even > the extension of it (the gamma function) is not defined at negative > integers. Although the factorial is often considered to have just the set of nonnegative integers as its domain, z! can also be considered to be _exactly the same_ as Gamma(z + 1). Furthermore, although the Gamma function is not finite at nonpositive integers, one can still make sense of the original question by just extending by limit. Doing so, the result is (-1)^n * Product(i=1..n, 2k/i + 1). > coefficient (the number of ways to choose n objects from -2k-1 objects, > were that possible), so perhaps you made an error when doing the work > leading up to this expression. There might have been an error, sure. But that's not necessarily the case. W. Cantrell === Subject: Re: How can I solve negative factorial that now i am sending with this title? no there is no mistake.my rpoblem is true but i couldnt solve it. The Qurqirish Dragon yazdi: > (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? No. The factorial function is not defined for numbers less than 0. Even > the extension of it (the gamma function) is not defined at negative > integers. coefficient (the number of ways to choose n objects from -2k-1 objects, > were that possible), so perhaps you made an error when doing the work > leading up to this expression. === Subject: Re: How can I solve negative factorial that now i am sending with this title? n,k > o Fem beginner level yazd[CapitalYAcute]: > (-2k-1)! / ( (n!) * (-2k-1-n)! ) can anyone help me? === Subject: Re: Modified Zhukovski function Kostyantyn Yusenko a e'crit : > Hi all > > I need some help in modified Zhukovski function. Zhukovski function > is defined as follows: > > w=1/2(z+1/z). > > I have modified function: > > w=1/a(z+b/z). > > Could you please write me some information or link to information. > > > > Kostya. > let's note w_ab(z) = 1/a(z+b/z) your modified function then : w_ab(z) = b^(1/2)/a (z/b^(1/2)+b^(1/2)/z) or w_ab(z) = 2 b^(1/2)/a w(z/b^(1/2)) with w(z) the Zhukovski function and b^(1/2) one of the complex roots of x^2 = b (that is +/-sqrt(b) if b is a non-negative real) this is of the kind w_ab(z) = u w(v z) that is the combination of : - the multiplication by the complex (real) value v= 1/b^(1/2) - the Zhukovsky (Joukowsky) transform w - the multiplication by the complex (real) value u= 2 b^(1/2)/a the case a=2 of your w_ab function is studied here : http://www.scs-europe.net/services/ecms2005/pdf/se-05.pdf concerning the Joukowski Airfoil itself see for example : http://jwilson.coe.uga.edu/olive/Joukowski.Web/Joukowski.Paper.html or the links here Hoping it helped, Raymond === Subject: nice,erdos,can. math bull show 2aCa doesnt divide 2nCn where a is natural no. bet. n/2 and n.nCr- n choose r === Subject: only theorem we can write all natural number(only bigger then Zero) two or two's base.I must find this theorem proof.But only child for 14-15 years. Example: 3=2^0+2^1 17=2^4+2^0 31=2^4+2^3+2^2+2^1+2^0 === Subject: Re: only theorem > we can write all natural number(only bigger then Zero) two or two's > base.I must find this theorem proof.But only child for 14-15 years. > Example: > 3=2^0+2^1 > 17=2^4+2^0 > 31=2^4+2^3+2^2+2^1+2^0 Find the largest power of two that divides into your number. It can only go once into the number. Substract that power of two from your number. Repeat until you have nothing left. while (n>0) { let m be largest natural number such that 2^m <= n; output(2^m); n = n - 2^m; } === Subject: Re: only theorem Originator: pouya@localhost [OrhanGOKCE ] >we can write all natural number(only bigger then Zero) two or two's >base.I must find this theorem proof.But only child for 14-15 years. >Example: >3=2^0+2^1 >17=2^4+2^0 >31=2^4+2^3+2^2+2^1+2^0 > You can prove this by induction. Let n be your number. The theorem is obviously true for n=1. Now assume that the theorem holds for all natural numbers < n. We either have n=2m or n=2m+1 for some m < n; so.... -- Pouya D. Tafti p dot d dot tafti at ieee dot org === Subject: Prime numbers I want to know your believes about this idea: We have to throw away all of those problems about the infinity sequences of prime numbers. For example: Goldbach Conjecture, Mersen numbers(2^p-1),... Omar.Hosseiny === Subject: Re: Prime numbers I want to know your believes about this idea: > We have to throw away all of those problems about the infinity > sequences of prime numbers. For example: Goldbach Conjecture, Mersen numbers(2^p-1),... > Everyone is free to do that. Some mathematicians study other things. The vast majority of humanity certainly has no interest in any of it. But some people pursue those problems anyway. Then one day, someone like Wiles spends 7 years working on FLT and not telling anyone ... and nails it. That's why mathematicians tackle the hard problem. For knowledge, sure. But also for glory. Star mathematicians have star egos. If someone told them not to work on a problem, that's exactly the problem they'd work on ... and they'd solve it. === Subject: Re: Prime numbers I want to know your believes about this idea: > We have to throw away all of those problems about the infinity > sequences of prime numbers. For example: Goldbach Conjecture, Mersen numbers(2^p-1),... You've been posting stuff like this here since August. Time to put up or shut up. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Prime numbers > I want to know your believes about this idea: > We have to throw away all of those problems about the infinity > sequences of prime numbers. For example: Goldbach Conjecture, Mersen numbers(2^p-1),... You've been posting stuff like this here since August. > Time to put up or shut up. Put up? I don't actually see any claims that he has anything to put up. I can't take issue with the alternative option, though. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We T, Inc./. === Subject: Re: Prime numbers Student of Math skrev: I want to know your believes about this idea: > We have to throw away all of those problems about the infinity > sequences of prime numbers. For example: Goldbach Conjecture, Mersen numbers(2^p-1),... Omar.Hosseiny I think it's a very silly idea. --- J K Haugland http://home.no.net/zamunda === Subject: HELP We call a square magic if the sum of the numbers in every row, column and diagonal is equal Now, given a Magic Square a b c d e f g h i i. e. a+b+c = d+e+f = g+h+i = a+e+i = c+e+g = a+d+g = b+e+h = c+f+i Proove that 2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 You can assume, that 2*(a+c+g+i)=b+d+f+h+4*e (I had prooved it, it's easy) === Subject: HELP We call a square magic if the sum of the numbers in every row, column and diagonal is equal Now, given a Magic Square a b c d e f g h i i. e. a+b+c = d+e+f = g+h+i = a+e+i = c+e+g = a+d+g = b+e+h = c+f+i Proove that 2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 You can assume, that 2*(a+c+g+i)=b+d+f+h+4*e (I had prooved it, it's easy) === Subject: Re: HELP Assume that the sum of every row, column and diagonal is z. You can then show that it must hold: a = 3x+3y-z b=4z-6x-3y c=3x d=2z-3y e=z f=3y g=2z-3x h=6x+3y-2z i=3z-3x-3y Plugging that into your equation should give the desired result Michael > We call a square magic if the sum of the numbers in every row, column > and diagonal is equal > Now, given a Magic Square > a b c > d e f > g h i > i. e. a+b+c = d+e+f = g+h+i = a+e+i = c+e+g = a+d+g = b+e+h = c+f+i > Proove that > 2*(a^3+c^3+g^3+i^3)=b^3+d^3+f^3+h^3+4*e^3 > You can assume, that > 2*(a+c+g+i)=b+d+f+h+4*e (I had prooved it, it's easy) > === Subject: non-injectivity of pullback here's one that i have been scratching my head over : view 4-dim affine space as a matrix: [a1 a2] [a3 a4] with entries in an alg closed field F. Call this affine variety Y, let's say. Define X = [x1 x2] [x3 x4] a matrix with indetrminate entires. Also define f : X - > Y f(X) = X^2; (matirx mult) show that the map is dominant (closure of f(X) = Y), but not finite (Aff(X) is not finitely generated as a f*(Aff(Y)), where f* is the pullback of Aff(Y)). Easy to see that f is dominat since any 2x2 matrix is a square of some 2x2 matrix in an algebraically closed field. The not finite part is not obvious at all. I tried pulling back thru f, getting : X^2 = [ (x1^2 + x2.x3) (x1.x2 + x2.x4)] [ (x1.x3 + x3.x4) (x2.x3 + x4^2) ] but i don't see which non-zero function g : Y -> F, is such that g(X^2) = 0; ideas would be helpful. === Subject: Re: non-injectivity of pullback I am having the impression that the notation is a bit screwed up here. So I suggest to straighten it out first. :-) > here's one that i have been scratching my head over : > view 4-dim affine space as a matrix: > [a1 a2] > [a3 a4] > with entries in an alg closed field F. Call this affine variety Y, > let's say. Define > X = [x1 x2] > [x3 x4] > a matrix with indetrminate entires. Also define f : X - > Y Maybe you mean f: Y -> Y here? > f(X) = X^2; (matirx mult) show that the map is dominant (closure of f(X) = Y), rather closure of f(Y) = Y ? > but not finite > (Aff(X) Aff(Y) ? > is not finitely generated as a f*(Aff(Y)), where f* is the > pullback of Aff(Y)). > Easy to see that f is dominat since any 2x2 matrix is a square of some > 2x2 matrix in an algebraically closed field. The not finite part is > not obvious at all. I tried pulling back thru f, getting : > X^2 = [ (x1^2 + x2.x3) (x1.x2 + x2.x4)] > [ (x1.x3 + x3.x4) (x2.x3 + x4^2) ] but i don't see which non-zero function g : Y -> F, is such that g(X^2) > = 0; > ideas would be helpful. J. === Subject: Re: non-injectivity of pullback <45576499.50108@web.de > I am having the impression that the notation is a bit screwed up here. > So I suggest to straighten it out first. :-) here's one that i have been scratching my head over : > view 4-dim affine space as a matrix: > [a1 a2] > [a3 a4] > with entries in an alg closed field F. Call this affine variety Y, > let's say. Define > X = [x1 x2] > [x3 x4] > a matrix with indetrminate entires. Also define f : X - > Y Maybe you mean f: Y -> Y here? f(X) = X^2; (matirx mult) show that the map is dominant (closure of f(X) = Y), rather closure of f(Y) = Y ? but not finite > (Aff(X) Aff(Y) ? is not finitely generated as a f*(Aff(Y)), where f* is the > pullback of Aff(Y)). > Easy to see that f is dominat since any 2x2 matrix is a square of some > 2x2 matrix in an algebraically closed field. The not finite part is > not obvious at all. I tried pulling back thru f, getting : > X^2 = [ (x1^2 + x2.x3) (x1.x2 + x2.x4)] > [ (x1.x3 + x3.x4) (x2.x3 + x4^2) ] but i don't see which non-zero function g : Y -> F, is such that g(X^2) > = 0; > ideas would be helpful. J. yep, yep, yep to all of your comments. Aff(Y) is the set of polynomial funtions on Y. === Subject: Re: non-injectivity of pullback >> I am having the impression that the notation is a bit screwed up here. >> So I suggest to straighten it out first. :-) > here's one that i have been scratching my head over : > view 4-dim affine space as a matrix: > [a1 a2] > [a3 a4] > with entries in an alg closed field F. Call this affine variety Y, > let's say. Define > X = [x1 x2] > [x3 x4] > a matrix with indetrminate entires. Also define f : X - > Y >> Maybe you mean f: Y -> Y here? > f(X) = X^2; (matirx mult) show that the map is dominant (closure of f(X) = Y), >> rather closure of f(Y) = Y ? > but not finite What does this mean? That, say, every fiber of f is finite? > (Aff(X) >> Aff(Y) ? > is not finitely generated as a f*(Aff(Y)), where f* is the > pullback of Aff(Y)). > Easy to see that f is dominat since any 2x2 matrix is a square of some > 2x2 matrix in an algebraically closed field. Are you sure that f is surjective? What about the preimage of the matrix with x1=x2=x4=1, x3=0 over a field of characteristic 2? Shouldn't we better find a Zariski open set U which is contained in f(Y)? Note that since Y is irreducible and if U is non-void, then U is dense, hence f dominant. > The not finite part is > not obvious at all. I tried pulling back thru f, getting : > X^2 = [ (x1^2 + x2.x3) (x1.x2 + x2.x4)] > [ (x1.x3 + x3.x4) (x2.x3 + x4^2) ] but i don't see which non-zero function g : Y -> F, is such that g(X^2) > = 0; > ideas would be helpful. >> J. yep, yep, yep to all of your comments. Aff(Y) is the set of polynomial > funtions on Y. Ok. Please see my questions above. HTH. J. === Subject: Problem I need help for this problem. Let 1+1/2+1/3+...+1/n=an/bn, where an/bn is (an, bn)=1. Proofe that there are infinitely many natural n, such that bn+1 < bn. Narek === Subject: Re: Problem > Let 1+1/2+1/3+...+1/n=an/bn, where an/bn is (an, bn)=1. Proofe that > there are infinitely many natural n, such that bn+1 < bn. Interesting. The table at http://www.research.att.com/~njas/sequences/b002805.txt shows this is true for n + 1 = 6, 18, 20, 21, 33, 42, 54, 63, 66, 77, 100, 110, 120, 156, 162, 189, and 198 table goes up to 200). Compare this to A074791 Numbers n such that n does not divide the denominator of the n-th Harmonic number. 6, 18, 20, 21, 33, 42, 54, 63, 66, 77, 100, 110, 120, 156, 162, 189, 198, 272, 294, 336, 342, 363, 377, 435, 486, 500, 506, 559, 567, 594, 600, 610, 629, 685, 703, 812, 847, 880, 924, 930, 957, 1067, 1166, 1210, 1243, 1247, 1287, 1320, 1332, 1458, 1590, 1640 More information at http://www.research.att.com/~njas/sequences/A074791 -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Program Synthesis and formallization? A: Your system. > * No proofs of correctness. > B: Coq (a Curry-Howard based proof checker). > * Rigourous underlying theory. >> Not really. Quite the opposite. It starts with this weird idea of a >> proof that has different rules than what proving means. >> That's constructive proof. Oh, it has a name? Ok, then it's good. (This actually brings up a relevant point - superficiality. If it's > published, if the author is well-known, if the author works for a > well-known school, then it's gold. Add to that, if it has a > buzz-word.) wake up and smell the coffee; if you want to, that is ... >> It says >> that Pv~P is not provable! >> You've learned something, then. I'm not sure if knowing that Pv~P is unprovable is all that useful, > to tell the truth. It's crucial to the claim being made, so just keep it in mind. >> Then we're supposed to believe that these >> rules produce a computer program from the assertion that (all x)(exists >> y)R(x,y) when this wff is not even necessarily true. >> But not enough -- >> you don't get the program from the assertion, or from the >> truth of the assrtion (in your terms). >> The program comes from the *constructive proof* of the assertion. > So Pv~P is not provable, but statements that aren't true must be (in > order to synthesize the program for x=(y*y))? No; > Could we reverse these > rules a bit? (Although that does show that a system can be consistent > but not sound.) > No offense, but I think that *constructive proof* is *warm and > squishy*. certainly CB's view of it is squishy. >> Your failure to provide even the hint of a reason why this latter >> claim is problematic is noted yet again. I guess you disposed of those objections handily! Your failure to provide even the hint of a reason why the claim above is problematic is noted yet again. > So what happened to the programs to decide and list factors? You're still quibbling about the simple examples given so far. C-B groente > -- Sander > -- >> Alan Smaill > -- Alan Smaill === Subject: Re: tensor product twister > >> If A and B are algebras (over an algebraically closed field F) with no >> nilpotent elements, can we say that A*B (tensor product) has no >> nilpotent elements? I thought that should be easy, but can not say that >> since linear combination of tensors can still be nilpotent. Is there a >> theorem that rules that out? Do there exist such algebras, apart from direct sums of copies of F? > I'm assuming you mean finite-dimensional algebras? > Isn't such an algebra with zero radical a direct sum of matrix rings? > In which case I would have thought there were always nilpotent elements, > unless they were all 1x1 matrices. I may be off-beam here - I haven't thought carefully about it. I just checked up on this, and I seem to be right. Your algebras are certainly semisimple. (An algebra is semisimple if it has no non-trivial nilpotent ideal.) Wedderburn showed that a semisimple algebra is a direct sum of simple algebras and that a simple algebra is a matrix algebra over a division ring (which in your case must be a field) . So it seems that an algebra with no nilpotent elements over an algebraically closed field F must be a direct sum of copies of F, in which case your result is immediate. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: tensor product twister > So it seems that an algebra with no nilpotent elements > over an algebraically closed field F > must be a direct sum of copies of F, > in which case your result is immediate. Is F[X,Y]/(Y-X^2) (integral domain) a good fit with your outcome? === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? [Tim Peters] > ... > The less interesting (and by far, if you ask me) part is then proving > that 7625597485199 (3^(3^3) + 8*3^3 - 4) is prime -- unless Christopher > has an elegant way of knowing that I'm not seeing. Of course it's > dead easy to prove that 7625597485199 is prime by using appropriate > software. Oops! That was Phil's variation. Christopher's original used p^(p^p) + 8*p^p + 26, so the tedious part of the original is proving that 7625597485229 is prime -- again unless he has an elegant way I'm not seeing. === Subject: Re: p and p^6 + 6008 prime => is p^(p^p) + 8 p^p + 26 necessarily prime? > [Tim Peters] > ... > The less interesting (and by far, if you ask me) part is then proving > that 7625597485199 (3^(3^3) + 8*3^3 - 4) is prime -- unless Christopher > has an elegant way of knowing that I'm not seeing. Of course it's > dead easy to prove that 7625597485199 is prime by using appropriate > software. Oops! That was Phil's variation. Christopher's original used p^(p^p) + > 8*p^p + 26, so the tedious part of the original is proving that > 7625597485229 is prime -- again unless he has an elegant way I'm not > seeing. The interesting part is confusing innocent bystanders on Usenet! Hoorah for me! Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We T, Inc./. === Subject: Re: metric spaces and cauchy days. My association with the Department is that of an alumnus. >On 2006-11-10 12:36:26 -0500, magidin@math.berkeley.edu (Arturo Magidin) said: Continuity is not enough. You need the spaces to be complete too. >> >> Nonsense. >(That's certainly sufficient, but not sure if that's necessary >> though). >> >> It's not sufficient either. Hmm. Suppose f:X->Y is continuous where X is complete. Isn't a Cauchy >sequence (x_n)_n in X would converge to a limit x in X too? Then >wouldn't (f(x_n))_n -> f(x) too? So, (f(x_n))_n is Cauchy in Y too? Oh, I remember now what I meant when I said nonsense. Since the question asked for an example where the continuous image of a Cauchy sequence was not a Cauchy sequence, I thought that your comment that continuity is not enough, you need the spaces to be complete meant that you needed the function to be continuous and the spaces to be complete in order to be able to construct an example. I admit it would be a rather silly assertion, and I should have realized that you meant the above instead. Anyway, lots of confusion, and I issue lots of apologies for my part in that confusion. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: infinteely many primes ... I came up with the following proof that the number of primes is infinite: ---------------------------------------------------------------------------- --- Suppose there are just finitely many primes, p_1, p_2, ..., p_n. First prove (to illustrate idea): the sum of all numbers of the form 1/(2^a * 3^b) is finite, where a and b are nonnegative integers. Let S = Sum[b:0-->inf] Sum[a:0-->inf] 1/(2^a * 3*b). Then S = Sum[b:0-->inf] (1/3^b) * Sum[a:0-->inf](1/2^a) = Sum[b:0-->inf] (1/3^b) * 2 = 2 * 3/2 = 3. QED. Then prove by essentially same method (iterated sums and the geometric series): the sum of all numbers of the form 1/(p_1^a_1 * p_2^a_2 * ... * p_n^a_n) is finite, where the a_i are nonnegative integers. But the sum of all numbers of the above form is just the harmonic series. The sum of the harmonic series is infinite. Therefore the original assumption, that there are only finitely many primes, is incorrect. QED. -------------------------------------------------------------------- I am guessing this is already a well-known proof and is contained in some weighty tome - does anyone know a reference? Marcus. === Subject: Re: infinteely many primes ... I am guessing this is already a well-known proof and is > contained in some weighty tome - does anyone know a > reference? This is Euler's proof of the infinitude of primes, and is mentioned here: http://scienceworld.wolfram.com/biography/Euler.html --- J K Haugland http://home.no.net/zamunda === Subject: What is the answer of the following integral? could somebody tell me the result of the following integral: int from 0 to infinitiy of (y^a)*exp(y+b*y^c) dy where a, b and c are fixed parameters Michael === Subject: Re: What is the answer of the following integral? could somebody tell me the result of the following integral: int from 0 to infinitiy of (y^a)*exp(y+b*y^c) dy where a, b and c are fixed parameters Michael > except c=0,1, or 2, there is probably no closed form === Subject: Re: What is the answer of the following integral? <121120061938349996%anniel@nym.alias.net.invalid could somebody tell me the result of the following integral: int from 0 to infinitiy of (y^a)*exp(y+b*y^c) dy where a, b and c are fixed parameters Michael > except c=0,1, or 2, there is probably no closed form There are closed forms for each positive integer a when c = 3 and b < 0. According to Maple: > int(exp(t*y + b*y^3),y=0..infinity) assuming b < 0; 1/9*Pi/(-b)^(1/3)*3^(1/2)*((-t)^(3/2)/(-b)^(1/2))^(1/3)*(1/3* (AngerJ(1/3,2/9*(-t)^(3/2)*3^(1/2)/(-b)^(1/2))- BesselJ(1/3,2/9*(-t)^(3/2)*3^(1/2)/(-b)^(1/2)))*3^(1/2)- BesselY(1/3,2/9*(-t)^(3/2)*3^(1/2)/(-b)^(1/2))-WeberE(1/3,2/9*(-t)^(3/2)*3^( 1/2)/(-b)^(1/2))) Expand in powers of t-1... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: What is the answer of the following integral? On 12 Nov 2006 12:40:14 -0800, Michael Haenlein could somebody tell me the result of the following integral: int from 0 to infinitiy of (y^a)*exp(y+b*y^c) dy where a, b and c are fixed parameters Michael Certainly you must have some conditions on the parameters. If they are all positive it diverges. -- === Subject: Re: What is the answer of the following integral? could somebody tell me the result of the following integral: int from 0 to infinitiy of (y^a)*exp(y+b*y^c) dy where a, b and c are fixed parameters Michael Depending on the values of the parameters, this may diverge. In the cases where it does converge, I would be surprised if there is a closed-form expression. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Information theory: entropy decrease on x from knowledge of x+y Reals x and y have been independently chosen according to known probability density functions X and Y. What is (in bit) the expected entropy variation on x due to disclosure of x+y ? I need something that extends, to continuous distributions, the expected variation of entropy sum(X(x)log2(1/(X(x)). Discrete example: x is 0 or 1 according to a coin flip, y is 1 to 6 according to a dice throw. Initial entropy on x is 1 bit. After x+y is revealed, x+y odds entropy on x 1 1/12 0 (x is known to be 0) 2 1/6 1 (x=1 y=1 and x=0 y=2 are equaly likely) 3 1/6 1 4 1/6 1 5 1/6 1 6 1/6 1 7 1/12 0 (x is known to be 1) Thus expected entropy decrease is 1/6 bit. My actual problem goes quite further. With Expected Entropy Decrease on x from knowledge of x+y denoted Eed(X,Y), - I'd like a tight upper bound Eed(Y) when it is only known that x is in range [0..1] (that is X(x)=0 if x<0 or x>1) - I want Y0 that minimize Eed(Y0) for a given range [0..y0] for y, and the matching Eed0(y0) - or, if that's easier, I want Y1 that minimize Eed(Y1) for y in R+ with given mean y1 for y, and the matching Eed1(y1) I focus on small Eed, thus big y0 and y1. Pointer to (preferably online) reference, formula, hint much appreciated. My information theory is limited and y, and the channel capacity theorem was stated to me without demonstration, and only for Gaussian noise and continous transmission. Fran.8dois Grieu === Subject: Re: A limit stumper >I was helping a student with calculus, and he posed me this problem >assigned by his professor: Determine the limit of [sqrt(6-x) - 2] / >[sqrt(3-x) - 1] as x approaches 2. Here's the catch: The problem is >from early in the book. Limits have been discussed, but the derivative >(let alone l'Hopital) has not been introduced yet. I don't see how to >do it wthout a notion of derivatives. Do you? >(I do see how to do it with only elementary facts about derivatives.) Use the standard device; let x = 2 + y. Then the expression becomes [sqrt(4-y) - sqrt(4)]/[(sqrt(1-y) - sqrt(1)], and using sqrt(a-y)-sqrt(a) = -a/[(sqrt(a-y)+sqrt(a)], the result is immediate. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: inner product spaces I am trying to show that if T is a linear map from V to W. Prove that T is surjective iff T* is injective I am having trouble proving this. I started off with the necessary condition i.e. Assume T is surjective and show that T* is injective. Since T is surjective this implies that ImT= W= (kerT*)' = W Taking Orthonogonal complements on both sides: ((kerT*)')'= W' kerT* = W'= {0} Does this imply T' is injective Now the converse, Assume T* is injective this implies kerT* = {0} KerT* = (rangeT)' = {0} Taking Orthogonal complements of both sides (kerT)' = range T= ((rangeT)')'= {0}'= V, this is where i am stuck please help and is the necessary way correct as well? === Subject: Re: inner product spaces >I am trying to show that if T is a linear map from V to W. Prove that T is surjective iff T* is injective I am having trouble proving this. > You don't have to make use of the inner product: if T is surjective and T*f=0, then f(Tv)=0 for all v in V, f(w)=0 for all w in W, and f=0. Thus T* is injective. On the other hand, if T is not surjective, pick an f mapping W to V such that f=0 on Im(T) but f(w) is not zero for some w not in Im(T). Then, T*f=0, but f is not identically zero, so T* is not injective. -- Daniel Mayost === Subject: Re: inner product spaces > I am trying to show that if T is a linear map from V to W. These being real or complex inner product spaces, I suppose, and T a continuous linear map? > Prove that T is surjective iff T* is injective I am having trouble proving this. ... which may be good, because it isn't true. > I started off with the necessary condition i.e. Assume T is surjective and show that T* is injective. Since T is surjective this implies that ImT= W= (kerT*)' = W ... where ' means orthogonal complement, I suppose... > Taking Orthonogonal complements on both sides: ((kerT*)')'= W' kerT* = W'= {0} Does this imply T' is injective T*, not T'. Injective is the same as having kernel {0}. > Now the converse, Assume T* is injective this implies kerT* = {0} > KerT* = (rangeT)' = {0} Taking Orthogonal complements of both sides (kerT)' = range T= ((rangeT)')'= {0}'= V, this is where i am stuck > please help and is the necessary way correct as well? This part is not correct. The orthogonal complement of Ker T* is the closure of Ran T, but is not necessarily equal to Ran T unless you happen to know that Ran T is closed. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Adjoints How do i prove that: 1. dimkerT* = dimkerT+dimW-dimV and also dimImT* = dimrangeT for T a linear map from V to W. I tried using, using dimV= dimImT+ dimkerT but too no avail any hints please 2. and also i was trying to find a formula for T*b for a fixed vector v in V and define T:V--->F by Tu= . where b is in F a field. Any hints please === Subject: Re: Adjoints >How do i prove that: 1. dimkerT* = dimkerT+dimW-dimV and also dimImT* = dimrangeT for T a >linear map from V to W. I tried using, using dimV= dimImT+ dimkerT but too no avail any hints please > To solve this, you have to use the dimension theorem for both T and T*. But the key is to establish an isomorphism between Im(T) and Im(T*) to show that dim(Im(T))=dim(Im(T*)). If Tv1, ..., Tvn is a basis for Im(T), let F(Tvi) = e_vi = T* e_Tvi (here, if x is part of a basis, then e_x(x)=1 while e_x(y)=0 for all elements y of the basis other than x). -- Daniel Mayost === Subject: Re: A problem in statistics >I am building a software application that could benefit from some > insight into this type of problem. Consider a poll conducted about heights in centimeters. The data of > this poll is collected such that 100 people are asked and answer > correctly the question, Are you at least x centimeters tall? where x > is 120, 130, 140, 150, 160, 170, 180, 190, 200, and 210. (A total of > 1,000 responses are collected. A different 100 people are asked at > each height, although always selected randomly from the population.) > How would you find the mean and standard deviation for the true height > in centimeters of this population? Assume it follows a bell curve. For each height x, you would get a percentage (Perc (x)) that were greater than that height. Greater than x 120 Perc (120) 130 Perc (130) 140 Perc (140) 150 Perc (150) etc. Perc of height x to x1 Range 120-130 Perc (120) - Perc (130) 130-140 Perc (130) - Perc (140) 140-150 Perc (140) - Perc (150) etc The smaller the interval and the larger number of people will give you the true distribution of the height of the population. Note that this distribution will only normally distributed if the distribution of the height of the population is normally distributed. The mean height of the population is approximately: ((Perc (120) - Perc (130) *125) + (Perc (130) - Perc (140))*135) + (Perc (140) - Perc (150))*145 + (Perc (150) - Perc (160) *155) + (Perc (160) - Perc (170))*165) + (Perc (170) - Perc (180))*175 + (Perc (180) - Perc (190) *185) + (Perc (190) - Perc (200))*195) + (Perc (200) - Perc (210))*205)/(Perc(120)-Perc(210)) This is basically a bar chart or histogram. According to http://www.gcseguide.co.uk/standard_deviation.htm the standard deviation is: Square root of the following: ((Perc (120) - Perc (130) * (125 - mean)^2 + (Perc (130) - Perc (140) * (135 - mean)^2 + (Perc (140) - Perc (150) * (145 - mean)^2 + (Perc (150) - Perc (160) * (155 - mean)^2 + (Perc (160) - Perc (170) * (165 - mean)^2 + (Perc (170) - Perc (180) * (175 - mean)^2 + (Perc (180) - Perc (190) * (185 - mean)^2 + (Perc (190) - Perc (200) * (195 - mean)^2 + (Perc (210) - Perc (200) * (205 - mean)^2)/(Perc (120) - Perc (210)) where mean is the mean height calculated above. This is approximate because clearly you don't know what the midpoints of the bars above 210 and below 120 are. I would suggest that you get some proper statistical advice before you actually put your software into use. I am a Chartered Statistician but I don't do consultancy over the Internet. I would check my results with someone before you go ahead. Nick === Subject: Validity of an equation Consider Equation (1) under the given conditions: p^r = (u/v)^p[rs(x^2 + y^2)]/[xy(r^2 + s^2)] (1) Conditions: p is a prime > 3, (u,v) = 1, (x,y) = 1, (r,s) = 1, integer r > p, p is prime to uv All the variables are integers , each > 0 and different from one another. Question: Under what conditions (1) will be valid? If no such conditions exist then (1) is not valid. Any comment and reference will be highly appreciated. === Subject: Re: Math Trek: Designer Decimals > Math Trek: Designer Decimals Ivars Peterson Calculate 100/89. You get the decimal expansion 1.1235955056 . . . Look closely, and you'll see that this fraction generates the first > five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other > digits. If you look even more closely, you'll see the rest of the Fibonaccis; 1 + .1 + .02 + .003 + .0005 + .00008 + .000013 + .0000021 + .... -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Distance between a point and y = ax^2 + bx + c <13087069.1162565670101.JavaMail.jakarta@nitrogen.mathforum.org> <20061108083735.942$ot@newsreader.com There's another approach which Magnus might prefer since it avoids complex > quantities entirely. I don't have time to write it up right now, but will > do so in a day or two. Are you thinking of the trigonometric substitution approach > to the Casus Irreducibilis (three real roots of an irreducible > cubic polynomial)? [Viete, 1593] Yes, if trigonometric includes both circular and hyperbolic. Did Viete use hyperbolic functions? > [Trigonometric Cubic Formula -- PlanetMath] > http://planetmath.org/?op=getobj&from=objects&name=ATrigonometricCubicFormula BTW, I noticed two errors there. One is funny: too real roots. The other is serious: The first part of their Case II says if -4q/3 >= 0, but their method fails if q = 0. [This needs to be handled as a special case, as in my previous post in this thread when my q (not their q) equals 1/2.] W. Cantrell === Subject: Re: Distance between a point and y = ax^2 + bx + c <14850095.1162995365291.JavaMail.jakarta@nitrogen.mathforum.org There's another approach which Magnus might prefer > since it avoids complex quantities entirely. I don't have > time to write it up right now, but will do so in a day or two. Cantrell I would be very interested indeed to see a new approach to the problem :-) The basic problem is to find the abscissa a of the point on the parabola y = x^2 which is closest to some given point (p,q). This was done in an essentially algebraic manner in the thread Point-to-Parabola Distance Formula (sci.math, 2001 Oct. 26) A potential drawback to my previous formula is that, although the result must obviously be real, intermediate steps sometimes require complex quantities. But such quantities can be avoided, as will be shown now: Define P = |p|/4 and Q = sqrt(|2q - 1|/6). If q = 1/2, then a = sgn(p) (2P)^(1/3); otherwise, a = 2 sgn(p) Q f(1/3 af(P/Q^3)) where sgn(p) = -1 if p < 0, {-1,+1} if p = 0, and +1 if p > 0 and the function f and its inverse af are, resp., sinh and asinh if q < 1/2, cosh and acosh if q > 1/2 and 27p^2 >= 2(2q - 1)^3, cos and acos if q > 1/2 and 27p^2 < 2(2q - 1)^3. Notes: 1. The signum function above, sgn, is intentionally bivalued when its argument is zero. (For points on the y-axis above 1/2, there are two closest points on the parabola.) 2. When 27p^2 = 2(2q - 1)^3, it's actually immaterial whether we use cosh or cos. (It would have been nice, eliminating a special case, if I could have said similarly that either sinh or cosh may be used when q = 1/2, but that doesn't work.) W. Cantrell === Subject: inner product spaces 2 i was trying to find a formula for T*b for a fixed vector v in V and define T:V--->F by Tu= . where b is in F a field. Any hints please === Subject: Ordinal based sets. Define: x is a subordinal <-> ........UUUUUUx = w w is Omega. Card(x) is the intersectional set of all ordinals,or subordinals bijectable to x. Zuhair === Subject: Re: Ordinal based sets. > Define: x is a subordinal <-> ........UUUUUUx = w A wonderful definition! No wonder stupid people like Jesse have such a hard time coming up with rebuttals. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Ordinal based sets. ........UUUUUUx = w A wonderful definition! No wonder stupid people like Jesse have such a > hard time coming up with rebuttals. -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus This guy wins the award for Most Likely To Be James Harris in 11 Years === Subject: Re: Ordinal based sets. >> A wonderful definition! No wonder stupid people like Jesse have such a >> hard time coming up with rebuttals. This guy wins the award for Most Likely To Be James Harris in 11 Years I have to wait eleven years? No fair! -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Ordinal based sets. A wonderful definition! No wonder stupid people like Jesse have such a >> hard time coming up with rebuttals. This guy wins the award for Most Likely To Be James Harris in 11 Years I have to wait eleven years? No fair! Well, you don't develop prime counting functions overnight. And you don't develop a radical new set theory that will change the world, improve the military, and give you all-powerful friends overnight either. === Subject: Re: Ordinal based sets. > This guy wins the award for Most Likely To Be James Harris in 11 Years >> I have to wait eleven years? No fair! Well, you don't develop prime counting functions overnight. I see my years of arrant pedantry in sci.logic count for nothing. What if I tell you there are errors - usually rather unimportant ones, admittedly - in all expositions of the incompleteness theorems I've ever encountered? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Ordinal based sets. Define: x is a subordinal <-> ........UUUUUUx = w w is Omega. Card(x) is the intersectional set of all ordinals,or subordinals > bijectable to x. Zuhair ....UUUUx is the set union of the set union of....of set union of x that the set union of ......UUUUx would yield .......UUUUx. for exampe Uw=w UUw=w Accordingly ....UUUUw = w. While for example x={ {} , { {} } , { {} , { {} } } Ux = { {} , { {} } } UUx = { {} } UUUx = { } UUUUx= { } UUUUUx= { } Therefore ........UUUx= { }. This works pritty beautiful for ordinals and all transitive sets, and some other where ...UUUx is {} or w. But what about other sets, I mean what is the bottom of these sets, would it be a member that is NOT a set? Zuhair === Subject: JSH: My prime counting, speed Posters lie to you all the time about my research and one of the most telling areas is with my prime counting function. I've talked recently about the sieve form giving the LaTex as that can give you a better look at it, if you paste it somewhere where you can process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this subject is that it is true that solving that expression out explicitly will give the same expression as Legendre's Method. It does. So no, the simple expression even in its sieve form is not going to be fast, but the first speed-up is to not iterate from i=1, as that covers evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Now I posted recently that the sieve form was fast, as I haven't played with this all for some time, and I always would correct out the evens, and it IS fast if you do that simple thing. But it doesn't stop there. It turns out that the iteration at i=2 can be given by floor((x-3)/6) so you can do yet another speed up by using: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) - sum_{i=3}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. And those of you who bother to try that out--hoping I got the math right on that last, I think I did--will find that it is far faster than Legendre's Method can be made to be. So what's the point here? I made some mistakes in talking about an area I haven't delved into for years, as I got bored with the speed issue, and some posters maintained one thing based on exploiting my mistakes, when the mathematical reality--the full story--wasn't too far away. So why do those slight changes make for very fast prime counting? They don't care. They don't care to let you know about those changes. They don't care what the mathematical reality is. My research covers a lot of ground, and I often forget details about it, as I go away from one particular area for years, so I do apologize for getting some of the facts wrong. But I'll come back and correct. I knew there were fast ways to count primes from my idea, so it was just a matter of getting the details right. The people I'm facing, well, I think for them it's just a game, where a lot of times you don't even know who they are as they're using pseudonyms, and clearly think that they can never be held accountable for what they say on Usenet. James Harris === Subject: Re: My prime counting, speed The people I'm facing, well, I think for them it's just a game, where a > lot of times you don't even know who they are as they're using > pseudonyms, and clearly think that they can never be held accountable > for what they say on Usenet. > I held ur moms ass accountable last night. === Subject: Re: JSH: My prime counting, speed [jstevh@msn.com] > I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. Well, duh. How many years is it that you've missed this, despite being shown it over and over? > So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Well, duh. I suppose you don't see how to do /the same thing/ with the similar spelling of Legendre's recurrence? I suppose you don't. > Now I posted> recently that the sieve form was fast, as I haven't > played with this all for some time, Yup. You don't test, you don't think, you don't listen, and you don't make progress. > and I always would correct out the evens, and it IS fast if you do > that simple thing. Fast/er/ than it was, yes. Still a bit slower than what you get if you do the equivalent thing to the similar spelling of Legendre's recurrence. > But it doesn't stop there. Not unless you stop there, true. Keep it up, and at this pace around the year 2050 you might catch up to the simpler bits of what's currently known about speeding Legendre's formula directly. > ... > And those of you who bother to try that out--hoping I got the math > right on that last, I think I did--will find that it is far faster than > Legendre's Method can be made to be. No. 100% false. > So what's the point here? Beats me. > ... [delusion] ... === Subject: Re: JSH: My prime counting, speed I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. Well, duh. How many years is it that you've missed this, despite being > shown it over and over? So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Well, duh. I suppose you don't see how to do /the same thing/ with the > similar spelling of Legendre's recurrence? I suppose you don't. > Then demonstrate. It is of mathematical interest to me. Also generalize to show how you can do so with successive primes, like how I showed the next step for 3: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) -sum_{i=3}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. And remember, each time I'm giving a complete prime counting function, as in if you code that you get a function that will, for instance, return 25 for P(100,4) as you only need the first four primes 2, 3, 5 and 7 to count the primes up to 100. Nothing is as fast and as short in the world. You cannot give a complete prime counting function in the same space, and yes, even if you use LaTex as I did, it just cannot be done. Concede now or demonstrate. James Harris === Subject: Re: JSH: My prime counting, speed [jstevh@msn.com] > I've talked recently about the sieve form giving the LaTex as that > can give you a better look at it, if you paste it somewhere where > you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out > explicitly will give the same expression as Legendre's Method. It does. [Tim Peters] >> Well, duh. How many years is it that you've missed this, despite >> being shown it over and over? > So no, the simple expression even in its sieve form is not going to > be fast, but the first speed-up is to not iterate from i=1, as that > covers evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. >> Well, duh. I suppose you don't see how to do /the same thing/ with >> the similar spelling of Legendre's recurrence? I suppose you don't. [jstevh@msn.com] > Then demonstrate. You could have done it yourself in seconds from what I posted before: The seive form of your formula, as executable Python: def P(x, a): return x-1 - sum(P(x // ps[i+1], i) - i for i in range(a)) def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return P(n, a) The seive form of Legendre's recurrence, as executable Python: def phi(x, a): return x - sum(phi(x // ps[i+1], i) for i in range(a)) def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return phi(n, a) + a - 1 I've explained these in detail before, including why they /don't/ bother (and don't need to bother) to ensure ps[a]^2 <= n on entry. Both speed up identically by doing so, but then neither is as concise. Note that both assume a global list ps[] exists, containing the primes at least through sqrt(n), where ps[1] = 2, ps[2] = 3, and so on. ps[0] is never referenced. Given a singe line to define ps, e.g., ps = [None, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] these are all complete, executable programs. All explained before. If you want to special-case a=1, it's trivial, but you did it incorrectly above. You're trying to unroll the a=1 piece of the summation in your original formula, and: P(x/2, 0) = floor(x/2) - 1 You want to fold that into the x-1 leading term of your original formula, giving correct combined term: x-1 - (floor(x/2)-1) = x - floor(x/2) = ceiling(x/2) = floor((x+1)/2) Making that correction, your formula with a=1 unrolled becomes: def P(x, a): if a == 0: # there is no a=1 to unroll in this case return x-1 else: return (x+1)//2 - sum(P(x // ps[i+1], i) - i for i in range(1, a)) def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return P(n, a) If you don't believe me, /run/ it, then simply observe that my form computes pi() correctly, while using your derivation it does not. Precisely similar analysis of the a=1 case of the original Legendre spelling gives: phi(x/2, 0) = floor(x/2) and combining that with its x original leading term gives: x - floor(x/2) = ceiling(x/2) = floor((x+1)/2) Same thing in the end (which is best seen as coincidence -- same thing in the end is specific to a=1). So the Legendre spelling with a=1 unrolled becomes: def phi(x, a): if a == 0: # there is no a=1 to unroll in this case return x else: return (x+1)//2 - sum(phi(x // ps[i+1], i) for i in range(1, a)) def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return phi(n, a) + a - 1 > It is of mathematical interest to me. Also generalize to show how you > can do so with successive primes, like how I showed the next step for > 3: No. If you really can't do this yourself after so much help, you're simply hopeless. More importantly, as I've explained before, a /much/ more effective way to speed these is to focus on primorial values of `a` instead (2, yes; 3, no -- leap to 6; then 30; then 210; etc), as you haven't yet understood when staring at equations [4] through [10] here: http://mathworld.wolfram.com/LegendresFormula.html It's not surprising you don't understood those, because MathWorld makes no attempt to /explain why/ they're important (read a book). Nevertheless, they're extremely important, allowing to evaluate phi(x, a) in /constant time/ whenever `a` is a primorial. A similar thing /could/ be done for your formula, although you haven't discovered that yet. > ... > And remember, each time I'm giving a complete prime counting function, > as in if you code that you get a function that will, for instance, > return 25 for P(100,4) as you only need the first four primes 2, 3, 5 > and 7 to count the primes up to 100. Of course. Here, pasted from an interactive Python session, using the last code for P() given above: > P(100, 4) 25 > Nothing is as fast and as short in the world. Complete nonsense. The Legendre code above is just as short, and actually a bit faster (for reasons explained many times before). > You cannot give a complete prime counting function in the same space, The above is just the most recent of the many times I've done this for you before. > and yes, even if you use LaTex as I did, it just cannot be done. Concede now or demonstrate. Fine. I'm done now. === Subject: Re: JSH: My prime counting, speed I've talked recently about the sieve form giving the LaTex as that > can give you a better look at it, if you paste it somewhere where > you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out > explicitly will give the same expression as Legendre's Method. It does. [Tim Peters] >> Well, duh. How many years is it that you've missed this, despite >> being shown it over and over? So no, the simple expression even in its sieve form is not going to > be fast, but the first speed-up is to not iterate from i=1, as that > covers evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. > Well, duh. I suppose you don't see how to do /the same thing/ with >> the similar spelling of Legendre's recurrence? I suppose you don't. [jstevh@msn.com] > Then demonstrate. You could have done it yourself in seconds from what I posted before: The seive form of your formula, as executable Python: > I didn't ask for Python. I asked for math. With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. You're asking people on these newsgroups to read Python code? And you are lying with it. Of course you'd go to what you know to lie. I want you to give MATH. The truth is you CANNOT give Legendre's in the same space! And you know it, so you go to code to hide the truth. And you CANNOT speed up Legendre's in the same way, like how I can just go to With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. As a way to speed up the count. That is mathematics. It's not a freaking scripting language. It's math. Show math or concede, and quit trying to lie by going to your field of expertise as if everyone on math newsgroups is supposed to know some scripting language. Show the math or concede. James Harris === Subject: Re: JSH: My prime counting, speed > Show the math or concede. > Ur mom conceded her ass to me last night. === Subject: Re: JSH: My prime counting, speed > Show the math or concede. > Ur mom conceder her ass to me last night. === Subject: Re: JSH: My prime counting, speed > I didn't ask for Python. I asked for math. Wow, you're more stupid than I thought. I don't know one practicing mathematician who hasn't written a program to help with their work. I do it all the time. But, you're not a mathematician so you wouldn't know. Dave === Subject: Re: JSH: My prime counting, speed mathematician who hasn't written a program to help with their work. I do it > all the time. But, you're not a mathematician so you wouldn't know. Dave He can obscure the inability to write a prime counting function as succinctly as can be done with mine by blurring things into code. That's the bottom-line. After all, I coded my prime counting function as well, so it's not about puzzling through code here but about a deliberate attempt at lying by using code. He betrays the spirit of Python and of the people who use it to try and lie with it. He is a traitor to his own research as well as that of others by trying to con with it. If not, then if he can give code, he can give math as well. James Harris === Subject: Re: JSH: My prime counting, speed mathematician who hasn't written a program to help with their work. I do it > all the time. But, you're not a mathematician so you wouldn't know. Dave He can obscure the inability to write a prime counting function as > succinctly as can be done with mine by blurring things into code. def phi(x, a): return x - sum(phi(x // ps[i+1], i) for i in range(a)) def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return phi(n, a) + a - 1 Looks pretty succint to me. Are you claiming that your prime counting function can be written in an even more succint way? - William Hughes === Subject: Re: JSH: My prime counting, speed [jstevh@msn.com] >> I didn't ask for Python. I asked for math. [Moran] > Wow, you're more stupid than I thought. I don't know one > practicing mathematician who hasn't written a program to help > with their work. I do it all the time. But, you're not a > mathematician so you wouldn't know. [jstevh@msn.com] >> He can obscure the inability to write a prime counting function as >> succinctly as can be done with mine by blurring things into code. [William Hughes] def phi(x, a): > return x - sum(phi(x // ps[i+1], i) for i in range(a)) def pi(n): > from math import sqrt > if n <= 1: > return 0 > a = pi(int(sqrt(n))) > return phi(n, a) + a - 1 Do note that this follows immediately from equation [3] at: http://mathworld.wolfram.com/LegendresFormula.html by iteratively applying [3] to its own left addend on the RHS until `a` reaches 0. If you can read math, and you're a programmer ;-), this is a thoroughly standard way to proceed. > Looks pretty succint to me. Are you claiming that your > prime counting function can be written in an even > more succint way? Yes, but he never gives the pi() part, just the very similar one-line P() summation, wishing away that to actually /use/ it you have to compute pi(sqrt(n)) first. The implementation I gave for his formula does not wish that away: def P(x, a): return x-1 - sum(P(x // ps[i+1], i) - i for i in range(a)) P() just above is the only part James ever gives, and is a dead-obvious translation of his LaTeX into Python. I also supplies an appropriate driver, else there simply /isn't/ an implemention of pi() here: def pi(n): from math import sqrt if n <= 1: return 0 a = pi(int(sqrt(n))) return P(n, a) Note that the definitions of the pi() functions are /almost/ identical between the two. They want to compute pi(sqrt(n)) first, but the pi() recursion won't terminate if n <= 1, so n <= 1 is special-cased first. Having computing pi(sqrt(n)), they both call the appropriate one-liner summation recurrence. That's the end of it for James's formula, but because phi(x, pi(sqrt(x))) = pi(x) - pi(sqrt(x)) + 1 the Legendre driver needs to add a-1 at the end. It's all very simple, although a correctness proof is not, largely because to make James's function even /more/ concise, I skipped the bit about reducing `a` so that p[a]^2 <= x inside the P() function. That's not know that (AFAICT, he doesn't understand any of this at a deep enough level). === Subject: Re: JSH: My prime counting, speed mathematician who hasn't written a program to help with their work. I do it > all the time. But, you're not a mathematician so you wouldn't know. Dave He can obscure the inability to write a prime counting function as > succinctly as can be done with mine by blurring things into code. That's the bottom-line. After all, I coded my prime counting function as well, so it's not > about puzzling through code here but about a deliberate attempt at > lying by using code. He betrays the spirit of Python and of the people who use it to try and > lie with it. He is a traitor to his own research as well as that of others by trying > to con with it. If not, then if he can give code, he can give math as well. > James Harris In what way did he try and con? He presented two snippets of code that represent the two functions. He did it in a very clean and precise manner. If you can't understand it, don't blame others. If you can understand it but don't like the coding, change it to something you feel more comfortable with it, and show his deception. Otherwise, you are just reinforcing the depth of your ignorant delusions. === Subject: Re: JSH: My prime counting, speed > I didn't ask for Python. I asked for math. >> Wow, you're more stupid than I thought. I don't know one practicing >> mathematician who hasn't written a program to help with their work. I do >> it >> all the time. But, you're not a mathematician so you wouldn't know. >> Dave He can obscure the inability to write a prime counting function as > succinctly as can be done with mine by blurring things into code. How do we know you haven't done that? That's the bottom-line. After all, I coded my prime counting function as well, so it's not > about puzzling through code here but about a deliberate attempt at > lying by using code. He betrays the spirit of Python and of the people who use it to try and > lie with it. How? He is a traitor to his own research as well as that of others by trying > to con with it. If not, then if he can give code, he can give math as well. > James Harris > Dave === Subject: Re: JSH: My prime counting, speed I've talked recently about the sieve form giving the LaTex as that > can give you a better look at it, if you paste it somewhere where > you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out > explicitly will give the same expression as Legendre's Method. It does. [Tim Peters] >> Well, duh. How many years is it that you've missed this, despite >> being shown it over and over? So no, the simple expression even in its sieve form is not going to > be fast, but the first speed-up is to not iterate from i=1, as that > covers evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. > Well, duh. I suppose you don't see how to do /the same thing/ with >> the similar spelling of Legendre's recurrence? I suppose you don't. [jstevh@msn.com] > Then demonstrate. You could have done it yourself in seconds from what I posted before: The seive form of your formula, as executable Python: > I didn't ask for Python. So convert it to Java. Anybody who claims to be a Java programmer ought to be able to translate Python. > I asked for math. Show me some Java. With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. You're asking people on these newsgroups to read Python code? Why not? Anybody with half a brain uses Python. And you are lying with it. Of course you'd go to what you know to lie. I want you to give MATH. The truth is you CANNOT give Legendre's in the same space! And you > know it, so you go to code to hide the truth. And you CANNOT speed up Legendre's in the same way, like how I can just > go to With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. As a way to speed up the count. Ok, show me the Big O analysis that proves your point. That is mathematics. It's not a freaking scripting language. No wonder you never learn anything. > It's math. Show math or concede, and quit trying to lie by going to your field of > expertise as if everyone on math newsgroups is supposed to know some > scripting language. Not just some, Python. Makes all the difference in the world. Python rules! Show the math or concede. Tim's code looks good to me, no need to conceed anything. > James Harris === Subject: Re: JSH: My prime counting, speed I just wanted to emphasize the important bit here: ... [jstevh@msn.com] >> You're asking people on these newsgroups to read Python code? [mensanator@aol.com] > Why not? Anybody with half a brain uses Python. ... >> Show math or concede, and quit trying to lie by going to your field >> of expertise as if everyone on math newsgroups is supposed to know >> some scripting language. > Not just some, Python. Makes all the difference in the world. Python rules! ... Everyone got it? If not, get it: http://www.python.org/ No charge, no registration, and James Harris will not call you unless you ignore this good advice ;-) For those who aren't dribbling, here's JSH's LaTeX: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} For implementation, it seemed prettier to me to subtract i instead of i-1 inside the implied summation loop, and adjusting the loop index to go from 0 through a-1 instead allows that replacement: :P(x,n) = x - 1 -sum_{i=0}^{a-1} {(P(x/p_{i+1},i) - i)} Then you really only need to know two non-/obvious/ things about Python to recreate my implementation: 1. range(n) returns a list of the first n integers starting with 0. 2. Infix // means floor division: Python i//j same-as mathematical floor(i/j). And that's it: def P(x, a): return x-1 - sum(P(x // ps[i+1], i) - i for i in range(a)) A more literal translation of the original LaTeX, skipping the index transformation, is computationally equivalent (but uglier to my eyes): def P(x, a): return x-1 - sum(P(x // ps[i], i-1) - (i-1) for i in range(1, a+1)) The Legendre recurrence is actually shorter than either of those, despite that phi is fatter than P: def phi(x, a): return x - sum(phi(x // ps[i+1], i) for i in range(a)) But I expect you need to be a genius to translate that into LaTeX ;-) === Subject: Re: JSH: My prime counting, speed > You're asking people on these newsgroups to read Python code? [mensanator@aol.com] > Why not? Anybody with half a brain uses Python. Hey! I resemble^H^H^H^H^H^H^H^H^H resent that remark! However, I was able to understand what was going on; the only thing that had me stumped for the moment was the // (which is integer division). It's actually pretty close to the pseudocode write-up, which ANY semi-competant programmer could read. (Which explains why JSH is confused by it. 8-)) > ... > Show math or concede, and quit trying to lie by going to your field >> of expertise as if everyone on math newsgroups is supposed to know >> some scripting language. Not just some, Python. Makes all the difference in the world. Python rules! ... Everyone got it? If not, get it: http://www.python.org/ So, what's the advantage of Python over, say, C, or Maple? IOW, what am I going to get in return for learning a completely new programming language, which I don't have already? > No charge, no registration, and James Harris will not call you unless you > ignore this good advice ;-) > For those who aren't dribbling, here's JSH's LaTeX: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} For implementation, it seemed prettier to me to subtract i instead of i-1 > inside the implied summation loop, and adjusting the loop index to go from 0 > through a-1 instead allows that replacement: :P(x,n) = x - 1 -sum_{i=0}^{a-1} {(P(x/p_{i+1},i) - i)} Then you really only need to know two non-/obvious/ things about Python to > recreate my implementation: 1. range(n) returns a list of the first n integers starting with 0. 2. Infix // means floor division: Python i//j same-as mathematical > floor(i/j). And that's it: def P(x, a): > return x-1 - sum(P(x // ps[i+1], i) - i for i in range(a)) A more literal translation of the original LaTeX, skipping the index > transformation, is computationally equivalent (but uglier to my eyes): def P(x, a): > return x-1 - sum(P(x // ps[i], i-1) - (i-1) for i in range(1, a+1)) The Legendre recurrence is actually shorter than either of those, despite > that phi is fatter than P: def phi(x, a): > return x - sum(phi(x // ps[i+1], i) for i in range(a)) But I expect you need to be a genius to translate that into LaTeX ;-) $$phi(x,a) = x - sum_{i=0}^a phileft(leftlfloor{x over ps_{i+1}}rightrfloor, iright).$$ All right, that's Plain TeX, but I think my status as a genius is istablished. 8-) --- === Subject: Re: JSH: My prime counting, speed You're asking people on these newsgroups to read Python code? I very much doubt that there are many people on sci.math who can't understand what the Python code means, even if they have no experience of Python. Can you, allegedly a professional programmer, really not figure it out? > And you are lying with it. Of course you'd go to what you know to lie. > The truth is you CANNOT give Legendre's in the same space! And you > know it, so you go to code to hide the truth. Programming languages specify algorithms. That is the whole point of programming languages. They specify algorithms. How the hell is going to code supposed to hide the truth? -Rotwang === Subject: Re: JSH: My prime counting, speed [jstevh@msn.com] > I didn't ask for Python. I asked for math. It's official, then: you really are too stupid to bother with. > [... complete idiocy ...] === Subject: Re: JSH: My prime counting, speed I didn't ask for Python. I asked for math. It's official, then: you really are too stupid to bother with. [... complete idiocy ...] You couldn't deliver. I knew you couldn't which is why I gave the challenge. But even when beaten totally you STILL just lie. If not, then GIVE THE MATH. Not some freaking scripting language as if math newsgroup people are supposed to all be programmers who read Python. Just give the math. And deleting out the mathematics I gave in your reply may be symbolic and important to you, but it doesn't take it away. Reality is people like you always lose when it really comes to the point on delivering on proving that I'm wrong, but then you just SAY that you don't, run away for a while, and come back later to lie. Give the math. Give a prime counting function as succinct as mine that is as fast using mathematics, not writing some code for people to puzzle through trying to figure out if you delivered or lied. James Harris === Subject: Re: JSH: My prime counting, speed I didn't ask for Python. I asked for math. It's official, then: you really are too stupid to bother with. [... complete idiocy ...] You couldn't deliver. I knew you couldn't which is why I gave the challenge. But even when beaten totally you STILL just lie. If not, then GIVE THE MATH. Not some freaking scripting language as if > math newsgroup people are supposed to all be programmers who read > Python. FWIW, undergrad maths students at Bristol university are all required to take courses on which they learn Python and Maple. -- mike. === Subject: Re: JSH: My prime counting, speed You couldn't deliver. I delivered to ur moms ass last night. === Subject: Re: JSH: My prime counting, speed I've talked recently about the sieve form giving the LaTex as that > can give you a better look at it, if you paste it somewhere where > you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out > explicitly will give the same expression as Legendre's Method. It does. [Tim Peters] >> Well, duh. How many years is it that you've missed this, despite >> being shown it over and over? So no, the simple expression even in its sieve form is not going to > be fast, but the first speed-up is to not iterate from i=1, as that > covers evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. > Well, duh. I suppose you don't see how to do /the same thing/ with >> the similar spelling of Legendre's recurrence? I suppose you don't. [jstevh@msn.com] > Then demonstrate. You could have done it yourself in seconds from what I posted before: The seive form of your formula, as executable Python: > I didn't ask for Python. I asked for math. With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. You're asking people on these newsgroups to read Python code? And you are lying with it. Of course you'd go to what you know to lie. I want you to give MATH. The truth is you CANNOT give Legendre's in the same space! And you > know it, so you go to code to hide the truth. And you CANNOT speed up Legendre's in the same way, like how I can just > go to With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. As a way to speed up the count. That is mathematics. It's not a freaking scripting language. It's > math. Show math or concede, and quit trying to lie by going to your field of > expertise as if everyone on math newsgroups is supposed to know some > scripting language. Show the math or concede. > James Harris He did show math in the form of Python code. You are just really being pathetic James. If you can't read simple Python code, then you shouldn't broadcast that fact to others. It just shows another area where you claim to have some experience(programming), but are sorely lacking in skill and understanding. === Subject: Re: JSH: My prime counting, speed I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. Well, duh. How many years is it that you've missed this, despite being > shown it over and over? So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Well, duh. I suppose you don't see how to do /the same thing/ with the > similar spelling of Legendre's recurrence? I suppose you don't. > Then demonstrate. It is of mathematical interest to me. Also generalize to show how you > can do so with successive primes, like how I showed the next step for > 3: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) -sum_{i=3}^n > {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. And remember, each time I'm giving a complete prime counting function, > as in if you code that you get a function that will, for instance, > return 25 for P(100,4) as you only need the first four primes 2, 3, 5 > and 7 to count the primes up to 100. Nothing is as fast and as short in the world. You cannot give a complete prime counting function in the same space, > and yes, even if you use LaTex as I did, it just cannot be done. Concede now or demonstrate. > James Harris Is he allowed to use a smaller font? === Subject: Re: JSH: My prime counting, speed > Posters lie to you all the time about my research and one of the most > telling areas is with my prime counting function. I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Now I posted > recently that the sieve form was fast, as I haven't played with this > all for some time, and I always would correct out the evens, and it IS > fast if you do that simple thing. But it doesn't stop there. It turns out that the iteration at i=2 can > be given by floor((x-3)/6) so you can do yet another speed up by using: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) - sum_{i=3}^n > {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. And those of you who bother to try that out--hoping I got the math > right on that last, I think I did--will find that it is far faster than > Legendre's Method can be made to be. So what's the point here? I made some mistakes in talking about an area I haven't delved into for > years, as I got bored with the speed issue, and some posters maintained > one thing based on exploiting my mistakes, when the mathematical > reality--the full story--wasn't too far away. So why do those slight changes make for very fast prime counting? They don't care. They don't care to let you know about those changes. > They don't care what the mathematical reality is. My research covers a lot of ground, and I often forget details about > it, as I go away from one particular area for years, so I do apologize > for getting some of the facts wrong. But I'll come back and correct. I knew there were fast ways to count > primes from my idea, so it was just a matter of getting the details > right. The people I'm facing, well, I think for them it's just a game, where a > lot of times you don't even know who they are as they're using > pseudonyms, and clearly think that they can never be held accountable > for what they say on Usenet. > James Harris So to sum up what you are trying to say: One of your examples about how mathematicains are corrupt has been how they lie about your prime counting function. Even over the past few days you have repeatedly called them liars because they said that your method was basically Legendre's mthod. But now you admit that is is based on legendre's method. I guess you are then admitting to calling people liars without any basis in fact. I didn't notice an apology in there though. I guess you must have made another mistake and accidently deleted that part of the post. It's alright James, we're all human. === Subject: Re: JSH: My prime counting, speed > Posters lie to you all the time about my research and one of the most > telling areas is with my prime counting function. I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Now I posted > recently that the sieve form was fast, as I haven't played with this > all for some time, and I always would correct out the evens, and it IS > fast if you do that simple thing. But it doesn't stop there. It turns out that the iteration at i=2 can > be given by floor((x-3)/6) so you can do yet another speed up by using: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) - sum_{i=3}^n > {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. And those of you who bother to try that out--hoping I got the math > right on that last, I think I did--will find that it is far faster than > Legendre's Method can be made to be. So what's the point here? I made some mistakes in talking about an area I haven't delved into for > years, as I got bored with the speed issue, and some posters maintained > one thing based on exploiting my mistakes, when the mathematical > reality--the full story--wasn't too far away. So why do those slight changes make for very fast prime counting? They don't care. They don't care to let you know about those changes. > They don't care what the mathematical reality is. My research covers a lot of ground, and I often forget details about > it, as I go away from one particular area for years, so I do apologize > for getting some of the facts wrong. But I'll come back and correct. I knew there were fast ways to count > primes from my idea, so it was just a matter of getting the details > right. The people I'm facing, well, I think for them it's just a game, where a > lot of times you don't even know who they are as they're using > pseudonyms, and clearly think that they can never be held accountable > for what they say on Usenet. > James Harris So to sum up what you are trying to say: One of your examples about how mathematicains are corrupt has been how > they lie about your prime counting function. Even over the past few > days you have repeatedly called them liars because they said that your > method was basically Legendre's mthod. But now you admit that is is > based on legendre's method. sieve form with the simplest implementation, but even a slight change: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. It is now far faster. And clearly NOT Legendre's. I guess you are then admitting to calling people liars without any > basis in fact. > They are liars as they use the reality at one level to make a claim that is easily shown to be false with a slight change. I'm fairly certain you can't make even a slight change like I just did with Legendre's. If you want to impress me, come back with a shift to his equations that is equivalent. If you are clever maybe you can do it--it's been a while since I went over this area so I'm not sure--but I'm certain you'll fail at the next prime, which of course is 3. While with my prime counting function it's easily done: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) -sum_{i=3}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including sqrt{x} then n is reset to that count. > I didn't notice an apology in there though. I guess you must have made > another mistake and accidently deleted that part of the post. It's > alright James, we're all human. The point is that people like you and other posters obscure the math. You want people to believe my prime counting function is just Legendre's so of course you don't mention simple speed-ups that have been discussed on math newsgroups before BECAUSE those speed-ups would show that my prime counting function is NOT Legendre's which cannot be sped up in the same way. So you are liars who manipulate people by carefully ignoring important mathematical facts. I DO get some things wrong as I go back to old territory that I found boring, but I can correct, and show what you people are doing. James Harris === Subject: Re: JSH: My prime counting, speed > Posters lie to you all the time about my research and one of the most > telling areas is with my prime counting function. I've talked recently about the sieve form giving the LaTex as that can > give you a better look at it, if you paste it somewhere where you can > process LaTex: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = x - 1 -sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. One correction I have to make to what I've said previously on this > subject is that it is true that solving that expression out explicitly > will give the same expression as Legendre's Method. It does. So no, the simple expression even in its sieve form is not going to be > fast, but the first speed-up is to not iterate from i=1, as that covers > evens. Here's a faster version: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. Now that makes it quite a bit more than twice as fast. Now I posted > recently that the sieve form was fast, as I haven't played with this > all for some time, and I always would correct out the evens, and it IS > fast if you do that simple thing. But it doesn't stop there. It turns out that the iteration at i=2 can > be given by floor((x-3)/6) so you can do yet another speed up by using: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) - sum_{i=3}^n > {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. And those of you who bother to try that out--hoping I got the math > right on that last, I think I did--will find that it is far faster than > Legendre's Method can be made to be. So what's the point here? I made some mistakes in talking about an area I haven't delved into for > years, as I got bored with the speed issue, and some posters maintained > one thing based on exploiting my mistakes, when the mathematical > reality--the full story--wasn't too far away. So why do those slight changes make for very fast prime counting? They don't care. They don't care to let you know about those changes. > They don't care what the mathematical reality is. My research covers a lot of ground, and I often forget details about > it, as I go away from one particular area for years, so I do apologize > for getting some of the facts wrong. But I'll come back and correct. I knew there were fast ways to count > primes from my idea, so it was just a matter of getting the details > right. The people I'm facing, well, I think for them it's just a game, where a > lot of times you don't even know who they are as they're using > pseudonyms, and clearly think that they can never be held accountable > for what they say on Usenet. > James Harris So to sum up what you are trying to say: One of your examples about how mathematicains are corrupt has been how > they lie about your prime counting function. Even over the past few > days you have repeatedly called them liars because they said that your > method was basically Legendre's mthod. But now you admit that is is > based on legendre's method. sieve form with the simplest implementation, but even a slight change: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) -sum_{i=2}^n {(P(x/p_i,i-1) - > (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. It is now far faster. And clearly NOT Legendre's. > I guess you are then admitting to calling people liars without any > basis in fact. > They are liars as they use the reality at one level to make a claim > that is easily shown to be false with a slight change. I'm fairly certain you can't make even a slight change like I just did > with Legendre's. If you want to impress me, come back with a shift to his equations that > is equivalent. If you are clever maybe you can do it--it's been a while since I went > over this area so I'm not sure--but I'm certain you'll fail at the next > prime, which of course is 3. While with my prime counting function it's easily done: With natural numbers x and n, where p_i is the i_th prime: :P(x,n) = floor(x/2) - floor((x-3)/6) -sum_{i=3}^n > {(P(x/p_i,i-1) - (i-1))} where if n is greater than the count of primes up to and including > sqrt{x} then n is reset to that count. I didn't notice an apology in there though. I guess you must have made > another mistake and accidently deleted that part of the post. It's > alright James, we're all human. The point is that people like you and other posters obscure the math. You want people to believe my prime counting function is just > Legendre's so of course you don't mention simple speed-ups that have > been discussed on math newsgroups before BECAUSE those speed-ups would > show that my prime counting function is NOT Legendre's which cannot be > sped up in the same way. So you are liars who manipulate people by carefully ignoring important > mathematical facts. I DO get some things wrong as I go back to old territory that I found > boring, but I can correct, and show what you people are doing. > James Harris Of course it is the same result. It is a prime counting function. Every prime counting function has the same result. Duh. Why would I want to impress you James? You are one of the top usenet cranks. I've never claimed to have made any breakthroughs in mathematics. That's what seperates me from you. I live in reality. That's not to say that I don't have some notable accomplishments. I just don't need to brag about them on usenet. You keep bragging about how you are using your real name on usenet. Do you really sure that it would be in your favor for a possible boss of yours to read what you post on these newsgroups? Think about it. === Subject: Axioms are preposterous Bucky Fuller said the axiomatic method has chosen axioms that have often not withstood the test of time (Euclid etc.), so, he recommended not using it. It is preposterous in Synergetics to start with less than the tetrahedron: pre: first, posterous: last A good teacher can remember learning the subject and reverse the order of presentation. I am a bad teacher, so I showed a preposterous presentation in http://users.adelphia.net/~cnelson9/ and http://library.wolfram.com/infocenter/MathSource/600/ and http://mathworld.wolfram.com/SynergeticsCoordinates.html The Mathematica notebooks should be modified to put first things first, second things second, third things third, etc. starting with the tetrahedron, like at: http://bfi.org/node/574 preposterous: I didn't here the name Beiderbecke until I was 48 years old in 1994 when I tuned in http://www.geocities.com/forwardintothepast/ Bix Beiderbecke died in 1931 when he was 28 years old, but he recorded a lot of what turned out to be my favorite music. My brothers and I played in the school band, my father was an Army band leader for twenty years, and all of my friends played instruments and we would have preferred Bix to Elvis when I was 14, but it was kept hidden. 1927 was the greatest year in the history of recorded music, but it was kept hidden. Like Bucky said: All history becomes suspect. http://www.bixbeiderbecke.com/ Cliff Nelson Dry your tears, there's more fun for your ears, Forward Into The Past 2 PM to 5 PM, Sundays, California time, http://www.geocities.com/forwardintothepast/ Don't be a square or a blockhead; see: http://bfi.org/node/574 http://www.rapidlyrotatingrecords.com http://www.bixbeiderbecke.com === Subject: JSH: Prime counting speed is boring So yeah, I know how to use some simple techniques to make algorithms from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I thought I figured out a few years back until I was informed it was known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used floor((x-4)/6) for the count of composites up to and including x that had 3 as a factor but were not even. That works too, but it's simpler to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. But so what?. My guess is that I could travel over a lot of ground already covered with previous people's ideas and end up with a separate effort using my own research that gets to about where people are already at, and it bored me, so I stop caring about speed and focused on more interesting areas like the partial difference equation. Now to you maybe it's a big deal like why wouldn't I just go ahead and make something that comes close in speed to what's already known to prove my point? Why should I have to? I've argued out the speed-ups to my prime counting function before! Regular posters who have argued with me for years already know about them. The math community isn't ignoring my research because it's wrong or not valuable. It's doing it because it doesn't like the impact of that research. And with prime counting, speed isn't the issue. Yes, you can use my ideas to re-hash over old ground and get to very fast prime counting algorithms that could equal those currently known, but I don't think they'd beat them, so why bother? But focus on the partial difference equation and you can maybe go after the Riemann hypothesis. Posters focus on speed. Given how important the Riemann hypothesis supposedly is, I think they focus on speed as they understand my disdain for it, and that getting working algorithms is enough work that it could take me a while, and that I'd probably not make anything faster than what's current--for some simple mathematical reasons--and they could always poke at such efforts for that reason. So speed is where they can feel confident they can always step back and claim my research isn't better, as at best it's likely to just match the fastest algorithms known. James Harris === Subject: Re:JSH: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. But > so what?. My guess is that I could travel over a lot of ground already covered > with previous people's ideas and end up with a separate effort using my > own research that gets to about where people are already at, and it > bored me, so I stop caring about speed and focused on more interesting > areas like the partial difference equation. Now to you maybe it's a big deal like why wouldn't I just go ahead and > make something that comes close in speed to what's already known to > prove my point? Why should I have to? I've argued out the speed-ups to my prime > counting function before! Regular posters who have argued with me for years already know about > them. The math community isn't ignoring my research because it's wrong or not > valuable. It's doing it because it doesn't like the impact of that > research. And with prime counting, speed isn't the issue. Yes, you can use my ideas to re-hash over old ground and get to very > fast prime counting algorithms that could equal those currently known, > but I don't think they'd beat them, so why bother? But focus on the partial difference equation and you can maybe go after > the Riemann hypothesis. Posters focus on speed. Given how important the Riemann hypothesis supposedly is, I think they > focus on speed as they understand my disdain for it, and that getting > working algorithms is enough work that it could take me a while, and > that I'd probably not make anything faster than what's current--for > some simple mathematical reasons--and they could always poke at such > efforts for that reason. So speed is where they can feel confident they can always step back and > claim my research isn't better, as at best it's likely to just match > the fastest algorithms known. > James Harris > I think all mathematicians at one point have to do things that they really don't want to do. My areas of interest are Differential Equations, Regression, Optimization, and Convexity Theory. What do you think my boss would say if I refused to do something outside of my areas of interest? I don't think I'd have a job much longer, if you ask me. Dave === Subject: Re: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. > Pwning ur moms ass isnt boring === So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. But > so what?. My guess is that I could travel over a lot of ground already covered > with previous people's ideas and end up with a separate effort using my > own research that gets to about where people are already at, and it > bored me, so I stop caring about speed and focused on more interesting > areas like the partial difference equation. Now to you maybe it's a big deal like why wouldn't I just go ahead and > make something that comes close in speed to what's already known to > prove my point? Why should I have to? I've argued out the speed-ups to my prime > counting function before! Regular posters who have argued with me for years already know about > them. The math community isn't ignoring my research because it's wrong or not > valuable. It's doing it because it doesn't like the impact of that > research. And with prime counting, speed isn't the issue. Yes, you can use my ideas to re-hash over old ground and get to very > fast prime counting algorithms that could equal those currently known, > but I don't think they'd beat them, so why bother? But focus on the partial difference equation and you can maybe go after > the Riemann hypothesis. Posters focus on speed. Given how important the Riemann hypothesis supposedly is, I think they > focus on speed as they understand my disdain for it, and that getting > working algorithms is enough work that it could take me a while, and > that I'd probably not make anything faster than what's current--for > some simple mathematical reasons--and they could always poke at such > efforts for that reason. So speed is where they can feel confident they can always step back and > claim my research isn't better, as at best it's likely to just match > the fastest algorithms known. > James Harris Do you know what a complex number is? === Subject: Re: JSH: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. I seriously doubt that you get anything that compares with the best algorithms that are known at the moment. > But > so what?. My guess is that I could travel over a lot of ground already covered > with previous people's ideas and end up with a separate effort using my > own research that gets to about where people are already at, As I say, I doubt you get up to where people are at at the moment. > and it > bored me, so I stop caring about speed and focused on more interesting > areas like the partial difference equation. Now to you maybe it's a big deal like why wouldn't I just go ahead and > make something that comes close in speed to what's already known to > prove my point? > It's not a big deal, I just won't accept your claim until you prove it. > Why should I have to? You don't. > I've argued out the speed-ups to my prime > counting function before! > But you haven't proved you can get anything that approaches even the Meissel-Lehmer method in efficiency. > Regular posters who have argued with me for years already know about > them. The math community isn't ignoring my research because it's wrong or not > valuable. No, I'm afraid that's exactly why they're ignoring it. > It's doing it because it doesn't like the impact of that > research. The impact of the research is nil. > And with prime counting, speed isn't the issue. > What is? > Yes, you can use my ideas to re-hash over old ground and get to very > fast prime counting algorithms that could equal those currently known, No, you can't. > but I don't think they'd beat them, so why bother? But focus on the partial difference equation and you can maybe go after > the Riemann hypothesis. > How? > Posters focus on speed. Given how important the Riemann hypothesis supposedly is, I think they > focus on speed as they understand my disdain for it, and that getting > working algorithms is enough work that it could take me a while, and > that I'd probably not make anything faster than what's current--for > some simple mathematical reasons--and they could always poke at such > efforts for that reason. > You have no prospect of making any progress on the Riemann hypothesis (you don't even understand what it says), and you have no prospect of coming up with an algorithm for evaluating the prime counting function which approaches even the Meissel-Lehmer algorithm in efficiency. > So speed is where they can feel confident they can always step back and > claim my research isn't better, as at best it's likely to just match > the fastest algorithms known. > All aspects of your research are totally insignificant, and indeed the great majority of the time they're just plain wrong. James Harris === Subject: Re: JSH: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. I seriously doubt that you get anything that compares with the best > algorithms that are known at the moment. > Oh yeah, always go back to the FASTEST IN THE WORLD as if I have to deliver at that level out of the gate. called PrimeCountH.java which could up to about 10^11 beat Mathematica. So that's to about 100,000,000,000 it was faster than commercial math software when it was just some Java code I'd written. But yeah, then Mathematica starts catching up and becomes much faster as you get to bigger numbers where the mathematical reasons aren't complicated. Any chance though you give a damn about the why's? Why does Mathematica beat it out at later ranges while getting beat with smaller numbers? James Harris === Subject: Re: JSH: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. I seriously doubt that you get anything that compares with the best > algorithms that are known at the moment. > Oh yeah, always go back to the FASTEST IN THE WORLD as if I have to > deliver at that level out of the gate. > I seriously doubt you can even get it to compete with the Meissel-Lehmer algorithm, asymptotically. > called PrimeCountH.java which could up to about 10^11 beat Mathematica. So that's to about 100,000,000,000 it was faster than commercial math > software when it was just some Java code I'd written. But yeah, then Mathematica starts catching up and becomes much faster > as you get to bigger numbers where the mathematical reasons aren't > complicated. Any chance though you give a damn about the why's? Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? > James Harris === Subject: Re: JSH: Prime counting speed is boring [Rupert, to JSH] > ... > I seriously doubt you can even get it to compete with the > Meissel-Lehmer algorithm, asymptotically. But I could :-) Remember that the /computationally sane/ versions of JSH's prime-counting formulas (the ones that don't recursively invoke themselves to determine /whether/ an integer is prime) are in fact minor variations of Legendre's formula, and because James's P(x, a) is very simply related to Legendre's phi(x, a). That cuts both ways: any way of speeding phi() computation maps to a similar way of speeding P() computation. There's no /point/ to that I can see, since it's already been done for phi(), but I don't see any theoretical problem in doing so. And there's always the silly way: P(x, a) = phi(x, a) + a - 1 That is, you can use that to compute P() almost exactly as fast as the fastest-known way to compute phi() as the centuries go by :-) === Subject: Re: JSH: Prime counting speed is boring <0dCdnemEdNKCQ8rYnZ2dnUVZ_vSdnZ2d@comcast.com [Rupert, to JSH] > ... > I seriously doubt you can even get it to compete with the > Meissel-Lehmer algorithm, asymptotically. But I could :-) Remember that the /computationally sane/ versions of JSH's > prime-counting formulas (the ones that don't recursively invoke themselves > to determine /whether/ an integer is prime) are in fact minor variations of > Legendre's formula, and because James's P(x, a) is very simply related to > Legendre's phi(x, a). That cuts both ways: any way of speeding phi() > computation maps to a similar way of speeding P() computation. There's no /point/ to that I can see, since it's already been done for > phi(), but I don't see any theoretical problem in doing so. And there's always the silly way: P(x, a) = phi(x, a) + a - 1 That is, you can use that to compute P() almost exactly as fast as the > fastest-known way to compute phi() as the centuries go by :-) Well, yeah, these are valid points and you're obviously a bit more up to speed in computer science than me, but what this illustrates is that when you make a vague claim like modifications of this method will yield efficient algorithms, you have to clarify what that means. James claims that somehow or other his description of the algorithm yields enough insight that efficient methods of computing pi(x) can be found with essentially no more insight required, but the question of what amounts to essentially no more insight is a vague one and you can't definitively resolve any disputes about it. === Subject: Re: JSH: Prime counting speed is boring [Rupert, to JSH] > ... > I seriously doubt you can even get it to compete with the > Meissel-Lehmer algorithm, asymptotically. [Tim Peters] >> But I could :-) Remember that the /computationally sane/ versions >> of JSH's prime-counting formulas (the ones that don't recursively >> invoke themselves to determine /whether/ an integer is prime) are in >> fact minor variations of Legendre's formula, and because James's >> P(x, a) is very simply related to Legendre's phi(x, a). That cuts >> both ways: any way of speeding phi() computation maps to a similar >> way of speeding P() computation. >> There's no /point/ to that I can see, since it's already been done for >> phi(), but I don't see any theoretical problem in doing so. >> And there's always the silly way: >> P(x, a) = phi(x, a) + a - 1 >> That is, you can use that to compute P() almost exactly as fast as the >> fastest-known way to compute phi() as the centuries go by :-) [Rupert] > Well, yeah, these are valid points and you're obviously a bit more > up to speed in computer science than me, Doesn't matter, except to the credibility of my confidence in being able to write a program backing up my claims (and even that has more to do with 35 years' intense programming experience than with comp sci theory): this is really more of a math point: P(x, a) = phi(x, a) + a - 1 Just keeping that in mind cuts through endless layers of bull here. James /can't/ make a claim about P() without (whether he knows it or not) making a simply-related claim about phi(), because there's such trivial difference between the functions (in the mathematical sense). > but what this illustrates is that when you make a vague claim like > modifications of this method will yield efficient algorithms, you > have to clarify what that means. Rest assured that if I ever make such a claim, I will clarify what it means :-) > James claims that somehow or other his description of the algorithm > yields enough insight that efficient methods of computing pi(x) can > be found with essentially no more insight required, While he no longer appears to do any work, a few years ago he did write at least one reasonably fast pi() program based on his ideas. It wasn't as those too get increasingly tedious to develop and program correctly. Of course James's programs of this kind were /much/ more gnarly than the one-line LaTeX he's been repeating recently too. > but the question of what amounts to essentially no more insight is > a vague one and you can't definitively resolve any disputes about it. I'm more inclined to the view that without clarification first, there's nothing /to/ dispute. James is a master at sucking people into arguing about nothing -- for example, what /did/ he disprove about standard teaching of Galois theory (or ideal or number theory, or anything else)? Nobody knows to this day ;-) === Subject: Re: JSH: Prime counting speed is boring > So yeah, I know how to use some simple techniques to make algorithms > from my prime counting function that are ever faster, but it's boring. What I was using in my previous post is some simple thing that I > thought I figured out a few years back until I was informed it was > known, where a key result is using floor((x-p_i)/(p_i*p_i-1*...2) where it's hard to write that but you subtract the prime from x and > divide by that prime and all the lesser primes. I had it wrong when I first did it as, for instance, I used > floor((x-4)/6) for the count of composites up to and including x that > had 3 as a factor but were not even. That works too, but it's simpler > to use floor((x-3)/6) and you can use that idea with more and more primes to keep speeding up > the algorithm, but there are more tricks that are available. Put them together and you can do some really fast prime counting. I seriously doubt that you get anything that compares with the best > algorithms that are known at the moment. > Oh yeah, always go back to the FASTEST IN THE WORLD as if I have to > deliver at that level out of the gate. called PrimeCountH.java which could up to about 10^11 beat Mathematica. So that's to about 100,000,000,000 it was faster than commercial math > software when it was just some Java code I'd written. But yeah, then Mathematica starts catching up and becomes much faster > as you get to bigger numbers where the mathematical reasons aren't > complicated. Any chance though you give a damn about the why's? Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? > James Harris Shouldn't you know why James? You seem to think you are the foremost researcher in the area. === Subject: Re: JSH: Prime counting speed is boring [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most /likely/ reason, a priori, is that Mathematica uses an asymptotically superior algorithm, whose overheads penalize it at smaller inputs. If you don't know what asymptotically superior algorithm means, read a book. === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most > /likely/ reason, a priori, is that Mathematica uses an asymptotically > superior algorithm, whose overheads penalize it at smaller inputs. > My guess is there is something like this at the head of the file 01/04/1847 Added an optimization that speeds this thing up by 30% for inputs around 10^14. This optimization really slows things down till about 10^11 or so. change the code so the older method is done for smaller inputs. -William Hughes === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most > /likely/ reason, a priori, is that Mathematica uses an asymptotically > superior algorithm, whose overheads penalize it at smaller inputs. My guess is there is something like this at the head > of the file 01/04/1847 Added an optimization that speeds this thing up by 30% for > inputs around 10^14. This optimization > really slows things down till about 10^11 or so. > change the code > so the older method is done for smaller inputs. -William Hughes Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. The reality is that optimizations can be done to my prime counting function easily that put you in the range of commercial math software with little effort. But that reality is not what you people want the groups to believe so you babble around it to promote your own point of view without the slightest interest in what the actual answer is. And that's where you diverge from most of humanity as you people do not want to know what is the truth--you want to convince other people of something without concern about how close to the truth you are. So you lack that fundamental human thing--curiosity. James Harris === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most > /likely/ reason, a priori, is that Mathematica uses an asymptotically > superior algorithm, whose overheads penalize it at smaller inputs. My guess is there is something like this at the head > of the file 01/04/1847 Added an optimization that speeds this thing up by 30% for > inputs around 10^14. This optimization > really slows things down till about 10^11 or so. > change the code > so the older method is done for smaller inputs. -William Hughes Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. > ask them what algorithm they used? > The reality is that optimizations can be done to my prime counting > function easily that put you in the range of commercial math software > with little effort. But that reality is not what you people want the groups to believe so > you babble around it to promote your own point of view without the > slightest interest in what the actual answer is. And that's where you diverge from most of humanity as you people do not > want to know what is the truth--you want to convince other people of > something without concern about how close to the truth you are. So you lack that fundamental human thing--curiosity. > James Harris === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most > /likely/ reason, a priori, is that Mathematica uses an asymptotically > superior algorithm, whose overheads penalize it at smaller inputs. My guess is there is something like this at the head > of the file 01/04/1847 Added an optimization that speeds this thing up by 30% for > inputs around 10^14. This optimization > really slows things down till about 10^11 or so. > change the code > so the older method is done for smaller inputs. -William Hughes Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. > ask them what algorithm they used? According to http://support.wolfram.com/mathematica/kernel/Symbols/System/PrimePi.html PrimePi and Prime use sparse caching and sieving for n less than about 10^8. For large n the Lagarias-Miller-Odlyzko algorithm is used for PrimePi, based on asymptotic estimates of the density of primes, and is inverted to give Prime[n]. Based on this information my best guess is that James made an error when he did his timing tests. - William Hughes === Subject: Re: JSH: Prime counting speed is boring [jstevh@msn.com] >> ... >> Why does Mathematica beat it out at later ranges while getting >> beat with smaller numbers? [Tim Peters] > Can't know for sure without having both implementations at hand. > The most /likely/ reason, a priori, is that Mathematica uses an > asymptotically superior algorithm, whose overheads penalize it at > smaller inputs. [William Hughes] >> My guess is there is something like this at the head >> of the file >> 01/04/1847 Added an optimization that speeds this thing up by >> 30% for inputs around 10^14. This optimization >> really slows things down till about 10^11 or so. >> change the code so the older method is done >> for smaller inputs. [jstevh@msn.com] > Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. [Rupert] >> and ask them what algorithm they used? [William Hughes] > According to http://support.wolfram.com/mathematica/kernel/Symbols/System/PrimePi.html PrimePi and Prime use sparse caching and sieving for n less than > about 10^8. For large n the Lagarias-Miller-Odlyzko algorithm is > used for PrimePi, based on asymptotic estimates of the density of > primes, and is inverted to give Prime[n]. Based on this information my best guess is that James made an > error when he did his timing tests. He was certainly using an older release of Mathematica at the time, so there's really no way to guess without knowing which version was used. Even then, your first guess can't be ruled out entirely short of seeing the implementation code. If there are N lines of code in Mathematica, there are at least N^2 ways a minor bug could cause a mysterious timing anomaly too ;-) === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com> > ... >> Why does Mathematica beat it out at later ranges while getting >> beat with smaller numbers? [Tim Peters] > Can't know for sure without having both implementations at hand. > The most /likely/ reason, a priori, is that Mathematica uses an > asymptotically superior algorithm, whose overheads penalize it at > smaller inputs. [William Hughes] >> My guess is there is something like this at the head >> of the file >> 01/04/1847 Added an optimization that speeds this thing up by >> 30% for inputs around 10^14. This optimization >> really slows things down till about 10^11 or so. >> change the code so the older method is done >> for smaller inputs. [jstevh@msn.com] > Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. [Rupert] >> and ask them what algorithm they used? > [William Hughes] > According to http://support.wolfram.com/mathematica/kernel/Symbols/System/PrimePi.html PrimePi and Prime use sparse caching and sieving for n less than > about 10^8. For large n the Lagarias-Miller-Odlyzko algorithm is > used for PrimePi, based on asymptotic estimates of the density of > primes, and is inverted to give Prime[n]. Based on this information my best guess is that James made an > error when he did his timing tests. He was certainly using an older release of Mathematica at the time, so > there's really no way to guess without knowing which version was used. Even > then, your first guess can't be ruled out entirely short of seeing the > implementation code. If there are N lines of code in Mathematica, there are > at least N^2 ways a minor bug could cause a mysterious timing anomaly too > ;-) Indeed. My guess (and it can't be seen as anything other than a very slighlty educated guess) was based on the thinking that while PrimePi might not be considered too important, Prime is. Also, I would expect Prime to be used most heavily in the range 1..10^8. So my conclusion was that one would expect good performance in that range. This may be wrong and we cannot of course exclude a bug or feature that affected only PrimePi. Still, on the balance I think you have to go with never underestimate James' incompetence. - William Hughes === Subject: Re: JSH: Prime counting speed is boring >> [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while > getting > beat with smaller numbers? >> [Tim Peters] >> Can't know for sure without having both implementations at >> hand. >> The most /likely/ reason, a priori, is that Mathematica uses >> an >> asymptotically superior algorithm, whose overheads penalize >> it at >> smaller inputs. >> [William Hughes] > My guess is there is something like this at the head > of the file 01/04/1847 Added an optimization that speeds this thing up > by > 30% for inputs around 10^14. This optimization > really slows things down till about 10^11 or so. > change the code so the older method is done > for smaller inputs. >> [jstevh@msn.com] >> Yeah babble like dimwits as if there isn't an answer. >> You know there are questions in life that can be answered. >> This is one of them. Speculation is not necessary. >> [Rupert] > and ask them what algorithm they used? >> [William Hughes] >> According to >> http://support.wolfram.com/mathematica/kernel/Symbols/System/PrimePi.html >> PrimePi and Prime use sparse caching and sieving for n less >> than >> about 10^8. For large n the Lagarias-Miller-Odlyzko algorithm >> is >> used for PrimePi, based on asymptotic estimates of the >> density of >> primes, and is inverted to give Prime[n]. >> Based on this information my best guess is that James made an >> error when he did his timing tests. >> He was certainly using an older release of Mathematica at the time, >> so >> there's really no way to guess without knowing which version was >> used. Even >> then, your first guess can't be ruled out entirely short of seeing >> the >> implementation code. If there are N lines of code in Mathematica, >> there are >> at least N^2 ways a minor bug could cause a mysterious timing >> anomaly too >> ;-) Indeed. My guess (and it can't be seen as anything other than a > very > slighlty educated guess) was based on the thinking that while > PrimePi > might not be considered too important, Prime is. Also, I would > expect > Prime to be used most heavily in the range 1..10^8. So my > conclusion > was that one would expect good performance in that range. This > may be wrong and we cannot of course exclude a bug or feature that > affected only PrimePi. Still, on the balance I think you have to > go > with > never underestimate James' incompetence. I am intrigued as to why you think PrimePi[x] (the number of primes <= x) might be considered less important that Prime[x] (the x'th prime). If I was forced to rate their importance, I would probably put them the other way around. -- Clive Tooth http://www.shutterstock.com/cat.mhtml?gallery_id=61771 === Subject: Re: JSH: Prime counting speed is boring <5K6dnU_V4seMWMrYnZ2dnUVZ_tGdnZ2d@comcast.com [jstevh@msn.com] > ... > Why does Mathematica beat it out at later ranges while getting beat > with smaller numbers? Can't know for sure without having both implementations at hand. The most > /likely/ reason, a priori, is that Mathematica uses an asymptotically > superior algorithm, whose overheads penalize it at smaller inputs. My guess is there is something like this at the head > of the file 01/04/1847 Added an optimization that speeds this thing up by 30% for > inputs around 10^14. This optimization > really slows things down till about 10^11 or so. > change the code > so the older method is done for smaller inputs. -William Hughes Yeah babble like dimwits as if there isn't an answer. You know there are questions in life that can be answered. This is one of them. Speculation is not necessary. > ask them what algorithm they used? > Not necessary. They post it on Wolfram's website. > The reality is that optimizations can be done to my prime counting > function easily that put you in the range of commercial math software > with little effort. But that reality is not what you people want the groups to believe so > you babble around it to promote your own point of view without the > slightest interest in what the actual answer is. And that's where you diverge from most of humanity as you people do not > want to know what is the truth--you want to convince other people of > something without concern about how close to the truth you are. So you lack that fundamental human thing--curiosity. I left that in to emphasize the problem with the people I'm facing, they do lack just basic human curiosity. There are so many interesting and amazing things associated around my research but none of it is of interest to people who spend so much time and effort to try and convince others that there is nothing of interest with what I have. They lack--curiosity. James Harris === Subject: Re: JSH: Prime counting speed is boring > They lack--curiosity. You dont lack booty stank. === Subject: Re: JSH: Prime counting speed is boring noise(t-1) / noise(t) < 1 === Subject: Re: JSH: Prime counting speed is boring > noise(t-1) / noise(t) < 1 Yeah it's noise when I refute your precious posters who lie to you all the time but you love it. You people sit back when I get things wrong and you feel satisfied, thinking that's all there is to it. But, it's the corrections that you want to cut off from because you don't give a damn about the math. And that's where mathematics foils you all, as yes, a person can get details wrong, and look stupid in the process, but they can also come back and correct to what is MATHEMATICALLY CORRECT and then your group processes can only deny. So you want to cut-off then. You have a contempt for the truth. James Harris === Subject: Re: JSH: Prime counting speed is boring The only mistake JSH made was trying to discuss mass with you absolute r-tards. All the errors you point out are pure bullmuffins IV. > noise(t-1) / noise(t) < 1 Yeah it's noise when I refute your precious posters who lie to you all > the time but you love it. You people sit back when I get things wrong and you feel satisfied, > thinking that's all there is to it. But, it's the corrections that you want to cut off from because you > don't give a damn about the math. And that's where mathematics foils you all, as yes, a person can get > details wrong, and look stupid in the process, but they can also come > back and correct to what is MATHEMATICALLY CORRECT and then your group > processes can only deny. So you want to cut-off then. You have a contempt for the truth. > James Harris === Subject: Re: JSH: Prime counting speed is boring > noise(t-1) / noise(t) < 1 Yeah it's noise when I refute your precious posters who lie to you all > the time but you love it. > You've never refuted anyone. > You people sit back when I get things wrong and you feel satisfied, > thinking that's all there is to it. > It is, more or less. It's pretty much all you ever do. > But, it's the corrections that you want to cut off from because you > don't give a damn about the math. And that's where mathematics foils you all, as yes, a person can get > details wrong, and look stupid in the process, but they can also come > back and correct to what is MATHEMATICALLY CORRECT and then your group > processes can only deny. > You've just about never asserted anything mathematically correct. > So you want to cut-off then. You have a contempt for the truth. > James Harris === Subject: Re: JSH: Prime counting speed is boring > noise(t-1) / noise(t) < 1 Yeah it's noise when I refute your precious posters who lie to you all > the time but you love it. You people sit back when I get things wrong and you feel satisfied, > thinking that's all there is to it. But, it's the corrections that you want to cut off from because you > don't give a damn about the math. And that's where mathematics foils you all, as yes, a person can get > details wrong, and look stupid in the process, but they can also come > back and correct to what is MATHEMATICALLY CORRECT and then your group > processes can only deny. So you want to cut-off then. You have a contempt for the truth. > James Harris The problem James isn't that you make mistakes. Everyone does from time to time. The problem is that you unleash your venom on everyone who points them out ot you, calling them liars and parasites. Then much later you admit to the mistakes, but make it seem like you are the one who realized them, not the 10 people who had pointed them out to you multiple times. Then you discount the mistakes as if they didn't metter, even though they usually totally undermine whatever point you were trying to make. It is really a pathetic cycle. === Subject: Re: JSH: Prime counting speed is boring > noise(t-1) / noise(t) < 1 Yeah it's noise when I refute your precious posters who lie to you all > the time but you love it. You people sit back when I get things wrong and you feel satisfied, > thinking that's all there is to it. But, it's the corrections that you want to cut off from because you > don't give a damn about the math. And that's where mathematics foils you all, as yes, a person can get > details wrong, and look stupid in the process, but they can also come > back and correct to what is MATHEMATICALLY CORRECT and then your group > processes can only deny. So you want to cut-off then. You have a contempt for the truth. > James Harris The problem James isn't that you make mistakes. Everyone does from time > to time. The problem is that you unleash your venom on everyone who > points them out ot you, calling them liars and parasites. Then much > later you admit to the mistakes, but make it seem like you are the one > who realized them, not the 10 people who had pointed them out to you > multiple times. Then you discount the mistakes as if they didn't > metter, even though they usually totally undermine whatever point you > were trying to make. It is really a pathetic cycle. No, this is: http://media.nothingtoxic.com/uploads/patheticbikewreck.jpg === Subject: Re: Any graduate program without Bachelor's degree? Most schools CAN admit someone for graduate study without a bachelors degree. But they only do it when the candidate is so exceptional that it makes sense. It is not a matter of lacking the requirement, but instead making an exception to it. === Subject: Re: Any graduate program without Bachelor's degree? In <1396463.1163040071613.JavaMail.jakarta@nitrogen.mathforum.org>, on 11/08/2006 at 09:40 PM, Mr.T said: >I would like to ask if there are some universities which have >graduate program in mathematics not requiring the Bachelor's degree >for entrance. There are universities that will accept students into a graduate program without a bachelor's degree. In every case that I'm aware of, it's an exception to normal procedures and requires demonstrating a strong background. >I have formally finished only the following courses in school: >Calculus 1, 2, and 3, Elementary Linear Algebra, and Differential >Equations. That's not nearly enough. What have you learned on your own? >the new sci.math system. There is no such thing. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: Any graduate program without Bachelor's degree? >fishfry said: >> I'm sure you'd be admitted anywhere if you could >> demonstrate sufficient >> mathematical skill and knowledge. It's not the >> degree, it's the >> knowledge that counts. >> In your case you do not have the prerequisites to >> enter grad school. You >> need to have taken at least real analysis and >> abstract algebra. Without >> those you would be totally lost in the grad-level >> math curriculum. >I have a friend at a tier two school which will be tier one next year getting a masters. Most of the people in his class did not have abstract algebra. So saying you have to have AT LEAST that is not true at all really. And, those people are doing fine, though of course it would be easier for them if they had had algebra first. This is too often the case, and even more so for real analysis. If you can show that you UNDERSTAND abstract algebra and real analysis, and know what is involved in constructing a proof, you should be a good candidate for a graduate assistantship at many good uviversities. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Connecting the dots > Mathematics is that important. Without this revolution there is no > future. > Theres no future for ur moms ass once im done pwning it. === Subject: Re: JSH: Connecting the dots >On 11 Nov 2006 13:04:37 -0800, William Hughes >>Jeeves. Sell the long term bonds. > >I have a question. It has little to do with math, but a lot with the >way James is ridiculed on a regular basis. I think most of us would >agree that making fun of somebody who is retarded is not right. In >fact, I think most of us would simply not make fun of someone who is >retarded at all. I think the reason for this is that the retarded >person cannot help the way they are. Now James is obviously (in a medical sense) not retarded, but from the >vast amount of posts (even if you only see the ones he hasn't deleted >yet) it is clear that he is unable to grasp that he runs into so much >trouble simply because he is wrong (most of the time anyway). As James >doesn't seem to be able to get this himself, why is it okay to make >fun of him? (Don't get me wrong, I think making fun of him is amusing >most of the time (although a bit easy). I'm just wondering why it is >different from making fun of a retard.) The people at the cafeteria in the Student Union who go around and clean up tables left in disorder by assholes who can't clean up after themselves are all somewhat retarded. Been that way for years - I've never checked, but they must have a policy of hiring such people for this job, which I think is a good thing, gives them something worthwhile to do, lets them mingle with a lot of people, etc. I've never seen anyone make fun of any of them even though they're obviously very slow - I know _I've_ certainly never made fun of them (not when they were around, anyway). One of them tends to join our little backgammon group outside to sit and watch the game. People say hi, nobody laughs at the stupid things he says and does (they do laugh when _I_ do something stupid, so it's not like they're all saints or anything; really doesn't seem fair, come to think of it. Sniffle.) When there's a pretty girl sitting by herself at a table this guy tends to sit down with her and strike up a conversation, obviously trying to get somewhere. I've never seen one of those pretty girls brush the guy off - they're always polite. However. It never happens that one of these people I'm talking about announces to me that he's the world's greatest number theorist. It never happens that one of them tells me that I'm incompetent. It never happens that one of them tells me that if I continue to teach my students lies about algebraic integers I should be fired for incompetence. Etc etc etc. If one of them behaved anything remotely like the way James behaves he'd be fired. ************************ === Subject: Math in dreams A friend of mine (Whyzard), has be going. He says he has developed mathematics to describe the nature of heaven & hell. And I thought, that's origonal. Math on earth is based on the objective reality we experience, and we don't really have a use for math in heaven yet. But math is subjective to reality first and foremost. So can you imagine what math would be like in dreams? Here is a little thought experiment for you. Imagine a cactus. Now cut the cactus in half, and plant both halfs in cactus soil. With good rooting hormone. Now how many cactuses do you have? The one cactus cut in half in a math text book should still make only one cactus (2 halfs). But both halfs root, and plant, and become two seperate cactuses. Now it's a philisophical debate only to say if we have one cactus that exists in two location (Called bio-location), or if we have two cactuses that exist seperately (Called self reproduction). If we strictly follow arithmetic then we only have one cactus. But in dreams we can devide a cactus into two halves, and believe we have create more than what we started with. Just go to sleep tonight, and try the experiment. I'm interested in hearing your conclusions. alt.magick.moderated is a MODERATED newsgroup. Consult http://www.alt-magick-moderated.org/ where you may locate the newest Posting Guidelines and Charter for the newsgroup before your first post. Contact the Moderation Team at moderators@alt-magick-moderated.org === Subject: Re: Math in dreams > Imagine a cactus. Now cut the cactus in half, and plant both halfs in > cactus soil. With good rooting hormone. Now how many cactuses do you have? The one cactus cut in half in a > math text book should still make only one cactus (2 halfs). But both > halfs root, and plant, and become two seperate cactuses. This is not a mathmatical puzzle, it is a semantic puzzle since it rests on how you are defining cactus R. alt.magick.moderated is a MODERATED newsgroup. Consult http://www.alt-magick-moderated.org/ where you may locate the newest Posting Guidelines and Charter for the newsgroup before your first post. Contact the Moderation Team at moderators@alt-magick-moderated.org === Subject: Re: Math in dreams > A friend of mine (Whyzard), has be going. He says he has developed > mathematics to describe the nature of heaven & hell. And I thought, > that's origonal. Math on earth is based on the objective reality we > experience, and we don't really have a use for math in heaven yet. But > math is subjective to reality first and foremost. So can you imagine > what math would be like in dreams? I already know. Here is a little thought experiment for you. Imagine a cactus. Now cut the cactus in half, and plant both halfs in > cactus soil. With good rooting hormone. Now how many cactuses do you have? The one cactus cut in half in a > math text book should still make only one cactus (2 halfs). But both > halfs root, and plant, and become two seperate cactuses. Now it's a philisophical debate only to say if we have one cactus that > exists in two location (Called bio-location), or if we have two > cactuses that exist seperately (Called self reproduction). If we strictly follow arithmetic then we only have one cactus. But in > dreams we can devide a cactus into two halves, and believe we have > create more than what we started with. Just go to sleep tonight, and try the experiment. I'm interested in > hearing your conclusions. I once dreamed I was signing the visitor log at a hospital. That's last week. month and day, writing dd/mm/yy instead of mm/dd/yy. But there is no possible date such that said transposition would result in a week (or less) difference! Conclusion: the subconscience mind is stupid. alt.magick.moderated is a MODERATED newsgroup. Consult http://www.alt-magick-moderated.org/ where you may locate the newest Posting Guidelines and Charter for the newsgroup before your first post. Contact the Moderation Team at moderators@alt-magick-moderated.org === Subject: Re: Math in dreams > I once dreamed I was signing the visitor log at a hospital. That's last week. > month and day, writing dd/mm/yy instead of mm/dd/yy. But there is no possible date such that said transposition > would result in a week (or less) difference! Conclusion: the subconscience mind is stupid. Dreaming of a stupid or careless nurse doesn't mean your subconscious is stupid. And at least your subconscious mind was sharp enough to write the date in a logical order rather than in the crazy Usanian order. The goddess Namagiri once appeared to Ramanujan in a dream and told him that pi was equal to the 4th root of 2143 / 22. alt.magick.moderated is a MODERATED newsgroup. Consult http://www.alt-magick-moderated.org/ where you may locate the newest Posting Guidelines and Charter for the newsgroup before your first post. Contact the Moderation Team at moderators@alt-magick-moderated.org === Subject: Re: Another Collatz-like function > I have just read something I did by 1995 about a Collatz related function. Collatz function, or 3x+1 problem, can be stated as the study of orbits > given by the function f(x)=x/2 if x odd, f(x)=(3x+1)/2 if x even. There are > plenty of studies done on this open problem. As this function can be written as f(x)=(x mod 2)*x+ceiling(x/2), I wondered > what if, instead, one took g(x)=(x mod 3)*x+ceiling(x/3). That is, g(x)=x/3 > if x=3t, g(x)=(4x+2)/3 if x=3t+1, g(x)=(7x+1)/3 if x=3t+2. Numerical experiments seem to lead to the existence of just two loops, one > with 2 as lowest member, one with 26. Also, there are some really wandering orbits. 101 eventually falls into a > loop after some 2600 steps, and reaching 19 digit values. 721 does so after > reaching 28 digit values; and 2396 after visiting 42 digit numbers. There are also some starting values that seem to escape. The first one whose > orbit reaches 500 digit numbers is 469. > Heuristically, since (1/3)(4/3)(7/3) = 28/27 > 1, I would expect things to escape. Many functions of this sort - piece-wise linear functions, the pieces depending on the congruence class of the argument for some modulus - have been studied. Not many conclusions Iterations, about this family of functions as a whole. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: AIDS KILLS TEN MILLION JEWS! ohhhh this hurts... people can be so stupid... it's unbelievable.. -- (Hudson on her role as Callisto) Hudson: Wouldn't you love it if you got to scream and yell and you got to fight people and you got a sword and no one yelled at you?! === Subject: Re: JSH: My prime counting, speed [jstevh@msn.com] >> I didn't ask for Python. I asked for math. [Tim Peters] > It's official, then: you really are too stupid to bother with. BTW, if anyone else feels like wasting their time on the trivial exercise of translating my teensy Python code back to equations, don't forget to ask James the proper way to measure the speed of LaTeX ;-) === Subject: Videos about chaos theory and fractals In 1989 and 1990 I produced a series of videos about chaos and dynamical systems with Bob Devaney and John Hubbard. These tapes are now available online - some free and some for only $3. They contain computer animations of dynamical systems and are currently distributed in VHS format by the American Mathematical Society. I want to find out if there is a market for these online and, if so, how best to market them. Here are some of the options: 1) Pay-per-download of a .mov or .wmv file for at least $3, but possibly as much as $10 2) Pay-per-stream at variable bit rates for $2 to $4 3) Free viewing of 10 minute segments via Revver.com which puts ads on the end of clips shares revenue 4) Sell DVD's for $20 per copy and only put some segments online. Programs include Chaos, Fractals and Dynamics http://scitv.com/cfd.html and Transition to Chaos http://scitv.com/ttc.html by Devaney The Beauty and Complexity of the Mandelbrot Set http://scitv.com/bcm.html by Hubbard Natural Minimal Surfaces http://scitv.com/nms.html by Hoffman The Art of Renaissance Science http://scitv.com/ars.html by Joseph W. Dauben I hope to produce more programs of this nature, but need to establish a viable business model. I encourage others to create programming and put it online via Revver, Google Video, YouTube and other venues. Gary Welz Science Television www.scitv.com === Subject: General Introduction of www.SoftwareEr.com General Introduction of www.SoftwareEr.com SoftwareEr.com (National Human Resources Public Service Platform of Software Industry) formerly was a trial project named HR Service Platform operated by Qingdao Software Park with assignment of the National Science and Technology Ministry in 2002-2005. With its successful operating experience, since 2005, it has been developing to all Torch Center certified Software Parks. SoftwareEr.com is one of the most important platforms established by the National Science and Technology Ministry to manage and serve the entire software industry. It is committed to providing the broadest and most effective human resources service network to Software enterprises having requirement on recruitment, IT talents, relevant eduational and training organizations etc located in 32 Software Parks and around. By sharing informations in and among 32 Software Parks, the platform will effectively resolve the HR bottleneck status in developing of software industry at present, and accelerate the development of China's software industry by leaps and bounds. Besides providing all-around professional IT position and recruiter information, SoftwareEr.com also provide various kinds of specialized services such as personnel testing, career planning, IT training, management consulting service to both enterprise and individual users, which are all designed and certified by experts from the Ministry of Science and Technology, Ministry of Education, Ministry of Information Industry, and other institutions. Its mission is to build up the largest and most authoritative recruitment network in terms of IT & software industry in China. Meanwhile, SoftwareEr.com undertake some research and support tasks for government organizatons, which will better assist them to macroscopically serve and manage enterprises and individuals of IT industry. Torch Center of the Ministry of Science and Technology is the top administrative and management department of SoftwareEr.com. And Qingdao JuShu Information & Technology Co., Ltd is authorized as the Operation & Management Center, which is responsible for extending and operating the platform throughout the entire country in coordinate ion with the local Software Parks. === Subject: Cut Points Upon having an opportunity to study previously downloaded material, I found the following theorem: Let X be a compact connected metric space with at least 2 points. Then for all x in X, X{x} has a non-cutpoint. This is clearly wrong. A circle is counter example. Is the theorem every compact connected metric space has a non-cut point or is there additional premises needed for the conclusion to be reached? Does anyone know how the theorem is supposed to read? Below is supposed proof and sketch for same results for compact connected multi-point Hausdorff spaces. Is there clue therein what the actual theorem is? -- Let X be a compact connected metric space with at least 2 points. Then for all x in X, X{x} has a non-cutpoint. Take a countable open base B_n (n in N), with B_0 = X. So suppose ALL points of X{x} are cutpoints. Define C_0 = {x}. We construct a chain of compact connected subsets of X, such that for all n: (1) C_n subset int(C_{n+1}) (2) C_n / B_n /= nulset (3) C_n /= X. Suppose that we already constructed C_n. If B_{n+1} C_n != nullset, then pick p in this set. Otherwise B_{n+1} subset C_n, then pick p from XC_n arbitrarily, which can be done by (3). By assumption, X{p} = U / V, where U and V are disjoint non-empty open sets of X, as p != x, and so is a cutpoint of X. We can assume wlog that x is in U. Note that U / {p} (and V/{p}) are closed and connected. [ closed, as X(U / {p}) = V, which is open. connected: assume that U / {p} = E / F, E and F relatively open and closed in U / {p}, and disjoint. We can assume that p is in E. Then F and V / E are disjoint and cover X, and are relatively closed-and-open. But X is connected, and V is non-empty (and so V / E as well), so F is empty. So A / {p} is connected.] Now, C_n is connected and a subset of X{p}; it intersects U (in x) so it cannot also intersect V, and hence C_n subset U. Then define C_{n+1} = U / {p}, and C_n subset U shows that (1) is satisfied. Also, C_{n+1} misses V so is not equal to X. If B_{n+1} was a subset of C_n, we are done with (2), but otherwise we chose p in B_{n+1}C_n subset B_{n+1}, so then we are ok too. As C_0 was as required, we can construct our sequence by induction. Now define C = /_n C_n. If C = X, then the interiors of the C_n cover X by (1), and are increasing. But then, as X is compact, there would be a finite subcover, and the largest C_n from this subcover must equal X, but this contradicts (3). So there is some y in XC. Again, y has to be a cutpoint by assumption, and so X{y} = E / F, for open non-empty sets E and F. Assume x is in E. All C_n are connected and miss y, so all C_n are a subset of E (as x is in all C_n). Hence C subset E. But F contains some basic open set B_k, and so C (and so C_k) misses B_k, and this contradicts (2). This final contradiction proves the claim. -- We can also prove this directly compact, Hausdorff, connected spaces. For this we do something similar: order all compact connected subsets of X by A <= B iff A subset int(B). To see that X{x} has a non-cutpoint, we take a maximal linearly ordered subset of all compact connected sets that contain {x}. and we consider the union of this subset... ---- === Subject: Infinite system of linear equations Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers, with i, j positive integers. Suppose also that we have a sequence {c_i} of reals. We wish to find a sequence {a_j} of reals so that the sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i. This is, in some sense, a collection of countably-many linear equations in countably-many variables. Are there any conditions on the coefficients b_(i, j) that would guarantee existence and/or uniqueness of a solution {a_j}? If there were only finitely-many equations and variables (the same number of each), then one could simply check that the determinant of the coefficient matrix is non-zero. Is there any analog of determinant for an omega-by-omega matrix? === Subject: Re: Infinite system of linear equations > Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers, > with i, j positive integers. Suppose also that we have a sequence > {c_i} of reals. We wish to find a sequence {a_j} of reals so that the > sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i. > This is, in some sense, a collection of countably-many linear equations > in countably-many variables. Are there any conditions on the > coefficients b_(i, j) that would guarantee existence and/or uniqueness > of a solution {a_j}? If there were only finitely-many equations and > variables (the same number of each), then one could simply check that > the determinant of the coefficient matrix is non-zero. Is there any > analog of determinant for an omega-by-omega matrix? Special cases of such infinite systems of linear equations have been investigated. In Multigrid Calculus it is proved that the infinite tri-diagonal system of linear equations: ........ -b 1 -a -b 1 -a -b 1 -a ......... is equivalent with a second order ordinary differential equation: d^2u/dx^2 - P.du/dx + (P^2-Q^2)/4.u = 0 where the matrix coefficients - with infinitesimal grid-spacings (dx) - are given by: a = exp(-(P-Q)/2.dx) ; b = exp(+(P+Q)/2.dx) And the general solution is of the form (with A,B arbitrary): u(x) = A.exp((P-Q)/2.x) + B.exp((P+Q)/2.x) The key reference is: http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf Han de Bruijn === Subject: Re: Infinite system of linear equations Am 13.11.2006 07:12 schrieb Jules: > Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers, > with i, j positive integers. Suppose also that we have a sequence > {c_i} of reals. We wish to find a sequence {a_j} of reals so that the > sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i. > This is, in some sense, a collection of countably-many linear equations > in countably-many variables. Are there any conditions on the > coefficients b_(i, j) that would guarantee existence and/or uniqueness > of a solution {a_j}? If there were only finitely-many equations and > variables (the same number of each), then one could simply check that > the determinant of the coefficient matrix is non-zero. Is there any > analog of determinant for an omega-by-omega matrix? > Just an idea: I actually don't know; what comes to mind is: - either the matrix is triangularizable by noniterative similarity transform - or you can find eigenvectors - and the associated (matrix of) Eigenvalues. Then the determinant is the limit of the product of the diagonal-entries. But I think this implies strict conditions for your infinite matrix. Gottfried Helms === Subject: Creation and Annihilation Operators in QM Suppose a self-adjoint operator F on a Hilbert space has discrete eigenvalues m_1 , m_2, m_3, etc. It is sometimes possible to find a creation operator C such that if phi is an eigenfunction of F for the eigenvalue m_n, then Cphi is an eigenfunction with eigenvalue m_{n+1}. Similarly an annihilation operator A does the opposite. Thus if phi has eigenvalue m_n then Aphi has eigenvalue m_{n-1}. The reason for the terminology is that one thinks of C and A as creating or annihilating quanta. Two classical examples of this in QM: 1) If L_z is the z-component of angular momentum (eigenvalues are hm for m=0,1,2) then one can show that L_x + iL_y and L_x - iL_y are creation and annihilation operators respectively. 2) For the harmonic oscillator with potential kx^2 one can show that C = cx + i dp and A = cx - dp are creation and annihilation operators for the Hamiltonian (x is position, p is momentum, and c and d are suitable constants. This is the most elegant way to mathematically derive the spectrum of energy eigenvalues. Question: Is there any theorem along the following lines. If an operator F has discrete spectrum and satisfies suitable hypotheses, then there exist creation and annihilation operators for F? I am no expert in functional analysis, but it seems to me that such a theorem should be possible. === SYNTHETIC TELEPATHY and PSYCHO-ELECTRONIC WEAPON ATTACKS on HUNDREDS of THOUSANDS of INNOCENT US CITIZENS by a combined 100,000 FBI and NSA FBI and NSA PSYCHOPATHS are conducting these NON-CONSENSUAL SECRET, ILLEGAL, SADISTIC, PERVERTED and PSYCHOPATHIC SYNTHETIC TELEPATHY EXPERIMENTS and PSYCHO-ELECTRONIC WEAPON ATTACKS on hundreds of thousands of innocent american citizens. VICTIMS and are being used as GUINEA PIGS by these FBI and NSA PERVERTS, TORTURERS and PSYCHOPATHS. The Pentagon and the U.S. intelligence agencies called the N.S.A. [ National Security Agency ], C.S.S. [ Central Security Service ], D.I.A. [ Defence Intelligence Agency ], D.A.R.P.A. [ Defense Advanced Reserch Projects Agency ], C.I.A. [ Central Intelligence Agency ], the D.H.S. [ Department of Homeland Security ], and the F.B.I. [ Federal Bureau of Investigation ] Cointelpro or Counter intelligence program unit has extremely top secret technology that can pick up the private thoughts given by individuals in the vibrations produced by the brains electrical impulses, and that these thoughts can be broadcast by means of microwave tranceivers, infrasound and ultrasound tranceivers, satellites, and any other form of technology which uses electromagnetic transfer, i.e. television, radio, the internet, and the telephone. The pentagon calls this mind reading technology SYNTHETIC TELEPATHY, although there is nothing paranormal about it at all. The N.S.A. or National Security Agency is a part of the U.S. Department of Defense, and a very secretive part at that. The N.S.A. has a black budget, and is always headed by a flag officer from one of the branches of the U.S. military. The top secret technologies available to the U.S. military is usually thirty years or maybe even more ahead of the technology which is available to the U.S. civilian economy. Who is to know among the general U.S. public what the level of technological development is within the secret laboratories of the N.S.A. and D.A.R.P.A. [ Defense Advanced Reserch Projects Agency ]. The headquarters of the National Security Agency is located at Fort George Meade in the U.S. state of Maryland. Today, there are microscopic computer chips based on nanotechnology which can be inserted into a person by means of an injection, capsule, or tooth filling, with or without the knowledge of a doctor or a pharmacist, and these computer chips can track the implanted person by radio or microwaves. These microscopic computer chips can also be made invisible to detection by means of C.A.T. [ Computerized Axial Tomography ], M.R.I. [ Magnetic Resonance Imaging ], and P.E.T. [ Positron Emission Tomography ] scans, by some means of electromagnetic degaussing. With the latest top secret, and rapid technological advancements available to the U.S. military intelligence agencies, the need for microscopic computer implants in conducting synthetic telepathy against targeted persons is obsolete by now. Rather, a system of subliminal, subconscious, wireless microwave remote control is used today against targeted individuals suffering from synthetic telepathy harrassment. Synthetic Telepathy is used for conducting torture training and/or experimentation against people as red herring good cop/bad cop or tough love acting, diversions, feints, decoys, and smokescreens, as smear campaigns by association, as openly secret intimidation campaigns against targeted dissidents, and as a means of espionage. Once it was believed that claims about C.I.A. agents secretly dumping radioactive material in the slums of U.S. cities during the 1950's as part of a research program were simply mass conspiracy theories, until the U.S. federal government in the 1990's publicly admitted that the allegation was true. FBI and NSA agents who are part of a secret US society are genetically inclined sociopaths and psychopaths who work as hit men, torturers, sexual abusers, stalkers, slanderers, and agent provocateurs involved in murders and torture of hundreds of thousands of innocent US citizens for years and years. At the last count there are 50,000 NSA spies who are businessmen, priests, community leaders and in all walks of life who are SPYING and keeping tabs on pretty much all americans. The synthetic telepathy espionage trainees, sociopaths, sadomasochistic torturers, slanderers, experimenters, and spies, in order to avoid developing any sympathy for their targeted victims, deliberately avoid actually feeling the emotions of their victims by following the squiggly EEG or Electroencephalograph lines on computer monitor screens, as well as the silent words thought by their victims which also appear on the computer monitor screens. Then again, who can feel all of the emotions and thoughts which any given person has undergone every second of their entire lifetime. The emotions and silent words of the victims of synthetic telepathy operators can be broadcast against their will to anyone the synthetic telepathy torturers choose to broadcast to, and they also have the power to block or censor any emotions and words of their victims which they do not want to be broadcast. The synthetic telepathy spies can also place their own criminal or antisocial, negative emotions and words into their victims by means of subliminal, subconscious, remote control microwave brainwashing technology, and they can change the voice behind their words so as to make it appear that it is the targeted victim and not themselves who are communicating by means of synthetic telepathy. They can also store emotions and silently thought words onto a supercomputer memory bank which they can rebroadcast at a later time. I believe that the synthetic telepathy operators, although having high I.Q.'s, are predisposed towards sociopathic behaviour because of a genetic defect or combination of genetic defects, and that the only way they can sustain enthusiasm in their lives is by engaging in an addictive pattern of sadomasochistic activities and fantasies. If you do not believe that synthetic telepathy exists, then you should look at these following web sites which reveal that part of electromagnetic psychological warfare and mind control which today is declassified: I urge the readers to print/save this column as well as the following link where the NSA and their PSYCHO-ELECTRONIC WEAPON ATTACKS and SADISTIC TORTURE of hundreds of thousands of innocent american citizens (guinea pigs and victims) is explained in detail with diagrams and technology. Covert Operations of the U.S. National Security Agency at http://www.mindcontrolforums.com/pro-freedom.co.uk/cov_us.html Readers are advised to print the document in the link above and study the pictures and technology. Important things from the above link for americans to educate and protect themselves and their loved ones from these secret, non-consensual and illegal psychopathic experiments, attacks and tortures by these 100,000 FBI and NSA secret society psychopaths and spies. 1) The NSA has records on all US citizens. The NSA gathers information on US citizens with over 50,000 NSA agents and spies (HUMINT). 2) NSA personnel in the community usually have cover identities such as social workers, priests, church leaders, lawyers and business owners. 3) NSA personnel can control the lives of hundreds of thousands of individuals in the US by using the NSA's domestic intelligence network and cover businesses. 4) This network (DOMINT) covers the entire US, involves tens of thousands of NSA personnel, and tracks millions of persons simultaneously . 5) NSA DOMINT has the ability to assassinate US citizens covertly or run covert psychological control operations to cause subjects to be diagnosed with ill mental health. 6) At the present time the NSA has nanotechnology computers that are 15 years ahead of present computer technology. 7) SYNTHETIC TELEPATHY and other psycho-electronic weapon chips can be inserted into the victims bodies painlessly without their knowledge in a matter of minutes by FBI and NSA spies. 8) NSA computer generated brain mapping can continuously monitor all of the electrical activity in the brain continuously. 9) For electronic surveillance purposes, electrical activity in the speech center of the brain can be translated into the subject's verbal thoughts. 10) NSA operatives can use this covertly to debilitate subjects by simulating auditory hallucinations characteristic of paranoid schizophrenia. 11) Without any contact with the subject, Remote Neural Monitoring can map out electrical activity from the visual cortex of a subject's brain and show images from the subject's brain on a video monitor. NSA operatives see what the surveillance subject's eyes are seeing. 12) Visual memory can also be seen. RNM can send images direct to the visual cortex, bypassing the eyes and optic nerves. NSA operatives can use this surreptitiously to put images into a surveillance subject's brain while they are in REM sleep for brain-programming purposes. 13) The NSA, Ft Meade has in place a vast two-way wireless RNM system which is used to track subjects and noninvasively monitor audio-visual information in their brains. This is all done with no physical contact with the subject. RNM is the ultimate method of surveillance and domestic intelligence. 14) Speech, 3D sound and subliminal audio can be sent to the auditory cortex of the subject's brain (bypassing the ears), and images can be sent into the visual cortex. RNM can alter a subject's perceptions, moods and motor control. RNM can alter a subject's perceptions, moods and motor control. 15) The NSA's Signals Intelligence has the proprietary ability to monitor remotely and non-invasively information in the human brain by digitally decoding the evoked potentials in the 30-50 Hz, 5 milliwatt electromagnetic emissions from the brain. 16) Remote monitoring/tracking of individuals in any location, inside any building, continuously, anywhere in the country. RNM can electronically identify individuals and track them anywhere in the US. 17) Tens of thousands of persons in each area working as spotters and neighbourhood/business place spies (sometimes unwittingly) following and checking on subjects who have been identified for covert control by NSA personnel. HOW FBI and NSA DISABLES, CRIPPLES and MURDERS its own innocent american citizens. God only knows how many of the unsolved murders and diseases are actually committed and spread by these 100,000 FBI and NSA agents and spies. 18) Chemicals and Drugs into Residential Buildings with Hidden NSA Installed and Maintained Plastic Plumbing lines. The NSA has kits for running lines into residential tap water and air ducts of subjects for the delivery of drugs (such as sleeping gas or brainwashing-aiding drugs). This is an outgrowth of CIA pharmapsychology (psychopharmacology). 19) Victims experience daily breakins in their homes by FBI and NSA secret society spies. Victims of this endless torture has no way to prove the breakins and poisoning of food since FBI and NSA spies have technology that can easily disable the video surveillance systems, phone lines and security alarm systems of the torture victims. 20) FBI and NSA spies threaten hundreds of thousands of these innocent torture victims in america with accidents, bio-terrorist attacks in the form of measles, chicken pox, small pox and other forms of viruses, life threatening diseases AIDS, debilitating diseases cystic fibrosis, alzheimers etc. FBI and NSA spies post these threatening messages on internet only the mind control victims can understand and knew those threats were meant for the victims. In some cases, unfortunately these torture victims dont even know they are torture victims. TORTURE VICTIMS Usually alone in his/her torture Has no privacy even for his/her private thoughts Cannot plan secretly, hold trade secrets or intellectual property Is subject to vicious physical and psychological attacks Does not know how harassment is happening or by whom Does not know why harassment is happening May be accused of mentall illness, called delusional Cannot get away, no matter where he/she goes Mind raped thought streams Bugged conversations Gossip, Rumours, Social and Job connections broken Microwave voices, clicks, pops, brain zapping, direct harassment Death and disease threats via internet messages from undercover FBI and NSA spies Mind control means influencing the behaviour of an individual. And to creating or remolding victims personality towards desired state. The desired model human for the elite of the world, or, the Illuminati is: Easily steerable mass-human without personal characteristics. A stupefied, dumbed down person. This secures the interests of the elite of the world. So they can lead our lives through changing times the way they want. Apparently growing out of earlier government mind control research programs such as MKULTRA, and government suppression-of-dissent programs such as COINTELPRO, today's mind control is covert, finely crafted, around the clock harassment perpetrated against citizens living in their homes and communities. Destruction of family and other relationships by way of lies, bribes, and threats is a goal of the phase of today's mind control. The current day mind control program has been carefully engineered so that if the target complains, their own words will instantly cause them to be labeled as mentally ill. The mind/body symptoms of current day mind control include: Excruciating PAIN(!) Exceptionally frequent blanking of recent memories, and truncation of new ideas - Very unnatural inability to sleep, as if large amounts of caffeine have been consumed. - Sudden forced awakening at precisely the same time in the middle of the night, continuing for at least months, and right on a clock time such as 4 am, zero minutes, zero seconds - Sudden clumsiness, which can result in spills, spoilage of precise work, or injury - Attacks of extreme fatigue, sometimes almost to the point of paralysis, when there is no reason for such attacks - Frequent powerful itching without rash, and which may start as a small electric shock - Artificial bee stings, especially while trying to get to sleep - Wildly racing heart without any cause - Sudden overheating, without any cause - Frequent flailing of arms and legs as you try to sleep - Fake sounds such as alarm clock going off when it shouldn't, telephone ringing when there is no incoming call, knocking on the door but no one is there - Voices, either very insulting, or telling you things that indicate you are under surveillance - Vibration of body parts when trying to sleep - High pitched tone in ears, which may change when switching electronic equipment on or off In some cases, statements by strangers indicating they know what you had for supper In some cases, statements by strangers indicating they can read your thoughts - Artificial and powerful sexual stimulation - Artificial and powerful PREVENTION of sexual stimulation. - Impotency caused by secret poisoning of the food by FBI and NSA secret spies. FBI and NSA spies post messages anonymously on usenet to indicate to the torture victims how they made the victims impotent. Ignore the title in the thread and read the subtle threatening text from FBI and NSA SPIES. One can imagine how many of the hundreds of thousands of innocent americans are made impotent by these disgusting, horrifying and sadistic FBI and NSA PSYCHOPATHS. This is THE EXTERIOR of mind control. The non-noticeable mind control is more dangerous one. That is the manipulation of our thoughts and emotions WITHOUT our notice of things. We think we lead our lives. But we do not. We have to wake up to take notice on the situation. And try to change the course of history. FBI and NSA SECRET SOCIETY imposes us the New world order using mind control. And we can`t shield ourselves from it. Which makes us as puppets on strings. Electromagnetic Weapons and Mind Control: from CNN's Special Assignment, about 1985, at http://www.mindcontrolforums.com/cnn-mc.htm , by Chuck De Caro, CNN Special Assignments, and List of mind control symptoms, in 2005 update: The convergence of the cold war history of mind control and electromagnetic weapons with new post cold war government neuroscience reserch programs, by Cheryl Welsh, Director, Mind Justice, 2005, at http://www.mindjustice.org/2005update.htm and http://www.mindjustice.org/symptoms.htm change lives: Imagine controlling a computer using only your thoughts. This technology already exists in the civilian economy as a means of playing video games, for example. See this link: Bazell, correspondent for NBC News, and was updated at 7:31p.m., December 6, activity of brainwaves, and by means of wireless technology, can be found in called Remote control brain sensor, at http://news.bbc.co.uk/2/hi/health/2361987.stm MANCHURIAN CANDIDATES 4 Elite Airborne commandos Return from a Special Operation in Afghanistan and suddenly torture and murder their wives Friday, July 26, 2002 1:33 p.m. EDT FORT BRAGG, N.C. (Reuters) - Four U.S. soldiers stationed at Fort Bragg, including three special operations servicemen who returned home this year after tours of duty in Afghanistan, allegedly killed their wives recently, military officials said on Friday. In two of the cases, the soldiers killed themselves after shooting their spouses, officials said. In the other two, the servicemen have been arrested and are facing charges. There has been a recent series of instances where soldiers have either committed murder or murder-suicide against their wives, said Maj. Gary Kolb, a spokesman for the Army Special Operations Command. I wonder how many of the murders and suicides in America were actually remotely manipulated, induced and orchestrated by these FBI and NSA secret mind control and mind manipulation experiments. http://www.projectfreedom.cng1.com/esp.html STARGATE is one of the many names given to the Army's classified clan of remote viewers. This project was hidden deeply within the government for close to two decades. The work of Harold Puthoff Ph.D., at SRI international http://www.sri.com/index/html joined with the work of others at the Cognitive Science Laboratory (http://www.lfr.org/csl) helped to supply the CIA with the framework first remote viewers were at work. It is rumored the cold war was responsible for the government's interest in remote viewing because Russia had spies not dissimilar to our remote viewers. Psychic Dictatorship in the U.S.A. Alex Constantine (1995) Another bombshell from the fecund furnace of Feral House Press. Constantine, a political researcher in the mold of Mae Brussell, has gathered together a well-documented survey. Bombing minds rather than bodies is the warfare of the new millennium... Funded under the euphemism of Non-Lethal Technology, the Pentagon has developed the ability to transmit voices, and inflict pain, madness, even death, with the push of a button. Hard to believe? Believe it! Official sources have publicly admitted to the existence of such technology. Topics covered: Telemetric mind control, hearing voices, false memory hoax, E. Howard Hunt's Death Squads, The GOP's Pink Triangle & the CIA, Johnny Carson and the S&L crisis, and let's not forget the chapter on NutraSweet as crowd control. 221 pages, PB PDU $12.95 The term mind control comes from former CIA director Allen Dulles. In 1953, Dulles, speaking before a national meeting of Princeton alumni, distinguished two fronts in the then-current battle for men's minds: a first front of mass indoctrination through censorship and propaganda, and a second front of individual brainwashing and brain changing. Before an audience of fellow Ivy Leaguers, Dulles skipped the usual pieties about democracy. The same year, Dulles approved the CIA's notorious MKULTRA project, and exempted it from normal CIA financial controls. A partial list of aggressive promoters of this new technology includes Oak Ridge National Lab, Sandia National Laboratories, Science Applications International Corporation, MITRE Corporation, Lawrence Livermore National Lab, and Los Alamos National Laboratory. In 1996, the US Air Force Scientific Advisory Board published a 14-volume study of future developments in weapons called New World Vistas. Tucked away on page 89 of an ancillary 15th volume are some hair-raising insights into the future 'coupling' of man and machine in a section dealing with 'Biological Process Control'. The author refers to an 'explosion' of knowledge in the field of neuroscience, adding, ominously: One can envision the development of electromagnetic energy sources, the output of which can be pulsed, shaped, and focused, that can couple with the human body in a fashion that will allow one to prevent voluntary muscular movements, control emotions (and thus actions), produce sleep, transmit suggestions, interfere with both short-term and long-term memory, produce an experience set, and delete an experience set. Translating the words 'experience set' from military jargon into plain English, this means, simply, that they envisage the ability to erase your life's memories and substitute a new, fictitious set. By projecting such developments into the future, the authors of New Vistas are camouflaging present day capabilities. A similar futuristic scenario with many references to mind manipulation is described in The Revolution in Military Affairs and Conflict Short of War (US Army War College, 1994). Authors Steven Metz and James Kievit declare: Behaviour modification is a key component of peace enforcement and The advantage of [using] directed energy systems is deniability. The authors ask: Against whom is such deniability aimed? The direct answer is the American people. Edward Tilton, President of Silent Sounds Inc., says this about S-quad in a letter dated 13 December, 1996: All schematics, however, have been classified by the US Government and we are not allowed to reveal the exact details. we make tapes and CDs for the German Government, even the former Soviet Union countries! All with the permission of the US State Department, of course. The system was used throughout Operation Desert Storm (Iraq) quite successfully. By using these computer-enhanced EEGs, scientists can identify and isolate the brain's low-amplitude emotion signature clusters, synthesise them and store them on another computer. In other words, by studying the subtle characteristic brainwave patterns that occur when a subject experiences a particular emotion, scientists have been able to identify the concomitant brainwave pattern and can now duplicate it. These clusters are then placed on the Silent Sound carrier frequencies and will silently trigger the occurrence of the same basic emotion in another human being! Microwaves can also alter the permeability of the body's blood-brain barrier,14 which can synergistically increase the effects of drugs, as the military is well aware. Using relatively low-level RFR, it may be possible to sensitise large military groups to extremely dispersed amounts of biological or chemical agents to which the unirradiated population would be immune.15 Sound can be transmitted even easier through the use of implants - cochlear implants, implants that send electrical signals into the fluid of the inner ear, or implants that transmit sound vibrations via bone conduction, such as the cases of dental fillings picking up audible radio signals. The stimoceiver, invented by Dr. Jose Delgado, consists of wires running from strategic points in the brain to a radio receiver/transmitter located entirely under the skin. Through this device, Delgado was able to stimulate raw emotions such as arousal, anxiety, and aggression with the turn of a knob. Of course, secret research by the US Government into microwaves and synthetic telepathy has moved on considerably since the end of the Cold War ... secret microwave radiation can be used to induce in unsuspecting victims: Headache, fatigue, perspiring, dizziness, menstrual disorders, irritability, agitation, tension, drowsiness, sleeplessness, depression, anxiety, forgetfulness, and the lack of concentration. Naval Research Lab Attempts To Meld Neurons And Chips: Studies May Produce Army of 'Zombies.' Future battles could be waged with genetically engineered organisms, such as rodents, whose minds are controlled by computer chips engineered with living brain cells.... The research, called Hippo-campal Neuron Patterning, grows live neurons on computer chips. This technology that alters neurons could potentially be used on people to create zombie armies, Lawrence Korb, a senior fellow at the Brookings Institution, said. See also National Security Agency in http://en.wikipedia.org/wiki/National_Security_Agency , Central Security Service in http://en.wikipedia.org/wiki/Central_Security_Service , ECHELON in http://en.wikipedia.org/wiki/ECHELON , Defense Intelligence Agency in http://en.wikipedia.org/wiki/Defense_Intelligence_Agency , and Defense Advanced Research Projects Agency in http://en.wikipedia.org/wiki/Defense_Advanced_Research_Projects_Agency The following links explain secret MKULTRA and COINTELPRO programs of FBI, CIA and NSA agencies. http://thirdworldtraveler.com/FBI/FBI_watch.html http://thirdworldtraveler.com/NSA/NSA_page.html http://thirdworldtraveler.com/CIA/CIA_ThirdWorld.html -- === Subject: Re: MACSYMA and AXIOM - the same failure pattern <45221DC3.9030100@tid.es> ................................................................. Today I visited again sci.math.symbolic, and with some genuine interest I discovered comments from Timothy Daly, Axiom Lead of Sep 26 2006 4:59 am Knowing that, according to the AXIOM book, it was Timothy Daly who was responsible for quality assurance matters (at least, he seems to be the ONLY contributor with explicitly defined QA function) I felt especial fun at the comparison of the next 2 paragraphs (the orthography is kept intact). ................................................................. TD> I've had the honor and privilege of working beside some of TD> the best and brightest computational mathematicians of the TD> 20th century. You should feel embarrased to measure such TD> wealth of science by the base coin of your testing efforts. ................................................................. TD> Oh, Axiom is loaded with bugs. More than even you can count. ................................................................. === Subject: Re: Small set theory. > >> Hi All. >> I would like to know what is against this small set theory? >> 1) Axiom of Extentiality: ( as in ZFC ). >> 2) Axiom of the Empty set: ( as in ZFC ). >> 3) Axiom of Dichotomy: A e A xor A !e A. >> In words the axiom of Dichotomy is: A set is either in itself or not, >> it cannot be both ,nor neither. Axiom (3) says nothing at all, since it is a tautology. Seemingly, your intention is the same as expressed by Hilbert: to maintain the belief on an illusion. I see Axiom (3) an attempt to exclude any possibility of having reals like void numbers. So your set of axioms squarely points to the most questionable aspects of set theory. === Subject: Re: Small set theory. <87zmazpegv.fsf@phiwumbda.org> Axiom (5). hmmm.....? give me a brake, I didn't think about that. You can have all the brakes, clutches, gears, and transmissions you want. I just thought you were claiming Exy(~x=y & xex & yey) -> Axiom (5). MoeBlee === Subject: Re: Small set theory. <87zmazpegv.fsf@phiwumbda.org> Axiom (5). hmmm.....? give me a brake, I didn't think about that. You can have all the brakes, clutches, gears, and transmissions you > want. I just thought you were claiming Exy(~x=y & xex & yey) -> Axiom > (5). MoeBlee Anyhow I already substituted your axiom instead of mine. this is finished. Zuhair === Subject: Re: Small set theory. >> 3) Axiom of Dichotomy: A e A xor A !e A. > As others have pointed out, your axiom of dichotomy is superfluous in > classical logic. And, if I'm not mistaken, as added to intuitionistic > logic, would just amount to having classical logic anyway. >> It wouldn't. Okay, so we'd have to add some sentential axioms other than excluded > middle. Adding excluded middle suffices to render the logic classical. However, excluded middle does not follow from excluded middle just for atomic sentences. As an illustration, pick your favourite Pi_1 conjecture AxP(x) - the Goldbach conjecture, say. Since P is primitive recursive, we have intuitionsitically that Ax(P(x) / ~P(x)), but from this we certainly don't get AxP(x) / Ex~P(x) - that would mean that from a proof of decidability of P we could construct either a proof that all x have the property P or an explicit counter-example. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. 3) Axiom of Dichotomy: A e A xor A !e A. > As others have pointed out, your axiom of dichotomy is superfluous in > classical logic. And, if I'm not mistaken, as added to intuitionistic > logic, would just amount to having classical logic anyway. >> It wouldn't. Okay, so we'd have to add some sentential axioms other than excluded > middle. Adding excluded middle suffices to render the logic classical. However, > excluded middle does not follow from excluded middle just for atomic > sentences. Yes, I misspoke my own intention. I didn't mean (though it did come out that way) that if we added the particular dichotomy mentioned then we would have classical logic over the system being disccussed. I actually meant that if we added an axiom schema for excluded middle, then we would have classsical logic. a bit. MoeBlee === Subject: Re: Small set theory. > Axiom5) ExEzAyey xex / (zex <->z!ey) . >>What are you trying to say here? it means that there exist more than one set that is a member in itself. Moe Blee version of this is the following: Axiom5) ExEy(~x=y & xex & yey) Anyhow I think the two versions are equivalent. Yeah. You think a lot of things, don't you? -- Jesse F. Hughes [I]t's the damndest thing. There's something wrong with every last one of you, and I *never* thought that was a possibility. But now I feel it's the only reasonable conclusion. --JSH sees some sorta light === Subject: Re: Small set theory. > Axiom5) ExEzAyey xex / (zex <->z!ey) . >>What are you trying to say here? it means that there exist more than one set that is a member in itself. > Moe Blee version of this is the following: > Axiom5) ExEy(~x=y & xex & yey) > Anyhow I think the two versions are equivalent. Yeah. You think a lot of things, don't you? Thinking is easier if you don't care if you can prove what you say. -- Marcus === Subject: Re: Small set theory. <87zmazpegv.fsf@phiwumbda.org> <87bqneznlo.fsf@phiwumbda.org> z!ey) . >>What are you trying to say here? it means that there exist more than one set that is a member in itself. > Moe Blee version of this is the following: > Axiom5) ExEy(~x=y & xex & yey) > Anyhow I think the two versions are equivalent. Yeah. You think a lot of things, don't you? Thinking is easier if you don't care if you can prove what you say. Correct! Zuhair -- > Marcus === Subject: Re: Small set theory. >> As it is, it seems to me that there's no problem with the set of >> well-founded sets in Zuhair's theory. This is, of course, because the >> theory is too damned weak to prove anything interesting. I am interested in this: why you say that there is not propblem with > the set of all well founded sets in this little theory. I think your line of thinking is as follows. The set of all regular sets ( i.e sets that are not in themselfs ) is > not a set in this theory. Wrong. I said (and thought) no such thing. > The set of all well founded sets is a proper subset of the set of all > regular set. Then the set of all well founded sets is a set, since a proper subset > of a proper class is a set. Wow. This is remarkably far from my reasoning. In fact, your version of my thought process is really, and I mean this in the most charitable manner possible, stupid. Since when is it a rule that the proper subclass of a class is a set? I think that this line of thinking is not all together right. Ya think? Forget the rest. What you said is simply silly. The fact is that I see no way in which your axioms plus the assumption that there is a set of non-self-membered (or, stronger, well-founded) sets yields a contradiction. -- All intelligent men are cowards. The Chinese are the world's worst fighters because they are an intelligent race[...] An average Chinese child knows what the European gray-haired statesmen do not know, that by fighting one gets killed or maimed. -- Lin Yutang === Subject: Re: Small set theory. > Forget the rest. What you said is simply silly. The fact is that I > see no way in which your axioms plus the assumption that there is a > set of non-self-membered (or, stronger, well-founded) sets yields a > contradiction. Strange. Just earlier, I showed MoeBlee that the set of non-self-membered sets *does* yield a contradiction. The previous post is just obviously butt-wrong. And, I think, there is also no set of well-founded sets. If there were such a set, then it would be well-founded (since every element of it is well-founded) and hence a member of itself and hence non-well-founded. Sorry. That post was just utterly confused. No idea what I was thinking (or not). -- Sorry, wakeup to the real world. You're on your own dependent on me as your guide. Luckily for you, I'm self-correcting to a large extent, so if the proof were wrong, I'd tell you. It's not wrong. --- James Harris confirms that his proof is correct. === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87r6wbp1b3.fsf@phiwumbda.org> <87fycqznp8.fsf@phiwumbda.org >> As it is, it seems to me that there's no problem with the set of >> well-founded sets in Zuhair's theory. This is, of course, because the >> theory is too damned weak to prove anything interesting. I am interested in this: why you say that there is not propblem with > the set of all well founded sets in this little theory. I think your line of thinking is as follows. The set of all regular sets ( i.e sets that are not in themselfs ) is > not a set in this theory. Wrong. I said (and thought) no such thing. The set of all well founded sets is a proper subset of the set of all > regular set. Then the set of all well founded sets is a set, since a proper subset > of a proper class is a set. Wow. This is remarkably far from my reasoning. In fact, your version > of my thought process is really, and I mean this in the most > charitable manner possible, stupid. Since when is it a rule that the > proper subclass of a class is a set? I know that, but since I always think that you are so stupid, that's why I thought that would be the kind of reasoning you have, I mentioned that this subclass of a proper class is not necessarily a set? stupid > I think that this line of thinking is not all together right. Ya think? Forget the rest. What you said is simply silly. The fact is that I > see no way in which your axioms plus the assumption that there is a > set of non-self-membered (or, stronger, well-founded) sets yields a > contradiction. > -- > All intelligent men are cowards. The Chinese are the world's worst > fighters because they are an intelligent race[...] An average Chinese > child knows what the European gray-haired statesmen do not know, that > by fighting one gets killed or maimed. -- Lin Yutang === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87r6wbp1b3.fsf@phiwumbda.org> <87fycqznp8.fsf@phiwumbda.org >> As it is, it seems to me that there's no problem with the set of >> well-founded sets in Zuhair's theory. This is, of course, because the >> theory is too damned weak to prove anything interesting. I am interested in this: why you say that there is not propblem with > the set of all well founded sets in this little theory. I think your line of thinking is as follows. The set of all regular sets ( i.e sets that are not in themselfs ) is > not a set in this theory. Wrong. I said (and thought) no such thing. who is talking about you? I am saying that the set of all sets that are not in themselfs is not a set in this theory. since it would be contradicitive. Zuhair The set of all well founded sets is a proper subset of the set of all > regular set. Then the set of all well founded sets is a set, since a proper subset > of a proper class is a set. Wow. This is remarkably far from my reasoning. In fact, your version > of my thought process is really, and I mean this in the most > charitable manner possible, stupid. Since when is it a rule that the > proper subclass of a class is a set? > I think that this line of thinking is not all together right. Ya think? Forget the rest. What you said is simply silly. The fact is that I > see no way in which your axioms plus the assumption that there is a > set of non-self-membered (or, stronger, well-founded) sets yields a > contradiction. > -- > All intelligent men are cowards. The Chinese are the world's worst > fighters because they are an intelligent race[...] An average Chinese > child knows what the European gray-haired statesmen do not know, that > by fighting one gets killed or maimed. -- Lin Yutang === Subject: Re: Small set theory. > No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. Man. You're simply incapable of fundamental reasoning. The fact that y e y -> y e y is *not* a contradiction. Today is either warm or not. If it is warm, then it is warm. If not, then not. Big deal. This fact is not contradicitive. -- The needs of the many outweigh the needs of the few [...] I must make the same choice as those who came before me without regard to the impact today, for the sake of the children of humanity, the children of tomorrow. -- JSH channels Spock and generic politicians everywhere === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org > No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. Man. You're simply incapable of fundamental reasoning. The fact that y e y -> y e y is *not* a contradiction. Today is either warm or not. If it is warm, then it is warm. If not, > then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the contradiction doesn't lie in yey -> yey, the contradiction lie in y being in itself and not in itself at the same time stupid. Zuhair > -- > The needs of the many outweigh the needs of the few [...] I must > make the same choice as those who came before me without regard to the > impact today, for the sake of the children of humanity, the children > of tomorrow. -- JSH channels Spock and generic politicians everywhere === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org > No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. Man. You're simply incapable of fundamental reasoning. The fact that y e y -> y e y is *not* a contradiction. Today is either warm or not. If it is warm, then it is warm. If not, > then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. Go back and look at your post. You'll see that you didn't prove what you claimed to have proved in that post, as I explained and as Jesse has explained to you. MoeBlee === Subject: Re: Small set theory. > No, it is contradictive. >> y={x|xex} >> if yey -> yey >> if y!ey -> y!ey >> therefore y is both regular and irregular at the same time. which is >> contradicitive. >> Man. You're simply incapable of fundamental reasoning. >> The fact that y e y -> y e y is *not* a contradiction. >> Today is either warm or not. If it is warm, then it is warm. If not, >> then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the >contradiction doesn't lie in yey -> yey, the contradiction lie in y >being in itself and not in itself at the same time stupid. Well, no, it's not a contradiction. Contradictions have the general form S->!S. What you have is something different, statements of the form S->S. (I'm not sure what the right tchnical term is.) The effect is that you cannot deduce the falsity or truth of S from the statement. In your case, y could either be true or false, but you lack sufficient information to say which. Your error of reasoning is to assume that the truth of the statement S->S implies the truth of S; it does not. HTH === Subject: Re: Small set theory. > Well, no, it's not a contradiction. Contradictions have the general > form S->!S. S -> !S is not a contradiction either. It is equivalent to !S. A contradiction has the form S & !S. -- I arrest anybody I think needs arresting, Mr. Carter, and I'm not in the habit of explaining why. There's a law about that --- You're in Dodge, Mr. Carter. -- Gunsmoke radio show / John Ashcroft === Subject: Re: Small set theory. On Sat, 11 Nov 2006 11:37:58 -0500, Jesse F. Hughes > Well, no, it's not a contradiction. Contradictions have the general >> form S->!S. S -> !S is not a contradiction either. It is equivalent to !S. A contradiction has the form S & !S. Quite true. My bad; I was thinking of a variant of the Jourdain card paradox. What he has is the statement (S ->S) & (!S -> !S) which is a tautology, i.e., it is true regardless of whether S is true. === Subject: Re: Small set theory. >> No, it is contradictive. >> y={x|xex} >> if yey -> yey >> if y!ey -> y!ey >> therefore y is both regular and irregular at the same time. which is >> contradicitive. >> Man. You're simply incapable of fundamental reasoning. >> The fact that y e y -> y e y is *not* a contradiction. >> Today is either warm or not. If it is warm, then it is warm. If not, >> then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. The sentences y e y -> y e y and y !e y -> y !e y are tautologies and hence do not contradict each other. Your statement is simply, stupidly false. And there is no contradiction to be had from assuming that the set { x | x e x } exists. At least none that I see. (Note: I'm speaking of Zuhair's teeny set theory and not some other theory.) -- You got more out of it than I put into it last night. Who were you thinking of when we were loving last night? -- Texas Tornadoes === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org >> No, it is contradictive. >> y={x|xex} >> if yey -> yey >> if y!ey -> y!ey >> therefore y is both regular and irregular at the same time. which is >> contradicitive. >> Man. You're simply incapable of fundamental reasoning. >> The fact that y e y -> y e y is *not* a contradiction. >> Today is either warm or not. If it is warm, then it is warm. If not, >> then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. The sentences y e y -> y e y and y !e y -> y !e y are tautologies and > hence do not contradict each other. Your statement is simply, stupidly false. And there is no > contradiction to be had from assuming that the set { x | x e x } > exists. At least none that I see. (Note: I'm speaking of Zuhair's teeny set theory and not some other > theory.) -- > You got more out of it > than I put into it last night. > Who were you thinking of when we were loving last night? > -- Texas Tornadoes Ok, I should develop axiom 6, the axiom that I have posted latelly contradicts logic as you said, but I am sure you know what I mean after development of this axiom. anyhow. Still nobody answered my simple question, if y={x|xex} is yey or y!ey. There is difference between Ontology and Epistemolgoy. y exists since there is no contradiction involved with its existance, but Epistemologically speaking we do not know if y is a member of itself or not. If I accept y to be a set in this theory, then I think it would be a unique set. y would be the only set which we do not know weather it is a member of itself or not. so I think tha y is unique, like how {} is unique. Anyhow , it is clear as I mentioned before that , the axioms of separation, replacement,power and choice are inconsistent with this theory( I mean these axioms as present in ZFC). I realize that separation is important, and I think that it should be changed in a manner as to be consistent with the other axioms. But still I didn't work it out. However this set theory will be as follow: Primitive e 1) Axiom of Extentiality 2) Axiom of Empty set 3) Axiom of Pairing 4) Axiom of Union 5) Axiom of Infinity 6) Axiom of Universe ExAy yex 7) Axiom of Multiplicity ExEzAy yey / xex / zex / z!ey 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in Z. Do anybody think that this axiomatic set theory is inconsistent? Zuhair === Subject: Re: Small set theory. > Still nobody answered my simple question, if y={x|xex} is yey or > y!ey. It seems to me that the statement y e y is independent of the axioms. There is no inconsistency if we assume y e y or assume y !e y. > There is difference between Ontology and Epistemolgoy. y exists since > there is no contradiction involved with its existance, but > Epistemologically speaking we do not know if y is a member of itself or > not. Anything not contradictory exists? That's an ontological principle I haven't heard. > If I accept y to be a set in this theory, then I think it would be a > unique set. y would be the only set which we do not know weather it is > a member of itself or not. so I think tha y is unique, like how {} is > unique. There are plenty of other sets that would be similar. For instance, consider the set { x | x e x and x != {x} }. If this set exists in your theory, then it's not clear whether it is self-membered. Anyhow , it is clear as I mentioned before that , the axioms of > separation, replacement,power and choice are inconsistent with this > theory( I mean these axioms as present in ZFC). No, it is not at all clear that power set and choice are inconsistent with your theory. And you have never given any argument to prove this. Why not? If it's so obvious, then surely you can provide a simple proof. > Primitive e 1) Axiom of Extentiality > 2) Axiom of Empty set > 3) Axiom of Pairing > 4) Axiom of Union > 5) Axiom of Infinity > 6) Axiom of Universe ExAy yex 7) Axiom of Multiplicity ExEzAy yey / xex / zex / z!ey I don't know what you're trying to say with this axiom, but I'm sure you got it wrong. As a consequence of this axiom, we can prove Ay yey. 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in Z. Do anybody think that this axiomatic set theory is inconsistent? It is inconsistent. Axiom 7 proves every set is self-membered, and we know that the empty set is not self-membered. -- Jesse F. Hughes How can I miss you when you won't go away? -- Dan Hicks and his Hot Licks === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org > Still nobody answered my simple question, if y={x|xex} is yey or > y!ey. It seems to me that the statement y e y is independent of the axioms. > There is no inconsistency if we assume y e y or assume y !e y. There is difference between Ontology and Epistemolgoy. y exists since > there is no contradiction involved with its existance, but > Epistemologically speaking we do not know if y is a member of itself or > not. Anything not contradictory exists? That's an ontological principle I > haven't heard. If I accept y to be a set in this theory, then I think it would be a > unique set. y would be the only set which we do not know weather it is > a member of itself or not. so I think tha y is unique, like how {} is > unique. There are plenty of other sets that would be similar. For instance, > consider the set { x | x e x and x != {x} }. If this set exists in > your theory, then it's not clear whether it is self-membered. > Anyhow , it is clear as I mentioned before that , the axioms of > separation, replacement,power and choice are inconsistent with this > theory( I mean these axioms as present in ZFC). No, it is not at all clear that power set and choice are > inconsistent with your theory. And you have never given any argument > to prove this. Why not? If it's so obvious, then surely you can > provide a simple proof. Jesse It is very clear that power is not consistent with this theory? very clear? Since separation is inconsistent with it. For example for the universe x in my theory were x={y|y=y} , what would be P(w), wouldn't it include z={k|k!ek} which is not a set in this theory. Same thing for choice, take all binary sets {a,b} were aea and b!eb , now c={ All b } don't you see that c ={ b|b!eb} , ie c is the set of all regular set, which is not a set in this theory. Replacement can be simple reduced to separation using the suitable function, I can define f(x)=y were x is any set, while y is a regular set. in such a manner that Range of f is the set of all regular sets, which is not a set. Regularity is cleary inconsistent with this theory. Have a nice time. Zuhair Primitive e 1) Axiom of Extentiality > 2) Axiom of Empty set > 3) Axiom of Pairing > 4) Axiom of Union > 5) Axiom of Infinity > 6) Axiom of Universe ExAy yex 7) Axiom of Multiplicity ExEzAy yey / xex / zex / z!ey I don't know what you're trying to say with this axiom, but I'm sure > you got it wrong. As a consequence of this axiom, we can prove > Ay yey. 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in Z. Do anybody think that this axiomatic set theory is inconsistent? It is inconsistent. Axiom 7 proves every set is self-membered, and we > know that the empty set is not self-membered. -- > Jesse F. Hughes How can I miss you when you won't go away? > -- Dan Hicks and his Hot Licks === Subject: Re: Small set theory. > Jesse It is very clear that power is not consistent with this theory? > very clear? Since separation is inconsistent with it. For example for > the universe x in my theory were x={y|y=y} , what would be P(w), > wouldn't it include z={k|k!ek} which is not a set in this theory. No, it is not very clear. If we don't have separation, then it is not clear that the set z exists. Now we might think that the power set axiom gives a different inconsistency, something like |P(x)| > |x| and x = P(x) (where x is the universe). But the proof of Cantor's theorem uses separation, too. So, no, Zuhair, it is not obvious. > Same thing for choice, take all binary sets {a,b} were aea and b!eb , > now c={ All b } > don't you see that c ={ b|b!eb} , ie c is the set of all regular set, > which is not a set in this theory. No, I don't see that at all. Are you sure you know what choice means? Anyway, how do we know that we can form the set of all pairs {a,b} as above? And if we can't form that set, we can't apply choice. -- Jesse F. Hughes The people who made up the words could have said 'newspaper' is 'trees'. -- Quincy P. Hughes, five-year-old Wittgensteinian (This comment came out of the blue at breakfast.) === Subject: Re: Small set theory. > Same thing for choice, take all binary sets {a,b} were aea and b!eb NOTE_______________________________^^^^ Were is a verb form. plural past tense of to be. So that using were where where should be is an error. === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <87slgomixw.fsf@phiwumbda.org > Jesse It is very clear that power is not consistent with this theory? > very clear? Since separation is inconsistent with it. For example for > the universe x in my theory were x={y|y=y} , what would be P(w), > wouldn't it include z={k|k!ek} which is not a set in this theory. No, it is not very clear. If we don't have separation, then it is not > clear that the set z exists. No. you are mixing between the meaning of subset and the axiom of separation, every regular set is a subset of the set of all sets x. but the axiom of separation entails that every subset of a set is a set. that is why it fails in this theory, because All regular sets is a subset of x but it is not a set . In a similar manner power sets fails, because P(w) will have the subset of x that is All regular sets and this is not a set. Now we might think that the power set axiom gives a different > inconsistency, something like |P(x)| > |x| and x = P(x) (where x is > the universe). But the proof of Cantor's theorem uses separation, > too. So, no, Zuhair, it is not obvious. Same thing for choice, take all binary sets {a,b} were aea and b!eb , > now c={ All b } > don't you see that c ={ b|b!eb} , ie c is the set of all regular set, > which is not a set in this theory. No, I don't see that at all. Are you sure you know what choice means? Anyway, how do we know that we can form the set of all pairs {a,b} as > above? And if we can't form that set, we can't apply choice. You know you have right regarding that, I was just thinking before few minutes that Every pair of sets in this theory is a regular set, I think this is a theorum in this set theory. Also I have another theorum that is the intersection of any two irregular sets is a set. and it is either one of the sets , or { }. Theorum: a={a},b={b}, a.b={x|xeb,xea} -> a.b= (a / b) xor { } . Theorum: AaAb & a=/=b & x= {a,b} -> x!ex Only if the last theorum is true then we cannot apply choice to the set that do not exist, that is the set of all pairs. However I still suspect this axiom to be consistent with this theory. Zuhair -- > Jesse F. Hughes > The people who made up the words could have said 'newspaper' is > 'trees'. -- Quincy P. Hughes, five-year-old Wittgensteinian > (This comment came out of the blue at breakfast.) === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <87slgomixw.fsf@phiwumbda.org > Jesse It is very clear that power is not consistent with this theory? > very clear? Since separation is inconsistent with it. For example for > the universe x in my theory were x={y|y=y} , what would be P(w), > wouldn't it include z={k|k!ek} which is not a set in this theory. No, it is not very clear. If we don't have separation, then it is not > clear that the set z exists. No. you are mixing between the meaning of subset and the axiom of > separation, every regular set is a subset of the set of all sets x. but > the axiom of separation entails that every subset of a set is a set. > that is why it fails in this theory, because All regular sets is a > subset of x but it is not a set . In a similar manner power sets fails, because P(w) will have the subset > of x that is All regular sets and this is not a set. Now we might think that the power set axiom gives a different > inconsistency, something like |P(x)| > |x| and x = P(x) (where x is > the universe). But the proof of Cantor's theorem uses separation, > too. So, no, Zuhair, it is not obvious. Same thing for choice, take all binary sets {a,b} were aea and b!eb , > now c={ All b } > don't you see that c ={ b|b!eb} , ie c is the set of all regular set, > which is not a set in this theory. No, I don't see that at all. Are you sure you know what choice means? Anyway, how do we know that we can form the set of all pairs {a,b} as > above? And if we can't form that set, we can't apply choice. I think this is wrong, the set of all pairs is a set in this theory, there is nothing against it. This choice thing is a little bit tricky, since you can say for example even if we have the set of all these binary sets, then a choice set is not necessarily a selection of only a regular set of each pair, of course this choice function picking only the regular sets of each pair is not a set, in reality it is even not a choice function since we can discriminate between a regular set and an irregular pair in each binary set, therefore there exist a rule determining the selection which is against choice, choice should be arbitrary, ie there should be no discriminative rule of picking a set from each binary set , Perhaps you are right, we can have choice on these bins, which then contains mixed types of sets ie regular and irregular sets. Anyhow I don't know perhaps choice is not inconsistent with this theory. But separation, regularity, replacement and power as mentioned in ZFC are inconsistent with this theory, unless we modify them in such a manner as to be consistent with the other axioms, they are until then to be considered as inconsistent with this theory. Zuhair You know you have right regarding that, I was just thinking before few > minutes that Every pair of sets in this theory is a regular set, I > think this is a theorum in this set theory. Also I have another theorum that is the intersection of any two > irregular sets is a set. and it is either one of the sets , or { }. Theorum: a={a},b={b}, a.b={x|xeb,xea} -> a.b= (a / b) xor { } . > Theorum: AaAb & a=/=b & x= {a,b} -> x!ex Only if the last theorum is true then we cannot apply choice to the set > that do not exist, that is the set of all pairs. However I still > suspect this axiom to be consistent with this theory. Zuhair -- > Jesse F. Hughes > The people who made up the words could have said 'newspaper' is > 'trees'. -- Quincy P. Hughes, five-year-old Wittgensteinian > (This comment came out of the blue at breakfast.) === Subject: Re: Small set theory. >> Jesse It is very clear that power is not consistent with this theory? >> very clear? Since separation is inconsistent with it. For example for >> the universe x in my theory were x={y|y=y} , what would be P(w), >> wouldn't it include z={k|k!ek} which is not a set in this theory. >> No, it is not very clear. If we don't have separation, then it is not >> clear that the set z exists. No. you are mixing between the meaning of subset and the axiom of > separation, every regular set is a subset of the set of all sets x. but > the axiom of separation entails that every subset of a set is a set. > that is why it fails in this theory, because All regular sets is a > subset of x but it is not a set . No, you are simply mistaken. If we cannot prove that z exists as a set, then we cannot prove it is in the powerset of x. > Also I have another theorum that is the intersection of any two > irregular sets is a set. and it is either one of the sets , or { }. Theorum: a={a},b={b}, a.b={x|xeb,xea} -> a.b= (a / b) xor { } . This is not a well-formed sentence. I guess you mean: For all sets a,b, (a = {a} and b = {b}) -> a.b = a v a.b = b v a.b = {}. It could be strengthened to read: For all sets a,b, (a = {a} and b = {b}) -> a.b = a v a.b = {}. In Aczel's anti-well-founded set theory, a stronger claim can be proved: For all sets a,b, (a = {a} and b = {b}) -> a = b. In any case, yes, this is a plainly obvious theorem. > Theorum: AaAb & a=/=b & x= {a,b} -> x!ex This is not likely a theorem. Show me the proof. > Only if the last theorum is true then we cannot apply choice to the set > that do not exist, that is the set of all pairs. However I still > suspect this axiom to be consistent with this theory. I don't see why this theorem has anything to do with your claim that choice is inconsistent with your theory. As far as I can guess, it seems that you could take this so-called theorem as an axiom without inconsistency, but it doesn't seem like a good idea. -- Jesse F. Hughes You know that view most people have of mathematicians as brilliant people? What if they're not? -- James S. Harris === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <87slgomixw.fsf@phiwumbda.org> <87d57smc1c.fsf@phiwumbda.org >> Jesse It is very clear that power is not consistent with this theory? >> very clear? Since separation is inconsistent with it. For example for >> the universe x in my theory were x={y|y=y} , what would be P(w), >> wouldn't it include z={k|k!ek} which is not a set in this theory. >> No, it is not very clear. If we don't have separation, then it is not >> clear that the set z exists. No. you are mixing between the meaning of subset and the axiom of > separation, every regular set is a subset of the set of all sets x. but > the axiom of separation entails that every subset of a set is a set. > that is why it fails in this theory, because All regular sets is a > subset of x but it is not a set . No, you are simply mistaken. If we cannot prove that z exists as a > set, then we cannot prove it is in the powerset of x. I am not mistaken, you are mistaken, go and see the definition of power set. z is a subset but it is not a set, according to ZFC which has the separation axiom ,every subset is a set that's why there is not problem of sethood of members of power sets, but here there is. the power set of x is not defined as the set of all subsets of x that are sets, it is rather defined as the set of all subsets of x. You are implicity mentioning another kind of power set axiom which I have just mentioned: P(x) is the set of all subsets of x that are sets. this axiom (which is not the same as that of ZFC) can hold in this theory. Also I have another theorum that is the intersection of any two > irregular sets is a set. and it is either one of the sets , or { }. Theorum: a={a},b={b}, a.b={x|xeb,xea} -> a.b= (a / b) xor { } . This is not a well-formed sentence. I guess you mean: For all sets a,b, > (a = {a} and b = {b}) -> a.b = a v a.b = b v a.b = {}. It could be strengthened to read: For all sets a,b, > (a = {a} and b = {b}) -> a.b = a v a.b = {}. In Aczel's anti-well-founded set theory, a stronger claim can be > proved: For all sets a,b, > (a = {a} and b = {b}) -> a = b. This is not Aczel's set theory. the above is not true for this theory. what I want to say in words is , the intersection of every two irregular sets ( i.e. sets that are in themselfs) is either one of them, or both of them ( in which case they are the same set ) or they are disjoint, ie there intersection is an empty set. Examples: y={x|x is an ordinal} and z={x|x is not an ordinal} , now yey and zez. but y.z={ } while y={x|x is an ordinal } and z={x|x=x} , here yCz , C means is a proper subset of. for singlton irregular sets a={a} and b={b} then a.b=b=a. Perhaps. However it is obvious that z.z were z={x|x=x} , is z. I hope now you understand what I meant by:a={a},b={b}, a.b={x|xeb,xea} -> a.b= (a / b) xor { } however I didn't right it well since a and b look pritty much singlton irregular sets which is not what I meant. better writtin, A aea, A beb , a.b={x|xeb,xea} -> a.b= (a / b) xor { } In any case, yes, this is a plainly obvious theorem. Theorum: AaAb & a=/=b & x= {a,b} -> x!ex This is not likely a theorem. Show me the proof. The prove is that x or y for x=/=y is always neither x nor y. so a lion of a dog is not the same as lion neither it is the same as do. the only ocation when x or y = x , or x or y = y is when x=y. but if x=/=y then it is always x or y is not x and x or y is not x. therefore {a,b} were a=/=b never fulfills its membership. therefore never in itself. Zuhair Only if the last theorum is true then we cannot apply choice to the set > that do not exist, that is the set of all pairs. However I still > suspect this axiom to be consistent with this theory. I don't see why this theorem has anything to do with your claim that > choice is inconsistent with your theory. As far as I can guess, it > seems that you could take this so-called theorem as an axiom without > inconsistency, but it doesn't seem like a good idea. -- > Jesse F. Hughes You know that view most people have of mathematicians as brilliant > people? What if they're not? -- James S. Harris === Subject: Re: Small set theory. > z is a subset but it is not a set By what manner of definitions can one have a subset of anything that is not itself set? It would be like having a subgroup which is not itself a group or a subspace not itself a space or a subring which is not itself a ring. === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org > Still nobody answered my simple question, if y={x|xex} is yey or > y!ey. It seems to me that the statement y e y is independent of the axioms. > There is no inconsistency if we assume y e y or assume y !e y. There is difference between Ontology and Epistemolgoy. y exists since > there is no contradiction involved with its existance, but > Epistemologically speaking we do not know if y is a member of itself or > not. Anything not contradictory exists? That's an ontological principle I > haven't heard. If I accept y to be a set in this theory, then I think it would be a > unique set. y would be the only set which we do not know weather it is > a member of itself or not. so I think tha y is unique, like how {} is > unique. There are plenty of other sets that would be similar. For instance, > consider the set { x | x e x and x != {x} }. If this set exists in > your theory, then it's not clear whether it is self-membered. > Anyhow , it is clear as I mentioned before that , the axioms of > separation, replacement,power and choice are inconsistent with this > theory( I mean these axioms as present in ZFC). No, it is not at all clear that power set and choice are > inconsistent with your theory. And you have never given any argument > to prove this. Why not? If it's so obvious, then surely you can > provide a simple proof. Primitive e 1) Axiom of Extentiality > 2) Axiom of Empty set > 3) Axiom of Pairing > 4) Axiom of Union > 5) Axiom of Infinity > 6) Axiom of Universe ExAy yex 7) Axiom of Multiplicity ExEzAy yey / xex / zex / z!ey I don't know what you're trying to say with this axiom, but I'm sure > you got it wrong. As a consequence of this axiom, we can prove > Ay yey. 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in Z. Do anybody think that this axiomatic set theory is inconsistent? It is inconsistent. Axiom 7 proves every set is self-membered, and we > know that the empty set is not self-membered. Yes that is right! Axiom 7) ExEzAy(yey) -> xex / zex / z!ey -- > Jesse F. Hughes How can I miss you when you won't go away? > -- Dan Hicks and his Hot Licks === Subject: Re: Small set theory. > Yes that is right! Axiom 7) ExEzAy(yey) -> xex / zex / z!ey What is this axiom supposed to say? Whatever it's suppose to say, it's inconsistent with the assumption that there is a self-membered set. Let x and z be given such that Ay(yey) -> xex / zex / z!ey Let's instantiate the quantifier, so that we have: xex -> xex / zex / z!ex. Suppose now that there is some set y such that y e y. Then it follows that x e x and hence z e x and z !e x. Thus, a consequence of this axiom is (A y)(y!ey). Again, what are you trying to assume here? -- Jesse F. Hughes I can't tell you how many times she left me. I lost count the very first time that she did. -- The Flatlanders, I Thought the Wreck Was Over === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <873b8onyyg.fsf@phiwumbda.org > Yes that is right! Axiom 7) ExEzAy(yey) -> xex / zex / z!ey What is this axiom supposed to say? Whatever it's suppose to say, it's inconsistent with the assumption > that there is a self-membered set. Let x and z be given such that Ay(yey) -> xex / zex / z!ey Let's instantiate the quantifier, so that we have: xex -> xex / zex / z!ex. Suppose now that there is some set y such that y e y. Then it follows > that x e x and hence z e x and z !e x. Thus, a consequence of this axiom is (A y)(y!ey). Again, what are you trying to assume here? -- > Jesse F. Hughes > I can't tell you how many times she left me. > I lost count the very first time that she did. > -- The Flatlanders, I Thought the Wreck Was Over Ok, let me tell you my intentions behind this axiom. I want to currenty that there exist more than one set that is a member of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will put it instead of 7) above so Axiom 7:) Exy(~x=y & xex & yey). Zuhair === Subject: Re: Small set theory. > I want to currenty that there exist more than one set that is a member > of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will > put it instead of 7) above Assumption of Exy(~x=y & xex & yey) will certainly convey that there is more that one set which is a member of itself. === Subject: Re: Small set theory. > Ok, let me tell you my intentions behind this axiom. I want to currenty that there exist more than one set that is a member > of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will > put it instead of 7) above so Axiom 7:) Exy(~x=y & xex & yey). There are two problems with this approach. First, just knowing that there are two self-membered sets doesn't tell you much of anything at all. Instead you want a principled manner to say *which* self-membered sets exist. Second, being self-membered isn't really what you want anyway. You want non-well-founded sets, but not every non-well-founded set is self-membered. But there are a lot smarter guys out there who have put some thought into anti-well-founded set theory. You should look into it rather than trying to haphazardly define your own axioms. The only problem is that these theories require a bit of mathematical maturity. It's not trivial to understand the set theory in /Vicious Circles/, for instance. But it's a lot more profitable to at least try to understand that theory than to come up with your own. Anyway, Axiom 7 says there are two self-membered sets. But I don't see that getting you very far. You have no idea what these two sets look like. -- I'm the theory guy. Other people are the experimental people. If you push me on details I get annoyed, as I'm the theory guy. I'm the theoretical amateur mathematician. --James S. Harris, poet === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <873b8onyyg.fsf@phiwumbda.org> <87lkmgmfjd.fsf@phiwumbda.org > Ok, let me tell you my intentions behind this axiom. I want to currenty that there exist more than one set that is a member > of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will > put it instead of 7) above so Axiom 7:) Exy(~x=y & xex & yey). There are two problems with this approach. First, just knowing that > there are two self-membered sets doesn't tell you much of anything at > all. Instead you want a principled manner to say *which* > self-membered sets exist. Though I am so tired today, that I have decided to close this subject but, two ideas came into my mind. This prinicipled manner you are talking about is a beautiful remark. ExAy ( yey<-> xey -> yey ) so this is where y is in itself, in words y is in itself if and only if it satisfies its membership. while, ExAy (y!ey <-> xey -> y!ey) so this is where y is not in itself, in words y is not in itself if and only if it doesnt' satisfy its membership. I don't know if I should axiomatize these two ( put them as axioms in this theory ). However, I think I reached to the axiomatization of what I always wanted to say, I always wanted to say that the set of all sets that are in themselfs is not a set, not because it is inconsistent, but because it is a tautology, and tautologies are not decisive, for example y={x|xex}, if I ask is yey or y!ey, the answer is unknown it might be in itself or not in itself. However to axiomatize removal of y from being a set in this theory, I can add an axiom like below: ~ExAy ( xey <-> yey ). since when x=y then x would be a tautology ( yey <-> yey). and accordingly this axiom will extenquish y={x|xex} from being a set in this theory. since this cannot be a theorum derived from the other axioms, then I should axiomatize it to force it. On the other hand I do not need to axiomatize ~ExAy(xey <-> y!ey) since it is clear that when x=y, then ( yey <-> y!ey) is logically contradictive and so since y={x|xex} would lead to (yey<->y!ey) then it is evident logically that this is contradictive, that's why I don't need to add it as an axiom, though adding it as an axiom will make its application more related to this theory,rather than being an application of general logic. At last I want to discuss the axiom of cautious separation, this axiom rises because this set has a universe that preventes unrestricted comprehension. EbAaAc: c e b <-> c e a / P(c) this is the axiom of separation in ZFC However this is in applicable in this theory, therefore I will modefy it to EbAaAc: c e b <-> c e a / P(c) / ( P(c) -> P(b) xor ~P(b)). what do I mean by ( P(c) -> P(b) xor ~P(b)). what I mean by it is if P(c) <-> P(b) then b is not a set in this theory, Also if P(c) <-> ~P(b) then b is clearly not a set in this theory. now according to this axiom , if b fullfils the axiom and P(c) -> P(b) then this means that beb, while if P(c) -> ~P(b) then this means that b!eb. Now I don't know if we can make a cautious replacement, cautious power or cautious choice, I don't know but perhaps this is possible. I am too tired , I should go to sleep. Zuhair Second, being self-membered isn't really what you want anyway. You > want non-well-founded sets, but not every non-well-founded set is > self-membered. But there are a lot smarter guys out there who have put some thought > into anti-well-founded set theory. You should look into it rather > than trying to haphazardly define your own axioms. The only problem is that these theories require a bit of mathematical > maturity. It's not trivial to understand the set theory in /Vicious > Circles/, for instance. But it's a lot more profitable to at least > try to understand that theory than to come up with your own. Anyway, Axiom 7 says there are two self-membered sets. But I don't > see that getting you very far. You have no idea what these two sets > look like. -- > I'm the theory guy. > Other people are the experimental people. > If you push me on details I get annoyed, as I'm the theory guy. > I'm the theoretical amateur mathematician. --James S. Harris, poet === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <873b8onyyg.fsf@phiwumbda.org> <87lkmgmfjd.fsf@phiwumbda.org > Ok, let me tell you my intentions behind this axiom. I want to currenty that there exist more than one set that is a member > of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will > put it instead of 7) above so Axiom 7:) Exy(~x=y & xex & yey). There are two problems with this approach. First, just knowing that > there are two self-membered sets doesn't tell you much of anything at > all. Instead you want a principled manner to say *which* > self-membered sets exist. Though I am so tired today, that I have decided to close this subject > but, two ideas came into my mind. This prinicipled manner you are > talking about is a beautiful remark. ExAy ( yey<-> xey -> yey ) so this is where y is in itself, in words y > is in itself if and only if it satisfies its membership. Correction I think it should be AxAy ( yey <-> xey -> yey ) while, ExAy (y!ey <-> xey -> y!ey) so this is where y is not in itself, in > words y is not in itself if and only if it doesnt' satisfy its > membership. Correction AxAy(y!ey<-> xey -> y!ey) I don't know if I should axiomatize these two ( put them as axioms in > this theory ). However, I think I reached to the axiomatization of what I always > wanted to say, I always wanted to say that the set of all sets that are > in themselfs is not a set, not because it is inconsistent, but because > it is a tautology, and tautologies are not decisive, for example > y={x|xex}, if I ask is yey or y!ey, the answer is unknown it might be > in itself or not in itself. However to axiomatize removal of y from being a set in this theory, I > can add an axiom like below: ~ExAy ( xey <-> yey ). since when x=y then x would be a tautology ( yey <-> yey). and > accordingly this axiom will extenquish y={x|xex} from being a set in > this theory. since this cannot be a theorum derived from the other > axioms, then I should axiomatize it to force it. On the other hand I do not need to axiomatize ~ExAy(xey <-> y!ey) since > it is clear that when x=y, then ( yey <-> y!ey) is logically > contradictive and so since y={x|xex} would lead to (yey<->y!ey) then it > is evident logically that this is contradictive, that's why I don't > need to add it as an axiom, though adding it as an axiom will make its > application more related to this theory,rather than being an > application of general logic. At last I want to discuss the axiom of cautious separation, this axiom > rises because this set has a universe that preventes unrestricted > comprehension. EbAaAc: c e b <-> c e a / P(c) this is the axiom of separation in ZFC However this is in applicable in this theory, therefore I will modefy > it to EbAaAc: c e b <-> c e a / P(c) / ( P(c) -> P(b) xor ~P(b)). what do I mean by ( P(c) -> P(b) xor ~P(b)). what I mean by it is if > P(c) <-> P(b) then b is not a set in this theory, Also if P(c) <- ~P(b) then b is clearly not a set in this theory. now according to this axiom , if b fullfils the axiom and P(c) -> P(b) > then this means that beb, while if P(c) -> ~P(b) then this means that > b!eb. Now I don't know if we can make a cautious replacement, cautious power > or cautious choice, I don't know but perhaps this is possible. I am too tired , I should go to sleep. Zuhair > Second, being self-membered isn't really what you want anyway. You > want non-well-founded sets, but not every non-well-founded set is > self-membered. But there are a lot smarter guys out there who have put some thought > into anti-well-founded set theory. You should look into it rather > than trying to haphazardly define your own axioms. The only problem is that these theories require a bit of mathematical > maturity. It's not trivial to understand the set theory in /Vicious > Circles/, for instance. But it's a lot more profitable to at least > try to understand that theory than to come up with your own. Anyway, Axiom 7 says there are two self-membered sets. But I don't > see that getting you very far. You have no idea what these two sets > look like. -- > I'm the theory guy. > Other people are the experimental people. > If you push me on details I get annoyed, as I'm the theory guy. > I'm the theoretical amateur mathematician. --James S. Harris, poet === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org> <873b8onyyg.fsf@phiwumbda.org> <87lkmgmfjd.fsf@phiwumbda.org > Ok, let me tell you my intentions behind this axiom. I want to currenty that there exist more than one set that is a member > of itself. Moe Blee has suggested the following: Exy(~x=y & xex & yey). Do you agree that this would convey what I am aiming at. if so I will > put it instead of 7) above so Axiom 7:) Exy(~x=y & xex & yey). There are two problems with this approach. First, just knowing that > there are two self-membered sets doesn't tell you much of anything at > all. Instead you want a principled manner to say *which* > self-membered sets exist. Second, being self-membered isn't really what you want anyway. You > want non-well-founded sets, but not every non-well-founded set is > self-membered. What do you mean isn't really what you want anyway. Who is you in your statement. If you mean me , then you are wrong, I am not interested in non-well founded sets, I am interested in sets that are members of themselfs. I think that x is well founded means that x!e ........Ux , for example: x={ {} , { {} } , { { {} } } } Ux= { {}, { {} } } UUx= { {} } UUUx= { } Now x!e......Ux = x!ex / x!e Ux / x!e UUx / x!eUUUx I am not interested in these set. By the way Frege's cardinals are not well founded sets. Anyhow , what I wanted to say is that I am not interested in such sets. what I am interested in is_ when a set is a member of itself? and when a set is no a member of itself? The whole of this set theory is made with this aim in mind, so that's why I should first axiomatize in a manner as to generate sets that are in themselves and sets that are not in themselfs, and set the different complex interaction between them. But I agree with you , this aim is so difficult , it needs logical and mathematical maturity. I think this subject should be closed. Zuhair But there are a lot smarter guys out there who have put some thought > into anti-well-founded set theory. You should look into it rather > than trying to haphazardly define your own axioms. The only problem is that these theories require a bit of mathematical > maturity. It's not trivial to understand the set theory in /Vicious > Circles/, for instance. But it's a lot more profitable to at least > try to understand that theory than to come up with your own. Anyway, Axiom 7 says there are two self-membered sets. But I don't > see that getting you very far. You have no idea what these two sets > look like. -- > I'm the theory guy. > Other people are the experimental people. > If you push me on details I get annoyed, as I'm the theory guy. > I'm the theoretical amateur mathematician. --James S. Harris, poet === Subject: Re: Small set theory. > If you mean me , then you are wrong, I am not interested in non-well > founded sets, I am interested in sets that are members of themselfs. I > think that x is well founded means that > x!e ........Ux , for example: No, that's not the meaning of the term. -- I've been thinking about my problems with getting any kind of admission that my math arguments showing the core error in mathematics are correct, so I've gone to marketing books. -- James S. Harris, on when mathematics isn't enough === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org> <877iy0o6eg.fsf@phiwumbda.org > Still nobody answered my simple question, if y={x|xex} is yey or > y!ey. It seems to me that the statement y e y is independent of the axioms. > There is no inconsistency if we assume y e y or assume y !e y. There is difference between Ontology and Epistemolgoy. y exists since > there is no contradiction involved with its existance, but > Epistemologically speaking we do not know if y is a member of itself or > not. Anything not contradictory exists? That's an ontological principle I > haven't heard. If I accept y to be a set in this theory, then I think it would be a > unique set. y would be the only set which we do not know weather it is > a member of itself or not. so I think tha y is unique, like how {} is > unique. There are plenty of other sets that would be similar. For instance, > consider the set { x | x e x and x != {x} }. If this set exists in > your theory, then it's not clear whether it is self-membered. > Anyhow , it is clear as I mentioned before that , the axioms of > separation, replacement,power and choice are inconsistent with this > theory( I mean these axioms as present in ZFC). No, it is not at all clear that power set and choice are > inconsistent with your theory. And you have never given any argument > to prove this. Why not? If it's so obvious, then surely you can > provide a simple proof. Primitive e 1) Axiom of Extentiality > 2) Axiom of Empty set > 3) Axiom of Pairing > 4) Axiom of Union > 5) Axiom of Infinity > 6) Axiom of Universe ExAy yex 7) Axiom of Multiplicity ExEzAy yey / xex / zex / z!ey I don't know what you're trying to say with this axiom, but I'm sure > you got it wrong. As a consequence of this axiom, we can prove > Ay yey. 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in Z. Do anybody think that this axiomatic set theory is inconsistent? It is inconsistent. Axiom 7 proves every set is self-membered, and we > know that the empty set is not self-membered. Yes that is right! Axiom 7) ExEzAy(yey) -> xex / zex / z!ey -- > Jesse F. Hughes How can I miss you when you won't go away? > -- Dan Hicks and his Hot Licks The theory should be like this: -Small set theory- Primitive: e 1) Axiom of Extentiality 2) Axiom of Empty set 3) Axiom of Pairing 4) Axiom of Union 5) Axiom of Infinity 6) Axiom of Universe : ExAy yex 7) Axiom of Multiplicity: ExEzAy(yey -> (xex / zex / ~zey)) 8) Axiom of Cautious separation. Not completed yet. 1)2)3)4)5) are as in ZFC. Is this theory inconsistent? Zuhair === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org> <87d57uxdgu.fsf@phiwumbda.org >> No, it is contradictive. >> y={x|xex} >> if yey -> yey >> if y!ey -> y!ey >> therefore y is both regular and irregular at the same time. which is >> contradicitive. >> Man. You're simply incapable of fundamental reasoning. >> The fact that y e y -> y e y is *not* a contradiction. >> Today is either warm or not. If it is warm, then it is warm. If not, >> then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. The sentences y e y -> y e y and y !e y -> y !e y are tautologies and > hence do not contradict each other. Your statement is simply, stupidly false. And there is no > contradiction to be had from assuming that the set { x | x e x } > exists. At least none that I see. (Note: I'm speaking of Zuhair's teeny set theory and not some other > theory.) Yes, you are speaking in the line of the theory that I have just presented. But I am speaking in the line of my intuitive desire, I don't desire this set to exist, therefore I am looking for an axiom which make it non existent, do u understand now. To make my idea somewhat clear, I think that every set should have a KNOWN status of being a member of itself or not. ie in my mind exist the following x is a set -> we know that xex xor we know that x!ex. To say that there is a set x which we don't know weather it is in itself or not, is something that I don't like. To make matters symbolic; if x is defined by intention in the following manner y={x|P(x) is true} is a set if and only if ( P(y) is true -> yey ) xor ( P(y) is false -> y!ey ). xor here implies that both statements cannot be true , neither both can be false. Example: y={x|x ={ } } , here we have (y={ } is true -> yey) is a false statement, because y=/={}. while ( y={ } is false -> y!ey ) is a true statement. Therefore y is a set. Now lets take y={x|x!ex}. here we have (y!ey is true -> yey ) is a false statement. And also ( ~y!ey is true -> y!ey ) which also a false statment, therefore y is not a set. Now we come to y={x|xex} here we have the following. (yey is true -> yey) a true statement, Also ( y!ey is true -> y!ey) a true statement. so we have both statements true. then y is not a set. Now let y={x|x=x} we have y=y -> yey a true statement, and y=/=y ->y!ey a false statement. therefore y is a set. I think I should axiomatize this by a sixth axiom and put it in my teeny theory. I will call it axiom of caucious comprehension. or caucious seperation. Zuhair -- > You got more out of it > than I put into it last night. > Who were you thinking of when we were loving last night? > -- Texas Tornadoes === Subject: Re: Small set theory. > To say that there is a set x which we don't know weather it is in > itself or not, is something that I don't like. To make matters symbolic; if x is defined by intention in the following > manner y={x|P(x) is true} is a set if and only if ( P(y) is true -> yey ) > xor ( P(y) is false -> y!ey ). xor here implies that both statements cannot be true , neither both can > be false. Right, but it is clear that both can't be true. And they also can't both be false. So xor doesn't help you here. Unless you mean some sense of possibly when you write can. [...] > I think I should axiomatize this by a sixth axiom and put it in my > teeny theory. I will call it axiom of caucious comprehension. or caucious > seperation. Good luck. And, I don't mean this as a spelling flame, but it should be cautious separation. -- Jesse F. Hughes I can't tell you how many times she left me. I lost count the very first time that she did. -- The Flatlanders, I Thought the Wreck Was Over === Subject: Re: Small set theory. No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. Man. You're simply incapable of fundamental reasoning. The fact that y e y -> y e y is *not* a contradiction. Today is either warm or not. If it is warm, then it is warm. If not, > then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. But your statements do not say that, so it is you being stupid. What you say you are saying is y e y / y !e y. What you are actually saying with if yey -> yey if y!ey -> y!ey is y e y / y !e y. === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87k622zo0l.fsf@phiwumbda.org> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. Man. You're simply incapable of fundamental reasoning. The fact that y e y -> y e y is *not* a contradiction. Today is either warm or not. If it is warm, then it is warm. If not, > then not. Big deal. This fact is not contradicitive. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. But your statements do not say that, so it is you being stupid. What you say you are saying is > y e y / y !e y. > What you are actually saying with > if yey -> yey > if y!ey -> y!ey > is y e y / y !e y. so were is the argument? u r so stupid even not to make an argument. Zuhair === Subject: Re: Small set theory. I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. But your statements do not say that, so it is you being stupid. What you say you are saying is > y e y / y !e y. > What you are actually saying with > if yey -> yey > if y!ey -> y!ey > is y e y / y !e y. so were is the argument? u r so stupid even not to make an argument. I do not have to make any argument when you provide the evidence of your own error so brazenly. You are claiming that the tautology, (y e y) / (y !e y) is the same as the contradiction (y e y) / (y !e y). A /or/ not A is a tautology. A /and/ not A is a contradiction. Those who cannot tell the difference are logically inept. === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. But your statements do not say that, so it is you being stupid. What you say you are saying is > y e y / y !e y. What you are actually saying with > if yey -> yey > if y!ey -> y!ey > is y e y / y !e y. so were is the argument? u r so stupid even not to make an argument. I do not have to make any argument when you provide the evidence of your > own error so brazenly. You are claiming that the tautology, (y e y) / (y !e y) > is the same as the contradiction (y e y) / (y !e y). A /or/ not A is a tautology. > A /and/ not A is a contradiction. Those who cannot tell the difference are logically inept. Of course I am not having the confusion you are referring to. But you are not understanding me. So that you can understand what I am saying I will confront you with the following simple question. Lety={x|xex} Is yey ? or y!ey? Do you KNOW? To me it appears that both are possible. since both do not lead to any contradiction. if yey then yey no contradiction if y!ey then y!ey no contradication But one of them should be true , this means that the correct answer is yey xor y!ey But to me this is not enough. I know very well that this is not a contradication, but at the same time it is not a final answer. To me I desire that every set x should have a clear regularity status, ie if x is a set then we KNOW for certain that if xex xor x!ex. ie the final answer should be something like xex , or should be something like x!ex, I will not accept an answer that is vague like xex xor x!ex. to make it simple for you, let me give you another example from logic, that is the barbor paradox. In city X , every citizen that do not shave himself should be shaved by Barbor G,were Barbor G do not ever shave a citizen of city X that shaves himself. The question is : Does barbor G shave himself or not? The correct answer, there exist no such a barbor. since its existance is logically contradictive. The Meta Barbor paradox ( designed by myself ) In city X, every citizen that do shave himself should be shaved by Barbor G, were Barbor G do not ever shave a citizen of city X that do not shave himself. Now the question is: Does Barbor G shave himself or not? Now it is obvious that the answer depends on Barbors G's free will. Such a barbor can exist, but we do not know if he shaves himself or not, do you understand me. Because if he chose to shave himself then he is shaving a citizen of city X that is shaving himself_ no contradiction, Similarily if decided not to shave himself, then he is not shaving a citizen which is not shaving himself_no contradiction. Now there is nothing in this question that help us say which decision the barbor will take, therefore we in reality do not know the answer to this question, the only thing we know is that such a barbor exist, but we don't know weather he shaves himself or not. This later barbor is quite similar to y={x|xex} . we don't know really weather the existant y is in itself or not. and that is what I do not desire. Every set should have a clear regularity (membership xor non membership in itself ) status. Since y doesn't have a clear regularity status, then y is not a set. I should add a sixth axiom to force this decision. I hope you now know what I am talking about. Zuhair === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> I want wonder why such a stupid man like you will not get my point! the > contradiction doesn't lie in yey -> yey, the contradiction lie in y > being in itself and not in itself at the same time stupid. But your statements do not say that, so it is you being stupid. What you say you are saying is > y e y / y !e y. What you are actually saying with > if yey -> yey > if y!ey -> y!ey > is y e y / y !e y. so were is the argument? u r so stupid even not to make an argument. I do not have to make any argument when you provide the evidence of your > own error so brazenly. You are claiming that the tautology, (y e y) / (y !e y) > is the same as the contradiction (y e y) / (y !e y). A /or/ not A is a tautology. > A /and/ not A is a contradiction. Those who cannot tell the difference are logically inept. Of course I am not having the confusion you are referring to. But you > are not understanding me. So that you can understand what I am saying I will confront you with > the following simple question. Lety={x|xex} Is yey ? or y!ey? Do you KNOW? To me it appears that both are possible. since both do not lead to any > contradiction. if yey then yey no contradiction if y!ey then y!ey no contradication But one of them should be true , this means that the correct answer is yey xor y!ey But to me this is not enough. I know very well that this is not a contradication, but at the same > time it is not a final answer. To me I desire that every set x should have a clear regularity status, > ie if x is a set then we KNOW for certain that if xex xor x!ex. ie the final answer should be > something like xex , or should be something like x!ex, I will not accept an answer > that is vague like xex xor x!ex. to make it simple for you, let me give you another example from logic, > that is the barbor paradox. In city X , every citizen that do not shave himself should be shaved by > Barbor G,were Barbor G do not ever shave a citizen of city X that > shaves himself. > The question is : Does barbor G shave himself or not? The correct answer, there exist no such a barbor. since its existance > is logically contradictive. The Meta Barbor paradox ( designed by myself ) In city X, every citizen that do shave himself should be shaved by > Barbor G, were Barbor G do not ever shave a citizen of city X that do > not shave himself. Now the question is: Does Barbor G shave himself or not? Now it is obvious that the answer depends on Barbors G's free will. Such a barbor can exist, but we do not know if he shaves himself or > not, do you understand me. Because if he chose to shave himself then he is shaving a citizen of > city X that is shaving himself_ no contradiction, Similarily if decided > not to shave himself, then he is not shaving a citizen which is not > shaving himself_no contradiction. Now there is nothing in this question that help us say which decision > the barbor will take, therefore we in reality do not know the answer to > this question, the only thing we know is that such a barbor exist, but > we don't know weather he shaves himself or not. This later barbor is quite similar to y={x|xex} . we don't know really > weather the existant y is in itself or not. and that is what I do not > desire. Every set should have a clear regularity (membership xor non > membership in itself ) status. Since y doesn't have a clear regularity status, then y is not a set. I should add a sixth axiom to force this decision. I hope you now know what I am talking about. Zuhair To me a set is not a set not only when its existence is a logical contradiction, to me a set that is of vague membership of itself status is not a set. I don't know how I put that in exact axiomatic frame, but I will try. Axiom 6:) ~ExAy xex xor x!ex since xex / x!ex is logically contradictive and also ~xex / ~x!ex is also logically contradictive, then the negation above should lead to a strict knoweldge of membership of x in itself. Perhaps? ( I said the above because I think that ~ A xor B = A / B xor ~A / ~B. ). Zuhair === Subject: Re: Small set theory. > I don't know how I put that in exact axiomatic frame, but I will try. Axiom 6:) ~ExAy xex xor x!ex since xex / x!ex is logically contradictive and also ~xex / ~x!ex is > also logically contradictive, then the negation above should lead to a > strict knoweldge of membership of x in itself. Er, the bound variable y doesn't occur in the rest of the formula. -- Jesse F. Hughes You're ketchup, so I'll put you on meatloaf! -- Quincy P. Hughes, age five, tries his hand at insults === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87odrdv0mo.fsf@phiwumbda.org > I don't know how I put that in exact axiomatic frame, but I will try. Axiom 6:) ~ExAy xex xor x!ex since xex / x!ex is logically contradictive and also ~xex / ~x!ex is > also logically contradictive, then the negation above should lead to a > strict knoweldge of membership of x in itself. Er, the bound variable y doesn't occur in the rest of the formula. Right, it should be: Axiom 6:) ~ExAy yex xor y!ex -- > Jesse F. Hughes You're ketchup, so I'll put you on meatloaf! > -- Quincy P. Hughes, age five, tries his hand at insults === Subject: Re: Small set theory. > Right, it should be: Axiom 6:) ~ExAy yex xor y!ex It is a theorem of logic that Ex Ay y e x xor y !e x. So axiom 6 makes your theory inconsistent *regardless* of what other axioms are available. Congrats. -- Jesse F. Hughes Contrariwise, continued Tweedledee, if it was so, it might be, and if it were so, it would be; but as it isn't, it ain't. That's logic! -- Lewis Carroll === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87odrdv0mo.fsf@phiwumbda.org > I don't know how I put that in exact axiomatic frame, but I will try. Axiom 6:) ~ExAy xex xor x!ex since xex / x!ex is logically contradictive and also ~xex / ~x!ex is > also logically contradictive, then the negation above should lead to a > strict knoweldge of membership of x in itself. Er, the bound variable y doesn't occur in the rest of the formula. Right, it should be: Axiom 6:) ~ExAy yex xor y!ex -- > Jesse F. Hughes You're ketchup, so I'll put you on meatloaf! > -- Quincy P. Hughes, age five, tries his hand at insults Axioms in ZFC that are inconsistent with this small set theory are: 1) Axiom of regularity 2) Axiom of separation 3) Axiom of replacement 4) Axiom of power set 5) Axiom of choice. On the other hand axioms of infinity and union can be added to this theory. And accordingly this theory will consist of 8 Axioms. However axiom 6 which that axiom of cautious separation is till now not developed fully. Zuhair === Subject: Re: Small set theory. As I said in another post, axiom 6 is a contradiction, so your set theory is already inconsistent. But let's pretend you didn't add axiom 6 and see what we have. Axioms in ZFC that are inconsistent with this small set theory are: 1) Axiom of regularity Yes. > 2) Axiom of separation Yes. > 3) Axiom of replacement You claim so but you've given no argument. I don't believe it. > 4) Axiom of power set > 5) Axiom of choice. I don't see any reason that these two axioms are inconsistent with your theory either. > On the other hand axioms of infinity and union can be added to this > theory. > And accordingly this theory will consist of 8 Axioms. However axiom 6 which that axiom of cautious separation is till now not > developed fully. Axiom 6 needs work. What are you trying to say with it? -- Jesse F. Hughes C is for Cookie. That's good enough for me. Cookie Monster === Subject: Re: Small set theory. On Sun, 12 Nov 2006 09:10:42 -0500, Jesse F. Hughes As I said in another post, axiom 6 is a contradiction, so your set >theory is already inconsistent. But let's pretend you didn't add >axiom 6 and see what we have. >> Axioms in ZFC that are inconsistent with this small set theory are: >> 1) Axiom of regularity Yes. > 2) Axiom of separation Yes. > 3) Axiom of replacement You claim so but you've given no argument. I don't believe it. In ZF replacement implies separation. Why would it be different here? > 4) Axiom of power set >> 5) Axiom of choice. I don't see any reason that these two axioms are inconsistent with >your theory either. >> On the other hand axioms of infinity and union can be added to this >> theory. >> And accordingly this theory will consist of 8 Axioms. >> However axiom 6 which that axiom of cautious separation is till now not >> developed fully. Axiom 6 needs work. What are you trying to say with it? -- >Jesse F. Hughes C is for Cookie. That's good enough for me. > Cookie Monster === Subject: Re: Small set theory. > On Sun, 12 Nov 2006 09:10:42 -0500, Jesse F. Hughes 3) Axiom of replacement [is inconsistent with Zuhair's theory] >>You claim so but you've given no argument. I don't believe it. In ZF replacement implies separation. Why would it be different > here? It does? That's news to me. Oh. Now that I look at the axiom as it's presented on Mathworld, it seems obvious. -- So I speak before a crowd of the damned, cursed to be unloved throughout time, with only their hatred and bile to comfort them now, having betrayed what should have been their one true lover: Mathematics. -- James Harris reaches a bit === Subject: Re: Small set theory. >> In ZF replacement implies separation. Why would it be different >> here? It does? That's news to me. Take a set A and a condition P(x). By replacement there is a set A' = {F(a) | a in A} where F(x) = {x} if P(x) and {} otherwise. Then {a in A | P(a)} is the union of A'. I want wonder why such a stupid man like you did not get that point! -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. > In ZF replacement implies separation. Why would it be different > here? >> It does? That's news to me. Take a set A and a condition P(x). By replacement there is a set A' = > {F(a) | a in A} where F(x) = {x} if P(x) and {} otherwise. Then {a in > A | P(a)} is the union of A'. I want wonder why such a stupid man like > you did not get that point! Me too. -- Jesse F. Hughes Anything was possible last night. That was the trouble with last nights. They were always followed by this mornings. -- Terry Pratchett, /Small Gods/ === Subject: Re: Small set theory. >> Take a set A and a condition P(x). By replacement there is a set A' = >> {F(a) | a in A} where F(x) = {x} if P(x) and {} otherwise. Then {a in >> A | P(a)} is the union of A'. I want wonder why such a stupid man like >> you did not get that point! Me too. What I want wonder is why zuhair thinks it is remarkable that a stupid man like you can't follow one of his cogent and clearly stated arguments. Usually, after all, stupidity does not enhance one's ability to recognize valid arguments. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. <87k622zo0l.fsf@phiwumbda.org> <87odrdv0mo.fsf@phiwumbda.org> <877iy0vjq5.fsf@phiwumbda.org> <45574727.597573312@news.sbtc.net> <87bqncoc6h.fsf@phiwumbda.org> In ZF replacement implies separation. Why would it be different >> here? It does? That's news to me. Take a set A and a condition P(x). By replacement there is a set A' = > {F(a) | a in A} where F(x) = {x} if P(x) and {} otherwise. Then {a in A > | P(a)} is the union of A'. I want wonder why such a stupid man like you > did not get that point! Yea , of course Jesse F. Hughes is stupid, I agree, he even didn't know this simple fact. Zuhair -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. > Yea , of course Jesse F. Hughes is stupid, I agree, he even didn't know > this simple fact. Compared to your magnificent brilliance Jesse is indeed abysmally stupid, but in all honesty I have to admit that I too sometimes find your arguments a bit difficult to follow. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org >> Do you mean the axioms you just gave don't entail the union axiom or do >> you mean that the union axiom is inconsistent with the axioms you just >> gave? Well regarding this axiom I really don't know. but what I meant was > that union axiom is inconsistent with the axioms I have just gave. > Anyhow I am not sure. Doesn't seem that union is inconsistent with your axioms. Neither is > infinity, choice or replacement, as far as I can tell. Separation, on the other hand, is clearly inconsistent with your axiom > that a universe exists. Separation,replacement and regularity are clearly inconsistent with this theory. Well in reality I don't know weather , union, infinity, power, choice are inconsistent with this theory, but I don't think adding them to this theory will make a difference, I will quote Moe Blee's summary of this little set theory. Primitive: e. Axiom (1) Axy(Az(zex <-> zey) -> x=y). Extensionality. Axiom (2) ExAy ~yex. Empty Set. Axiom (3) AxyEzAt(tez <-> (t=x v t=y)). Pairing. Axiom (4) ExAy yex. Universal set. Axiom (5) ExEzAy(yey -> (xex & (zex <-> ~zey))). Definition (6): x is regular <-> ~xex. Definition (7): x is irregular <-> ~ x is regular. Now if I add axioms_union, infinity,power and perhaps choice, what difference they will make ? I deliberately want this set theory to contain the least number of axioms. for example the axiom of infinity is not needed for this kind of theory, neither do power or choice , so I tend to think that putting them as axioms would be redundant. However the axiom of union is the only axiom that might be important to put ( in case it doesn't contradict the other axioms of course). Zuhair -- > [T]he Cantorian pseudomathematicians are defending a religion, and > they really can't see what monsters they have become. What the > Cantorians are doing is nothing less than a crime against > humanity. What they are doing is evil. -- Petry, victim. === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org Now if I add axioms_union, infinity,power and perhaps choice, what > difference they will make ? They'll entail a bunch of theorems that are not now theorems. What exactly is it that you want to know? MoeBlee === Subject: Re: Small set theory. > Do you mean the axioms you just gave don't entail the union axiom or do > you mean that the union axiom is inconsistent with the axioms you just > gave? >> Well regarding this axiom I really don't know. but what I meant was >> that union axiom is inconsistent with the axioms I have just gave. >> Anyhow I am not sure. >> Doesn't seem that union is inconsistent with your axioms. Neither is >> infinity, choice or replacement, as far as I can tell. >> Separation, on the other hand, is clearly inconsistent with your axiom >> that a universe exists. Separation,replacement and regularity are clearly inconsistent with > this theory. No. Replacement is not clearly inconsistent with your teeny theory. At least it's not clear to me. > Well in reality I don't know weather , union, infinity, power, choice > are inconsistent with this theory, but I don't think adding them to > this theory will make a difference, I will quote Moe Blee's summary of > this little set theory. You don't know if adding axioms will make a difference to your theory? Axioms that are pretty clearly independent? Wow. You *do* need to learn basic logic. Not set theory, but *logic*. (And after that, set theory) > Primitive: e. > Axiom (1) Axy(Az(zex <-> zey) -> x=y). Extensionality. > Axiom (2) ExAy ~yex. Empty Set. > Axiom (3) AxyEzAt(tez <-> (t=x v t=y)). Pairing. > Axiom (4) ExAy yex. Universal set. > Axiom (5) ExEzAy(yey -> (xex & (zex <-> ~zey))). What is this supposed to mean? > Definition (6): x is regular <-> ~xex. > Definition (7): x is irregular <-> ~ x is regular. Now if I add axioms_union, infinity,power and perhaps choice, what > difference they will make ? Um, there will be more sets? And more theorems? > I deliberately want this set theory to contain the least number of > axioms. for example the axiom of infinity is not needed for this > kind of theory, neither do power or choice , so I tend to think that > putting them as axioms would be redundant. However the axiom of > union is the only axiom that might be important to put ( in case it > doesn't contradict the other axioms of course). Before you say what axioms are important, perhaps you ought to consider one of the following: What basic intuitions are you trying to capture? What theorems do you want to be true? -- Jesse F. Hughes Mama: Do you need help? Sir Quincy: Nay, I'm a big knight. Knights don't need help with pee-pee. === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org> <87odrezo51.fsf@phiwumbda.org No. Replacement is not clearly inconsistent with your teeny theory. If I'm not mistaken, it is, since it entails separation. MoeBlee === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org> <87odrezo51.fsf@phiwumbda.org > Do you mean the axioms you just gave don't entail the union axiom or do > you mean that the union axiom is inconsistent with the axioms you just > gave? >> Well regarding this axiom I really don't know. but what I meant was >> that union axiom is inconsistent with the axioms I have just gave. >> Anyhow I am not sure. >> Doesn't seem that union is inconsistent with your axioms. Neither is >> infinity, choice or replacement, as far as I can tell. >> Separation, on the other hand, is clearly inconsistent with your axiom >> that a universe exists. Separation,replacement and regularity are clearly inconsistent with > this theory. No. Replacement is not clearly inconsistent with your teeny theory. At least it's not clear to me. Well in reality I don't know weather , union, infinity, power, choice > are inconsistent with this theory, but I don't think adding them to > this theory will make a difference, I will quote Moe Blee's summary of > this little set theory. You don't know if adding axioms will make a difference to your theory? > Axioms that are pretty clearly independent? Wow. You *do* need to learn basic logic. Not set theory, but > *logic*. (And after that, set theory) Primitive: e. > Axiom (1) Axy(Az(zex <-> zey) -> x=y). Extensionality. > Axiom (2) ExAy ~yex. Empty Set. > Axiom (3) AxyEzAt(tez <-> (t=x v t=y)). Pairing. > Axiom (4) ExAy yex. Universal set. > Axiom (5) ExEzAy(yey -> (xex & (zex <-> ~zey))). What is this supposed to mean? Definition (6): x is regular <-> ~xex. > Definition (7): x is irregular <-> ~ x is regular. Now if I add axioms_union, infinity,power and perhaps choice, what > difference they will make ? Um, there will be more sets? And more theorems? No , it will not. Zuhair > I deliberately want this set theory to contain the least number of > axioms. for example the axiom of infinity is not needed for this > kind of theory, neither do power or choice , so I tend to think that > putting them as axioms would be redundant. However the axiom of > union is the only axiom that might be important to put ( in case it > doesn't contradict the other axioms of course). Before you say what axioms are important, perhaps you ought to > consider one of the following: What basic intuitions are you trying to > capture? What theorems do you want to be true? -- > Jesse F. Hughes > Mama: Do you need help? > Sir Quincy: Nay, I'm a big knight. Knights don't need help with > pee-pee. === Subject: Re: Small set theory. >> Now if I add axioms_union, infinity,power and perhaps choice, what >> difference they will make ? >> Um, there will be more sets? And more theorems? No , it will not. The only possible way that I could be wrong is if these axioms are in fact theorems of your teeny theory. So, show me the proofs. If you cannot prove that the union of any set is also a set, then clearly adding the union axiom would create new theorems. And if you are too stupid to understand that very obvious fact, perhaps you should learn some logic before revolutionizing set theory. -- And God Himself won't help you if this goes bad as despite your beliefs I can assure you that angry people will against all law if necessary tear their rage out of your hides if it goes badly. -- James S. Harris, on the dangers of criticizing his mathematics === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org> <87odrezo51.fsf@phiwumbda.org> <87k622xdro.fsf@phiwumbda.org >> Now if I add axioms_union, infinity,power and perhaps choice, what >> difference they will make ? >> Um, there will be more sets? And more theorems? No , it will not. The only possible way that I could be wrong is if these axioms are in > fact theorems of your teeny theory. So, show me the proofs. If you cannot prove that the union of any set is also a set, then > clearly adding the union axiom would create new theorems. And if you are too stupid to understand that very obvious fact, > perhaps you should learn some logic before revolutionizing set theory. I am not revolutionizing set theory, this is another missunderstanding of yours. I am only testing the axioms in ZFC, and see why other axioms might direct thought to. Anyhow weather I am ignorant or not I should always learn a lot of logic before I attempt to make a frank serious trial to revolutionize set theory. But I want to tell you something : not all axioms leads to increament in the number of theorums as you think , indeed some of them might be added to ristrict too many theorums, for example the axiom of regularity and the axiom of choice are in a way restricting axioms. anyhow. Zuhair -- > And God Himself won't help you if this goes bad as despite your > beliefs I can assure you that angry people will against all law if > necessary tear their rage out of your hides if it goes badly. > -- James S. Harris, on the dangers of criticizing his mathematics === Subject: Re: Small set theory. > But I want to tell you something : not all axioms leads to > increament in the number of theorums as you think , indeed some of > them might be added to ristrict too many theorums, for example the > axiom of regularity and the axiom of choice are in a way restricting > axioms. anyhow. Really, this is an utterly stupid claim. I mean this with all due respect: Adding to the axioms can not restrict too many theorems. Really. More axioms mean more theorems (unless the new axioms are not independent --- but even then, it means *no fewer* theorems). More axioms mean more proofs. The fact you don't realize this is really remarkable. -- Jesse F. Hughes Women aren't that unpredictable. Well, I can't guess what you're getting at, honey. -- Hitchcock's _Rear Window_ === Subject: Re: Small set theory. > More axioms mean more proofs. The fact you don't realize this is really remarkable. Given zuhair's track record - not really. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Small set theory. > But I want to tell you something : not all axioms leads to increament > in the number of theorums as you think , indeed some of them might be > added to ristrict too many theorums, for example the axiom of > regularity and the axiom of choice are in a way restricting axioms. > anyhow. If the new axiom is not provable from the previous axioms, then if you add the axiom, you get *more* theorems and *fewer* models. I think you are confusing theorems and models. It is impossible to add an axiom and get fewer theorems. The reason is that every valid proof that didn't use the new axiom is still a valid proof after you add the new axiom. In fact, this proves what I just said. Have you read any books on mathematical logic or taken any courses on the subject? -- Marcus === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org> <87odrezo51.fsf@phiwumbda.org> <87k622xdro.fsf@phiwumbda.org> in the number of theorums as you think , indeed some of them might be > added to ristrict too many theorums, for example the axiom of > regularity and the axiom of choice are in a way restricting axioms. > anyhow. If the new axiom is not provable from the previous axioms, then if you > add the axiom, you get *more* theorems and *fewer* models. I think you > are confusing theorems and models. That is correct, I admit that. Zuhair It is impossible to add an axiom and get fewer theorems. The reason is > that every valid proof that didn't use the new axiom is still a valid > proof after you add the new axiom. In fact, this proves what I just > said. Have you read any books on mathematical logic or taken any courses on > the subject? -- > Marcus === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> <87ejsc3pv3.fsf@phiwumbda.org> <87odrezo51.fsf@phiwumbda.org > Do you mean the axioms you just gave don't entail the union axiom or do > you mean that the union axiom is inconsistent with the axioms you just > gave? >> Well regarding this axiom I really don't know. but what I meant was >> that union axiom is inconsistent with the axioms I have just gave. >> Anyhow I am not sure. >> Doesn't seem that union is inconsistent with your axioms. Neither is >> infinity, choice or replacement, as far as I can tell. >> Separation, on the other hand, is clearly inconsistent with your axiom >> that a universe exists. Separation,replacement and regularity are clearly inconsistent with > this theory. No. Replacement is not clearly inconsistent with your teeny theory. Yes it is stupid. At least it's not clear to me. Well in reality I don't know weather , union, infinity, power, choice > are inconsistent with this theory, but I don't think adding them to > this theory will make a difference, I will quote Moe Blee's summary of > this little set theory. You don't know if adding axioms will make a difference to your theory? > Axioms that are pretty clearly independent? Wow. You *do* need to learn basic logic. Not set theory, but > *logic*. (And after that, set theory) Primitive: e. > Axiom (1) Axy(Az(zex <-> zey) -> x=y). Extensionality. > Axiom (2) ExAy ~yex. Empty Set. > Axiom (3) AxyEzAt(tez <-> (t=x v t=y)). Pairing. > Axiom (4) ExAy yex. Universal set. > Axiom (5) ExEzAy(yey -> (xex & (zex <-> ~zey))). What is this supposed to mean? Definition (6): x is regular <-> ~xex. > Definition (7): x is irregular <-> ~ x is regular. Now if I add axioms_union, infinity,power and perhaps choice, what > difference they will make ? Um, there will be more sets? And more theorems? > I deliberately want this set theory to contain the least number of > axioms. for example the axiom of infinity is not needed for this > kind of theory, neither do power or choice , so I tend to think that > putting them as axioms would be redundant. However the axiom of > union is the only axiom that might be important to put ( in case it > doesn't contradict the other axioms of course). Before you say what axioms are important, perhaps you ought to > consider one of the following: What basic intuitions are you trying to > capture? What theorems do you want to be true? -- > Jesse F. Hughes > Mama: Do you need help? > Sir Quincy: Nay, I'm a big knight. Knights don't need help with > pee-pee. === Subject: Re: Small set theory. > > Separation, on the other hand, is clearly inconsistent with your axiom > that a universe exists. >> Separation,replacement and regularity are clearly inconsistent with >> this theory. >> No. Replacement is not clearly inconsistent with your teeny theory. Yes it is stupid. Did you mean Yes, it is stupid that I said this, or Yes, replacement *is* inconsistent, stupid? If the latter, show me why replacement is inconsistent with your teeny theory. -- I was driving down the interstate through Winslow, Arizona, I had Seven Vices on my mind -- Sloth and Avarice, Fornication, Television, Whiskey, Beer and Wine. -- Austin Lounge Lizards === Subject: Re: Small set theory. > It is more than four times in this thread that I made it clear and very > clear what I meant by saying regular. x is a regular set <-> x!ex > x is irregular set <-> xex Just don't say i didn't mention this many times in this thread. Well, I missed it. But I don't read every post. Anyway, that's a strange notion of regular. According to you, a set satisfying X = { {}, {X}} is regular. But never mind. Let's take that as our definition. -- [Criticizing JSH's mathematics will result in] one of the worst debacles in the history of the world. It is foretold in most mythologies and religions. And yes, you are the ones, the cursed ones, who destroy the world. --James S. Harris reads from the Aztec Book of the Damned Mathematicians === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> member of itself. That does not entail that the set is in itself and > not in itself. It is in itself. And that is not a contradiction, unless > you add axioms that it would contradict. No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. There y would be the set of all and only those sets that are members of > themselves. And that is not necessarily the same as the set of all > sets. And you have not shown that EyAx(xey <-> xex) entails a contradiction. Your two formulas, yey -> yey > ~ yey -> ~yey are just two tautologies. They don't entail a contradiction. All you showed is: If y is regular then y is regular, and if y is not regular then y is > not regular. That is not a contradiction. Rather, it is validity of sentential logic > alone. so you think that y={x|xex} is a set in this theory. But still I don't know what is y. is it regular or is it irregular? from what you are saying this mean that there should be an axiom like saying ExAy yex xor ~yex , but this axiom is not in this theory. 2) and 4) are so decisive while this is not decisive, now since this axiom is not present in this theory, then y={x|xex} is not a set. Perhaps I am mistaken I don't know. But I want a clear answer about this. should I add an axiom like this: ~ExAy yex xor ~yex . Zuhair You haven't shown, as you claimed: y is regular & ~ y is regular. That is, you haven't shown: EyAx(xey <-> xex) -> Ey(y is regular & y is irregular). MoeBlee === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> member of itself. That does not entail that the set is in itself and > not in itself. It is in itself. And that is not a contradiction, unless > you add axioms that it would contradict. No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. There y would be the set of all and only those sets that are members of > themselves. And that is not necessarily the same as the set of all > sets. And you have not shown that EyAx(xey <-> xex) entails a contradiction. Your two formulas, yey -> yey > ~ yey -> ~yey are just two tautologies. They don't entail a contradiction. All you showed is: If y is regular then y is regular, and if y is not regular then y is > not regular. That is not a contradiction. Rather, it is validity of sentential logic > alone. so you think that y={x|xex} is a set in this theory. NO, again, please don't extrapolate beyond what I said. I said that I have not seen a proof in your theory of ~EyAx(xey <-> xex). That does not entail that I think EyAx(xey <-> xex) is a member of your theory. > But still I don't > know what is y. is it regular > or is it irregular? If EyAx(xey <-> xex), then it still might be undecided whether such a y is such that yey. > from what you are saying this mean that there should be an axiom like > saying ExAy yex xor ~yex , I see no need for such an axiom since, unless I'm spacing out right now, it is already a theorem of logic. > but this axiom is not in this theory. 2) and 4) > are so decisive while this is not decisive, now since this axiom is not > present in this theory, then y={x|xex} is not a set. Wrong. Please get that logic book you keep saying you'll get. > Perhaps I am mistaken I don't know. But I want a clear answer about > this. should I add an axiom like this: ~ExAy yex xor ~yex . No. It would make your theory contradict basic predicate logic. MoeBlee === Subject: Re: Small set theory. <87mz70bw5e.fsf@phiwumbda.org> member of itself. That does not entail that the set is in itself and > not in itself. It is in itself. And that is not a contradiction, unless > you add axioms that it would contradict. No, it is contradictive. y={x|xex} if yey -> yey > if y!ey -> y!ey > therefore y is both regular and irregular at the same time. which is > contradicitive. There y would be the set of all and only those sets that are members of > themselves. And that is not necessarily the same as the set of all > sets. And you have not shown that EyAx(xey <-> xex) entails a contradiction. Your two formulas, yey -> yey > ~ yey -> ~yey are just two tautologies. They don't entail a contradiction. All you showed is: If y is regular then y is regular, and if y is not regular then y is > not regular. That is not a contradiction. Rather, it is validity of sentential logic > alone. so you think that y={x|xex} is a set in this theory. NO, again, please don't extrapolate beyond what I said. I said that I > have not seen a proof in your theory of ~EyAx(xey <-> xex). That does > not entail that I think EyAx(xey <-> xex) is a member of your theory. But still I don't > know what is y. is it regular > or is it irregular? If EyAx(xey <-> xex), then it still might be undecided whether such a y > is such that yey. from what you are saying this mean that there should be an axiom like > saying ExAy yex xor ~yex , I see no need for such an axiom since, unless I'm spacing out right > now, it is already a theorem of logic. but this axiom is not in this theory. 2) and 4) > are so decisive while this is not decisive, now since this axiom is not > present in this theory, then y={x|xex} is not a set. Wrong. Please get that logic book you keep saying you'll get. I do really want to bye it, but I can't , I don't have a credit card yet , though I have Amazon certificate, but the book at Amazon is expensive, about 53$ for used ones. I don't know if I can pay the other site which has the 37$ price, by paypal, but even them my paypal account is 0$. Anyhow, I should find a way of transfering money from CashU account to Paypal account, so that I can buy this book. It is a pitty I can't buy this book, At Amazons there is a nice introducation to his book, I do really want to buy it, but I don't know who I can do so. Zuhair Perhaps I am mistaken I don't know. But I want a clear answer about > this. should I add an axiom like this: ~ExAy yex xor ~yex . No. It would make your theory contradict basic predicate logic. MoeBlee === Subject: Re: Hello Everyone-- I need a little help proving 2^n > n^2 for all n > 4 > Could someone give me some hints to continue after: > 2^(k+1) > (k+1)^2 > 2*2^k > k^2 + 2k + 1 >> try applying logarithms to both sides of the inequality > n.log 2 > 2.log n What now? prove that n > 2 (log n)/(log 2) which will always be true for all n > 4 -- Vempire u144241 Games that I like to play Multiplayer Online Games Strategy Games Unification Wars Massive Multiplayer Online Games Galactic Conquest Strategy Games Runescape Kings of chaos === Subject: Re: Periodic solution of differential equation <32336368.1163186749316.JavaMail.jakarta@nitrogen.mathforum.org Did you know that your equation admits trivial depression of order: x'=p(x) pdp/dx=-ax+bxp this equation is integrated explicitly by the separation of variables couldn't find an explicit solution because of the term bxp. Am I missing something here? TIA === Subject: Re: Periodic solution of differential equation <5ad42$4555dc25$82a1bcd4$5273@news2.tudelft.nl > I have a second order differential equation on x(t) whose solution > cannot be expressed in algebraic terms. However, I do know (from > computer simulation) that the solution is periodic. Is there a way to > find out more about the frequency of the solution? > In other words: since x(t) is too complex, can I at least exploit the > fact that x(t) = x (t+T) in order to find T? Any suggestions? > TIA PS: In case you are interested, the equation is x'' = - a x + b x x' The two parameters can be eliminated from the equation (which moves them to > the IC's, which is much more pleasant), by the transformation tt = t / sqrt(a) and > y = b x / sqrt(a) Then you only need to solve one system without free parameters: y'' + y = y y' PS2: Is there a way to prove that the solution is periodic? Yes there is a way, if you can find a region in the phase space where all > trajectories are going out (x dot n > 0 for all x on S) and another region > where all trajectories are inward (x dot n < 0 for all x on S) then you > have a limit cycle. See e.g. Strogatz' nonlinear dynamics and chaos or look > up the Poincare-Bendixson theorem. No limit cycles here. The equivalent first-order system is y' = v v' = y (v - 1) Note the symmetry t -> -t y -> y v -> -v Thus any trajectory starting on the axis y=0 that returns to y=0 forms a closed loop with the reflection of this trajectory across y=0. It is easy to prove that all trajectories in the region v < 1 must eventually return to y=0, thus forming a closed loop, while those in v >= 1 go off to infinity. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Periodic solution of differential equation <5ad42$4555dc25$82a1bcd4$5273@news2.tudelft.nl > I have a second order differential equation on x(t) whose solution > cannot be expressed in algebraic terms. However, I do know (from > computer simulation) that the solution is periodic. Is there a way to > find out more about the frequency of the solution? > In other words: since x(t) is too complex, can I at least exploit the > fact that x(t) = x (t+T) in order to find T? Any suggestions? > TIA PS: In case you are interested, the equation is x'' = - a x + b x x' The two parameters can be eliminated from the equation (which moves them to > the IC's, which is much more pleasant), by the transformation tt = t / sqrt(a) and > y = b x / sqrt(a) Then you only need to solve one system without free parameters: y'' + y = y y' PS2: Is there a way to prove that the solution is periodic? Yes there is a way, if you can find a region in the phase space where all > trajectories are going out (x dot n > 0 for all x on S) and another region > where all trajectories are inward (x dot n < 0 for all x on S) then you > have a limit cycle. See e.g. Strogatz' nonlinear dynamics and chaos or look > up the Poincare-Bendixson theorem. No limit cycles here. The equivalent first-order system is y' = v > v' = y (v - 1) Note the symmetry t -> -t > y -> y > v -> -v Thus any trajectory starting on the axis y=0 that returns to y=0 forms > a closed loop > with the reflection of this trajectory across y=0. It is easy to prove that all trajectories in the region v < 1 must > eventually return to y=0, > thus forming a closed loop, while those in v >= 1 go off to infinity. As for finding the period: with the help of Maple, we get an equation between v and y: v = W(C exp(y^2/2 - 1)) + 1 where W is the Lambert W function and C is an arbitrary constant. If y = y_0 when v = 0, we get C = - exp(-y_0^2/2), so v = W(- exp((y^2 - y0^2)/2 - 1)) + 1 On a closed orbit there will be two such v's for each y, one (for the part of the trajectory with y increasing) corresponding to the principal branch W_0 and the other (for y decreasing) to the -1 branch W_{-1} of the Lambert W function. I then get the following expression for the period: T = int_{-y_0}^{y_0} dy (1/(W_0( - exp((y^2 - y0^2)/2 - 1)) + 1) - 1/(W_{-1}(- exp((y^2 - y0^2)/2 - 1)) + 1)) However, this may not be very helpful for numerical computation as it is because of the nasty singularities at y = (+/-) y_0. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Periodic solution of differential equation >> >> >I have a second order differential equation on x(t) whose solution >>cannot be expressed in algebraic terms. However, I do know (from >>computer simulation) that the solution is periodic. Is there a way to >>find out more about the frequency of the solution? >>In other words: since x(t) is too complex, can I at least exploit the >>fact that x(t) = x (t+T) in order to find T? >>Any suggestions? >>TIA >>PS: In case you are interested, the equation is x'' = - a x + b x x' >>PS2: Is there a way to prove that the solution is periodic? There is a way to prove it's periodic in some cases, if you are lucky. >For example, if you were able to show there was a conserved >quantity like total energy or some such, then you are golden. >It means that it has to come back to x=0 eventually, and when >it does, it has to have the same velocity every time. >> >> >> Huh??? >> >> Consider the differential equation x' = x. It's not too hard >> to show that x'/x is a conserved quantity. Now it follows >> that the solutions to the equation are periodic? This is for 2nd order equations, which can be rewritten as a pair of 1st >order equations: x'=f(x,y) >y'=g(x,y) The idea is to find a conserved quantity F(x,y) whose level sets are >compact, 1 dimensional submanifolds of the plane. Then any solution >must find its way back to a place it started. Well you need some other >hypotheses as well (maybe (f,g) is never zero on the level set, >uniqueness of solutions of the ODE), but you get the idea. we want to use this idea in a _proof_ that the solutions to some DE are periodic we need to state those hypotheses and then verify that they hold... ************************ === Subject: Re: Periodic solution of differential equation > >I have a second order differential equation on x(t) whose solution >cannot be expressed in algebraic terms. However, I do know (from >computer simulation) that the solution is periodic. Is there a way to >find out more about the frequency of the solution? >In other words: since x(t) is too complex, can I at least exploit the >fact that x(t) = x (t+T) in order to find T? Any suggestions? >TIA PS: In case you are interested, the equation is x'' = - a x + b x x' >PS2: Is there a way to prove that the solution is periodic? >>There is a way to prove it's periodic in some cases, if you are lucky. >>For example, if you were able to show there was a conserved >>quantity like total energy or some such, then you are golden. >>It means that it has to come back to x=0 eventually, and when >>it does, it has to have the same velocity every time. >Huh??? Consider the differential equation x' = x. It's not too hard >to show that x'/x is a conserved quantity. Now it follows >that the solutions to the equation are periodic? >>This is for 2nd order equations, which can be rewritten as a pair of 1st >>order equations: >>x'=f(x,y) >>y'=g(x,y) >>The idea is to find a conserved quantity F(x,y) whose level sets are >>compact, 1 dimensional submanifolds of the plane. Then any solution >>must find its way back to a place it started. Well you need some other >>hypotheses as well (maybe (f,g) is never zero on the level set, >>uniqueness of solutions of the ODE), but you get the idea. > we want to use this idea in a _proof_ that the solutions to > some DE are periodic we need to state those hypotheses and > then verify that they hold... Yes. But the poster who made the original comment about finding a conserved quantity was kind of speaking colloquially. He was trying to convey the idea in an inprecise fashion. In a way its a cultural thing - I have found that harmonic analysts like more precision in their language than, say, dynamical systems people - and on the other hand harmonic analysists tend to be less precise in the speech than mathematical logicians. The more precise approach leads to less mistakes, but can also lead to less leaps of the imagination. Stephen === Subject: Re: Periodic solution of differential equation > Contd ...( x - x' ) graph must be a closed loop.This is a phase > portrait. For a circle u^2 + v^2 = const, du/dv = - v/u . So the > simplest among them is hence d x'/ dx = x ''/x' = - const x/x' , or > x'' + const x = 0 , the sine curve set. So every closed loop graph > drawn between axes (u - v) or ( x - x' ) supplies one essential > condition for periodicity. Contd.. My numerical solutions of this problem show by trial and error that the (a,b) graph between oscillatory and divergent motion is a closed loop > containing points (a,b) = (1,-18.3) , ( 0.1, -1.95).i.e., > for a = 1, b > -18.3; for a = 0.1 , b > -1.95. The upper limit of b > has not been found. It will be very helpful if a critirion to establish this boundary loop > between stable and unstable motions is found.It may have to do with > roots of some characteristic equation becoming complex, a determinant > zero etc., but this is just a guess. Whether the solution diverges or is periodic also depends on the initial contitions x(0) and x'(0). === Subject: Re: Periodic solution of differential equation <5ad42$4555dc25$82a1bcd4$5273@news2.tudelft.nl > I have a second order differential equation on x(t) whose solution > cannot be expressed in algebraic terms. However, I do know (from > computer simulation) that the solution is periodic. Is there a way to > find out more about the frequency of the solution? > In other words: since x(t) is too complex, can I at least exploit the > fact that x(t) = x (t+T) in order to find T? Any suggestions? > TIA PS: In case you are interested, the equation is x'' = - a x + b x x' The two parameters can be eliminated from the equation (which moves them to > the IC's, which is much more pleasant), by the transformation tt = t / sqrt(a) and > y = b x / sqrt(a) Then you only need to solve one system without free parameters: y'' + y = y y' PS2: Is there a way to prove that the solution is periodic? Yes there is a way, if you can find a region in the phase space where all > trajectories are going out (x dot n > 0 for all x on S) and another region > where all trajectories are inward (x dot n < 0 for all x on S) then you > have a limit cycle. See e.g. Strogatz' nonlinear dynamics and chaos or look > up the Poincare-Bendixson theorem. Is there some way to find the w = 2 pi f, the circular frequency and the loop graph between (a,b) for stability?