mm-1068 The set of all complex numbers with the usual topology is both separable and 2nd countable. ------------------------------------------------ this is true. because, f : C -> R X R, f(x+yi) = (x,y) so, C and R^2 is homeomorphic. and countable basis of R^2 is {U X V | U=(a,b),V=(c,d), a,b,c,d in Q} so, 2nd countable. so, separable.... um.......right ?? thank you very much for your advice. > ------------------------------------------------ > this is true. > because, f : C -> R X R, f(x+yi) = (x,y) > so, C and R^2 is homeomorphic. > and countable basis of R^2 is > {U X V | U=(a,b),V=(c,d), a,b,c,d in Q} This set is not a basis. but this set is dense in R^2 and countable. Therefore, this set shows that R^2 is separable. Furthermore, Since R^2 is metrizable, R^2 is 2nd countable. (for it's known that a metrizable and separable space is second countable.) > so, 2nd countable. so, separable.... > um.......right ?? > thank you very much for your advice. pointed out. I misreaded the set as something else. mina_world's argument is right. >>------------------------------------------------ >>this is true. >>because, f : C -> R X R, f(x+yi) = (x,y) >>so, C and R^2 is homeomorphic. >>and countable basis of R^2 is >>{U X V | U=(a,b),V=(c,d), a,b,c,d in Q} > This set is not a basis. but this set is dense in R^2 and countable. > Therefore, this set shows that R^2 is separable. > Furthermore, Since R^2 is metrizable, R^2 is 2nd countable. (for it's known > that a metrizable and separable space is second countable.) >>so, 2nd countable. so, separable.... >>um.......right ?? >>thank you very much for your advice. How is this set not a basis? this is the set of all products of open intervals with rational endpoints. if B1 is a basis for X and B2 is a basis for Y then the set {AxB|A is in B1,B is in B2} is a basis for XxY. >>------------------------------------------------ >>this is true. >>because, f : C -> R X R, f(x+yi) = (x,y) >>so, C and R^2 is homeomorphic. >>and countable basis of R^2 is >>{U X V | U=(a,b),V=(c,d), a,b,c,d in Q} > This set is not a basis. but this set is dense in R^2 and countable. > Therefore, this set shows that R^2 is separable. > Furthermore, Since R^2 is metrizable, R^2 is 2nd countable. (for it's known > that a metrizable and separable space is second countable.) Perhaps it's been too long and my recollection is faulty, but if (x,y) is in R^2, and U is an open set of the plane containing (x,y), then isn't there a product (a,b)x(c,d) of intervals with rational endpoints, within U, that contains (x,y)? Surely, if there is a ball around (x,y) with positive radius within U, there is a box with rational vertices as well. Doesn't that count as forming a basis? >>so, 2nd countable. so, separable.... >>um.......right ?? >>thank you very much for your advice. Dale > hello.....doctor~ > The set of all complex numbers with the usual topology > is both separable and 2nd countable. > ------------------------------------------------ > this is true. > because, f : C -> R X R, f(x+yi) = (x,y) > so, C and R^2 is homeomorphic. > and countable basis of R^2 is > {U X V | U=(a,b),V=(c,d), a,b,c,d in Q} > so, 2nd countable. so, separable.... Your proof of second countability is correct. Separability means the existence of a countable dense subset. For R^2, Q^2 naturally embedded is both dense and countable. Igor === Subject: Function naming I'm doing some study on a perticular type of functions (in my spare time just for fun). The problem is however that I don't know how these functions are called in English. Could someone provide me with the correct english term and point me perhaps to some more info on this topic. The functions are of the form; y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 This function has two input variables and has an order of two. The number of variables and order can vary accordingly ofcourse. Is there a general term for this kind of functions? === Subject: Re: Function naming > I'm doing some study on a perticular type of functions (in my spare time > just for fun). The problem is however that I don't know how these functions > are called in English. Could someone provide me with the correct english > term and point me perhaps to some more info on this topic. > The functions are of the form; > y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 > This function has two input variables and has an order of two. The number of > variables and order can vary accordingly ofcourse. Is there a general term > for this kind of functions? Darius, This is a quadratic (what you call order 2 but in English is usually called degree two) polynomial in two variables. If you vary the number of variables (inputs) then they are called polynomials in several variables. It is impossible to tell your level of mathematical education, so it is not clear where to direct you for further information. Achava === Subject: Re: Function naming function group as I described polynomials. I have some knowledge of math, but I was always under the impression that a polynome was defined as a one variable function, and that these were merely a special case of the multivariable case. Probably has to do with my understanding ;-) Achava Nakhash, the Loving Snake schreef in bericht > I'm doing some study on a perticular type of functions (in my spare time > just for fun). The problem is however that I don't know how these functions > are called in English. Could someone provide me with the correct english > term and point me perhaps to some more info on this topic. > The functions are of the form; > y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 > This function has two input variables and has an order of two. The number of > variables and order can vary accordingly ofcourse. Is there a general term > for this kind of functions? > Darius, > This is a quadratic (what you call order 2 but in English is > usually called degree two) polynomial in two variables. If you vary > the number of variables (inputs) then they are called polynomials in > several variables. It is impossible to tell your level of > mathematical education, so it is not clear where to direct you for > further information. > Achava === Subject: Re: Function naming >function group as I described polynomials. I have some knowledge of math, >but I was always under the impression that a polynome was defined as a one >variable function, and that these were merely a special case of the >multivariable case. Probably has to do with my understanding ;-) (Please don't post upside down.) A good resource for definitions is http://mathworld.wolfram.com -- the definition at http://mathworld.wolfram.com/Polynomial.html says A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. I understand that this wouldn't help you when you didn't know the term, but when you do want a definition of a specific term (or facts about a term or concept) it's a great starting point. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com A: Maybe because some people are too annoyed by top-posting. Q: Why do I not get an answer to my question(s)? A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? === Subject: Re: Function naming >function group as I described polynomials. I have some knowledge of math, >but I was always under the impression that a polynome was defined as a one >variable function, and that these were merely a special case of the >multivariable case. Probably has to do with my understanding ;-) >I'm doing some study on a perticular type of functions (in my spare time >just for fun). The problem is however that I don't know how these > >functions >are called in English. Could someone provide me with the correct english >term and point me perhaps to some more info on this topic. >The functions are of the form; >y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 >This function has two input variables and has an order of two. The > >number of >variables and order can vary accordingly ofcourse. Is there a general > >term >for this kind of functions? > To be more specific, I would call it a bivariate quadratic function. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Function naming > function group as I described polynomials. I have some knowledge of math, > but I was always under the impression that a polynome was defined as a one > variable function, and that these were merely a special case of the > multivariable case. Probably has to do with my understanding ;-) You can call them polynomials indeed, but in most cases this is (again indeed) as involving a single variable. To alleviate you could use polynomials with multiple variables for the group of functions. (I have seen the term multinomial used for it, but this creates a different kind of confusion.) -- 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: Function naming > I'm doing some study on a perticular type of functions (in my spare time > just for fun). The problem is however that I don't know how these functions > are called in English. Could someone provide me with the correct english > term and point me perhaps to some more info on this topic. > The functions are of the form; > y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 > This function has two input variables and has an order of two. The number of > variables and order can vary accordingly ofcourse. Is there a general term > for this kind of functions? It's a quadratic (= your order of two) polynomial in two indeterminates (= your input variables). If you don't want to be tied to order and number of indeterminates just call it a polynomial. Actually _as functions_ these things are better called polynomial functions rather than just polynomials, but that may not matter to you. You may also want to Google binary quadratic form. === Subject: Re: Function naming > I don't know how these functions >are called in English. Could someone provide me with the correct english >term and point me perhaps to some more info on this topic. >The functions are of the form; >y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 This particular one is a quadratic in two variables. (The last _three_ terms are all second degree.) >This function has two input variables and has an order of two. The number of >variables and order can vary accordingly ofcourse. Is there a general term >for this kind of functions? More generally, a polynomial of degree N in M variables. degree of 1 -- linear degree of 2 -- quadratic degree of 3 -- cubic degree of 4 -- quartic degree of 5 -- quintic degree of 6 -- sextic etc. But above quadratic the special terms are progressively less used. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com A: Maybe because some people are too annoyed by top-posting. Q: Why do I not get an answer to my question(s)? A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? === Subject: Re: Function naming > I'm doing some study on a perticular type of functions (in my spare time > just for fun). The problem is however that I don't know how these > functions are called in English. Could someone provide me with the correct > english term and point me perhaps to some more info on this topic. > The functions are of the form; > y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 > This function has two input variables and has an order of two. The number > of variables and order can vary accordingly ofcourse. Is there a general > term for this kind of functions? I'd call it quadratic. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Function naming >> I'm doing some study on a particular type of function (in my spare time >> just for fun). The problem is however that I don't know how these >> functions are called in English. Could someone provide me with the >> correct English term and point me perhaps to some more info on >> this topic? >> The functions are of the form; >> y = c0 + c1x1 + c2x2 + c3x1x2 + c4x1^2 + c5x2^2 >> This function has two input variables and has an order of two. >> The number of variables and order can vary accordingly of course. The functions are polynomial functions in several variables. >> Is there a general term for these kind of functions? > I'd call it quadratic. The example you gave is of a quadratic polynomial function, whose graph is a quadric surface. === Subject: Re: prime numbers and 111111..... > Hi there. > When doing a math problem involving numbers with only 1s, like 11 and > 1111111 etc (which could be any base really), i noticed that n digit 1s > divide by m digit 1s if and only if m is a factor of n. > But I guess this is reducable to prime numbers and is not sagnificant. > Martin Johansen Similar to the previous respondent, A number of repeatable digits, 1's, e.g. 11111 in any base b is representable as the sum of n terms of the geometric progression (b^n-1)/(b - 1) = b^0 + b^1 + b^2 + .. b^(n-1) The polynomial 'f(b)n' defined as f(b)n = b^0 + b^1 + b^2 + .. b^(n-1) i.e. such that f(b)n * (b-1) = (b^n-1) is a cyclotomic polynomial, see Wolfram. It has some very interesting factor properties. Most importantly, if n is composite, the polynomial is composite. Furthermore, it will only have factors of the form 2ln+1 for odd n. e.g. if n = 5, 11111 = 41 * 271, both are of the form 2*l*5 + 1, for 41 l = 4, for 271 l = 27 You can see that a number x such as x=11111 is always such that, when n=5, base 10, 2n|(x-1) and you will get to your interesting observation for some n. f(b)n is one hell of an interesting polynomial. Richard Miller === Subject: Re: third order stationarity behavior of random process >The first order stationarity behavior of random process is that all first >order pdf/pmfs are time-independent; >The second order stationarity behavior of random process is that all joint >pdf/pmfs of any two samples are only dependent on the difference of the two >sampling times, i.e., (t1-t2). >>I don't think those are correct. At least they're not consistent with >>the definitions I've seen. For example, in Cox & Miller, The Theory >>of Stochastic Processes: >> ... a process is stationary of order p if all moments up to order p have >> the stationarity property. >> It depends on how one defines moments. If one only uses >> products of the random variables, it is woefully inadequate. >> Moments might not exist; this does not affect stationarity. >Inadequate for what? Are you denying that this is the definition >of stationary of order p in Cox & Miller? Similarly in Bartlett, >An Introduction to Stochastic Processes, section 6.1: > More generally, a process with finite moments will be called stationary > to the rth order if all the moments of order r or less are invariant > under translation. >Or Grimmett and Stirzaker, Probability and Random Processes, sec. 8.2: > X = {X(t): t >= 0} is weakly (or second-order or covariance) stationary if > E(X(t_1)) = E(X(t_2)) > and > cov(X(t_1),X(t_2)) = cov(X(t_1+h),X(t_2+h)) > for all t_1, t_2 and h > 0. The term weakly is very important here. Just as uncorrelated is much weaker than independent, so is covariance stationarity much weaker than stationary. Also, for Gaussian processes, the two are equivalent in both cases, again because joint normal distributions are determined by their moments of order 1 and 2. It is this form of stationarity which is needed for the Loeve-Karhunen representation. Stationary of order k can be defined as above, or as joint distributions at k or fewer time points are preserved under time translation. -- 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: Symbolic solution of quadratic matrix equations I need to find a symbolic solution of a quadratic matrix equation X^2 * A + X * B + A' = 0 where A' is the transpose of (kxk) nonsingular matrix A and B is a (kxk) symmetric matrix. I've found some papers about numeric solution of such kind of matrix equation but I couldn't find anything about the symbolic solution. If somebody helps me, I will appreciate a lot. === Subject: Re: Symbolic solution of quadratic matrix equations >I need to find a symbolic solution of a quadratic matrix equation > X^2 * A + X * B + A' = 0 >where A' is the transpose of (kxk) nonsingular matrix A and B is a >(kxk) symmetric matrix. I've found some papers about numeric solution >of such kind of matrix equation but I couldn't find anything about the >symbolic solution. If somebody helps me, I will appreciate a lot. This is not going to be easy, except in the case where all the matrices commute. You can treat it as a system of k^2 quadratic polynomials in the entries of X, and use try Groebner basis techniques, but I'd guess it's likely to be rather complicated unless k is very small. A 2 x 2 example should work pretty well, but even 3 x 3 might be very ugly. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Symbolic solution of quadratic matrix equations >>I need to find a symbolic solution of a quadratic matrix equation >> X^2 * A + X * B + A' = 0 >>where A' is the transpose of (kxk) nonsingular matrix A and B is a >>(kxk) symmetric matrix. >This is not going to be easy, except in the case >where all the matrices commute. You can treat it >as a system of k^2 quadratic polynomials in the entries >of X, and use try Groebner basis techniques, but I'd guess >it's likely to be rather complicated unless k is very small. >A 2 x 2 example should work pretty well, but even 3 x 3 >might be very ugly. Of course Robert is correct; this is going to be a mess. But it looks like there IS some structure to the answer; I don't have an explanation for it but I offer my findings so someone else can explain the theory making this work. I tried some 2x2 and 3x3 examples, talking to Maple like this: X:=matrix(3,3,[z,y,x,w,v,u,t,s,r]); A:=matrix(3,3,[seq(rand() mod 100, i=1..9)]); B:=matrix(3,3,[seq(rand() mod 100, i=1..9)]); for i to 3 do for j to i-1 do B[i,j]:=B[j,i]:od:od:print(B); evalm(X&*X&*A + X&*B + transpose(A)); eq:={seq(seq(%[i,j],j=1..3),i=1..3)}; lprint(%); and then to Magma like this: Q:=RationalField(); P:=PolynomialRing(Q,9); I:=ideal; Groebner(I); Write(SeeMe,I); The 2x2 case is finished right away; the 3x3 case takes a minute or two. The 3x3 solution looks like this: each of z, y, x, ... can be expressed as a degree-19 polynomial in r with coefficients which are ratios of roughly 150-digit numbers. So there is one solution matrix X for each value of r. And there are 20 possible values of r, namely the roots of a polynomial of degree 20 with integer coefficients. (In the 2x2 case, replace 20 by 6.) The only surprise (to me) is that the polynomial always seems to factor: in the 2x2 case it's the product of a quadratic and a quartic with coefficients in the 7- and 15-digit range, respectively. In the 3x3 case there's a degree-8 factor (60-digit coefficients) and a degree-12 factor (90-digit coefficients). Moreover, these polynomials are special: the quartic has a dihedral Galois group, and the two factors in the 3x3 case have galois groups of order just 48. So it appears there are two types of solutions to the original quadratic equation, and that at least for these low degrees, the different solutions can be obtained by some easy root extractions. I admit I don't understand why there is this much of a pattern to what should be a more random-looking polynomial. dave === Subject: Re: Symbolic solution of quadratic matrix equations In answer to Dave's question below, I believe I can shed a little more light on this. Rewrite the matrix equation as A.X^2 + B.X + C = 0 (1) Let k and v be an eigenvalue and eigenvector of X so X.v = k.v Then multiplying (1) by v gives (A.k^2 + B.k + C).v = 0 (2) so that det(A.k^2 + B.k + C) = 0 (3) This is then a polynomial equation of degree 2.n for the n eigenvalues of X. Now we take n solutions k_i of (3) we can then solve (2) for v_i. Then X is given by the eigen decomposition formula X = V.K.V^-1 where K = diagonal matrix k_i V = matrix of vectors v_i Which n solutions of (3) should be chosen I don't know. Maybe they all give valid solutions, maybe only some of them do ... >>I need to find a symbolic solution of a quadratic matrix equation >> X^2 * A + X * B + A' = 0 >>where A' is the transpose of (kxk) nonsingular matrix A and B is a >>(kxk) symmetric matrix. >This is not going to be easy, except in the case >where all the matrices commute. You can treat it >as a system of k^2 quadratic polynomials in the entries >of X, and use try Groebner basis techniques, but I'd guess >it's likely to be rather complicated unless k is very small. >A 2 x 2 example should work pretty well, but even 3 x 3 >might be very ugly. > Of course Robert is correct; this is going to be a mess. > But it looks like there IS some structure to the answer; > I don't have an explanation for it but I offer my findings > so someone else can explain the theory making this work. > I tried some 2x2 and 3x3 examples, talking to Maple like this: > X:=matrix(3,3,[z,y,x,w,v,u,t,s,r]); > A:=matrix(3,3,[seq(rand() mod 100, i=1..9)]); > B:=matrix(3,3,[seq(rand() mod 100, i=1..9)]); > for i to 3 do for j to i-1 do B[i,j]:=B[j,i]:od:od:print(B); > evalm(X&*X&*A + X&*B + transpose(A)); > eq:={seq(seq(%[i,j],j=1..3),i=1..3)}; lprint(%); > and then to Magma like this: > Q:=RationalField(); > P:=PolynomialRing(Q,9); > I:=ideal; > Groebner(I); > Write(SeeMe,I); > The 2x2 case is finished right away; the 3x3 case takes a minute or two. > The 3x3 solution looks like this: each of z, y, x, ... can be expressed > as a degree-19 polynomial in r with coefficients which are ratios of > roughly 150-digit numbers. So there is one solution matrix X for > each value of r. And there are 20 possible values of r, namely the > roots of a polynomial of degree 20 with integer coefficients. > (In the 2x2 case, replace 20 by 6.) > The only surprise (to me) is that the polynomial always seems to factor: > in the 2x2 case it's the product of a quadratic and a quartic with > coefficients in the 7- and 15-digit range, respectively. > In the 3x3 case there's a degree-8 factor (60-digit coefficients) > and a degree-12 factor (90-digit coefficients). Moreover, these > polynomials are special: the quartic has a dihedral Galois group, > and the two factors in the 3x3 case have galois groups of order just 48. > So it appears there are two types of solutions to the original > quadratic equation, and that at least for these low degrees, > the different solutions can be obtained by some easy root extractions. > I admit I don't understand why there is this much of a pattern to > what should be a more random-looking polynomial. > dave === Subject: Re: basic covering space question at 05:50 PM, nojb@fibertel.com.ar (Nicolas Ojeda Baer) said: >Let X -> B, Y -> B be two covering maps. Then X x_B Y -> B is a >covering map. (X x_B Y is the fiber product of X and Y). I know this >is basic, but I can't seem to get it right. Consider these questions: 1. What is the inverse image of a point in B under X x_B Y -> B? 2. How do you define the topology of X x_B Y? Then apply the definition of a covering map. -- 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 === Subject: Re: In what sense does this vector converge to zero? === >Subject: In what sense does this vector converge to zero? None; convergence is a property of sequences. >Let M be an nxn symmetric positive semidefinite matrix. Suppose v >is an n-tuplet that satisfy 2 conditions (only) in the limit that n >goes to infinity: > v' v -> 1 > v' M v -> 0. The above has no meaning. You need to replace it with statements about sequences and their limits. Presumably you mean that {vn} and {Mn} are sequences such that vn and Mn are an n-tuplet and an n*n symmetric positive semidefinite matrix and vn' vn -> 1 vn' Mn vn -> 0. Similarly, you don't have a definition for w but rather a sequence {wn}, wn=Mn vn. In order to talk about limits you'l need to imbed all of your vectors into a common vector space and specify what topology you're using on it, at which point you can ask whether {wn} has a limit. -- 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 === Subject: Cardinality of an Infinite set, which is smaller than Alef 0? Without the Axiom of Choice, can anyone prove that Alef 0, is the smallest cardinality of an infinite set? Please note: Assuming that for any pair of non empty sets, the smaller set can be mapped one-to-one to a subset of the larger set, is equivalent to the Axiom of Choice. http://math.vanderbilt.edu/~schectex/ccc/choice.html === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? No; is is the only smallest cardinality of an infinite set, in every other infinite cardinality must have a smaller cardinality. But there can be incomparable infinite cardinalities. >Please note: Assuming that for any pair of non empty sets, the >smaller set can be mapped one-to-one to a subset of the larger >set, is equivalent to the Axiom of Choice. That is part of the definition. What is equivalent ot the Axiom of Choice is that of two sets, on is larger than the other. -- 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: Cardinality of an Infinite set, which is smaller than Alef 0? > Without the Axiom of Choice, can anyone prove that Alef 0, is the > smallest cardinality of an infinite set? It depends what you mean. On the one hand if a set A has cardinality less than or equal to aleph_0, then by definition there is an injection f:A -> omega. This means A is wellorderable. If it is also infinite it must have cardinality aleph_0. On the other hand without AC there may be sets which are not comparable with aleph_0 in cardinality. So aleph_0 is minimal among infinite cardinals but not necessarily the minimum without AC. It is the minimum among the cardinals of infinite wellorderable sets. === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? One does not need AC to prove the following: Any subset of a countably infinite set is countable. However, without the axiom of choice, one cannot prove that an infinite set necessarily has a countably infinite subset. A set is Dedekind infinite if and only if it has a countably infinite subset, and it consistent with ZF+not AC that there exist infinite Dedekind finite sets. Thus, countably infinite is a minimal infinite cardinality, but without AC it is not necessarily the only one. Note that aleph0 has a very precise definition to many authors: It is the set of all finite ordinals; the axiom of infinity guarantees its existence. Thus, one does not need the axiom of choice to show that if an ordinal (or cardinal) a < aleph0, then a is finite. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? > One does not need AC to prove the following: Any subset of a countably > infinite set is countable. > However, without the axiom of choice, one cannot prove that an infinite > set necessarily has a countably infinite subset. A set is Dedekind > infinite if and only if it has a countably infinite subset, and it > consistent with ZF+not AC that there exist infinite Dedekind finite sets. I have always been massively confused by this, and I figure this is as good a time to ask as any, since it has been explicitly stated here. The obvious question is, why can't I just pick a countable series of elements from the infinite set? What is it that's preventing me from doing that? Let A be the infinite set, then we know that there exists a member x of the set. Put that x into a subset S. Then we know that there exists a y in A such that y != x, else it would be finite. Put that into a subset S. Continuing, finding, for instance, a z such that z != x, y, you have a countable set. Is it the statements we know that there exists a member ... of the set A that prevent this construction? > Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? > One does not need AC to prove the following: Any subset of a countably > infinite set is countable. > However, without the axiom of choice, one cannot prove that an infinite > set necessarily has a countably infinite subset. A set is Dedekind > infinite if and only if it has a countably infinite subset, and it > consistent with ZF+not AC that there exist infinite Dedekind finite > sets. > I have always been massively confused by this, and I figure this is as good > a time to ask as any, since it has been explicitly stated here. The obvious > question is, why can't I just pick a countable series of elements from the > infinite set? What is it that's preventing me from doing that? > Let A be the infinite set, then we know that there exists a member x of the > set. Put that x into a subset S. Then we know that there exists a y in A > such that y != x, else it would be finite. Put that into a subset S. > Continuing, finding, for instance, a z such that z != x, y, you have a > countable set. > Is it the statements we know that there exists a member ... of the set A > that prevent this construction? > Stephen J. Herschkorn sjherschko@netscape.net shows that this construction is impossible without the axiom of choice? === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >>Without the Axiom of Choice, can anyone prove that Alef 0, is the >>smallest cardinality of an infinite set? ..................... >> Let A be the infinite set, then we know that there exists a member x of >the >> set. Put that x into a subset S. Then we know that there exists a y in A >> such that y != x, else it would be finite. Put that into a subset S. >> Continuing, finding, for instance, a z such that z != x, y, you have a >> countable set. >> Is it the statements we know that there exists a member ... of the set A >> that prevent this construction? >shows that this construction is impossible without the axiom of choice? It at most requires the countable axiom of choice, and I believe it is weaker than that. -- 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: Cardinality of an Infinite set, which is smaller than Alef 0? |shows that this construction is impossible without the axiom of choice? When Paul Cohen proved that if ZF is consistent, then so is ZF+the axiom of choice is false, along with that he also proved that if ZF is consistent then so is ZF+there exists a set S which is infinite but has no countably infinite subset. The proof is in his famous book if you're curious. Here the definition of infinite is that S is infinite if there is a family F of subsets of S that contains the empty set, and such that if X is in F and s is in S, then the union of X with {s} is in F too. This corresponds to the usual definition of finite which is most closely related to the principle of mathematical induction on the natural numbers. Sometimes authors give a different definition of infinite which is called Dedekind infinite which is equivalent to containing a countably infinite subset (S is called Dedekind infinite if there exists a 1-1 correspondence between S and a proper subset of S.) Assuming the axiom of choice the two definitions are equivalent. While not assuming the axiom of choice I believe the first definition I gave is preferred. Keith Ramsay === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >> >However, without the axiom of choice, one cannot prove that an infinite >set necessarily has a countably infinite subset. A set is Dedekind >infinite if and only if it has a countably infinite subset, and it >consistent with ZF+not AC that there exist infinite Dedekind finite > >>sets. >> >>I have always been massively confused by this, and I figure this is as >> >good >>a time to ask as any, since it has been explicitly stated here. The >> >obvious >>question is, why can't I just pick a countable series of elements from the >>infinite set? What is it that's preventing me from doing that? >> >shows that this construction is impossible without the axiom of choice? start, though an explicit model of ZF where there exists a Dedekind set is not given there. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >>However, without the axiom of choice, one cannot prove that an infinite >>set necessarily has a countably infinite subset. A set is Dedekind >>infinite if and only if it has a countably infinite subset, and it >>consistent with ZF+not AC that there exist infinite Dedekind finite >> >sets. >I have always been massively confused by this, and I figure this is as good >a time to ask as any, since it has been explicitly stated here. The obvious >question is, why can't I just pick a countable series of elements from the >infinite set? What is it that's preventing me from doing that? >Let A be the infinite set, then we know that there exists a member x of the >set. Put that x into a subset S. Then we know that there exists a y in A >such that y != x, else it would be finite. Put that into a subset S. >Continuing, finding, for instance, a z such that z != x, y, you have a >countable set. >Is it the statements we know that there exists a member ... of the set A >that prevent this construction? Logical conjunction is a binary operation, so you can use induction to form a subset of any arbitrary *finite* size. Without the axiom of choice, a problem arises here because you want to conjunct infinitely many times. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? ... > However, without the axiom of choice, one cannot prove that an infinite > set necessarily has a countably infinite subset. A set is Dedekind > infinite if and only if it has a countably infinite subset, and it > consistent with ZF+not AC that there exist infinite Dedekind finite > sets. > I have always been massively confused by this, and I figure this is as good > a time to ask as any, since it has been explicitly stated here. The obvious > question is, why can't I just pick a countable series of elements from the > infinite set? What is it that's preventing me from doing that? My gut feeling is that it is the axiom of choice that allows you to do it. If there is no axiom of choice you can not pick such a countable series of elements. -- 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: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? > One does not need AC to prove the following: Any subset of a countably > infinite set is countable. > However, without the axiom of choice, one cannot prove that an infinite > set necessarily has a countably infinite subset. A set is Dedekind > infinite if and only if it has a countably infinite subset, and it > consistent with ZF+not AC that there exist infinite Dedekind finite > sets. > I have always been massively confused by this, and I figure this is as good > a time to ask as any, since it has been explicitly stated here. The obvious > question is, why can't I just pick a countable series of elements from the > infinite set? What is it that's preventing me from doing that? > Let A be the infinite set, then we know that there exists a member x of the > set. Put that x into a subset S. Then we know that there exists a y in A > such that y != x, else it would be finite. Put that into a subset S. > Continuing, finding, for instance, a z such that z != x, y, you have a > countable set. > Is it the statements we know that there exists a member ... of the set A > that prevent this construction? > Stephen J. Herschkorn sjherschko@netscape.net The problem is that to choose these countably many elements, you use at least the countable AC. In various (topos) models of set theory, you can even have infinite (that is non-finite) subsets of finite sets. In fact, there are a number of distinct definitions of finite. A regular morass, that disappears under AC. === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >Without the Axiom of Choice, can anyone prove that Alef 0, is the >smallest cardinality of an infinite set? > One does not need AC to prove the following: Any subset of a countably > infinite set is countable. > However, without the axiom of choice, one cannot prove that an infinite > set necessarily has a countably infinite subset. A set is Dedekind > infinite if and only if it has a countably infinite subset, and it > consistent with ZF+not AC that there exist infinite Dedekind finite > sets. > I have always been massively confused by this, and I figure this is as good > a time to ask as any, since it has been explicitly stated here. The obvious > question is, why can't I just pick a countable series of elements from the > infinite set? What is it that's preventing me from doing that? > Let A be the infinite set, then we know that there exists a member x of the > set. Put that x into a subset S. Then we know that there exists a y in A > such that y != x, else it would be finite. Put that into a subset S. > Continuing, finding, for instance, a z such that z != x, y, you have a > countable set. > Is it the statements we know that there exists a member ... of the set A > that prevent this construction? > Stephen J. Herschkorn sjherschko@netscape.net You're making infinitely many choices. Basically, you are assuming a choice function for the nonempty subsets of A. === Subject: Re: Cardinality of an Infinite set, which is smaller than Alef 0? >>Without the Axiom of Choice, can anyone prove that Alef 0, is the >>smallest cardinality of an infinite set? >> One does not need AC to prove the following: Any subset of a countably >> infinite set is countable. >> However, without the axiom of choice, one cannot prove that an infinite >> set necessarily has a countably infinite subset. A set is Dedekind >> infinite if and only if it has a countably infinite subset, and it >> consistent with ZF+not AC that there exist infinite Dedekind finite >sets. >I have always been massively confused by this, and I figure this is as good >a time to ask as any, since it has been explicitly stated here. The obvious >question is, why can't I just pick a countable series of elements from the >infinite set? What is it that's preventing me from doing that? >Let A be the infinite set, then we know that there exists a member x of the >set. Put that x into a subset S. Then we know that there exists a y in A >such that y != x, else it would be finite. Put that into a subset S. >Continuing, finding, for instance, a z such that z != x, y, you have a >countable set. So how do you keep choosing the elements? You can choose a finite number of them, but ... . >Is it the statements we know that there exists a member ... of the set A >that prevent this construction? >> Stephen J. Herschkorn sjherschko@netscape.net -- 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: Determinants and commuting matrices (old Hoffman and Kunze Exercise) The last exercise in section 5.4 of Hoffman and Kunze's Linear Algebra asks the reader to prove that, given 4 commuting n x n matrices A, B, C, and D, the determinant of the 2n x 2n matrix A B C D is given by det(AD - BC). I'm feeling dense, but I don't see why this should be. Any insight would be appreciated... === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > The last exercise in section 5.4 of Hoffman and Kunze's Linear Algebra > asks the reader to prove that, given 4 commuting n x n matrices A, B, C, > and D, the determinant of the 2n x 2n matrix > A B > C D > is given by det(AD - BC). > I'm feeling dense, but I don't see why this should be. Any insight would > be appreciated... If they all commute (including AD-BC), then they have the same invariant subspaces. Try to find a basis where the matrix [ A, B; C, D ] becomes block-diagonal. The determinant of a block diagonal matrix is the product of determinants of each block. Hope this helps. Igor === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) Perhaps it can be done a bit more directly. det ( A, B ; C, D ) = det(A)*det(C)*det(I,A^-1*B; I, C^-1*D) = det (A)*det(C)*det(C^-1*D - A^-1*B) = det(A) * det( D - A^-1 * B * C ) = det(AD - BC) det-ails of why commutativity justifies these steps left to the reader. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > Perhaps it can be done a bit more directly. > det ( A, B ; C, D ) = det(A)*det(C)*det(I,A^-1*B; I, C^-1*D) I don't see how what's below follows from what's above. How do you reduce the determinant of a 2nx2n matrix to the determinant of an nxn matirx? You also get into trouble if A or C is not invertible. Igor > = det (A)*det(C)*det(C^-1*D - A^-1*B) > = det(A) * det( D - A^-1 * B * C ) > = det(AD - BC) > det-ails of why commutativity justifies these steps left to the reader. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) Apologies if this turns into a double-post; Google Groups 2 beta kicked back the first attempt. det( I, M; I, N) = det( I, M; 0, N-M) Subtract the upper rows from the lower ones. To generalize slightly what you pointed out, determinant of a block triangular matrix is the product of the determinants of the diagonal blocks. Yes, you get in trouble if A or C is not invertible, but this can be handled by a generic perturbation argument. Add a small multiple of the identity matrix, sufficient to avoid the singularity, then take the limit as the multiple tends to zero. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) I M I N Subtract the upper rows from the lower ones. Determinant of a block triangular matrix is .... Yes, you get in trouble if A or C is not invertible, but this can be handled by a generic perturbation argument. Add a small multiple of the identity matrix, sufficient to avoid the singularity, then take the limit as the multiple tends to zero. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > The last exercise in section 5.4 of Hoffman and Kunze's Linear Algebra > asks the reader to prove that, given 4 commuting n x n matrices A, B, > C, and D, the determinant of the 2n x 2n matrix > A B > C D > is given by det(AD - BC). > I'm feeling dense, but I don't see why this should be. Any insight would be appreciated... If you had a block matrix of the form | P Q | =M | Z R | where Z is the appropriate sized zero matrix, then any term of the det of M expanded about the 1st column of P must contain a zero term if it contains an element or elements of Q. So DetM comprises priducts of the form product n elements of P times product n elements of R i.e. Det(P)Det(R) = Det(PR). If you multiply |A B| by | I -(A^(-1)*B | |C D| | Z I | The determinant of the 2nx2n matrix is unaltered. The product equals | A Z | | C D - CA^(-1)B| So Det(M) = det(A)Det(D - CA^(-1)B = Det(AD -AC A^(-1)B) = Det(AD -CB) for matrices that commute. This assumes that A is invertible but if both A and D were not invertible Det(M)=0 I don't know if this would be the approach envisaged for the excercise === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > This assumes that A is invertible but if > both A and D were not invertible Det(M)=0 Consider A = D = 0, B = C = I. I prefer a perturbation argument to overcome singularity of (for example) A. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > This assumes that A is invertible but if > both A and D were not invertible Det(M)=0 > Consider A = D = 0, B = C = I. Yes, but you are still assuming that the matrices on a diagonal are invertible. You would then swap columns or choose a different multiplying matrix. In any case, it seems that you would have to assume that at least one matrix was non-singualar. > I prefer a perturbation argument to overcome singularity of (for > example) A. Yes, but I suppose you are going beyond elementary methods. I am not familiar with the textbook from which the excercise was taken. Knowing what topics had been covered might suggest how the problem should be solved using commutivity alone or with a peturbation argument if that had been covvered. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) > This assumes that A is invertible but if > both A and D were not invertible Det(M)=0 > Consider A = D = 0, B = C = I. Yes, but you are still assuming that the matrices on a diagonal are invertible. You would then swap columns or choose a different multiplying matrix. In any case, it seems that you would have to assume that at least one matrix was non-singualar. > I prefer a perturbation argument to overcome singularity of (for > example) A. Yes, but I suppose you are going beyond elementary methods. I am not familiar with the textbook from which the excercise was taken. Knowing what topics had been covered might suggest how the problem should be solved using commutivity alone or with a peturbation argument if that had been covvered. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) Just to clarify what tools a reader of the textbook might reasonably bring to bear on the exercise, perturbation arguments and even invariant subspaces have not yet been introduced. Topics covered thus far include elementary row operations, vector spaces, bases, and the idea of dimension, linear transformations and their representation by matrices, dual spaces and dual bases, elementary polynomial algebra (ideals of polynomials and some simple results depending therefrom), and determinants of matrices. No canonical forms of any sort have been discussed, though similarity insofar as it relates to changes of bases has been covered. My suspicion, after reading through this thread, is that the solution they're looking for involves something along the lines of first assuming that both A and D are invertible, proving the result based on something similar I.M. Davidson's line of reasoning, and then showing that it still holds even if A and/or D are both singular. === Subject: Re: Determinants and commuting matrices (old Hoffman and Kunze Exercise) >> This assumes that A is invertible but if >> both A and D were not invertible Det(M)=0 >> Consider A = D = 0, B = C = I. >Yes, but you are still assuming that >the matrices on a diagonal are invertible. >You would then swap columns or choose a >different multiplying matrix. >In any case, it seems that you would have to assume >that at least one matrix was non-singualar. >> I prefer a perturbation argument to overcome >singularity of (for >> example) A. >Yes, but I suppose you are going beyond >elementary methods. Not really. The determinant of a matrix being a polynomial in its elements is continuous. As the determinant of A+tI is a polynomial in t, it must be non-singular for all non-zero t which are sufficiently small, and hence A+tI is non-singular. Now pass to the limit as t -> 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: Is x^n - x^{n - 1} - ... - x - 1 irreducible? >For n = 1, 2, ... let f_n(x) = x^n - x^{n - 1} - ... - x - 1 >= (x^{n + 1} - 2x^n + 1)/(x - 1). Is f_n always irreducible >over Z? Leafing further through my sheaf of photocopies (I've a bad habit of collecting stuff and not reading it), I found that this has been posed and solved as an American Mathematical Monthly problem: Elementary Problem E 3008 (Vol. 90, No. 7, Aug. - Sep., 1983) Solution of Elementary Problem E 3008 (Vol. 96, No. 2, Feb., 1989) Editorial comment. M. J. DeLeon noted that the result of the problem is contained in the following theorem of Alfred Brauer: if a_1, a_2, ..., a_n are integers with a_1 >= a_2 >= ... > = a_n > 0, then the polynomial x^n - a_1x^{n - 1} - a_2x^{n - 2} - ... - a_n is irreducible over the irrationals. Cf. Alfred Brauer, On algebraic equations with all but one root in the interior of the unit circle, Math. Nachrichten, 4 (1950/51), 250--257. The positive root of the polynomial considered in this problem (or more generally of the polynomial considered in Brauer's Theorem) is an example of what is known as a PV-number, which is a real algebraic integer greater than 1 all of whose conjugates lie inside the unit circle. Cf. J. W. S. Cassells, _An Introduction to Diophantine Approx- imation_, Cambridge University Press, 1957, Chapter VIII. -- Angus Rodgers Contains mild peril === Subject: Re: Is x^n - x^{n - 1} - ... - x - 1 irreducible? > is irreducible over the irrationals. a misprint, right? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Is x^n - x^{n - 1} - ... - x - 1 irreducible? >> is irreducible over the irrationals. >a misprint, right? I have discovered a whole new field of mathematics, disproving the ravings of those upstarts Galois and Cantor. Bow down and worship me! Woe to those who dare to suggest that my Word is not Law! Death to the infidel Edgar! -- Angus Rodgers Contains mild peril === Subject: Re: Is x^n - x^{n - 1} - ... - x - 1 irreducible? >> For n = 1, 2, ... let f_n(x) = x^n - x^{n - 1} - ... - x - 1 >> = (x^{n + 1} - 2x^n + 1)/(x - 1). Is f_n always irreducible over Z? > Before anyone wastes too much time on this: the answer is yes > (although I don't know why). > Ernst S. Selmer, On the irreducibility of certain trinomials, > Math. Scand. 4 (1956), 287--302. > I actually had a photocopy of this paper lying around (because > of a question on irreducibility that came up in sci.math a few > years back), but hadn't got around to reading it. > An important criterion, typical of one approach to the problem, > is given by Perron [8]: The polynomial (with integer coefficients) > x^n + a_1x^{n - 1} + a_2x^{n - 2} + ... + a_{n - 1}x + a_n > is irreducible if > |a_1| > 1 + |a_2| + |a_3| + ... + |a_n|. > Applied to f(x) = x^n + ax +- 1, where we substitute x = 1/z, > this shows that f(x) is irreducible for |a| >= 3. When |a| = > 2 and f(+-1) [is not equal to] 0, we can still conclude > irreducibility according to Perron. When |a| = 2 and f(x) has > a rational factor x + 1 or x - 1, the second factor of f(x) > will be irreducible (this is not contained among Perron's > statements, but follows easily from his method). > I don't have access to Perron's paper, but the reference is: > 8. O. Perron, Neue Kriterien fuer die Irreduzibilitat > algebraischer Gleichungen, J. reine angew. Math. 132 (1907), > 288--307. This was discussed on sci.math.research on 1998/03/28, see [1]. For an extensive treatment of irreducibility criteria see Filaseta's online book [2]. --Bill Dubuque [2] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/latexbook/ === Subject: Re: Is x^n - x^{n - 1} - ... - x - 1 irreducible? Filaseta's online book [2]. > [2] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/latexbook/ I wasn't able to access this. -- 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: Is x^n - x^{n - 1} - ... - x - 1 irreducible? >> For an extensive treatment of irreducibility criteria see >> Filaseta's online book [2]. >> [2] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/latexbook/ > I wasn't able to access this. In the past it was accessible without a password, so perhaps if you email Filaseta he'll tell you how to access it. --Bill Dubuque === Subject: Re: Is x^n - x^{n - 1} - ... - x - 1 irreducible? > Ernst S. Selmer, On the irreducibility of certain trinomials, > Math. Scand. 4 (1956), 287--302. > An important criterion, typical of one approach to the problem, > is given by Perron [8]: The polynomial (with integer coefficients) > x^n + a_1x^{n - 1} + a_2x^{n - 2} + ... + a_{n - 1}x + a_n > is irreducible if > |a_1| > 1 + |a_2| + |a_3| + ... + |a_n|. >[...] I don't have access to Perron's paper [...] ... but of course there's Google: (Both references note the necessity of the additional condition that a_n is nonzero.) -- Angus Rodgers Contains mild peril === Subject: JSH: Being me I know, many of you probably think it's horrible being me, with all these people calling me name, putting up nasty webpages, and spending so much time talking bad about me. Some of you are in your own little world and maybe still think I'm wrong, while some of you realize that I'm right and STILL wouldn't want to be me considering how much opposition my results have faced and are likely to face. But hey, it's actually fun! Like I haven't just done arguing with people on math or even just math research as I have an open source project on SourceForge, and I have made some friends (believe it or not) in a few Internet communities. I *thought* I could use the Internet in a groundbreaking way to introduce major results but here I am WAITING ON A JOURNAL, when I said in the past that I wouldn't even use journals! :-) Live and learn. In any event, I also get to go over my own mathematical work, and enjoy talking about it as you might have noticed with me starting new threads--yet again--going over the arguments in different ways. Different looks. Perspective. But, on to something practical, as I've been looking at BitTorrent clients while I'm doing downloads, and I see the download speeds hopping all over the place as the clients--I guess--use various algorithms to try and figure out what's the best connection to make. Anyone here know anything about those algorithms? I'm thinking maybe I might make my own BitTorrent client, or go in and fiddle with the algorithms on the one I have as I have Azureus. But it's an idle thought, so why not toss it on sci.math? Any of you know anything about the algorithms being used in BitTorrent clients? James Harris === Subject: Re: JSH: Being me > I know, many of you probably think it's horrible being me, with all > these people calling me name, putting up nasty webpages, and spending > so much time talking bad about me. I have to be honest, if being you is horrible, those are not the causes I would have listed for it. You receive very little name-calling, the websites mainly say you are obstinately wrong. I've seen people treated far worse in the public schools. You've got nothing on some of the politicians out there. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: Being me >I know, many of you probably think it's horrible being me, with all >these people calling me name, putting up nasty webpages, and spending >so much time talking bad about me. Why in the world would you think anyone cares? And just out of curiosity, who are you adressing here? This happens every once in a while, you seem to be talking to all your other readers, the ones who are not part of the vast conspiracy. What makes you think that they exist? >Some of you are in your own little world and maybe still think I'm >wrong, while some of you realize that I'm right and STILL wouldn't >want to be me considering how much opposition my results have faced >and are likely to face. Like, what gives you the idea that there are people out there who think you're right? Nobody ever _says_ they think you're right. >But hey, it's actually fun! >Like I haven't just done arguing with people on math or even just math >research as I have an open source project on SourceForge, and I have >made some friends (believe it or not) in a few Internet communities. That _is_ hard to believe. Why do you keep coming back here? >I *thought* I could use the Internet in a groundbreaking way to >introduce major results but here I am WAITING ON A JOURNAL, when I >said in the past that I wouldn't even use journals! You've also said in the past that the internet didn't matter, all that mattered was the fact that you were about to be published in a journal. Given just about anything you've ever said, you've said something directly contraditory in the past. ************************ David C. Ullrich === Subject: Re: JSH: Being me > Like, what gives you the idea that there are people out there > who think you're right? Nobody ever _says_ they think you're > right. Wasn't there that guy from the genius club? What ever happened to him? === Subject: Re: JSH: Being me > Like, what gives you the idea that there are people out there > who think you're right? Nobody ever _says_ they think you're > right. > Wasn't there that guy from the genius club? What ever happened to him? I don't recall Quinn Tyler-Jackson ever saying Harris was right, he objected to how Harris was treated. I don't recall A. Beckwith ever saying Harris was right, he just tried to steal the credit for his paper. === Subject: Re: JSH: Being me Discussion, linux) > Like, what gives you the idea that there are people out there > who think you're right? Nobody ever _says_ they think you're > right. Just think. Sci.math has *lots* of readers. What are the odds that they're *all* wrong about James's work? -- 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: JSH: Being me > Like, what gives you the idea that there are people out there > who think you're right? Nobody ever _says_ they think you're > right. > Just think. > Sci.math has *lots* of readers. What are the odds that they're *all* > wrong about James's work? Well, I think sci.math readers are stupid. Besides, I do remind you that I'm just here goofing off, while I wait. It makes me no never mind to play with you people and see just what I can see. I trace out your neural pathways this way. Like, tomorrow I will come to see who replies to this post, and read information bounced around in their heads. I do this enough that I can build a map, and a model, then I simply test the model. If it fails to respond as you would, then I test again, until it responds as you do. So I build a world and then I can simply test that world, moving the pieces in it as I see fit. And then I know how you will move, as I see fit. I am mostly done. James Harris === Subject: Re: JSH: Being me > Well, I think sci.math readers are stupid. > Besides, I do remind you that I'm just here goofing off, while I wait. > It makes me no never mind to play with you people and see just what I > can see. > I trace out your neural pathways this way. > Like, tomorrow I will come to see who replies to this post, and read > information bounced around in their heads. > I do this enough that I can build a map, and a model, then I simply > test the model. > If it fails to respond as you would, then I test again, until it > responds as you do. > So I build a world and then I can simply test that world, moving the > pieces in it as I see fit. And then I know how you will move, as I > see fit. > I am mostly done. > James Harris It appears you have managed to elevate your delusions of grandeur to delusions of divinity. Keep it up. There are special places for you -- even here on earth! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: JSH: Being me > Like, what gives you the idea that there are people out there > who think you're right? Nobody ever _says_ they think you're > right. > Just think. > Sci.math has *lots* of readers. What are the odds that they're *all* > wrong about James's work? > Well, I think sci.math readers are stupid. You read sci.math, right? === Subject: Re: JSH: Being me > I know, many of you probably think it's horrible being me, with all > these people calling me name, putting up nasty webpages, and spending > so much time talking bad about me. The unswerving self-belief (in the face of a mountain of contradictory evidence) must be some comfort, I imagine. > Some of you are in your own little world Project much? > and maybe still think I'm > wrong, while some of you realize that I'm right and STILL wouldn't > want to be me considering how much opposition my results have faced > and are likely to face. > But hey, it's actually fun! Must have an almost narcotic effect, that level of self-belief. > Like I haven't just done arguing with people on math or even just math > research as I have an open source project on SourceForge, and I have > made some friends (believe it or not) in a few Internet communities. You should be careful making friends on the Internet James, there's a lot of weirdos out there. > I *thought* I could use the Internet in a groundbreaking way to > introduce major results but here I am WAITING ON A JOURNAL, when I > said in the past that I wouldn't even use journals! You'll be waiting a while, I'd wager. > :-) > Live and learn. If only you would! > In any event, I also get to go over my own mathematical work, and > enjoy talking about it as you might have noticed with me starting new > threads--yet again--going over the arguments in different ways. You? Starting new threds? Can't say I'd spotted any, no... > Different looks. Perspective. > But, on to something practical, as I've been looking at BitTorrent > clients while I'm doing downloads, and I see the download speeds > hopping all over the place as the clients--I guess--use various > algorithms to try and figure out what's the best connection to make. > Anyone here know anything about those algorithms? I'm thinking maybe > I might make my own BitTorrent client, or go in and fiddle with the > algorithms on the one I have as I have Azureus. But it's an idle > thought, so why not toss it on sci.math? > Any of you know anything about the algorithms being used in BitTorrent > clients? You have Internet access, learn how to use it. Oh wait, I forgot, you're pathologically incapacle of learning. -- Larry Lard Replies to group please === Subject: Re: JSH: Being me > I know, many of you probably think it's horrible being me, > But hey, it's actually fun! Would you describe it as . . . bliss? === Subject: Re: JSH: Being me > But, on to something practical, as I've been looking at BitTorrent > clients while I'm doing downloads, and I see the download speeds > hopping all over the place as the clients--I guess--use various > algorithms to try and figure out what's the best connection to make. > Anyone here know anything about those algorithms? I'm thinking maybe > I might make my own BitTorrent client, or go in and fiddle with the > algorithms on the one I have as I have Azureus. But it's an idle > thought, so why not toss it on sci.math? > Any of you know anything about the algorithms being used in BitTorrent > clients? Sure. What do you want to know? It's a much more naive algorithm than you seem to think it is. http://www.bittorrent.com/protocol.html 'cid 'ooh === Subject: Re: JSH: Being me jstevh@msn.com says... > I know, many of you probably think it's horrible being me, with all > these people calling me name, putting up nasty webpages, and spending > so much time talking bad about me. > Some of you are in your own little world and maybe still think I'm > wrong, while some of you realize that I'm right and STILL wouldn't > want to be me considering how much opposition my results have faced > and are likely to face. > But hey, it's actually fun! > Like I haven't just done arguing with people on math or even just math > research as I have an open source project on SourceForge, and I have > made some friends (believe it or not) in a few Internet communities. > I *thought* I could use the Internet in a groundbreaking way to > introduce major results but here I am WAITING ON A JOURNAL, when I > said in the past that I wouldn't even use journals! > :-) > Live and learn. > In any event, I also get to go over my own mathematical work, and > enjoy talking about it as you might have noticed with me starting new > threads--yet again--going over the arguments in different ways. > Different looks. Perspective. > But, on to something practical, as I've been looking at BitTorrent > clients while I'm doing downloads, and I see the download speeds > hopping all over the place as the clients--I guess--use various > algorithms to try and figure out what's the best connection to make. > Anyone here know anything about those algorithms? I'm thinking maybe > I might make my own BitTorrent client, or go in and fiddle with the > algorithms on the one I have as I have Azureus. But it's an idle > thought, so why not toss it on sci.math? > Any of you know anything about the algorithms being used in BitTorrent > clients? > James Harris Bi-Polar much? === Subject: Re: JSH: Being me Frightening thought. === Subject: Re: JSH: Being me > Frightening thought. Must be hell in there. Dirk Vdm === Subject: Simple Group Theory Question - AGAIN I can't seem to find any way to solve these problems! Let G be a cyclic group of order p when p is prime. Find all the automorphisims of G. === Subject: Re: Simple Group Theory Question - AGAIN >I can't seem to find any way to solve these problems! >Let G be a cyclic group of order p when p is prime. >Find all the automorphisims of G. First, can you tell us which elements of G _generate_ G? ************************ David C. Ullrich === Subject: Re: Simple Group Theory Question - AGAIN > I can't seem to find any way to solve these problems! > Let G be a cyclic group of order p when p is prime. > Find all the automorphisims of G. An automorphism f:G -> G is a homomorphism which maps generators to generators. So fix a generator g so that G = . Since p is a prime, g |-> g^n is an automorphism for 0=floor(e^x) So n(1)=2, n(2)=5, n(3)=9, n(4)=10, ect. What is the maximum value of n(x) on the interval (1,2)? At what value(s) of x is n(x) largest? Rich === Subject: Re: A question about floor(e^x) >Let n(x) be the least integer such that: >1+x+x^2/2+...+x^n/n!>=floor(e^x) >So n(1)=2, n(2)=5, n(3)=9, n(4)=10, ect. I think you're off by 1 on all of these, e.g. floor(e^3) = 20 and sum_{j=0}^8 3^j/j! > 20. >What is the maximum value of n(x) on the interval (1,2)? At what value(s) of x >is n(x) largest? n(ln(k)) does not exist if k is an integer > 1, and n(x) -> infinity as x -> ln(k)+. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: A question about floor(e^x) >>Let n(x) be the least integer such that: >>1+x+x^2/2+...+x^n/n!>=floor(e^x) >>So n(1)=2, n(2)=5, n(3)=9, n(4)=10, ect. >I think you're off by 1 on all of these, e.g. >floor(e^3) = 20 and sum_{j=0}^8 3^j/j! > 20. Yes. I was thinking of n(x) as the number of terms of e^x needed to compute floor(e^x) and did not make the adjustment when I posted the (slightly) different question. >>What is the maximum value of n(x) on the interval (1,2)? At what value(s) >of x >>is n(x) largest? >n(ln(k)) does not exist if k is an integer > 1, and n(x) -> infinity >as x -> ln(k)+. Rich === Subject: Re: A question about floor(e^x) > Let n(x) be the least integer such that: > 1+x+x^2/2+...+x^n/n!>=floor(e^x) > So n(1)=2, n(2)=5, n(3)=9, n(4)=10, ect. > What is the maximum value of n(x) on the interval (1,2)? At what value(s) of x > is n(x) largest? How about x = ln(3)? Asger. === Subject: adjoining elements to rings What happens when we adjoin the inverse of 2 to the ring Z_12 i.e. what is the field Z_12[x]/(2x-1) isomorphic to? I have reason to believe that it is simply the field of 3 elements, because of the following: In Z_12[x]/(2x-1) we have that 12=0 and 2x-1=0. Therefore 12x-6=0 ==> 6=0. Similarly we have 6x-3=0 ==> 3=0. This makes me believe that we are talking about F_3 here. However, I cannot prove this. I tried setting up the following maps: j:Z ---> Z_12 pi:Z_12 ---> Z_12[x]/(2x-1) i:Z ---> Z_3 If ker i = ker (pi o j) then we have that Z_12[x]/(2x-1) is isomorphic to Z_3 by the first isomorphism theorem. I know that ker j = 12Z ker pi = (2x-1) ker i = 3Z But for some reason I cannot compute the kernel of pi o j. Is this even the right track to go on? Similarly I am trying to describe Z[i]/(2+i). Since 2+i=0 we must have 5=0, which leads me to believe that Z[i]/(2+i) is simply Z_5. I know that 2+i is prime in Z[i] and therefore the quotient should be a field. I tried a similar approach to the above but did not get far. I must be going in the wrong direction for these problems. Can someone tell me what is going on? === Subject: Re: adjoining elements to rings >What happens when we adjoin the inverse of 2 to the ring Z_12 i.e. what >is the field Z_12[x]/(2x-1) isomorphic to? I have reason to believe >that it is simply the field of 3 elements, because of the following: >In Z_12[x]/(2x-1) we have that 12=0 and 2x-1=0. Therefore 12x-6=0 ==6=0. Similarly we have 6x-3=0 ==> 3=0. This makes me believe that we >are talking about F_3 here. >However, I cannot prove this. I tried setting up the following maps: >j:Z ---> Z_12 >pi:Z_12 ---> Z_12[x]/(2x-1) >i:Z ---> Z_3 >If ker i = ker (pi o j) then we have that Z_12[x]/(2x-1) is isomorphic >to Z_3 by the first isomorphism theorem. I know that >ker j = 12Z >ker pi = (2x-1) >ker i = 3Z >But for some reason I cannot compute the kernel of pi o j. Is this even >the right track to go on? Well, you know that the kernel contains (3). Since (3) is a maximal ideal of Z, the only possibilities are for the kernel to be all of Z, or else to be just equal to (3). So the question then becomes: is 1 in the kernel? This will happen if and only if 1 = 0 (mod (2x-1)) in Z_{12}[x]. That is, if and only if there exists a polynomial f(x) in Z_{12}[x] such that f(x)(2x-1) = 1. Write f(x) = a_n*x^n + ... + a_1*x + a_0 with a_i in Z. Then (2x-1)f(x) = 2a_n*x^{n+1} + (2a_{n-1}-a_n)x^n + ... (2a_0-a_1)x - a_0. So you must have a_0 = - 1 (mod 12) 2a_{i-1} - a_i = 0 (mod 12) for i=1,...,n 2a_n = 0 (mod 12). So, from a_0 = -1, we have -2 - a_1 = 0 (mod 12), or a_1 = -2. Then 2a_1 - a_2 = 0 (mod 12), so -4 = a_2 (mod 12). 2a_2 - a_3 = 0 (mod 12) so a_3 = -8 (mod 12). Continuing in this way, a_{i} = -2^i (mod 12). Since we also need 2a_n = 0 (mod 12), that means that -2^{n+1}=0 (mod 12). But this never happens. So no such polynomial exists. Therefore, 1 is not in the kernel of the map. Since the kernel already contains 3, and is not everything, the kernel of pi o j must be (3). HOWEVER: you have not proven that Z_{12}[x]/(2x-1) is isomorphic to F_3; you've only shown that it CONTAINS F_3; you would also have to prove that the induced map from Z is surjective in order to prove that. (I mean, what if it is some other field of characteristic 3?) Slightly easier: you have already shown that 3 is in (2x-1), so the quotient Z_{12}[x]/(2x-1) factors through Z_{3}[x]/(2x-1). And since Z_3[x]/(2x-1) is isomorphic to Z_3, you are done. >Similarly I am trying to describe Z[i]/(2+i). This is a bit simpler since Z[i] is not only a domain, but a UFD. > Since 2+i=0 we must have >5=0, which leads me to believe that Z[i]/(2+i) is simply Z_5. I know >that 2+i is prime in Z[i] and therefore the quotient should be a field. It's maximal, and therefore the quotient should be a field (prime and maximal are equivalent in Z[i], but not in general). >I tried a similar approach to the above but did not get far. Yes, it is isomorphic to Z/5Z. Certainly, it is a field a characteristic 5, since a similar approach to the above will establish that the kernel of the map Z -> Z[i] -> Z[i]/(2+i) is just (5), by noting that 1 is not in (2+i). But you would still have to show that this map is surjective, which is not too hard: you can easily show that every element of Z[i] can be written as (a+bi)(2+i) + r, with r = 0, 1, 2, 3, or 4. This because 5 = (2+i)(2-i). So, given x+yi, with x and y in Z, then we have x-2y = (x+yi) - y(2+i). Now let x-2y = 5t + r, with t an integer, r=0,1,2,3, or 4; Then x - 2y = (2t-ti)(2+i) + r So (2t-ti)(2+i) + r = (x+yi)-y(2+i) hence (x+yi) = (2t+y - ti)(2+i) + r. Thus, every element in Z[i]/(2+i) is congruent modulo 2+i to 0, 1, 2, 3, 4, or 5, so the map Z->Z[i]->Z[i]/(2+i) is surjective, kernel (5), and you are done. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: adjoining elements to rings >What happens when we adjoin the inverse of 2 to the ring Z_12 i.e. what >is the field Z_12[x]/(2x-1) isomorphic to? I have reason to believe >that it is simply the field of 3 elements, because of the following: >In Z_12[x]/(2x-1) we have that 12=0 and 2x-1=0. Therefore 12x-6=0 >6=0. Similarly we have 6x-3=0 ==> 3=0. This makes me believe that we >are talking about F_3 here. >However, I cannot prove this. I tried setting up the following maps: >j:Z ---> Z_12 >pi:Z_12 ---> Z_12[x]/(2x-1) >i:Z ---> Z_3 >If ker i = ker (pi o j) then we have that Z_12[x]/(2x-1) is isomorphic >to Z_3 by the first isomorphism theorem. I know that >ker j = 12Z >ker pi = (2x-1) >ker i = 3Z >But for some reason I cannot compute the kernel of pi o j. Is this even >the right track to go on? > Well, you know that the kernel contains (3). Since (3) is a maximal > ideal of Z, the only possibilities are for the kernel to be all of Z, > or else to be just equal to (3). So the question then becomes: is 1 in > the kernel? > This will happen if and only if 1 = 0 (mod (2x-1)) in Z_{12}[x]. That > is, if and only if there exists a polynomial f(x) in Z_{12}[x] such > that f(x)(2x-1) = 1. > Write f(x) = a_n*x^n + ... + a_1*x + a_0 with a_i in Z. Then > (2x-1)f(x) = 2a_n*x^{n+1} + (2a_{n-1}-a_n)x^n + ... (2a_0-a_1)x - a_0. > So you must have a_0 = - 1 (mod 12) > 2a_{i-1} - a_i = 0 (mod 12) for i=1,...,n > 2a_n = 0 (mod 12). > So, from a_0 = -1, we have -2 - a_1 = 0 (mod 12), or a_1 = -2. > Then 2a_1 - a_2 = 0 (mod 12), so -4 = a_2 (mod 12). > 2a_2 - a_3 = 0 (mod 12) so a_3 = -8 (mod 12). > Continuing in this way, a_{i} = -2^i (mod 12). > Since we also need 2a_n = 0 (mod 12), that means that -2^{n+1}=0 (mod > 12). But this never happens. So no such polynomial exists. Therefore, > 1 is not in the kernel of the map. Since the kernel already contains > 3, and is not everything, the kernel of pi o j must be (3). > HOWEVER: you have not proven that Z_{12}[x]/(2x-1) is isomorphic to > F_3; you've only shown that it CONTAINS F_3; you would also have to > prove that the induced map from Z is surjective in order to prove > that. (I mean, what if it is some other field of characteristic 3?) > Slightly easier: you have already shown that 3 is in (2x-1), so the > quotient Z_{12}[x]/(2x-1) factors through Z_{3}[x]/(2x-1). And since > Z_3[x]/(2x-1) is isomorphic to Z_3, you are done. >Similarly I am trying to describe Z[i]/(2+i). > This is a bit simpler since Z[i] is not only a domain, but a UFD. > Since 2+i=0 we must have >5=0, which leads me to believe that Z[i]/(2+i) is simply Z_5. I know >that 2+i is prime in Z[i] and therefore the quotient should be a field. > It's maximal, and therefore the quotient should be a field (prime and > maximal are equivalent in Z[i], but not in general). >I tried a similar approach to the above but did not get far. > Yes, it is isomorphic to Z/5Z. Certainly, it is a field a > characteristic 5, since a similar approach to the above will establish > that the kernel of the map Z -> Z[i] -> Z[i]/(2+i) is just (5), by > noting that 1 is not in (2+i). But you would still have to show that > this map is surjective, which is not too hard: you can easily show > that every element of Z[i] can be written as (a+bi)(2+i) + r, with r = > 0, 1, 2, 3, or 4. > This because 5 = (2+i)(2-i). So, given x+yi, with x and y in Z, then > we have x-2y = (x+yi) - y(2+i). Now let x-2y = 5t + r, with t an > integer, r=0,1,2,3, or 4; Then > x - 2y = (2t-ti)(2+i) + r > So > (2t-ti)(2+i) + r = (x+yi)-y(2+i) > hence > (x+yi) = (2t+y - ti)(2+i) + r. > Thus, every element in Z[i]/(2+i) is congruent modulo 2+i to 0, 1, 2, > 3, 4, or 5, so the map Z->Z[i]->Z[i]/(2+i) is surjective, kernel (5), > and you are done. > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu first. While messing with the second I came across a question that I could not answer. Do you get a different object if you adjoin i to C? I know that R[x]/(x^2-1)=C, but does C[x]/(x^2+1)=C or do you get something else? I am guessing you get something else since a ring mod an ideal is a field if an only if the ideal is maximal... however (x^2+1) is not maximal in C[x]. But what do you get? === Subject: Re: adjoining elements to rings >What happens when we adjoin the inverse of 2 to the ring Z_12 i.e. what >is the field Z_12[x]/(2x-1) isomorphic to? I have reason to believe >that it is simply the field of 3 elements, because of the following: >In Z_12[x]/(2x-1) we have that 12=0 and 2x-1=0. Therefore 12x-6=0 >6=0. Similarly we have 6x-3=0 ==> 3=0. This makes me believe that we >are talking about F_3 here. >However, I cannot prove this. I tried setting up the following maps: >j:Z ---> Z_12 >pi:Z_12 ---> Z_12[x]/(2x-1) >i:Z ---> Z_3 >If ker i = ker (pi o j) then we have that Z_12[x]/(2x-1) is isomorphic >to Z_3 by the first isomorphism theorem. I know that >ker j = 12Z >ker pi = (2x-1) >ker i = 3Z >But for some reason I cannot compute the kernel of pi o j. Is this even >the right track to go on? > Well, you know that the kernel contains (3). Since (3) is a maximal > ideal of Z, the only possibilities are for the kernel to be all of Z, > or else to be just equal to (3). So the question then becomes: is 1 in > the kernel? > This will happen if and only if 1 = 0 (mod (2x-1)) in Z_{12}[x]. That > is, if and only if there exists a polynomial f(x) in Z_{12}[x] such > that f(x)(2x-1) = 1. > Write f(x) = a_n*x^n + ... + a_1*x + a_0 with a_i in Z. Then > (2x-1)f(x) = 2a_n*x^{n+1} + (2a_{n-1}-a_n)x^n + ... (2a_0-a_1)x - a_0. > So you must have a_0 = - 1 (mod 12) > 2a_{i-1} - a_i = 0 (mod 12) for i=1,...,n > 2a_n = 0 (mod 12). > So, from a_0 = -1, we have -2 - a_1 = 0 (mod 12), or a_1 = -2. > Then 2a_1 - a_2 = 0 (mod 12), so -4 = a_2 (mod 12). > 2a_2 - a_3 = 0 (mod 12) so a_3 = -8 (mod 12). > Continuing in this way, a_{i} = -2^i (mod 12). > Since we also need 2a_n = 0 (mod 12), that means that -2^{n+1}=0 (mod > 12). But this never happens. So no such polynomial exists. Therefore, > 1 is not in the kernel of the map. Since the kernel already contains > 3, and is not everything, the kernel of pi o j must be (3). > HOWEVER: you have not proven that Z_{12}[x]/(2x-1) is isomorphic to > F_3; you've only shown that it CONTAINS F_3; you would also have to > prove that the induced map from Z is surjective in order to prove > that. (I mean, what if it is some other field of characteristic 3?) > Slightly easier: you have already shown that 3 is in (2x-1), so the > quotient Z_{12}[x]/(2x-1) factors through Z_{3}[x]/(2x-1). And since > Z_3[x]/(2x-1) is isomorphic to Z_3, you are done. >Similarly I am trying to describe Z[i]/(2+i). > This is a bit simpler since Z[i] is not only a domain, but a UFD. > Since 2+i=0 we must have >5=0, which leads me to believe that Z[i]/(2+i) is simply Z_5. I know >that 2+i is prime in Z[i] and therefore the quotient should be a field. > It's maximal, and therefore the quotient should be a field (prime and > maximal are equivalent in Z[i], but not in general). >I tried a similar approach to the above but did not get far. > Yes, it is isomorphic to Z/5Z. Certainly, it is a field a > characteristic 5, since a similar approach to the above will establish > that the kernel of the map Z -> Z[i] -> Z[i]/(2+i) is just (5), by > noting that 1 is not in (2+i). But you would still have to show that > this map is surjective, which is not too hard: you can easily show > that every element of Z[i] can be written as (a+bi)(2+i) + r, with r = > 0, 1, 2, 3, or 4. > This because 5 = (2+i)(2-i). So, given x+yi, with x and y in Z, then > we have x-2y = (x+yi) - y(2+i). Now let x-2y = 5t + r, with t an > integer, r=0,1,2,3, or 4; Then > x - 2y = (2t-ti)(2+i) + r > So > (2t-ti)(2+i) + r = (x+yi)-y(2+i) > hence > (x+yi) = (2t+y - ti)(2+i) + r. > Thus, every element in Z[i]/(2+i) is congruent modulo 2+i to 0, 1, 2, > 3, 4, or 5, so the map Z->Z[i]->Z[i]/(2+i) is surjective, kernel (5), > and you are done. > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu first. While messing with the second I came across a question that I could not answer. Do you get a different object if you adjoin i to C? I know that R[x]/(x^2-1)=C, but does C[x]/(x^2+1)=C or do you get something else? I am guessing you get something else since a ring mod an ideal is a field if an only if the ideal is maximal... however (x^2+1) is not maximal in C[x]. But what do you get? === Subject: Re: adjoining elements to rings days. My association with the Department is that of an alumnus. >first. >While messing with the second I came across a question that I could not >answer. Do you get a different object if you adjoin i to C? Depends what you mean by adjoin i! If you add the complex number i to a ring that already has it, then you just get the same ring back. If you mean taking quotients of certain polynomial rings, that's something else: >I know that >R[x]/(x^2-1)=C, but does C[x]/(x^2+1)=C or do you get something else? I >am guessing you get something else since a ring mod an ideal is a field >if an only if the ideal is maximal... however (x^2+1) is not maximal in >C[x]. Exactly! So C[x]/(x^2+1) CANNOT be a field. > But what do you get? Notice that (x^2+1) = (x+i)(x-i); that (x+i) + (x-i) = 1. So the that C[x]/(x^2+1) is isomoprhic to ( C[x]/(x+i) ) x ( C[x]/(x-i) ) which in turn is isomorphic to C x C, the product of two copies of the complex numbers. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Prime ideal proof hello all, I have been racking my brain for hours trying to prove the following. P is a prime ideal of a ring R iff R / P is a prime ring I wish I could say that I had a good starting point but I don't. Any help would be greatly appreciated. TJ === Subject: Re: Prime ideal proof days. My association with the Department is that of an alumnus. >hello all, >I have been racking my brain for hours trying to prove the following. >P is a prime ideal of a ring R iff R / P is a prime ring What is the definition of Prime Ring? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prime ideal proof A ring R is called a prime ring if (0) is a prime ideal of R. -Tj -- >hello all, >I have been racking my brain for hours trying to prove the following. >P is a prime ideal of a ring R iff R / P is a prime ring > What is the definition of Prime Ring? > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: Prime ideal proof days. My association with the Department is that of an alumnus. >A ring R is called a prime ring if (0) is a prime ideal of R. In the commutative case, that's called an integral domain... >>I have been racking my brain for hours trying to prove the following. >>P is a prime ideal of a ring R iff R / P is a prime ring P is a prime ideal if and only if for every x,y in R, xy in P -> x in P or y in P. ==> Let x+P, y+P in R/P be such that xy + P in (0) in R/P. That means that xy is in P in R. Which means that... <== Let x, y in R such that xy in P. Then (x+P)(y+P) = 0+P in R/P. Therefore... -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prime ideal proof > A ring R is called a prime ring if (0) is a prime ideal of R. > -Tj > -- That's normally called an integral domain. Prove that a ring is a prime ring if and only if it has no zero divisors (i.e. whenever xy=0, either x=0 or y=0). Now prove that P is prime if and only if R/P has no zero divisors. === Subject: Re: Prime ideal proof days. My association with the Department is that of an alumnus. >> A ring R is called a prime ring if (0) is a prime ideal of R. >> -Tj >> -- >That's normally called an integral domain. In the commutative case. But there was no assumption that R was a commutative ring. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Delphi or VB Column generation ? === >Subject: Re: Delphi or VB Column generation ? >Message-id: cutting, it stores the orders and runs the optimization routine in VBA. >The problem is as I stated, it appears that the code that runs to generate >the valid cutting patterns is malformed or incomplete and excludes some >valid logic that we think should be included. >I need a clean starting point for a simple one dimensional stock cutting >program. Ok, but that still means nothing to me. You need to explain what this actually is. Where I buy lunch, they have 6, 8 and 10 sandwiches they make from 36 loaves of French bread. Would this be a simple example of one dimensional stock cutting? You would not want to make four 8 sandwiches off one loaf because you would be left with an unuseable 4 remnant. >All of the solvers out there have their own way and code to generate the >valid cutting plans / patterns, I want a generic pattern generation / >cutting plan generator that I can modify with our own logic so we can >analyze the impact on adding certain attributes and variables to the mix. But you also said earlier that you want to feed the results to a solver. Do you just want a program/database that creates patterns that will then be optimized, or are you going to do the the whole ball of wax: generate all possible patterns with custom criteria that you will then process with your own optimization algorithm? >might be we need to chat on aol or msn messenger for me to convey my needs >more completely as I am not a mathematician, I am a programmer. Chat sessions might be a problem. There's scheduling and I've found live chats to be less productive for technical stuff. I am not a professional mathemetician nor a professional programmer, so keeping the discussion public might be better. And don't be concerned about how complicated this gets. Sci.math has endless topics concerning pure crap so a long thread on a legitimate topic should not be a problem. >Yes I can convert C code to VB Good, because I can't. Seriously, I know VB and Access and Python so we shouldn't have any problems discussing algorithms. === >>Subject: Re: Delphi or VB Column generation ? >>Message-id: The current output is generated simply by putting together all possible >>permutations, even the ones that for obvious reasons (i.e. material >>utilization, etc..) are not good ones therefore increasing the number of >>combinations exponentially and allowing non operationally optimal >patterns >>to be included and selected for usage. >>I want a clean starting point that we can go through and INTENTIONALLY >>modify and enhance with LOGIC, not guesswork as it appears to have been >done >>in the spaghetti code in the current application. >>I have been reading everything I could get my hands on about the subject >>linear optimization, linear algebra, combinatorics and 1d stock cutting >>solutions but it seems that nobody has ever published a good, >>straightforward way to generate columns / permutations using a computer >for >>submission to a solver of some kind like lp solve. >>Unfortunately I cannot post the code as I do not own the copyright, only >>license to use/modify, sorry guys. >> Ok, I thought you said you wanted to fix an existing Access/VB system, >> but later you mentioned growing your own. >> Are you still wanting this done in Access? Access can easily generate >> permutations - provided you want a reasonable scope. For instance, I've >> got a database query that generates all possible 4 letter words. The >456,976 >> record query is handled easily on modern PCs. But asking for all possible >> 8 letter words gets you in trouble because the query would return 208 >billion >> records. >> And logic is fairly easy to add. I can take my 4 letter word query and ask >> to return only permutations in which the first and last letters are >consonants >> and the middle two letters are vowels. That reduces the the record count >from >> 456,976 to 11,025. But again, trying this with 8 letter words can still >result >> in 121 million records. >at a >> time >> would only be limited by disk space. VB in Access could handle this. If >your >> goal is to prepare a file for some other program, there are probably >better >> choices than VB. >> Let me ask again, what exactly are you looking for? How to generate >> permutations in Access? How to generate them in some programming >> language? What languages do you have available? How good a programmer >> are you? Could you translate a C algorithm into VB? A sample of what you >> want could help identify the scope of the problem. >> === >Subject: Delphi or VB Column generation ? >Message-id: <_LIpd.48$6K5.1@newsread2.news.atl.earthlink.net >I am doing some research into fixing a simple optimizer written in VB >Access for a friends company. The product is a few years old and does not exactly provide the best >results. After some studying of this code it appears that the way that the >column >generation is being done is probably the start of the difficulties. I have looked but cannot locate a code snippit or function for column >generation that I could start with and modify for my needs or learn >how >others are generating columns and grow my own version. What do you mean by column generation? Can you provide an example of what it is currently output vs. what you > would like it to output? Can you quote the code in question so that >those > with some experience in Access/VB don't have to try to read your mind? > It may be that what you're looking for is not in the VB portion but in > the SQL code of the queries involved. Or it could be both. I've seen cases where VB programs build temporary > tables with the column generation taking place dynamically based on > a query, which is itself genereated by a SQL statement created inside > a different VB program. This could get way too complicated for a newsgroup discussion, and no, > you cannot post a copy of the database. But you _can_ post VB code and > if necessary, you can show the queries by opening them in design mode > and selecting SQL view which can be posted. >Stephen Hauck > -- > Mensanator > Ace of Clubs >> -- >> Mensanator >> Ace of Clubs -- Mensanator Ace of Clubs === Subject: Re: Infinite number of people toss a coin infinite times >> I said there used to be a theory that the universe was already infinite >> as in no expansion it was _already_ infinite. >> Expansion undermined this theory not because it provided evidence >> of the finiteness of the universe, but because if the universe is under >> expansion there can't be a working completed infinity of the universe >> theory. > I don't see why not. > But then I'm not a physicist, Jim, just a simple Texas > set theorist. If there's some reason according to currently > understood physics why an already-infinite universe can't > be expanding, someone please enlighten me. I described this as used to be a theory because I couldn't remember the name of it. I thought at first is was steady-state. But I couldn't reconclile the presence of expansion in steady-state with what I had learned about this. So I kept digging and found: http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/cml_newt.htm Newtonian Space and Time Cosmology For Newton, space and time are absolute, i.e., the same throughout the Universe and unchanging or unchangeable Absolute space, in its own nature, without relation to anything external, remains always similar and immovable... Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external... Newtonian space time - space and time are absolutes and exist independent of the existence of matter in the Universe; space and time precede the material Universe's existence and will survive its demise Decartes and Gottfried Whilhelm Liebniz (1646-1716) did not support Newton's belief in an absolute space and time Liebniz: I hold space to be something purely relative, as time is. Newtonian Cosmology Copernican Principle (Aristarchus, Copernicus, Bondi) - we are not privileged occupants of the universe, so that there is nothing exceptional about our existence and we are consequently not at the Universe's center Newtonian Universe - gravity demands that the universe be centerless, edgeless, and of infinite extent in all directions SH: I kept trying to apply the static mathematical Newton infinity to the sold-state theory and it didn't quite fit. Newton's absolute space was unchanging and therefore not undergoing expansion. It seems to me there are traces of Newton's cosmology in the steady-state theory. Stephen === Subject: Re: Infinite number of people toss a coin infinite times > An infinite number of people each toss a coin infinite times, > can you guarantee a new sequence of heads and tails? > Herc Actually, there may be an infinite number of new sequences since: inf + inf = inf Thus, your premise of infinite people and infinite tosses does not guarantee that you are missing an infinite number of people to exaust all possible sequences. Paradoxes of infinity giving finite minds a hard time:) (especially those that look at infinity as a number. lol) Mike === Subject: Re: Infinite number of people toss a coin infinite times right, its possible you can make a new sequence (everyone tossed heads that day), but can you GUARANTEE it? Herc === Subject: Re: Infinite number of people toss a coin infinite times > right, its possible you can make a new sequence (everyone tossed heads > that day), but can you GUARANTEE it? > Herc There is only one thing that you can guarantee in this world with P=1: death some time after birth. Watch out governments that guarantee public debt issues and guaranteed performance mutual funds.:) Mike === Subject: Re: Infinite number of people toss a coin infinite times > Stephen Harris says... >>I said there used to be a theory that the universe was already infinite >>as in no expansion it was _already_ infinite. > Yes, I know. That's *still* the dominant theory. In an expanding The steady-state theory is not still dominant and I posted some reference quotes to Mike Oliver. > (open) universe, the universe is spatially infinite from the beginning. I accept that this is the explanation. But, I have trouble accepting the fact that it is also claimed that Big Bang originated from an infinitesimal point and then expanded/exploded without the volume of the universe changing. I seem to recall them saying things like after .00000000000000000001 seconds the diameter of the universe was only a few hundred light years. > It isn't that it's size is unbounded, but that it is actually > infinite from the beginning. As I said, there are basically two I suppose it is Hawking who uses the finitely unbounded terminology. > models (if we set the cosmological constant to zero, and ignore > weird topologies such as a toroidal universe): (1) the universe is > spatially infinite at all times, and will expand forever, (2) the > universe is spatially finite, and will eventually stop expanding > and recontract to a big crunch. > The possibility that you are considering, that the universe is > finite in size but will expand forever, might be possible by > adjusting the cosmological constant and the matter density. But > expanding does not imply finiteness. > -- > Daryl McCullough > Ithaca, NY === Subject: Re: Infinite number of people toss a coin infinite times Stephen Harris says... >The steady-state theory is not still dominant and I posted some >reference quotes to Mike Oliver. I'm sorry---I meant the idea that the universe is infinite. >> (open) universe, the universe is spatially infinite from the beginning. >I accept that this is the explanation. But, I have trouble accepting >the fact that it is also claimed that Big Bang originated from >an infinitesimal point and then expanded/exploded without the volume >of the universe changing. In the unbounded cosmologies, the Big Bang was not a time when the universe was at a single point---it was a time when the universe had infinite density. You might want to look at the cosmology faq entry here: http://www.astro.ucla.edu/~wright/infpoint.html -- Daryl McCullough === Subject: Re: Infinite number of people toss a coin infinite times > Stephen Harris says... >>The steady-state theory is not still dominant and I posted some >>reference quotes to Mike Oliver. > I'm sorry---I meant the idea that the universe is infinite. The infinity of the steady-state theory SST is infinitely old and infinitely large. There is no beginning of time as in the Big Bang model. I said the infinity of the steady-state theory is not the same type of infinity as described by the Big Bang model, the universe begins with time. The word infinite is used in both cases but has two different meanings. The SST meaning has a static, absolute, completed sense, which reminds me of how a mathematical infinite plane is stated into existence, no process of expansion of the entire universe but expansion within volumes of the universe that do no exceed their original boundaries. The Big Bang uses a time dependent function for density and size of the universe. This infinity seems relative to me, there is process. And I think that is why Einstein's theory impacted the plausibility of SST. > (open) universe, the universe is spatially infinite from the beginning. >>I accept that this is the explanation. But, I have trouble accepting >>the fact that it is also claimed that Big Bang originated from >>an infinitesimal point and then expanded/exploded without the volume >>of the universe changing. > In the unbounded cosmologies, the Big Bang was not a time when the > universe was at a single point---it was a time when the universe had > infinite density. I'm comparing SSt to the Big Bang. There is no infinite density in SSt. Because the same word infinite in both theories does not mean the concept that the word represents is the same in both theories. This answers part of my question within the Big Bang theory context. Note that in the above paragraphs I have been careful to use the term observable Universe rather than Universe. The Universe itself, or the maximum amount of space that we will eventually be able to see given an infinite amount of time, may well be infinite. In quoting a size of the Universe we infer how far we can see in one direction (15 billion light years), and how far we can see in the other direction (15 billion light years) and add the two to get a size (30 billion light years). An age of 15 billion light years in each direction therefore leads us to infer that we are at the centre of a sphere with radius 15 billion light-years, and hence that the Universe is 30 billion light-years across. The trick, however, is that because the Universe is homogeneous and isotropic, every observer must measure a size of the Universe that is 30 billion light years... even ones that are at the edge of our observable Universe! This means that either the Universe is sufficiently curved that space doubles back on itself (like on the surface of a sphere), or that the actual Universe is much larger than the observable one. We currently think that the latter possibility is the case. This quote says the at the moment of creation, space and time were created so that there was a physical dimension. 13.7 billion years ago, the entirety of our universe was compressed into the confines of an atomic nucleus. Known as a singularity, this is the moment before creation when space and time did not exist. According to the prevailing cosmological models that explain our universe, an ineffable explosion, trillions of degrees in temperature on any measurement scale, that was matter and energy but space and time itself. Cosmology theorists combined with the observations of their astronomy colleagues have been able to reconstruct the primordial chronology of events known as the big bang. > You might want to look at the cosmology faq entry here: > http://www.astro.ucla.edu/~wright/infpoint.html > -- > Daryl McCullough There used to be a theory in astronomy/cosmology that the universe was infinite. Then evidence that the universe was expanding caused this theory to be dropped. *** This means the theory SST was dropped with its intepretation of infinity. It does not mean that infinity was dropped from the Big Bang theory. I think the interpretation of infinity in BB is different from SSt becasue SSt used a universe of infinite age/infinitely large while the BB uses an infinity which has a beginning in time. I don't see the state, whatever it was before the big bang as equivalent to the universe as infinitely old. When you challenged my *** quote I clarified with: I said there used to be a theory that the universe was already infinite as in no expansion it was _already_ infinite. Expansion undermined this theory not because it provided evidence of the finiteness of the universe, but because if the universe is under expansion there can't be a working completed infinity of the universe theory. I'm distinguishing between the senses of infinity used. Einstein did his work around 1915. He thought the universe was fixed, no expansion. I'm saying that there was a theory like SSt which said this fixed size universe was also infinite. This is before Big Bang theory or evidence from Hubble that the universe was expanding; expansion is part of the Big Bang theory tied to a beginning in time. SSt has no beginning in time. I'm saying there are two types of infinity involved in this discussion. Actually, the evidence that the universe is expanding is not evidence that it is finite. I did not say that because the universe is expanding it is finite. I said that evidence for expansion meant that the sense of meaning of infinite in the Big Bang theory (which is tied to the beginning of time and space) was evidence to conclude that the sense of meaning of infinite in a proto-SS theory (which ties time and space in a context of _no_ beginning) was no longer as plausible as the Big Bang theory which replaced it. My comment was about the type of conceptual infinities involved in a comparison between the two theories, not an attribution of future finiteness to one. Their difference is static to dynamic. The BB theory is often described as having a finite beginning and is then infinite. The Newtonian cosmological idea has a universe infinitely old. It has no finite beginning nor finite ending (is infinite). The finiteness assertion I'm making has to do with the beginning of the universe which seems defintional in both theories. Expansion favors the BB theory. The BB theory that I read has a physical universe evolution with a finite beginning. Contra, an infintely old universe does not have a finite beginning in time evolving into an infinite expression. So I've said infinity has different meaning between the two theories. To challenge that, one would need to reconcile the meaning of infinity between the two theories. I've explained my difference in terms of the beginning of time which implies that time was/is not infiite. Explaining why Big Bang also rightly uses the term infinity does not explain how the meanings of infinity are equivalent between the theories. Infinity to me means no beginning and no end. The BB theory says the universe is 13.7 billion years old, not infinitely old. Stephen === Subject: Re: Infinite number of people toss a coin infinite times Stephen Harris says... >The Big Bang uses a time dependent function for density and size of >the universe. This infinity seems relative to me, there is process. And >I think that is why Einstein's theory impacted the plausibility of SST. My point was that the size of the universe *isn't* a function of time in an open Big Bang cosmology. The universe starts out with infinite volume, and always has infinite volume. In a closed Big Bang universe, then yes, the volume of the universe is a function of time. -- Daryl McCullough Ithaca, NY === Subject: [Another Successful Troll by Herc] (no longer mentions): Infinite number of people toss a coin infinite times http://mygate.mailgate.org/mynews/comp/comp.theory/47f9ac33199d60366a9c1d786 a f8e916.48257%40mygate.mailgate.org Stephen Harris |- This answers part of my question within the Big |- Bang theory context. Note that in the above |- paragraphs I have been careful to use the term |- observable Universe rather than Universe. The |- Universe itself, or the maximum amount of space |- that we will eventually be able to see given an |- infinite amount of time, may well be infinite. In |- quoting a size of the Universe we infer how far we |- can see in one direction (15 billion light years), |- and how far we can see in the other direction (15 |- billion light years) and add the two to get a size |- (30 billion light years). An age of 15 billion |- light years in each direction therefore leads us |- to infer that we are at the centre of a sphere |- with radius 15 billion light-years, and hence that |- the Universe is 30 billion light-years across. |- The trick, however, is that because the Universe |- is homogeneous and isotropic, every observer must |- measure a size of the Universe that is 30 billion |- light years... even ones that are at the edge of |- our observable Universe! This means that either |- the Universe is sufficiently curved that space |- doubles back on itself (like on the surface of a |- sphere), or that the actual Universe is much |- larger than the observable one. We currently think |- that the latter possibility is the case. Um, no, not at all. Observers on the edge of _our_ observable universe are at, or very near, the epoch of the big bang. _Their_ observable universe is much, much smaller, and so stuff there is much closer together. In the limit of our possible observation, their universe collapses to something The universe may be homogeneous and isotropic, but that is merely a hypothesis, not a theory subject to falsifiability, even _in_ theory, since we can't _see_ *that* universe. The one _we_ see grows denser and younger with distance. [It also isn't the least bit isotropic; it has structure at every scale.] xanthian. It's always valuable to remember this recent interchange before wasting a lot of Net bandwidth on another of Herc's inane trolls: Its like talking to someone with amnesia, forget Kent he's a moron who takes antipsychotics all day. -- Herc (gotch@beauty.com) Well, I just did a bit of a search through some of your recentish posts (just a few months ago) in comp.theory. [...] Seems Kent Paul Dolan was right about you, and that it would just be a waste of time for me to continue 'discussing' your 'solution(s)' with you. -- Simon G Best (s.g.best@btopenworld.com) -- === Subject: Re: Infinite number of people toss a coin infinite times > In the unbounded cosmologies, the Big Bang was not a time when the > universe was at a single point---it was a time when the universe had > infinite density. Except it wasn't really a *time*, was it? I mean, there *are* times that get closer and closer to the BB, but according to current cosmology it doesn't make sense to say there was a time *at* which the BB occurred, right? (Be gentle, I'm *way* out of my depth here...) Chris Menzel === Subject: Re: Infinite number of people toss a coin infinite times >> Mike Oliver says... >If there's some reason according to currently >understood physics why an already-infinite universe can't >be expanding, someone please enlighten me. It is not current, but I think part of the inconsistent steady-state theory. >> No, there is no reason. > That's what I thought. Stephen, the point on > which you may be confused is this: In the infinite > case, an expanding universe doesn't actually mean > the universe is getting bigger, just that everything's > moving farther apart. > If you want to know how that can happen without the > universe getting bigger, someone's holding an impromptu > seminar on it down at the Hilbert Hotel. All the > conference rooms are booked, but management says > that won't be a problem.... I don't think I remember this incorrectly, but just in case, I will proceed cautiously. I said there was an older SST theory than the modern (Big Bang) which was discarded because of expansion among other things. I think the SST had a static infinity. http://www.pbs.org/wnet/hawking/universes/html/univ_steady.html Proposed in 1948 by Hermann Bondi, Thomas Gold, and Fred Hoyle, the steady-state theory was based on an extension of something called the perfect cosmological principle. This holds that the universe looks essentially the same from every spot in it and at every time. (This applies only to the universe at large scales; obviously planets, stars, and galaxies are different from the space between them.) Obviously, for the universe to look the same at all times, there could have been no beginning or no end. This struck a philosophical chord with a number of scientists, and the steady-state theory gained many adherents in the 1950s and 1960s. How could the universe continue to look the same when observations show it to be expanding, which would tend to thin out its contents? Supporters of this cosmology balanced the ever-decreasing density that results from the expansion by hypothesizing that matter was continuously created out of nothing. The amount required was undetectably small-about a few atoms for every cubic mile each year. http://www.counterbalance.net/cq-fab/stead-frame.html Statistically speaking, a steady-state universe presents the same face to observers over all time. There is no Big Bang; the Universe existed infinitely far into the past and will exist infinitely far into the future. Galaxies expand out of a given volume, to be replaced by young galaxies newly formed from matter spontaneously generated within the same volume. ------------------------------------------------------------- Olber's Paradox If the universe is infinitely old, infinitely large, and uniform why isn't the sky bright? answer - the universe is not infinitely old, infinitely large, and uniform because expansion can also solve this problem http://phyld.ucr.edu/astronomy/A-28.htm ...Explains and presents supportive evidence for the two opposing theories of the origin of the universe: the big bang theory, which proposes that it began suddenly from nothing, and the steady state theory, which holds it to be infinitely old and large. The time since the Big Bang is the expansion age of the Universe, which is simply how long the Universe has had to expand in order to get to its present size. This is a BB description. If you plug in infinite time and size as used in SST you get: The universe expanded for an infinitely long time (infinitely old) and reached _infinite_ size (infinitely large). The infinitely old means a completed or static concept of time. There is no beginning. Thus there is no start to a universe that grows larger over a time that grows infinitely larger. You can't add to infinity. If the universe is infinitely large, there is no force (gravity) which can reach the end of infinity and start a contraction of the infinitely large universe. Does the universe infinitely age? Which brings up the question of whether time ends upon heat death of the universe or if time continues because there is no end or beginning to infinity--how can something be infinitely old and yet have a finite ending in time. The SSt does not seem to me to have the notion of expanding into infinity, but that it is already infinite. I looked at this situation as a 3-d version of an infinitely flat plane doesn't grow to infinity, it begins at infinity. Maybe the mathematical infinity is a bit more objective/absolue than the subjective/relative physical universe infinity. Stephen === Subject: Re: Infinite number of people toss a coin infinite times >>That's what I thought. Stephen, the point on >>which you may be confused is this: In the infinite >>case, an expanding universe doesn't actually mean >>the universe is getting bigger, just that everything's >>moving farther apart. >>If you want to know how that can happen without the >>universe getting bigger, someone's holding an impromptu >>seminar on it down at the Hilbert Hotel. All the >>conference rooms are booked, but management says >>that won't be a problem.... > I don't think I remember this incorrectly, but just in case, I will > proceed cautiously. I said there was an older SST theory than the > modern (Big Bang) which was discarded because of expansion > among other things. I think the SST had a static infinity. I may be out of my depth here, but I don't think that's exactly right. AIUI the steady-state theory *did* account for expansion; that's not the difference between SS and BB. In the steady-state picture, as galaxies moved away from one another (expansion), new hydrogen atoms were born ex nihilo and formed new galaxies to fill the intervening space. According to the big-bang, infinite universe version, the volume of the universe is (already) infinite, but its age is finite, and there are no new baby hydrogen atoms to replace the ones that move away. Rather, the density of the universe simply diminishes as galaxies move apart. An aside--there are two wonderful vignettes in Calvino's Le Cosmicomiche, one on each of these themes. The Big Bang one, Tutto in un punto, is IMO perhaps the best in the whole collection, or perhaps comes in behind I colori. The book has been translated into English under the title Cosmicomics and I highly recommend it. === Subject: Re: Infinite number of people toss a coin infinite times >That's what I thought. Stephen, the point on >which you may be confused is this: In the infinite >case, an expanding universe doesn't actually mean >the universe is getting bigger, just that everything's >moving farther apart. >If you want to know how that can happen without the >universe getting bigger, someone's holding an impromptu >seminar on it down at the Hilbert Hotel. All the >conference rooms are booked, but management says >that won't be a problem.... >> I don't think I remember this incorrectly, but just in case, I will >> proceed cautiously. I said there was an older SST theory than the >> modern (Big Bang) which was discarded because of expansion >> among other things. I think the SST had a static infinity. > I may be out of my depth here, but I don't think that's exactly > right. AIUI the steady-state theory *did* account for expansion; > that's not the difference between SS and BB. In the steady-state > picture, as galaxies moved away from one another (expansion), new > hydrogen atoms were born ex nihilo and formed new galaxies to > fill the intervening space. Right, I was floundering before I found the appropriate old theory. I will repost my reply. Maybe I will make a literaty excurison into The book has been translated into English under the title Cosmicomics and I highly recommend it. Repost: >> I said there used to be a theory that the universe was already infinite >> as in no expansion it was _already_ infinite. >> Expansion undermined this theory not because it provided evidence >> of the finiteness of the universe, but because if the universe is under >> expansion there can't be a working completed infinity of the universe >> theory. > I don't see why not. I described this as used to be a theory because I couldn't remember the name of it. I thought at first is was steady-state. But I couldn't reconclile the presence of expansion in steady-state with what I had learned about this. So I kept digging and found: http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/cml_newt.htm Newtonian Space and Time Cosmology For Newton, space and time are absolute, i.e., the same throughout the Universe and unchanging or unchangeable Absolute space, in its own nature, without relation to anything external, remains always similar and immovable... Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external... Newtonian space time - space and time are absolutes and exist independent of the existence of matter in the Universe; space and time precede the material Universe's existence and will survive its demise Decartes and Gottfried Whilhelm Liebniz (1646-1716) did not support Newton's belief in an absolute space and time Liebniz: I hold space to be something purely relative, as time is. Newtonian Cosmology Copernican Principle (Aristarchus, Copernicus, Bondi) - we are not privileged occupants of the universe, so that there is nothing exceptional about our existence and we are consequently not at the Universe's center Newtonian Universe - gravity demands that the universe be centerless, edgeless, and of infinite extent in all directions SH: I kept trying to apply the static mathematical Newton infinity to the sold-state theory and it didn't quite fit. Newton's absolute space was unchanging and therefore not undergoing expansion. It seems to me there are traces of Newton's cosmology in the steady-state theory. Stephen === Subject: Re: Infinite number of people toss a coin infinite times >>That's what I thought. Stephen, the point on >>which you may be confused is this: In the infinite >>case, an expanding universe doesn't actually mean >>the universe is getting bigger, just that everything's >>moving farther apart. >>If you want to know how that can happen without the >>universe getting bigger, someone's holding an impromptu >>seminar on it down at the Hilbert Hotel. All the >>conference rooms are booked, but management says >>that won't be a problem.... > I don't think I remember this incorrectly, but just in case, I will > proceed cautiously. I said there was an older SST theory than the > modern (Big Bang) which was discarded because of expansion > among other things. I think the SST had a static infinity. >> I may be out of my depth here, but I don't think that's exactly >> right. AIUI the steady-state theory *did* account for expansion; >> that's not the difference between SS and BB. In the steady-state >> picture, as galaxies moved away from one another (expansion), new >> hydrogen atoms were born ex nihilo and formed new galaxies to >> fill the intervening space. > Right, I was floundering before I found the appropriate old theory. > I will repost my reply. Maybe I will make a literaty excurison into > The book has been translated into English under the > title Cosmicomics and I highly recommend it. > Repost: > I said there used to be a theory that the universe was already infinite > as in no expansion it was _already_ infinite. > Expansion undermined this theory not because it provided evidence > of the finiteness of the universe, but because if the universe is under > expansion there can't be a working completed infinity of the universe > theory. >> I don't see why not. > I described this as used to be a theory because I couldn't > remember the name of it. I thought at first is was steady-state. > But I couldn't reconclile the presence of expansion in steady-state > with what I had learned about this. So I kept digging and found: > http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/cml_newt.htm > Newtonian Space and Time Cosmology > For Newton, space and time are absolute, i.e., the same throughout the > Universe and unchanging or unchangeable > Absolute space, in its own nature, without relation to anything external, > remains always similar and immovable... Absolute, true and mathematical > time, of itself, and from its own nature, flows equably without relation > to > anything external... > Newtonian space time - space and time are absolutes and exist independent > of > the existence of matter in the Universe; space and time precede the > material > Universe's existence and will survive its demise > Decartes and Gottfried Whilhelm Liebniz (1646-1716) did not support > Newton's belief in an absolute space and time > Liebniz: I hold space to be something purely relative, as time is. > Newtonian Cosmology > Copernican Principle (Aristarchus, Copernicus, Bondi) - we are not > privileged occupants of the universe, so that there is nothing exceptional > about our existence and we are consequently not at the Universe's center > Newtonian Universe - gravity demands that the universe be centerless, > edgeless, and of infinite extent in all directions http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/cml_newt.htm Paradoxes of Newtonian Universe [SH: continued] Olber's paradox (Heinrich Olbers, 1823) - an infinite universe would produce an infinite amount of light at our position, so why is the night sky dark? Probably known since Kepler's time; certainly Cheseaux and Halley Probably Newton aware of paradox Gravity paradox (Newton) - an infinite universe would also produce an infinite gravitational attraction at our position Newton assumed that the resolution of the gravity paradox was that stars effects (very unlikely) Solution to both paradoxes comes in modern time > SH: I kept trying to apply the static mathematical Newton infinity to > the sold-state theory and it didn't quite fit. Newton's absolute space > was unchanging and therefore not undergoing expansion. It seems to > me there are traces of Newton's cosmology in the steady-state theory. > Stephen === Subject: Re: Infinite number of people toss a coin infinite times Originator: tchow@MATHSTATION029.MIT.EDU.mit.edu (Timothy Chow) >It is not current, but I think part of the inconsistent steady-state theory. Steady-state was abandoned because of the cosmic microwave background radiation and the lack of any evidence that matter/energy was being created ex nihilo, not because of any mathematical inconsistency. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: Infinite number of people toss a coin infinite times >>It is not current, but I think part of the inconsistent steady-state >>theory. > Steady-state was abandoned because of the cosmic microwave background > radiation and the lack of any evidence that matter/energy was being > created > ex nihilo, not because of any mathematical inconsistency. There is no argument against these being valid reasons. There is also a hydrogen argument which supports abandonment of SSt. > -- Does steady-state use the idea of a universe infinitely old and large? Does Big Bang use the idea of the universe which has a finite beginning in time and probably an infinite continuation? I said those ideas of infinity (meanigs) are not consistent. Einstein fudged his results in order to obtain a static universe. Hubbel found evidence for expansion of the universe. I said this undermined SST. Because: Obler asked why there wasn't more starlight if the universe was infinitely old and large. Expansion provides an explanation which challenges the SSt assumption of an infinitely old and large universe. That does not assert that expansion shifts the static infinity of SSt to the dynamic infinity of Big Band. That is done by the interprettion SSt uses of time as having no beginning (infinitely old) vs. having a beginning, BB. That does not seem like a mathematical inconsistency so much as a by choice theoretical lack of agreement. Are you saying those are equivalent? Stephen === Subject: Re: Infinite number of people toss a coin infinite times > -After n tosses, the chance that any two people will have generated > -the same sequence is 2^-n. Since lim n -> oo. 2^-n = 0 there is > -no chance that any two people in the set will generate the same > -sequence indefinitely. > Incorrect. P=0 does not mean no chance, by that logic any infinite > sequence is impossible. there is 1 chance in oo. 1/oo = 0. That's what the concept of probability density is for when dealing with continuous probability functions. -- Ralph === Subject: Re: Infinite number of people toss a coin infinite times > But your statement that 2 people cannot form the same sequence is > flawed. > 1 person forming any sequence is P=0, is there no chance of that? Funnily enough, the answer is yes. To illustrate, let me change the problem statement to something equivalent, but easiser to work with: a player randomly chooses a real number in the range [0, 1); for any number n in this range, what is the probability P(n) that the player picks n? (This is the same as the coin tossing problem if we write n down in binary). If P(n) is non-zero then the sum of the probabilities for the different outcomes must be greater than 1 since there are an infinite number of such n. Therefore P(n) must be zero. What you need is a probability density function P(x, dx) which gives the probability of the player choosing a number in the range [x, x + dx). -- Ralph === Subject: Re: Sum of values of cards. Hard. > IÇm developing a game with 60 cards. Each card has a small integer value > (values up to 20, for example). There are groups of cards sharing the same > value, obviously. > This is a multiplayer game. It can be played by 2, 3, 4, 5 or 6 players. The > full deck of cards is delaed so that... > 1. Each player has the same number of cards (30 for a 2 player game, 20 for > a 3 player game...). > 2. The sum of the values of the cards of each player musy be the same!!! > (Trivial case where all values are the same is not permited). > Haw can I obtain such value distribution? How about having four cards of each denomination up to 15? The 60 cards then add up to 480, and I'm sure that with a bit of trial and error you can split them into two sets of 30, each summing to 240, or three sets of 20, each summing to 160, or ... or 6 sets of 10, each summing to 80. Another idea: have 3 of each denomination up to 20, except don't have any elevens, have 6 ones instead. Now the sum of all the cards is 600, and again some trial and error should get you the deals you want. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Prime ideals in Z[x] How do prime ideals in Z[x] look like? I know Z[x] is noetherian, but I couldn't find any prime ideal that would have more than two generators, so... maybe there isn't any? :-) sirix. === Subject: Re: Prime ideals in Z[x] days. My association with the Department is that of an alumnus. >How do prime ideals in Z[x] look like? I know Z[x] is noetherian, but I >couldn't find any prime ideal that would have more than two generators, >so... maybe there isn't any? :-) Here is a post with an answer to that question, from Bill Dubuque: You can safely ignore my quoted response, which is harder than it should be. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prime ideals in Z[x] >>How do prime ideals in Z[x] look like? I know Z[x] is noetherian, but I >>couldn't find any prime ideal that would have more than two generators, >>so... maybe there isn't any? :-) > Here is a post with an answer to that question, from Bill Dubuque: > You can safely ignore my quoted response, which is harder than it > should be. a curiosity, just to show what will we be working on after geting through Atiyah-McDonald) something like that: If the ring has a big Krull dimension then it is not a UFD. As a matter of fact, bigger the Krull dim is, worse things may happen with any kind of uniqueness. What precisely did he mean? I thought that there is something like Ring has Krull dim=1 iff Ring is UFD but it's not the case, as in Z[x] (x) subset (x, 2) and both (x) and (x,2) are prime... sirix. === Subject: Re: Prime ideals in Z[x] days. My association with the Department is that of an alumnus. >How do prime ideals in Z[x] look like? I know Z[x] is noetherian, but I >couldn't find any prime ideal that would have more than two generators, >so... maybe there isn't any? :-) >> Here is a post with an answer to that question, from Bill Dubuque: >> You can safely ignore my quoted response, which is harder than it >> should be. >a curiosity, just to show what will we be working on after geting >through Atiyah-McDonald) something like that: If the ring has a big >Krull dimension then it is not a UFD. As a matter of fact, bigger the >Krull dim is, worse things may happen with any kind of uniqueness. What >precisely did he mean? I thought that there is something like Ring has >Krull dim=1 iff Ring is UFD but it's not the case, as in Z[x] (x) >subset (x, 2) and both (x) and (x,2) are prime... No, the Krull dimension is no guarantee; you have there an example of a UFD which has Krull dimension greater than 1; and Z[sqrt(-5)] is an example where the Krull dimension is 1 but the ring is not a UFD. I'm not entirely sure what he meant; you can get arbitrarily high Krull dimension and still have a UFD, simply by taking things like Z[x1,....,xn]. Then you have the chain 0< (x1) < (x1,x2) < (x1,x2,x3) < ... < (x1,...,xn) < (2,x1,...,xn) so the dimension is at least n+1. On the other hand, all but one of those prime ideals come from the transcendence degree. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prime ideals in Z[x] > No, the Krull dimension is no guarantee; you have there an example of > a UFD which has Krull dimension greater than 1; and Z[sqrt(-5)] is an > example where the Krull dimension is 1 but the ring is not a UFD. > I'm not entirely sure what he meant; you can get arbitrarily high > Krull dimension and still have a UFD, simply by taking things like > Z[x1,....,xn]. Then you have the chain > 0< (x1) < (x1,x2) < (x1,x2,x3) < ... < (x1,...,xn) < (2,x1,...,xn) > so the dimension is at least n+1. On the other hand, all but one of > those prime ideals come from the transcendence degree. More generally, a given integral domain R is a UFD if and only if all prime ideals of height 1 are principal. It is easy to show that any prime ideal of height r>0 must be generated by at least r elements. The above tells us is that the converse holds for r=1 <=> R is a UFD. === Subject: Re: Root Finder 12 > You people are savage, and there's no call for it. I suppose > you are one of the anal perfectionist obsessed with keeping on > course down to the Angstrom to offset the Gudermanian. > Newton's Method is an invention of genius. Why not use it? > Newton just didn't come up with the approximations to plug into > it... > or did he? No, but they have been done, and in mathematics, it doesn't matter who does it, just that it is true and can be used. Remember, and you have been told this repeatedly: YOU ARE NOT ALLOWED TO USE APPROXIMATIONS. You also say that Newton's method is great. However, there are established results saying where a good place is to start. So what you have done here is not new, and what you claim to do (to FIND roots, not just The reason that Root Finder N will be wrong for all values of N is that you have a bunch of equations of the form t^n = A_n t^(n-1) = A_(n-1) t^(n-2) = A_(n-2) etc. Now, if you can find a value of t that satisfies ALL equations SIMULTANEOUSLY, then you indeed have found a root of the original polynomial. However, if there is no solution, your claim that you have found a root is not automatic; it may be a root, or it may not. In fact, you may also be close to a root. But this is like firing a bullet at a wall, then drawing the target around it. -- Christopher Heckman P.S. I can use cut and paste, too. > I thought you already posted the last word on this subject. Are you > suffering from some kind of attention deficit disorder, or are you > being deliberately misleading? If there's a third possibility, I'd > welcome your explanation. > -- > There are two things you must never attempt to prove: the unprovable > -- and the obvious. > -- > Democracy: The triumph of popularity over principle. > -- > http://www.crbond.com > === Subject: Re: Root Finder 12 > ex. > t^2+2t-3=0 > N=(2,1) |N|^2=5 Q=(3/5)(2,1) D=(1,2) |D|^2=5 Q*D=12/5 > (mD-Q)*D=0 > m|D|^2=Q*D m=(Q*D)/|D|^2 = 12/25 > T=mD = (12/25)(1,2) > t^2=24/25 ~ 1 So, you're saying that t=sqrt(24/25) is a root of x^2+2t-3=0. Remember, and you have been told this repeatedly: YOU ARE NOT ALLOWED TO USE APPROXIMATIONS. You also say that Newton's method is great. However, there are established results saying where a good place is to start. So what you have done here is not new, and what you claim to do (to FIND roots, not just The reason that Root Finder N will be wrong for all values of N is that you have a bunch of equations of the form t^n = A_n t^(n-1) = A_(n-1) t^(n-2) = A_(n-2) etc. Now, if you can find a value of t that satisfies ALL equations SIMULTANEOUSLY, then you indeed have found a root of the original polynomial. However, if there is no solution, your claim that you have found a root is not automatic; it may be a root, or it may not. In fact, you may also be close to a root. But this is like firing a bullet at a wall, then drawing the target around it. -- Christopher Heckman >>Root Finder 12 >>by Jon Giffen. >>This solution was so simple that I couldn't believe it. >>But I tried it, and it works. >>It is found that the roots to the polynomial, >>a[0]+a[1]t+a[2]t^2+...+a[n]t^n=0 where >>T=(t,t^2,t^3,..,t^n) and >>N=(a[1],a[2],a[3],...,a[n]) are given by, >> -a[0] >>T= -----D >> D*N >>D=(1,2,3,4,...,n) > Ok then, attempt to construct the quadratic formula for your method: > at^2 + bt + c = 0 > T = (t, t^2) > N = (b, a) > D = (1, 2) > T = -c/(b+2a) * (1,2) = (t, t^2) > t = -c/(b+2a) > t^2 = -2c/(b+2a) > Clearly you are dead wrong, since the correct t's are > t = (-b + sqrt[b^2-4ac])/(2a) > and > t = (-b - sqrt[b^2-4ac])/(2a) > If you can not reconstruct the quadratic formula, then you are wrong. > Any polynomials with a root approximated by your method are just > coincidences. === Subject: Re: Root Finder 12 > ex. > t^2+2t-3=0 > N=(2,1) |N|^2=5 Q=(3/5)(2,1) D=(1,2) |D|^2=5 Q*D=12/5 > (mD-Q)*D=0 > m|D|^2=Q*D m=(Q*D)/|D|^2 = 12/25 > T=mD = (12/25)(1,2) > t^2=24/25 ~ 1 No, this says that t = sqrt(24/25), which is not a root of t^2+2t-3. My claim that your method can't solve quadratic equations is thus proven by your example. Remember, and you have been told this repeatedly: YOU ARE NOT ALLOWED TO USE APPROXIMATIONS. You also say that Newton's method is great. However, there are established results saying where a good place is to start. So what you have done here is not new, and what you claim to do (to FIND roots, not just The reason that Root Finder N will be wrong for all values of N is that you have a bunch of equations of the form t^n = A_n t^(n-1) = A_(n-1) t^(n-2) = A_(n-2) etc. Now, if you can find a value of t that satisfies ALL equations SIMULTANEOUSLY, then you indeed have found a root of the original polynomial. However, if there is no solution, your claim that you have found a root is not automatic; it may be a root, or it may not. In fact, you may also be close to a root. But this is like firing a bullet at a wall, then drawing the target around it. > ex. > at^2+bt+c=0 > N=(b,a) |N|^2=b^2+a^2 Q=(-c/[b^2+a^2])(b,a) D=(1,2) |D|^2=5 > Q*D=(-c/[b^2+a^2])(b+2a) > (mD-Q)*D=0 > m=(Q*D)/|D|^2 = (1/5)(-c/[b^2+a^2])(b+2a) > T=mD=(1/5)(-c/[b^2+a^2])(b+2a)(1,2) > t^2 =(2/5)(-c/[b^2+a^2])(b+2a) > b+/-{b^2-4ac}^(1/2) > t ={(2/5)(-c/[b^2+a^2])(b+2a)}^(1/2)=-------------------- > 2a > solve and find the required correction What is this correction? A fudge factor, maybe? -- Christopher Heckman >>Root Finder 12 >>by Jon Giffen. >>This solution was so simple that I couldn't believe it. >>But I tried it, and it works. >>[...] >>Solve the 6th degree polynomial, >>f(t)=t^6 + t - 10=0 >>a[0] = -10 N=(1,0,0,0,0,1) D=(1,2,3,4,5,6) >>a[0]=-10 D*N=7 >> -a[0] 10 >>T = -----D = ---(1,2,3,4,5,6) >> D*N 7 >>t =10/7 t=1.42835 f(1.42835)=-0.071 > 10/7 is not a root of t^6 + t - 10 = 0. The Rational Root Test says > that any rational solutions to this polynomial are +/-1, +/-2, +/5, > or +/10. It can only be an approximation. > Strike Twelve. > (The Rational Root Theorem -- a.k.a. the Rational Zero Theorem -- > can be found at http://mathworld.wolfram.com/RationalZeroTheorem.html . >>Solve the 49th degree polynomial, >>f(t)=t^49 + t^16 + 40t^5 - 6000 = 0 >>a[0]=-6000 N=(1,0,0,0,40,0,..,1,0,...,1) D=(1,2,3,..,49) >>D*N=1+200+16+49=266 >> -a[0] 6000 >>T = -----D = ------(1,2,3,..,49) >> D*N 266 >>t^49=294000/266=1105.26 t=1.15375 f(1.15375)=-3575.99 > Once again, the Rational Root Theorem says that the only possible rational > roots are integers. >>Applying Newton's Method, > Ah, so. You aren't finding roots after all, only approximations to them. > You've been told repeatedly that this isn't the same as finding the roots. >>t=1.15375-(-3575)/[49(1.15375^48)+16(1.15375^15)+200(1.15375)^4] >> =1.15375 again, so the answer must depend on distant decimal >>places. > The problem here is you don't have enough precision to make Newton's > Method work. >>[...] >>t^2 + 2t - 3 = 0 >>a[0]=-3 N=(2,1) D=(1,2) D*N=4 >> -a[0] 3 >>T = -----D =---(1,2) t^2=3/4 t=0.866 ~ 1 >> D*N 4 > What? Your method can't even solve a quadratic equation? That's when > you know it's really bad. > -- Christopher Heckman === Subject: Re: Root Finder 12 Dodging the issue, heh? I proved your method failed for the general formula, and all you did was post one of the coincidences I mentioned. Why don't you even try to discuss the fact I proved that the general quadratic can not be factored by your method? > ex. > t^2+2t-3=0 > N=(2,1) |N|^2=5 Q=(3/5)(2,1) D=(1,2) |D|^2=5 Q*D=12/5 > (mD-Q)*D=0 > m|D|^2=Q*D m=(Q*D)/|D|^2 = 12/25 > T=mD = (12/25)(1,2) > t^2=24/25 ~ 1 >>Root Finder 12 >>by Jon Giffen. >>This solution was so simple that I couldn't believe it. >>But I tried it, and it works. >>It is found that the roots to the polynomial, >>a[0]+a[1]t+a[2]t^2+...+a[n]t^n=0 where >>T=(t,t^2,t^3,..,t^n) and >>N=(a[1],a[2],a[3],...,a[n]) are given by, >> -a[0] >>T= -----D >> D*N >>D=(1,2,3,4,...,n) > Ok then, attempt to construct the quadratic formula for your method: > at^2 + bt + c = 0 > T = (t, t^2) > N = (b, a) > D = (1, 2) > T = -c/(b+2a) * (1,2) = (t, t^2) > t = -c/(b+2a) > t^2 = -2c/(b+2a) > Clearly you are dead wrong, since the correct t's are > t = (-b + sqrt[b^2-4ac])/(2a) > and > t = (-b - sqrt[b^2-4ac])/(2a) > If you can not reconstruct the quadratic formula, then you are wrong. > Any polynomials with a root approximated by your method are just > coincidences. === Subject: Re: Root Finder 13 > Root Finder 13 Strike Thirteen. > Jon Giffen > Another approach is considered, along with a possibility for > finding the root to an Infinite Series. > It is discovered that the property of the nth degree polynomial, > a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 where > N=(a[1],a[2],a[3],...,a[n]) > T=(t.t^2.t^3,t^4,..., t^n ) The reason that Root Finder N will be wrong for all values of N is that you have a bunch of equations of the form t^n = A_n t^(n-1) = A_(n-1) t^(n-2) = A_(n-2) etc. Now, if you can find a value of t that satisfies ALL equations SIMULTANEOUSLY, then you indeed have found a root of the original polynomial. However, if there is no solution, your claim that you have found a root is not automatic; it may be a root, or it may not. In fact, you may also be close to a root. But this is like firing a bullet at a wall, then drawing the target around it. -- Christopher Heckman > [non sequitor part of the post has been cut] === Subject: Re: Root Finder 13 > Root Finder 13 > Jon Giffen > Another approach is considered, along with a possibility for > finding the root to an Infinite Series. > It is discovered that the property of the nth degree polynomial, Your discovery is wrong, I will explain a few more lines down. > a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 where > N=(a[1],a[2],a[3],...,a[n]) > T=(t.t^2.t^3,t^4,..., t^n ) > Is, > |C|^2 > |T|^2=(-a[0])^2{-------------------} > |N|^2|C|^2-(N*C)^2 > where > C=(a[1],2a[2],3a[3],4a[4],...,na[n]) > f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t - 100 = 0 Try changing the constant term of the polynomial to 1000. You will see that your method fails miserably. > a[0]=-100 N=(-1,1,5,4,-1,1) C=(-1,2,15,16,-5,6) > |C|^2=547 |N|^2=48 N*C=153 > |N|^2 |C|^2 - (N*C)^2 = 48(547)-153^2=2847 > |C|^2 D > T =(-a[0]){----------------------}^(1/2) --- > |N|^2 |C|^2 - (N*C)^2 |D| where This equation is your method in a nutshell. If it is wrong, your whole method is wrong. Lets say we have two polynomials who differ only in the constant term. Lets call the polynomials W and V and their respective constant terms w and v. Since the equation above has a scalar term of a[0], that equation then implies that if (x*w) is a root of W then that (x*v) is a root of V. That is clearly false, just compare the two polynomials x^5 - 2x - 28 = 0 and x^5 - 2x - 237 = 0. Assume that the root of the first polynomial, which is 2, was found by your method. By the fact that a[0] is scalar in the equation the root was derived from, your method would say the second polynomial would have a root of 237/14=16.93, which is clearly false, since the root is 3. So if you choose any polynomial where your method created an accurate answer, I can find an infinite number polynomials where your method fails miserably by simply changing the constant term. Because of this, your method is fatally flawed and totally wrong. > D=(1,2,3,4,5,6) and > 100 547 > T = -------- {-----}^(1/2) (1,2,3,4,5,6) = 4.594933(1,2,3,4,5,6) > 91^(1/2) 2847 > t^6=27.5696 t=1.738097 > t^5=22.9746 t=1.871758 > ---------- > t=1.856637 is the correct root > Notice that the root to a polynomial that is so long, that > it is virtually an infinite Power Series; is found by using > the solution to the Geometric Series, > |C|^2 > |T|^2=t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{------------------} > |N|^2|C|^2-(N*C)^2 > adding 1 to both sides, > |C|^2 > 1+t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{------------------}+1 > |N|^2|C|^2-(N*C)^2 > then the sum S=1/(1-t^2) but > |C|^2 > S=(-a[0])^2{------------------}+1 (1-t^2)=1/S t^2=1-1/S and > |N|^2|C|^2-(N*C)^2 > t ={1 - 1/S}^(1/2) where > N=(a[1],a[2],a[3],...,a[n]) > T=(t,t^2,t^3,t^4,..., t^n ) > C=(a[1],2a[2],3a[3],4a[4],...,na[n]) > to the nth degree power series, > a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 > Development > Suppose the polynomial, > a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 > Is expressed as, > a[0]+a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n=0 > Where c is almost 1 then dividing the two, > a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n= -a[0] > --------------------------------------------------- > a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n = -a[0] > Suppose K=(a[1]c,a[2]c^2,a[3]c^3,...a[n]c^n) then simply, > T*(K-N)=0 so (K-N) is orthogonal to T. Consequently, > N*(K-N) > T is parallel to N - ------- K and > |K-N|^2 > N*(K-N) > [m(N - -------- K) - Q]*N=0 solve this for m. Then > |K-N|^2 > N*(K-N) > T= m(N - -------- K) > |K-N|^2 > Substitute m in the above and take the square of the > magnitude of both sides. Then > (K-N)*(K-N) 0 > |T|^2=Lim (-a[0])^2 ---------------------------- = --- > c->1 |N|^2(K-N)*(K-N)-[N*(K-N)]^2 0 > applying L'Hopital two times with respct to c, > |C|^2 > |T|^2=(-a[0])^2{-------------------} > |N|^2 |C|^2-(N*C)^2 > where > d > C = Lim ---- K = =(a[1],2a[2],3a[3],4a[4],...,na[n]) > c->1 dc > E.O.P. > Jon Giffen > http://mypeoplepc.com/members/jon8338/polynomial/id7.html === Subject: Re: Root Finder 13 >>f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t - 100 = 0 > Try changing the constant term of the polynomial to 1000. You will > see that your method fails miserably. f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t + 1000 = 0 a[0]=1000 N=(-1,1,5,4,-1,1) D=(1,2,3,4,5,6) N*D=33 t^6 = (-1000/33)(6) = -2000/11 (2k+1)(pi) (2k+1)(pi) t = {2000/11}^(1/6)[cos ---------- + i sin -----------] 6 6 Binomial Equation, where k=0,1,2,3,4,5 > Lets say we have two polynomials who differ only in the constant term. > Lets call the polynomials W and V and their respective constant terms > w and v. Since the equation above has a scalar term of a[0], that > equation then implies that if (x*w) is a root of W then that (x*v) is > a root of V. > That is clearly false, just compare the two polynomials x^5 - 2x - 28 > = 0 and x^5 - 2x - 237 = 0. Assume that the root of the first > polynomial, which is 2, was found by your method. By the fact that > a[0] is scalar in the equation the root was derived from, your method > would say the second polynomial would have a root of 237/14=16.93, > which is clearly false, since the root is 3. > So if you choose any polynomial where your method created an accurate > answer, I can find an infinite number polynomials where your method > fails miserably by simply changing the constant term. Because of > this, your method is fatally flawed and totally wrong. x^5 - 2x - 28 a[0]=-28 N=(-2,0,0,0,1) D=( 1,2,3,4,5) -a[0]/(N*D)=28/3 (t,t^2,t^3,t^4,t^5)=(28/3)(1,2,3,4,5) t^5 = (28/3)(5) t=(140/3)^(1/5)=2.1567 (should be 2) x^5 - 2x - 237 a[0]=-237 N=(-2,0,0,0,1) D=( 1,2,3,4,5) -a[0]/(N*D)=237/3 (t,t^2,t^3,t^4,t^5)=(237/3)(1,2,3,4,5) t^5 = (237/3)(5) t=(1185/3)^(1/5)=3.3061 (should be 3) In general, the root to the nth degree polynomial a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n where T=(t,t^2,t^3,...,t^n) N=(a[1],a[2],a[3],...,a[n]) D=(1,2,3,...,n) is approximated by the identity, -a[0] T = ----- D decoding, N*D -a[0] t = -------------------------- a[1]+2a[2]+3a[3]+...+na[n] -2a[0] t^2= -------------------------- a[1]+2a[2]+3a[3]+...+na[n] -3a[0] t^3= -------------------------- a[1]+2a[2]+3a[3]+...+na[n] . . . -na[0] t^n= -------------------------- a[1]+2a[2]+3a[3]+...+na[n] >>http://mypeoplepc.com/members/jon8338/polynomial/id7.html === Subject: Re: Root Finder 13 Yes I see that squaring all terms leads to a lack of inheritance of the property of negatives in the series. N*C destroys them as well. Root Finder 13 appears to be inferior. Root Finder 12, though, still seem to hold promise. >>Root Finder 13 >>Jon Giffen >>Another approach is considered, along with a possibility for >>finding the root to an Infinite Series. >>It is discovered that the property of the nth degree polynomial, > Your discovery is wrong, I will explain a few more lines down. >>a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 where >>N=(a[1],a[2],a[3],...,a[n]) >>T=(t.t^2.t^3,t^4,..., t^n ) >>Is, >> |C|^2 >>|T|^2=(-a[0])^2{-------------------} >> |N|^2|C|^2-(N*C)^2 >>where >>C=(a[1],2a[2],3a[3],4a[4],...,na[n]) >>f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t - 100 = 0 > Try changing the constant term of the polynomial to 1000. You will > see that your method fails miserably. >>a[0]=-100 N=(-1,1,5,4,-1,1) C=(-1,2,15,16,-5,6) >>|C|^2=547 |N|^2=48 N*C=153 >>|N|^2 |C|^2 - (N*C)^2 = 48(547)-153^2=2847 >> |C|^2 D >>T =(-a[0]){----------------------}^(1/2) --- >> |N|^2 |C|^2 - (N*C)^2 |D| where > This equation is your method in a nutshell. If it is wrong, your > whole method is wrong. > Lets say we have two polynomials who differ only in the constant term. > Lets call the polynomials W and V and their respective constant terms > w and v. Since the equation above has a scalar term of a[0], that > equation then implies that if (x*w) is a root of W then that (x*v) is > a root of V. > That is clearly false, just compare the two polynomials x^5 - 2x - 28 > = 0 and x^5 - 2x - 237 = 0. Assume that the root of the first > polynomial, which is 2, was found by your method. By the fact that > a[0] is scalar in the equation the root was derived from, your method > would say the second polynomial would have a root of 237/14=16.93, > which is clearly false, since the root is 3. > So if you choose any polynomial where your method created an accurate > answer, I can find an infinite number polynomials where your method > fails miserably by simply changing the constant term. Because of > this, your method is fatally flawed and totally wrong. >>D=(1,2,3,4,5,6) and >> 100 547 >>T = -------- {-----}^(1/2) (1,2,3,4,5,6) = 4.594933(1,2,3,4,5,6) >> 91^(1/2) 2847 >>t^6=27.5696 t=1.738097 >>t^5=22.9746 t=1.871758 >> ---------- >>t=1.856637 is the correct root >>Notice that the root to a polynomial that is so long, that >>it is virtually an infinite Power Series; is found by using >>the solution to the Geometric Series, >> |C|^2 >>|T|^2=t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{------------------} >> |N|^2|C|^2-(N*C)^2 >>adding 1 to both sides, >> |C|^2 >>1+t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{------------------}+1 >> |N|^2|C|^2-(N*C)^2 >>then the sum S=1/(1-t^2) but >> |C|^2 >>S=(-a[0])^2{------------------}+1 (1-t^2)=1/S t^2=1-1/S and >> |N|^2|C|^2-(N*C)^2 >>t ={1 - 1/S}^(1/2) where >>N=(a[1],a[2],a[3],...,a[n]) >>T=(t,t^2,t^3,t^4,..., t^n ) >>C=(a[1],2a[2],3a[3],4a[4],...,na[n]) >>to the nth degree power series, >>a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 >>Development >>Suppose the polynomial, >>a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 >>Is expressed as, >>a[0]+a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n=0 >>Where c is almost 1 then dividing the two, >>a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n= -a[0] >>--------------------------------------------------- >> a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n = -a[0] >>Suppose K=(a[1]c,a[2]c^2,a[3]c^3,...a[n]c^n) then simply, >>T*(K-N)=0 so (K-N) is orthogonal to T. Consequently, >> N*(K-N) >>T is parallel to N - ------- K and >> |K-N|^2 >> N*(K-N) >>[m(N - -------- K) - Q]*N=0 solve this for m. Then >> |K-N|^2 >> N*(K-N) >>T= m(N - -------- K) >> |K-N|^2 >>Substitute m in the above and take the square of the >>magnitude of both sides. Then >> (K-N)*(K-N) 0 >>|T|^2=Lim (-a[0])^2 ---------------------------- = --- >> c->1 |N|^2(K-N)*(K-N)-[N*(K-N)]^2 0 >>applying L'Hopital two times with respct to c, >> |C|^2 >>|T|^2=(-a[0])^2{-------------------} >> |N|^2 |C|^2-(N*C)^2 >>where >> d >>C = Lim ---- K = =(a[1],2a[2],3a[3],4a[4],...,na[n]) >> c->1 dc >>E.O.P. >>Jon Giffen >>http://mypeoplepc.com/members/jon8338/polynomial/id7.html === Subject: Root Finder: Angle Subtending Arc and Chord ANGLE SUBTENDING ARC AND CHORD On a circle of unknown radius, arc A and chord B subtend angle x, which is obtained by the formula, x=5.062792913(1-B/A)^(1/2) B/A x from FORMULA x SHOULD BE 0.995892735 0.324463996 0.314159265 0.983631643 0.647728053 0.628318531 0.963397762 0.968598927 0.942477796 0.935489284 1.28589674 1.256637061 0.900316316 1.598461583 1.570796327 0.858393691 1.905160043 1.884955592 0.810331958 2.204891605 2.199114858 0.756826729 2.496594918 2.513274123 0.698646585 2.77925389 2.827433388 0.636619772 3.051903588 3.141592654 1 <= B/A <= 2/pi Development Let x=angle (unknown) A=arc B=chord then from trigonometry, B x - - = sin(x/2) A 2 and letting t=(x/2)^2, 1 1 1 1 (1-B/A)- ---t + ---t^2 - ---t^3 + ---t^4 - .... 3! 5! 7! 9! then N=(-1/3! , 1/5! , -1/7! , 1/9! , .... ) and -a[0] T = ----- Root Approximation Formula D*N where a[0]=(1-B/A) D=(1,2,3,4,5..) N=(-1/3! , 1/5! , -1/7! , 1/9! , .... ) then oo n 1 2 3 4 D*N = Sum(-1)^n ------- = - --- + --- - --- + --- - .... n=1 (2n+1)! 3! 5! 7! 9! Carrying out the calculation up to n=7, D*N= -0.1505843395 1/(D*N)=-6.640796802 (t,t^2,t^3,t^4,t^5,t^6,t^7)=6.640796802(1-B/A)(1,2,3,4,5,6,7) Selecting the first component, t=6.640796802(1-B/A)(1) but since t=(x/2)^2 , x=2(t^1/2) and x=5.062792913(1-B/A)^(1/2) This sheds merit on -a[0] T = ----- Root Approximation Formula D*N proving its utility in arriving within enough accuracy to require only a few iterations of Newton's Method to obtain a very precise (and accurate) result. Jon Giffen === Subject: Re: Root Finder: Angle Subtending Arc and Chord > ANGLE SUBTENDING ARC AND CHORD > On a circle of unknown radius, arc A and chord B > subtend angle x, which is obtained by the formula, > x=5.062792913(1-B/A)^(1/2) > B/A x from FORMULA x SHOULD BE > 0.995892735 0.324463996 0.314159265 > 0.983631643 0.647728053 0.628318531 > 0.963397762 0.968598927 0.942477796 > 0.935489284 1.28589674 1.256637061 > 0.900316316 1.598461583 1.570796327 > 0.858393691 1.905160043 1.884955592 > 0.810331958 2.204891605 2.199114858 > 0.756826729 2.496594918 2.513274123 > 0.698646585 2.77925389 2.827433388 > 0.636619772 3.051903588 3.141592654 > 1 <= B/A <= 2/pi What this table tells me is that FORMULA, whatever it is, is wrong. Check out the difference between pi and (2143/22)^(1/4). pi = 3.14159265353... (2143/22)^(1/4) = 3.14159265258... Just because two things are close doesn't mean they're equal. -- Christopher Heckman === Subject: Re: Root Finder: Angle Subtending Arc and Chord > ANGLE SUBTENDING ARC AND CHORD > On a circle of unknown radius, arc A and chord B > subtend angle x, which is obtained by the formula, > x=5.062792913(1-B/A)^(1/2) > B/A x from FORMULA x SHOULD BE > 0.995892735 0.324463996 0.314159265 > 0.983631643 0.647728053 0.628318531 > 0.963397762 0.968598927 0.942477796 > 0.935489284 1.28589674 1.256637061 > 0.900316316 1.598461583 1.570796327 > 0.858393691 1.905160043 1.884955592 > 0.810331958 2.204891605 2.199114858 > 0.756826729 2.496594918 2.513274123 > 0.698646585 2.77925389 2.827433388 > 0.636619772 3.051903588 3.141592654 Using your approximation formula, the worst |relative error| is roughly 3%. > 1 <= B/A <= 2/pi That's not possible, and so I suppose that you intended to write 2/pi <= B/A <= 1 instead. > Development > Let > x=angle (unknown) > A=arc > B=chord > then from trigonometry, > B x > - - = sin(x/2) > A 2 > and letting t=(x/2)^2, > t=6.640796802(1-B/A)(1) but since > t=(x/2)^2 , x=2(t^1/2) and > x=5.062792913(1-B/A)^(1/2) I fail to see how the coefficient 5.062792913 was obtained. Indeed, it seems clear from the end of your development that we should instead have had x = 2*t^(1/2) = 2*(6.640796802*(1 - B/A))^(1/2) = 5.1539487*(1 - B/A)^(1/2) Using that approximation, relative error exceeds 5% when B/A is near 1. > This sheds merit on > -a[0] > T = ----- Root Approximation Formula > D*N > proving its utility in arriving within enough accuracy to > require only a few iterations of Newton's Method to obtain > a very precise (and accurate) result. I'm not sure that it sheds merit. How did 5.06... arise, rather than 5.15...? It is easy to establish that a simple approximation is x = 2*Sqrt(6*(1 - B/A)). The relative error in that approximation is worst, about -6%, when B/A = 2/pi, and its |relative error| decreases to 0 as B/A approaches 1. Instead of the coefficient 2*Sqrt(6), which is about 4.9, if we use a somewhat larger coefficient, then we can reduce worst |relative error|, assuming of course that B/A is restricted to the interval [2/pi, 1]. Specifically, to minimize worst |relative error|, it can be shown that the coefficient should be 5.0504... In any event, it's not clear how you arrived at 5.062792913 for the coefficient. Please explain. David Cantrell === Subject: C code for Whittle Estimator I'm wondering, does anybody have a C code for Whittle Estimator ? Dmitry === Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin) > Mathematics doesn't always erase confusion in our intuition by > conforming to it; it also does it by _informing_ it. > It seems unintuitive (at least to me) at first that, even in > principle, you can't trisect every angle using the usual straight edge > and compass; but it is something you can prove; and eventually it > becomes intuitive. > I disagree with this formalist interpretation of mathematics. That is your perogative. But to return to the topic of the thread, it should have no bearing on the correctness or incorrectness of Zenkin's argument - his argument fails because it doesn't follow logically; not because he espouses a formalist or other philosophy. > Your intuition was simply wrong at first. That was my point. Your intuition regarding size of a set appears to be wrong as well - at least when you use the standard definition of size of a set. If you want a definititon of size of a set which conforms to your intuitions, you'll need to be more specific about what your intuitions require of the term, size of a set. > It's muddled to think that > you just get used to some formal relation. If you don't understand it, > it's just syntax. Until you understand it, it's just syntax. When you understand it, it becomes incorporated in your new, _correct_ intuition about the result in question. Or don't you think one can acquire new intuitions which extend and correct one's old ones? > (So, I don't like that statement of Von Neumann at > all...) That's another argument, though, and I don't think it's > necessary for me to make it. > Why should the _name_ of the things > we define have any significance in the logical systems we construct? > I never said the names matter. What matters is that they have > reference. I think that should be obvious to anybody who took a logic > class. You have claimed (without particular proof) that cardinality is antimonius - i.e., leads to a _logical_ contradiction. Then if we define the term chasanality identically to cardinality, and list its properties, then it should create a _logical_ contradiciton iff cardinality creates a _logical_ contradicition. I have seen no evidence of such a contradicition. It seems the antimony you are claiming is not a _logical_ contradiction, but instead offends your preconception of what size of a set _should_ mean. The onus is then on you to resolve this for yourself via a different definititon of size of a set. The standard one seems to work well for most mathematicians; in particular, as others have pointed out, it is consistent with one's intuition regarding finite sets, unlike the alternate definition you mentioned. === Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin) > It seems the antimony you are claiming is not a _logical_ > contradiction, but instead offends your preconception of what size of > a set _should_ mean. > The onus is then on you to resolve this for yourself via a different > definititon of size of a set. The standard one seems to work well > for most mathematicians; in particular, as others have pointed out, it > is consistent with one's intuition regarding finite sets, unlike the > alternate definition you mentioned. -- Eray Ozkural === Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin) >> Since there is the observed antinomy of the infinitely big, > unfortunately size of a set is far from being an obvious concept. > What antinomy is that? >> I'm sure you will find a few philosophers who have better command of >> English than I have. > Can't find anything. Can you name some names? Give a hint of what >> the actual antinomy is? Give some reference more specific than >> Google? > http://ls.poly.edu/~jbain/philmath/philmathlectures/M01.Intro.pdf > There are probably better expositions than these slides, but that's > the best thing I could find. > Certainly, you should check out the rest of the site. That way, > you'll see that his solution to the *apparent* paradoxes of infinite > sets is to follow Cantor. As far as I can tell, he presents the slide > you refer to for historical reasons and to motivate the later > discussion on Cantor. > See > . I've read all the notes, yes. Cantor proposes a solution. Maybe Galileo's paradox still persists, though. > In any case, let's not confuse lecture notes with philosophy of > mathematics research. Why not point me to a recent publication on > paradoxes of infinitely big written by a working philosopher of > mathematics? > (Not that Bain is or isn't a working philosopher of mathematics, but > lecture notes aren't a good indication of current issues in > philosophy.) Lecture notes, inspired by textbooks he teaches. Philosophy is not caution: I should be looking in the textbooks myself. If the current poster did not think that there could be more preferable formalizations of infinity, he would not suggest looking in Galileo's paradox. He would have deemed it solved. -- Eray Ozkural === Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin) <87zn187lzj.fsf@phiwumbda.org> <87fz2zfzid.fsf@phiwumbda.org> <87d5y1vm2f.fsf@phiwumbda.org> <87wtw7pa62.fsf@phiwumbda.org> Discussion, linux) >> See >> . > I've read all the notes, yes. Cantor proposes a solution. > Maybe Galileo's paradox still persists, though. Yeah, maybe. Except, of course, it doesn't in any reasonable sense. Cantor's set theory disposes of Galileo's paradox to the same degree that Frege's logical foundations overthrew the ontological proof of the existence of God. >> In any case, let's not confuse lecture notes with philosophy of >> mathematics research. Why not point me to a recent publication on >> paradoxes of infinitely big written by a working philosopher of >> mathematics? >> (Not that Bain is or isn't a working philosopher of mathematics, but >> lecture notes aren't a good indication of current issues in >> philosophy.) > Lecture notes, inspired by textbooks he teaches. Philosophy is not > caution: I should be looking in the textbooks myself. Philosophy is indeed not like technology. For the philosophy instructor, it is imperative that he puts the issues in context by showing the questions that needed to be answered. Even the mathematician doesn't have to motivate Cantor in the same historical manner as the philosopher. It is simply stupid to look to lecture notes in a philosophy class to measure what the current philosophical questions are---especially when the questions you're looking at are later resolved *in the same notes*. > If the current poster did not think that there could be more > preferable formalizations of infinity, he would not suggest looking > in Galileo's paradox. He would have deemed it solved. Current poster? Which current poster? Why should I care about him? -- 'Every man who has ever lived holds tight to the belief that for him you will marry Guinevere. You do not want advice --- only agreement.' Merlin sighed... -- John Steinbeck === Subject: Category Theory books Hi all, I'm intrested in buying a good book about category theory. I would like to buy (only)one of the following: 1)Categories for the Working Mathematician Saunders Mac Lane; 2)Conceptual Mathematics: A First Introduction to Categories F. William Lawvere, Stephen H. Schanuel . I need a good piece of advice... I would really appreciate it. TIA. PS: other books about categories are welcome. === Subject: Re: Category Theory books > Hi all, > I'm intrested in buying a good book about category theory. I would like to > buy (only)one of the following: > 1)Categories for the Working Mathematician > Saunders Mac Lane; > 2)Conceptual Mathematics: A First Introduction to Categories > F. William Lawvere, Stephen H. Schanuel . > I need a good piece of advice... I would really appreciate it. > TIA. > PS: other books about categories are welcome. It is difficult to give advice here without knowing your current state of mathematical experience. I am glad to have both of them, but in my opinion the are aimed at different audiences. The book (1) is the standard introduction for those students who have already met enough mathematics to be able to appreciate the examples. In other words, the title should be taken seriously to some degree. It will provide you with most of the technical tools you ever need. In contrast (2) can be digested by first year students or sufficiently bright high-school kids. The examples and illustrations are taken from finite sets and variants thereof. It cannot go so deep into technical details, but explains a lot and in a way teaches to 'think deeply about the simple things'. Perhaps the most useful advice is: try to obtain these books through a nearby (university) library and take a look at them before making a decision. Marc === Subject: Re: Category Theory books >Hi all, >I'm intrested in buying a good book about category theory. I would like to >buy (only)one of the following: >1)Categories for the Working Mathematician > Saunders Mac Lane; >2)Conceptual Mathematics: A First Introduction to Categories > F. William Lawvere, Stephen H. Schanuel . >I need a good piece of advice... I would really appreciate it. >TIA. I don't know about the second, but the first is the standard classic on the subject written by one of the founders, and it has had a recent second edition. OTOH I remember vaguely finding online J. Ad.87mek, H. Herrlich and G.E. Strecker -- Abstract and Concrete Categories But I don't have the url. And I don't like it very much -- it is well written but IMHO the emphasis is misplaced. Check also: http://www.geocities.com/alex_stef/mylist.html there are some interesting links there. G. Rodrigues === Subject: Re: Category Theory books >>I'm intrested in buying a good book about category theory. I would like to >>buy (only)one of the following: >>1)Categories for the Working Mathematician >> Saunders Mac Lane; >>2)Conceptual Mathematics: A First Introduction to Categories >> F. William Lawvere, Stephen H. Schanuel . ... >I don't know about the second, but the first is the standard classic >on the subject written by one of the founders, and it has had a recent >second edition. The second seems to be written for beginning mathematics students. It starts with simple examples of what sets and maps between sets are. This leads to the idea of what morphisms are and gradually introduces the ideas underlying categories. At one time I think I remember it was published as a notebook of loose pages, or maybe it was bound with those plastic fingers that you fit into perforations down the side of the page, perhaps to make this as inexpensive for students as possible. Then Cambridge sprang for a new version that is paperbound. My copy is buried somewhere and I can't find it at the moment to give a summary of the sections in the book. But for the audience that I think it was intended for I believe it could be a good book. For something that can be as slippery and intangible to a beginner as category theory easily is I think this could lead them to a place where they could begin seeing everything as categories. === Subject: Re: Category Theory books > Hi all, > I'm intrested in buying a good book about category theory. I would like to > buy (only)one of the following: > 1)Categories for the Working Mathematician > Saunders Mac Lane; > 2)Conceptual Mathematics: A First Introduction to Categories > F. William Lawvere, Stephen H. Schanuel . > I need a good piece of advice... I would really appreciate it. > TIA. > PS: other books about categories are welcome. While I'm not familiar with the latter book, the first is a classic IMO and is one I've used. (and one I probably ought to buy) === Subject: Re: help finding a closure of a set under som operations > hi... > need some help here... > my world consists of all finite sets of integers. > an inerval [a,b] is the set of all integers between a and b > (inclusive). > given two sets one can define three operations: interestion, addition, > and multiplication by constant. > intersection is the simple set intersection. > the addition of two sets is the set consisting of all sums of one > integer in the first set and the other in the second. > multiplication by a constant results in a set in which every element > is times an integer from the original set. > the family of all intervals is closed under intersections and > additions, as one can easily verify. > the family of all finite arithmetic sequences is closed under > intersections and multiplication by constants. > can you help me(send a ref, search keyword, share ideas, or give me > the result if known) finding the smallest family of finite integer > subset that is closed under all three operations, and includes all > intervals? > amit. Try this (sketch). You are looking at the set of finite subsets of N generated by the intervals and closed under pointwise sum, intersection and pointwise scalar multiplication. Call those sets constructible. I claim it consists of all finite subsets. Suppose every k element set is constructible. Let {a_1,...,a_{k+1)} be a set with a_1 < ... < a_{k+1}. Let n = a_{k+1} - a_1. Then the inductive hypothesis implies that {a_1,...,a_k} is constructible and certainly {n} is. Their pointwise sum is {a_1,...,a_k,a_{k+1}=a_1+n,a_2+n,...}. Now intersect the latter set with the interval [1,...,a_{k+1}]. === Subject: Re: help finding a closure of a set under som operations > hi... > need some help here... > my world consists of all finite sets of integers. > an inerval [a,b] is the set of all integers between a and b > (inclusive). > given two sets one can define three operations: interestion, addition, > and multiplication by constant. > intersection is the simple set intersection. > the addition of two sets is the set consisting of all sums of one > integer in the first set and the other in the second. > multiplication by a constant results in a set in which every element > is times an integer from the original set. > the family of all intervals is closed under intersections and > additions, as one can easily verify. > the family of all finite arithmetic sequences is closed under > intersections and multiplication by constants. > can you help me(send a ref, search keyword, share ideas, or give me > the result if known) finding the smallest family of finite integer > subset that is closed under all three operations, and includes all > intervals? > amit. > Try this (sketch). You are looking at the set of finite subsets of N > generated by the intervals and closed under pointwise sum, > intersection and pointwise scalar multiplication. Call those sets > constructible. I claim it consists of all finite subsets. Suppose > every k element set is constructible. Let {a_1,...,a_{k+1)} be a set > with a_1 < ... < a_{k+1}. Let n = a_{k+1} - a_1. Then the inductive > hypothesis implies that {a_1,...,a_k} is constructible and certainly > {n} is. Their pointwise sum is {a_1,...,a_k,a_{k+1}=a_1+n,a_2+n,...}. > Now intersect the latter set with the interval [1,...,a_{k+1}]. That {n} should have been {0,n}. What if you allow only positive integers? Gets much more complicated. You cannot get the set {1,3} for example. === Subject: simple math question. In my book, They resolve that SQRT(56) is equal to 2*SQRT(14). Correct, but what method would you use to come to that conclusion. crzzy1. === Subject: Re: simple math question. > In my book, > They resolve that > SQRT(56) is equal to 2*SQRT(14). > Correct, but what method would you use to come to that conclusion. > crzzy1. sqrt(56) = sqrt (2*2*2*7) = sqrt (2*2) * sqrt (2*7) = 2 * sqrt (14) General approach: To simplfy sqrt(n), express n as a product of prime factors. Wherever some prime factor appears two or more times (like 2 in this case), move out an even number of them under their own square root and simplify. For example, sqrt (2 * 3^7 * 5^2 * 17^3) - whatever that number is - will be = sqrt(3^6 * 5^2 * 17^2) * sqrt(2 * 3 * 17) = 3^3 * 5 * 17 * sqrt(102) === Subject: Re: simple math question. > In my book, > They resolve that > SQRT(56) is equal to 2*SQRT(14). > Correct, but what method would you use to come to that conclusion. > crzzy1. > sqrt(56) = sqrt (2*2*2*7) > = sqrt (2*2) * sqrt (2*7) > = 2 * sqrt (14) > General approach: > To simplfy sqrt(n), express n as a product of prime factors. Wherever some > prime factor appears two or more times (like 2 in this case), move out an > even number of them under their own square root and simplify. > For example, sqrt (2 * 3^7 * 5^2 * 17^3) - whatever that number is - will be > = sqrt(3^6 * 5^2 * 17^2) * sqrt(2 * 3 * 17) > = 3^3 * 5 * 17 * sqrt(102) Excellent explanation. crzzy1. === Subject: Re: SR consistency is crap. > Shame on you: > http://arxiv.org/abs/astro-ph/0304006 > http://arxiv.org/abs/astro-ph/0303346 > Why don't you take a break from these ng's. You post crap. It's > Kopeikin by the way. 'Kopkein' is what you think you have heard like > the other stuff you post here. > Mike > Could mike be trying to insinuate that the above reference say that the > speed of gravitational wave propagation is *not* the same as the speed of > light? If so shame on Mike! I'd like to quibble (it's fun). I think you should say c not speed of light, since in a medium light moves less than c. c is the speed of light in a vacuum. === Subject: Re: SR consistency is crap. >> J.E.: >> >> >> Then you do not even know what the PoR is. >> > >I'm sure you MEANT to say Principle of Relativity which means that >> >the laws of physics have the same form in all inertial frames of >> >reference. It's a stupid principle which the modern theory makes >> >unnecissary. It's a waste of time and just confuses people like you. >> >> Except for the part about androsleaze being confused, I disagree and >> I would say that your comment below is inconsistent with your comment about >> inertial frames. >What IS there to be inconsistent. I don't USE the PoR. I use >Minkowski geometry, so there are EVEN two things TO contradict each >other, just ONE thing, Minkowksi geometry. >> [...] >> >I was saying that Minkowksi geometry is based on set theory. SR is >> >based on Minkowksi geometry. >> Sure, but without the principle of relativity to connect that geometry >> to physics, all you have is minkowski geometry with no physical meaning. >> I could equally well say that any geometry is based on set theory, but >> in order to decide which one applies to the universe, I also need a >> connection to physical phenomena. That connection is the principle of >> relativity, since what it requires is that physics be unchanged by an >> infinitesimal displacement of the spacetime variables. >I disagree wholeheartedly and completely. What you need is a mapping >between physical events and minkowski geometry, and then a procedure >for computing future observables in the geometry, which can then be >mapped back to observations in reality. PoR is redundant, unecissary >and confusing. Drop it. The symmetries of Minkowski geometry do all >the work FOR you, you don't NEED to do anything else. Did Terrance's >clock tick? Find the event. Did Stella see it? Trace a line to find >out what event corresponds to Stella seeing it. What OTHER events >that Stella saw did see observe it BETWEEN, that is ALL you need, the >rest is fluff and if fluff confuses people, bag it. Body bag it. >[snip disparagements against Androcles] But that won't obviously tell you whether the resulting theory of physics is one that our universe feels compelled to obey. In Minkowski geometry, an inertial object's path is a straight line through the metric, and the properties of that line are the same regardless of whether or not such an object actually exists. The validity of the exercise depends on the idea that the existence of a real object, or the motion of a real object, has no effect on the lightbeam geometry. light along in their immediate vicinities (Fizeau), and even a reduction in lightspeeds (refractive index). Special relativity does not claim validity for particulate media ... be a particulate medium of some sort. If you are an astronaut, your moving twin ought to be able to declare that the speed of light is reduced where it begins to encounter the glass of your helmet visor, and that that moving glass also produces an offset in the nearby speed of light in your direction of motion. So, while Minkowski geometry might well describe how the principle of relativity would apply to mathematical non-particulate observers exchanging signals, the real geometry of real particulate objects exchanging signals would seem to be different. Mechanims are in play that alter local lightspeeds around real-life observers, that do not exist in special relativity. Start with an empty perfect Minkowski metric and throw a rock through the region, and the rock ought to distort the metric and leave a sort of streak running through the block of spacetime, like an optical imperfection in a block of glass. relativity, then we need a theory that works consistently in curved-spacetime, and a metric for the description of inertial physics that is /not/ purely Minkowskian. Sure, we probably want to start with a flat background, but we need the physics played out against that flat background to incorporate localised velocity-dependent distortions. Q: Does SR cope robustly with this sort of velocity-dependent curvature? A: No, apparently not Q: Does GR currently incorporate these effects? A: Apparently no, not yet. Because people insist that it has to reduce to SR. Q: What research do we currently have on methods of amending SR and the Minkowski metric to incorporate velocity-dependent curvature? A: Apparently it can't be done, we seem to need a new type of metric and a new associated theory of how the deviations from the SR metric can also be relativistic. Q: What progress have we made so far on identifying these new relativistic rules and constructing the next-generation theory? A: Apparently none. Instead, we find it easier to insist that we know that spacetime is flat, and that SR/Minkowski is therefore known to be correct (even though experiment seems to say otherwise) ,and that these issues therefore do not need to be addressed. So, I think that we can play around with Minkowskian geometry to our heart's content, but what we discover still won't necessarily be real physics, because in real life, that's not how real matter seems to behave. Saying that these deviations form the idealised metric are small and don't need to be considered is not a legitimate response, IMO, that would be like saying that small deviations from energy conservation in a theory would be unimportant. If there are localised velocity-dependent deviations from flat spacetime with real physical matter, we have to know how to deal with them relativistically. With the Minkowski geometry generated by SR, we are using an idealised description of reality whose form we know breaks down when the idealisation is not perfect (usually a sign of a pathological theory), and the usual sanity test that we'd do in this situation -- exploring how the theory would come out if the idealisation was lost -- does not seem to have been done. We can attempt to come up with a more realistic theory of relativity by discarding the Minkowski geometry and relationships and deriving a new theory of relativity from scratch -- but if we followed your suggestion of keeping the Minkowskian geometry and discarding the PoR as superfluous, then we'd appear to be cutting ourselves off from all hope of progress in this area. IMO we'd then be locking ourselves into a theory that seems to be geometrically incompatible with real-life physics, and throwing away the key. IMO, if there's a possibility that we might have made a mistake in adopting SR so wholeheartedly, then we are obligated to check -- either to put SR onto a new stronger footing, or to isolate the problem areas and the parts of physics that would have to be changed in the successor theory. Until that's done, we don't know for sure whether SR is deserves to be considered as hardcore physics, or whether its just a sidebranch that gives a Euclidean thumbnail sketch of some intrinsically non-Euclidean physics, and which provides a crude approximation of parts of the final theory, without being a full subset of it. Minkowskian geometry may be self-consistent as a piece of abstract geometry, but that does not automatically make it physics. It also does not automatically make it consistent with more general correct physics theory, in which some of the idealisations applied within Minkowski geometry might be incompatible with the more general principles being applied (eg GR says that localised energy concentrations warp spacetime, SR requires an arbitrarily-high metric -- one could argue that this should not be a legal situation, on principle, and that perhaps a full general theory ought not to reduce to SR, on principle). Physics theory based upon Minkowski's metric may still turn out to be consistently right in some respects, but consistently wrong in others. To find out, we probably have to step outside the closed Euclidean mindset and see how the problem might look from outside. =Erk= (Eric Baird) : Just look at him. Square. The shape of evil. : -- Plankton, SpongeBob SquarePants === Subject: Re: SR consistency is crap. >> J.E.: >> >> Then you do not even know what the PoR is. >> > >I'm sure you MEANT to say Principle of Relativity which means that >> >the laws of physics have the same form in all inertial frames of >> >reference. It's a stupid principle which the modern theory makes >> >unnecissary. It's a waste of time and just confuses people like you. >> >> Except for the part about androsleaze being confused, I disagree and >> I would say that your comment below is inconsistent with your comment about >> inertial frames. >What IS there to be inconsistent. I don't USE the PoR. I use >Minkowski geometry, so there are EVEN two things TO contradict each >other, just ONE thing, Minkowksi geometry. >> [...] > >I was saying that Minkowksi geometry is based on set theory. SR is >> >based on Minkowksi geometry. > Sure, but without the principle of relativity to connect that geometry >> to physics, all you have is minkowski geometry with no physical meaning. >> I could equally well say that any geometry is based on set theory, but >> in order to decide which one applies to the universe, I also need a >> connection to physical phenomena. That connection is the principle of >> relativity, since what it requires is that physics be unchanged by an >> infinitesimal displacement of the spacetime variables. >I disagree wholeheartedly and completely. What you need is a mapping >between physical events and minkowski geometry, and then a procedure >for computing future observables in the geometry, which can then be >mapped back to observations in reality. PoR is redundant, unecissary >and confusing. Drop it. The symmetries of Minkowski geometry do all >the work FOR you, you don't NEED to do anything else. Did Terrance's >clock tick? Find the event. Did Stella see it? Trace a line to find >out what event corresponds to Stella seeing it. What OTHER events >that Stella saw did see observe it BETWEEN, that is ALL you need, the >rest is fluff and if fluff confuses people, bag it. Body bag it. > >[snip disparagements against Androcles] > But that won't obviously tell you whether the resulting theory of > physics is one that our universe feels compelled to obey. There simply is no way to THAT. The best you can hope is the do experiments the distinguish between model/theory pairs that make different predictions, and then everyone picks their favoriate model/theory pair that DOES match all observations. Anything more is impossible and so shouldn't be a standard. > In Minkowski geometry, an inertial object's path is a straight line > through the metric, and the properties of that line are the same > regardless of whether or not such an object actually exists. > The validity of the exercise depends on the idea that the existence of > a real object, or the motion of a real object, has no effect on the > lightbeam geometry. > light along in their immediate vicinities (Fizeau), and even a > reduction in lightspeeds (refractive index). Do you not understand the concept of a vacuum or are you just joking? > Special relativity does not claim validity for particulate media ... > be a particulate medium of some sort. If you are an astronaut, your > moving twin ought to be able to declare that the speed of light is > reduced where it begins to encounter the glass of your helmet visor, > and that that moving glass also produces an offset in the nearby speed > of light in your direction of motion. You can model light through non-vaccums in terms of straight lines in Minkowski geometry too, you just need more than one line. > So, while Minkowski geometry might well describe how the principle of > relativity would apply to mathematical non-particulate observers > exchanging signals, the real geometry of real particulate objects > exchanging signals would seem to be different. Um, does the phrase to within experimental erros mean anything to you. We are talking about 10W signals verus 6W signals, that's a huge difference, there is room for much errors, like extended bodies, not quite instanteous turnaround, etc. > Mechanims are in play > that alter local lightspeeds around real-life observers, that do not > exist in special relativity. That's nonsense. SR theory doesn't make ANY predictions without a material model. And if you make the RIGHT material model, then OF COURSE you can models to effects of glass, non instant turn arounds, extended bodies, etc. And obviously if you make a SIMPLER model that ignores those things then it won't be 100% accurate. So what? No big deal. Besides, right now I was discussing INTERNAL inconsistencies. > Start with an empty perfect Minkowski > metric and throw a rock through the region, and the rock ought to > distort the metric and leave a sort of streak running through the > block of spacetime, like an optical imperfection in a block of glass. You are a big GR believer aren't you? Yes, things affects things, for instance you can't have a laser be absorbed and emitted and yet be a single wavelength, so all my equal separation stuff is impossible, but GET THIS, for large amounts of time, the BIG effect 10 > 6 CAN be observed EVEN dispite the simplicity of the model because we can RESTRICT ourselves to the cases where the other effects are very small, get it? > relativity, then we need a theory that works consistently in > curved-spacetime, I was write you do believe in GR. Well get this, we can ALSO have a background spacetime flat. > and a metric for the description of inertial physics > that is /not/ purely Minkowskian. Sure, we probably want to start with > a flat background, but we need the physics played out against that > flat background to incorporate localised velocity-dependent > distortions. Of EITHER a field OR the metric, I know which I'd prefer, ... whichever has the eeasier mathematics of course. > Q: Does SR cope robustly with this sort of velocity-dependent > curvature? > A: No, apparently not Fallacy: Confusing models and theories. > Q: Does GR currently incorporate these effects? > A: Apparently no, not yet. Because people insist that it has to reduce > to SR. Interesting, but I've never EVER heard that before, so please elaborate and/or provide references. > Q: What research do we currently have on methods of amending SR and > the Minkowski metric to incorporate velocity-dependent curvature? > A: Apparently it can't be done, we seem to need a new type of metric > and a new associated theory of how the deviations from the SR metric > can also be relativistic. I keep imagining everything you say about curvature being about magnetism, you know that velocity dependant force. Rememeber how they made a FIELD theory out of magnetism. Too bad we can't do the same for gravity, wait! We CAN. Whew, you had me worried for a second. > Q: What progress have we made so far on identifying these new > relativistic rules and constructing the next-generation theory? > A: Apparently none. Instead, we find it easier to insist that we know > that spacetime is flat, and that SR/Minkowski is therefore known to be > correct (even though experiment seems to say otherwise) ,and that > these issues therefore do not need to be addressed. More references to experimental verification of SR from a person who can't distinguish a model from a theory. Go ahead, try and support your claims. > So, I think that we can play around with Minkowskian geometry to our > heart's content, but what we discover still won't necessarily be real > physics, because in real life, that's not how real matter seems to > behave. Yeah real electrons don't have the hyperfine corrections predicted by AFT, how silly of me. > Saying that these deviations form the idealised metric are small and > don't need to be considered is not a legitimate response, IMO, that > would be like saying that small deviations from energy conservation > in a theory would be unimportant. Saying that they have to be large enough to observe and predict IS a requirement, since we can't MAKE a theory match your PERSONAL FEELINGS about deviations. > If there are localised > velocity-dependent deviations from flat spacetime with real physical > matter, we have to know how to deal with them relativistically. The ONLY hard core proof of curvature (non-flatness) as opposed to field interaction would be a non-euclidean topology. Do you have evidence for that? > With the Minkowski geometry generated by SR, we are using an idealised > description of reality whose form we know breaks down when the > idealisation is not perfect (usually a sign of a pathological theory), > and the usual sanity test that we'd do in this situation -- exploring > how the theory would come out if the idealisation was lost -- does not > seem to have been done. People have done it, their theories are more complicated, and frankly since I've never heard of a necissity to USE them, I've never much studied them. > We can attempt to come up with a more realistic theory of relativity > by discarding the Minkowski geometry and relationships and deriving a > new theory of relativity from scratch -- but if we followed your > suggestion of keeping the Minkowskian geometry and discarding the PoR > as superfluous, then we'd appear to be cutting ourselves off from all > hope of progress in this area. Yeah, we'd never make a field theory based on observations, that didn't happen over ten years ago where the flat Minkoski metric is UNobservable and ONLY exists to MAKE THE MATH EASY. I must have imagined that. I better alert Cambridge that their paper don't exist I do wonder what those speakers were REALLY saying at those conflicts. wouldn't have known. Either that, or you are WRONG and just attacking a strawman because you don't understand the difference between a model and a theory. > IMO we'd then be locking ourselves into a theory that seems to be > geometrically incompatible with real-life physics, and throwing away > the key. Or making math easy. Sheese, since I allow CURVES and ARBITRARY lines, you could KEEP my model and use a different theory, and you could keep the theory and use another model. Sheesh, the advantage of the Minkowski geometry is to CLASSIFY and QUANTIFY certain lines in certain ways. So it's just to make the math easy. > IMO, if there's a possibility that we might have made a mistake in > adopting SR so wholeheartedly, then we are obligated to check -- > either to put SR onto a new stronger footing, or to isolate the > problem areas and the parts of physics that would have to be changed > in the successor theory. I won't stop you, if you want to do things that make the math harder that don't affect any observations or experiments we have done or plan to do, go for it. If it turns out we need it, I'll be grateful. But please stop harassing me and making unfounded allegations about my inability to add fields to a model to have more detail, because you're wrong, and I can do it. > Until that's done, we don't know for sure whether SR is deserves to be > considered as hardcore physics, or whether its just a sidebranch that > gives a Euclidean thumbnail sketch of some intrinsically non-Euclidean > physics, and which provides a crude approximation of parts of the > final theory, without being a full subset of it. I have no idea why you mention Euclidean anything with reference to Minkowksi based models. Do you not know what the words mean? Is this a typo? Or did I just miss something? > Minkowskian geometry may be self-consistent as a piece of abstract > geometry, but that does not automatically make it physics. Didn't say it did. But others were saying that certain SR predictions that were predictions of an ELEMENTARY Minkowki model WERE inconsistent, hence the motivation for this VERY SIMPLE model. > It also > does not automatically make it consistent with more general correct > physics theory, in which some of the idealisations applied within > Minkowski geometry might be incompatible with the more general > principles being applied (eg GR says that localised energy > concentrations warp spacetime, SR requires an arbitrarily-high > metric -- one could argue that this should not be a legal situation, > on principle, and that perhaps a full general theory ought not to > reduce to SR, on principle). Metric-fixation. You don't OBSERVE the metric, sorry to disappoint you. > Physics theory based upon Minkowski's metric may still turn out to be > consistently right in some respects, but consistently wrong in > others. To find out, we probably have to step outside the closed > Euclidean mindset and see how the problem might look from outside. No, to find out each side has to make models and theories and then we have to compare to observations. I choose to make models and theoires where the math is EASY for ME to do, if it's so hard that I can't make either then I don't accomplish much of ANYTHING. So GO AHEAD and do the math that is HARD for ME, and let's COMPARE to observation. But saying A PRIORI how metric must be related to observation in MY models and MY theories is CHEATING. Make your best model, and compare to my best model, anything else is less than honest. === Subject: Re: SR consistency is crap. >>I'd rather teach QM from the start, and get special relativity as a >>limiting case of that, and then Newtonian dynamics as a special case >>of that. >> How do you expect to get special relativity as a limiting case of QM? >> David >comes as a limiting case. Many of the so-called quantum corrections >are actually present in classical SR, so in a sense QM has always been >SR theory. When it comes to gravitational horizons, it seems that the NM energy relationships lead to a classical model that agrees (qualitatively, perhaps also quantitatively) with QM, whereas the SR relationships lead to GR's current description of the horizon as being inescapable, which is apparently not compatible with QM. So in this sense, it does rather seem as though perhaps QM has always been /incompatible/ with SR theory. ;) (SR's concept of how a lightspeed barrier works seems to be too clean to be compatible with QM, GR then inherits certain SR conventions and converts that clean lightspeed barrier into a clean event horizon, which makes the conflict more obvious -- according to QM, information should be able to bleed outwards through the horizon, current GR says that it can't). =Erk= (Eric Baird) : You can't see me, I have my eyes shut! === Subject: Re: SR consistency is crap. >I'd rather teach QM from the start, and get special relativity as a >>limiting case of that, and then Newtonian dynamics as a special case >>of that. > How do you expect to get special relativity as a limiting case of QM? > David >comes as a limiting case. Many of the so-called quantum corrections >are actually present in classical SR, so in a sense QM has always been >SR theory. > When it comes to gravitational horizons, it seems that the NM energy > relationships lead to a classical model that agrees (qualitatively, > perhaps also quantitatively) with QM, whereas the SR relationships > lead to GR's current description of the horizon as being inescapable, > which is apparently not compatible with QM. I have absolutely no idea what you are talking about. First, because of time dilation I've never actually seen anything cross an event horizon and I doubt can, so by Hawking radition the black hole evaporates before you reach it. So the only information inside an event horizon is from the initial collapse and that information is sent out during the collapse. Plus you can get gravitational theories in a flat SR metric, and the fields can move at any speed relative to the background space, so I don't get your concerns about that either. > So in this sense, it does rather seem as though perhaps QM has always > been /incompatible/ with SR theory. Have you studied a serious SR based model of QM with gravity, there are more than a few to pick from you know, and I don't know any published ones that create problems like you mention. > (SR's concept of how a lightspeed barrier works seems to be too > clean to be compatible with QM, GR then inherits certain SR > conventions and converts that clean lightspeed barrier into a > clean event horizon, which makes the conflict more obvious -- > according to QM, information should be able to bleed outwards through > the horizon, current GR says that it can't). Current GR is a HORRIBLE place to do QM, so use a quantum gravitational theory instead. There are quantum gravitational theories that match all present data AND are even more restrictive than GR since they restrict topologies. === Subject: Re: SR consistency is crap. >> >> I disagree. SR is about a proposed symmetry of nature. >> Explain the twin paradox then. By the way, introducing accelerations >> or boosts is a departure from SR and a move into the Dynamics domain. >I'd be happy to explain the twin paradox. Start with the financial >twin paradox that I just explained to eleaticus. And I disagree >about accelerations and boosts being outside SR, altough I understand >why people ignore it. SR is a theory about how the symmetries of >spacetime are similar to the symmetries of Minkowksi geometry. That's >why the financial twin paradox using an indefinite quadratic form to >determine the rate at which money is sucked out our your pocket and >sent to someone else. >> You got a problem right from the start. SR came to >> challenge the Copernican view of the world (see Hans Reichenbach, The >> Philosophy of Space and Time) the cornerstone of Newton's dynamics. >> Ptolemaic and Copernican systems are kinematically equivalent >> according to Relativity. Yet, while admitting SR, the educational >> status quo refuses to drop Newtonian dynamics, claiming there is ample >> emprirical evidence to support it. Actually, sicne Newton's laws are >> mere tautologies, in non-relativistic limits they will always conform >> to experiment. >> >> Newtonian dynamics are not dropped. They are derived locally, you >> already conceded that below that they are local tautologies. A >> dynamic isn't much without a law about elementary forces. >> >> I took Physics 101 and Modern Phycsi 101 together during the same >> semester as soon as I started undergraduate. The mentor protested and >> argued I should take Modern Physics (Relativity) after I take >> Classical Mechanics, because that was a prerequisite. I protested, I >> argued it must be my choice not theirs. I won. I got a A- in Classical >> and an A+ in Modern. >> >> Good for you. I didn't even realize I liked physics until Modern >> Physics (which for us covered QM, solid state, Stat mech, etc. too) I >> just took it as an easy class since I was good at math. >> Yesy, interesting to see that while it is an easy subject for some >> people it is beyond comprehension for others. >I think it's bad teaching. If I had a good background to easily >understand a fixed way of teaching the subject, that doesn't make the >subject inherently hard for others or easy for me, in means by >background and the teaching lined up well. Funnily enough, I found that it was actually very easy to explain GR principles to complete physics newbies, with a suitable choice of words. You just say things like, increasing the strength of gravity in a region seems to make makes space seem more dense and time more rarefied, so that clocks there tick more slowly ... so if we want to create a map of the light-distances in a slice through the region, so that the distances in the map correspond to the distances in the region, we have to extrude the map in order ot be able to cram in the extra space (produce diagram of gravity-well, with a flourish). People seem to get that. You might get an occasional Gravity makes local time appear to go by slower? Really? Yup! Oh, Okay then But try to explain SR time dilation to the same person, and they'll usually get upset and insist that the thing is rubbish. So untrained people seem to be intuitively happier with supposedly advanced ideas like spacetime curvature than simple ones like SR's Minkowski metric (I found). They seem to find most of the GR-ey principles easier to visualise and accept, stars bend light or gravity slows time or rotating stars pull stuff around with them seem to be easier concepts to take onboard than moving astronauts age slower than each other. Maybe schools should teach the principles of general relativity first, and leave SR to the more advanced students. ;) >> I think this is the way to go. Teach students everything together as >> competing theories and not relativity as an advance extension to >> Newtonian Dynamics. >> >> I'd rather teach QM from the start, and get special relativity as a >> limiting case of that, and then Newtonian dynamics as a special case >> of that. >> Well, you are talking 1000 years from now when we will understand >> better how quantum uncertainty gives rise to macrocosmic certainty. >> For now, such approach can result only in confusion. >You seem to have strong opinions about things you obviously don't >know. Bohm did a good job of explaining maro-certaintanty as a >was decades ago. Teaching is always behind the cutting edge. I was taught at school that Newton's prism experiment proved that light was composed of seven colours, and we were made to memorise the seven colours and conduct the experiment ourselves and see the seven colours ourselves. Then we were given coloured filters to play with and were taught that white light was actually composed of three distinct colours, and we were shown how to conduct expeirments to prove /that/. No wonder the poor kids were a bit bewildered. I survived those classes by being arrogant enough to assume that if something sounded wrong it was probably bull. Hopefully anyone going through that syllabus with a real talent for physcs woudl have realised that thge problem was not with them but with the teaching, but I do sometimes wonder how many adepts leave physcs because they think,wrongly, that their misgivings about certain subjects are becuase they arenlt good at the subject. If the people taking up the subject are predominantly people who are more prepared to suspend disbelief than the norm, then that might not be good for the subject. I suppose the people who already have a grounding in physcs before they take the class (eg family background), or those who are pig-headed enough to believe that they are right and the system is wrong, may still get through. I do notice that a strangely high percentage of physics people seem to have physicists or teachers as parents or as elder siblings etc, Perhaps a support network makes it easier for one to survive introductory physics classes without having one's brain scrambled. >> But remember, you still have a problem. SR+GR are >> axiomatic systems. No different from Euclidean geometry in that >> respect. You gain understanding of the theories early on but also you >> raise the doubt in their foundations. >> >> Huh? You want to avoid doubt? Science has doubt, predictions are >> made, the predictions can be compared to data, they might pass or >> fail. >> >> Eventually, someone will lose. >> >> Only theories can lose, not people, please explain more what you mean. >> I meant that popularization of SR, GR will turn out more questioning >> and eventually abandoment. This is the fate of every theory that >> becomes pupolarized. >I seriously doubt that. I think that when the correct SR theory is >finally popularized the incorrect alleged SR theories will FINALLY >be adandoned, and not a moment too soon, yeak! I think that perhaps part of why SR is such a bad subject is that there are probably a lot of pro-SR people strenuously insisting that what they were taught is right, even when it isn't. I think the Penrose/Terrell case illustrates this nicely, we had a situation where professional physcists had supposedly been pushing a wrong result for decades, even though it disagreed with the math, because they had been /taught/ that moving objects are seen to be contracted under SR. Social conditionaing overrode mathematics and geometry. Penrose was a mathematician who snuck his result out as a non-peer-revirewed letter in an obscure local journal, Terrell was an undergraduate who struggled for years to get his paper through peer review. So the matter was eventually tackled by a newbie and a mathematican, not by mature physics people. For some reason, these things always seem to end up being corrected by outsiders, the highly-trained mainstream don't seem to be willing or able to do it themselves. >> Those that are afraid of ending up with a loser keep SR+GR as advance >> subjects. >> >> I'd love to have GR and SR as grade-school subjects, does that mean >> I'm not afraid of losing? I don't know what you mean by afraid of >> losing or lose, I do hope my predictions match the data, that's >> because if it's easy to be wrong, so why bother learning a theory if >> it makes wrong predictions? >> >> Remember what happened to Euclidean geometry? >> >> What happened? It's still around! >> >> As soos as >> they started teaching it, thousands attempted to disprove the Playfair >> axiom (parallel lines never meet). The result was a >> relative-consistency with spherical and hypoerbolic geometries. >> >> Elemetary euclidean geometry was proved consistent, you can drop >> relative-consistency and just say consistent if what you mean is >> that elementary hyperbolic geometry is as consistent as elementary >> Euclidean geometry. And if you mean something other than elementary, >> then that useally means bringing in sets and once you've done that >> then all become equiconsistent with set theory. >> I agree. But I hope you recall that the consistency of Euclidean >> geometry was under question for some time. >Well, the same grounding to elementary euclidean geometry applies to >elementary hyperbolic geometry, and I can do SR predictions with >ELEMENTARY hyperbolic geometry. If you want not instantaneous >accelerations, then I need FULL hyperbolic geometry, which is as >consistent as FULL euclidean geometry. Mmmm, but the GR experience has hopefully taught everyone that mathematical consistency is not necessarily enough -- in physics, a structure also has to be appropriate. If basic Euclidean geometry is consistent, it doesn't neccessarily mean that it is appropriate or sufficient for describing lightbeam geometry in the presence of gravitational fields or accelerating masses (until you start adding dimensions), or even relativiely moving masses. So a theory or model can be completely consistent in its own (artificial) context, but physcally wrong when it comes to attempts to use it to model the behaviour of the real world. Mathematical theorems can appear to be rock-solid, but still be hopelessly wrong (or misunderstood) in the context of attempts to construct physical theory. Nuances of language can be incredibly important. >> The >> same will happen to SR+GR as soon as they became a target of the >> masses. >> >> What will happen to SR+GR (which is GR, unless you meant the same >> will happen to SR and the same will happen to GR as soon as each >> becomes a target of the masses)? And what masses are currently kept >> away from it? >> >> The majority of people fails math and you should know that. The >> majority works using a common sense basis. Anything as advanced as GR >People fail at math because of math is taught badly. Did you know >that you can multiply vectors? Life is easier with that, and hard >without it. I was a math whizz as a pre-adolescent. I nearly failed my eleven plus test because it contained a set of letter-number substitution problems, and I couldn't crack the rolling encryption model they were using. Without having any math background I spent that test filling sheets of paper running through the different conceivable encryption systems that they might have been using (fractional number bases, etc) until I finally came up with the idea of using prime number sequences as a key, and I spent a quarter of the test period working on the problem of how to arbitrarily quantise a range of coordinate-free number surfaces representing different forms of number system before realising that it was going to take me at least another half hour and there were only ten minutes left. So I cheated with the first example (CAT=XXX, DOG=XXX, COG=???), by just doing a dumb substitution, and then realised that every other example worked the same way. I distinctly remember feeling panicky and inadequate, looking around at a hall filled with other elevenyearolds scratching away, and thinking I was the only one there who didn't know how to crack the output of a a prime number encoding system. Anyhow, I only mention this because after starting secondary school and spending years laboriously practicing cross-multiplication day after day (and being accused of cheating when I didn't include working out, which was an alien concept to me at the time), the math part of my brain shrivelled up until I eventually had such a strong mental block to do with anything mathy that I had to drop out of school because I could no longer even add and subtract reliably. My brain was just stalling on me. I had used to go through science subjects just writing down the first number that came into my head and getting the answers right. Then the magic stopped working. The power of aversion therapy I guess. :( (I still can't do math any more, but I have some dim memories of what it was lke when I could, and I remember being distinctly unimpressed with what was inthe textbooks). > will create a reaction and eventually will be dropped, creating more >> problems than the intended solution. Another solution recently >> proposed is to make physics a subject for selected few. >I will concede that fixing math eduation is paramount to fixing >physics education, there isn't a strong reason not to do both by >fixing the education of geometry to be incorporated the cutting edge >from the 19th century, which frankly we have yet to do and that's just >sad sad sad. >> The current approach is to keep the theory for the few and >> feed the supposed prediction to the masses via the media. Silly but >> works due to hype. >> Mike >> >> I actively try to teach physics at as early a stage as possible, and >> books on relatively are freely sold in stores, there is no conspiracy >> going on. Anyone can learn. Heck, I'm on Usenet because I want to >> help people learn SR, among other reasons, this is a public forum >> where anyone can read it. >> You are wasting your time. If 95% of your student can remember all >> three Newton's laws 2 years after graduation from any level exept grad >> school then you have accomplished your task. You probably aiming for a >> small % while frustrating the rest of the class and ending up with the >> opposite that what you aim, i.e. making people dislike physics. >Physics education should start out with modeling in general, and >comparing models to data, with computers you can bury most of the math >until a later epistological phase that COULD be reserved for the few >that care, where the class discusses the foundations of the models >used earlier. If someone isn't going to remember the basics of how to >make or test a model, then you shouldn't BOTHER teaching science at >all, just drill job skills instead. IMO most people publishing material on SR testing don't seem to display much ability when it comes to being able to correctly compare models. When you see experiment after experiment looking for transverse redshifts, finding them, and then declaring that this result is a significant proof of SR : ... because classical theory does not predict transverse redshifts , one is driven to despair. In the experiments involved, almost every old theory in the books would have predicted a redshift, they just wouldn't normally have /called/ a transverse shift (I think Oliver Lodge referred to the aether theories' transverse effects as something like false Doppler) So ... those experimenters needed to be tapped on the shoulder and quietly told that either the supposedly null transverse predictions for other theories didn't relate to the sorts of experiments that they were actually carrying out, or, if we used the word transverse to mean transverse as measured in the lab frame, the only way that the statement would be correct would be if classical theory was defined as being a very limited range of theories that arguably did not include the major C19th theories, or even Newtonian mechanics. Which would make these experiments slightly limited in what they could tell us about how SR compared to other theories. People who think linguistically are possibly more prone to making mistakes when working across different theories with different understood meaning to words, an equation is not physics without that additional context. The critical legalistic small print is usually not actually written down, it's instead often assumed that the physics experts already know enough to interpret the understood meanings and usages appropriately without screwing the thing up, and sometimes they get it wrong. Consider this logic trap: - Theory Proponent: : We know that the standard of living is far higher in the US than : in France, because it is known, as a fact, that car ownership is : high in the US, but that nobody in France owns or drives a car. Sceptic: : But I have seen photographs of French roads filled with cars! Theory Proponent: : But those are not really cars! In France, they are more correctly : referred to as voitures, or autos. So I stand by my statement, : car ownership in France is zero, or at least negligible, and the : original statement holds. , and compare that misunderstanding with: SR Proponent: : We know that SR is the right theory, because SR predicts : transverse redshifts, no other theory predicts transverse : redshifts, and transverse redshifts have been found : experimentally. Sceptic: : But almost every older theory would have predicted some sort : of redshift effect in those SR experiments! SR Proponent: : Ah, but those would not have been properly referred to as : /transverse/ redshifts, because according to test theory, in : order to measure a transverse frequency under those other : theories, one should point the detector in a different direction. : So the original statement stands: We conducted our experiment : according to SR protocols, we found a transverse redshift, and : since no earlier theory predicts transverse redshifts, our result : shows that SR is right and earlier theory was wrong. It's like standing up at a food industry conference and stating that The research proves that Italy is the only place in the world where they make Bolognese sauce for pasta, and then when someone produces a can of the stuff made by Heinz in the US, saying, well that's not really bolognese sauce, because in order to earn that name, it has to be made in Bologna, Italy. It makes the original statement a bit pointless. With SR testing this sort of illegal switching of implied definitions in mid-argument seems to be quite common. --------- BTW, This is one of the reasons why I'm still an SR sceptic (even though I count myself as a harcore relativist): apart from the fact that many or most of the experimental SR proofs seem to have been badly compromised, and that there still seems to me to be at least one major theoretical loophole that hasn't yet been dealt with (and which IMO should have been tackled decades ago), it's the sheer badness of most of the analysis. I don't honestly believe that physics people are usually this bad, without good reason. If SR was wrong, and the community was heroically struggling along with a reference theory that didn't work properly, then perhaps those analyses might be the best that could be achieved without exposing apparent conflicts with SR, and perhaps then we'd have a logical reason why the analyses seem to be so consistently compromised. If SR is /not/ a correct physcal theory, then perhaps these repeated screwups make some sort of sense and tell a story. Otherwise, if SR really is correct, I suppose the explanation would have to be that the whole community is just hopeless at these sorts of calculations, period. :( I prefer the heroic interpretation. :) =Erk= (Eric Baird) : Time is money. Time is not money. : Space and time are interchangeable. Space and time are not interchangeable. : Special relativity is true. Special relativity is not true. === Subject: Re: SR consistency is crap. >> I took Physics 101 and Modern Phycsi 101 together during the same >> semester as soon as I started undergraduate. The mentor protested and >> argued I should take Modern Physics (Relativity) after I take >> Classical Mechanics, because that was a prerequisite. I protested, I >> argued it must be my choice not theirs. I won. I got a A- in Classical >> and an A+ in Modern. > Good for you. I didn't even realize I liked physics until Modern >> Physics (which for us covered QM, solid state, Stat mech, etc. too) I >> just took it as an easy class since I was good at math. > Yesy, interesting to see that while it is an easy subject for some >> people it is beyond comprehension for others. >I think it's bad teaching. If I had a good background to easily >understand a fixed way of teaching the subject, that doesn't make the >subject inherently hard for others or easy for me, in means by >background and the teaching lined up well. > Funnily enough, I found that it was actually very easy to explain GR > principles to complete physics newbies, with a suitable choice of > words. > You just say things like, increasing the strength of gravity in a > region seems to make makes space seem more dense and time more > rarefied, so that clocks there tick more slowly ... so if we want to > create a map of the light-distances in a slice through the region, so > that the distances in the map correspond to the distances in the > region, we have to extrude the map in order ot be able to cram in the > extra space (produce diagram of gravity-well, with a flourish). I'd be interested in seeing that work in practise, you can really get newbies to see that a parabola in space is a geodesic in spacetime? > People seem to get that. You might get an occasional Gravity makes > local time appear to go by slower? Really? Yup! Oh, Okay then > But try to explain SR time dilation to the same person, and they'll > usually get upset and insist that the thing is rubbish. But how do you know you aren't using the wrong words? By thesis is that SR is taught badly. > So untrained people seem to be intuitively happier with supposedly > advanced ideas like spacetime curvature than simple ones like SR's > Minkowski metric (I found). They seem to find most of the GR-ey > principles easier to visualise and accept, stars bend light or > gravity slows time or rotating stars pull stuff around with them > seem to be easier concepts to take onboard than moving astronauts age > slower than each other. And there is an example of bad teaching. Astronauts don't age more slowly than each other, you should discuss what the astronauts see, and if they see each age slower then faster than themself, then who aged more depends on what percentage of the observations were slower versus faster. > Maybe schools should teach the principles of general relativity first, > and leave SR to the more advanced students. ;) Maybe schools should teach things properly. >> I think this is the way to go. Teach students everything together as >> competing theories and not relativity as an advance extension to >> Newtonian Dynamics. > I'd rather teach QM from the start, and get special relativity as a >> limiting case of that, and then Newtonian dynamics as a special case >> of that. > Well, you are talking 1000 years from now when we will understand >> better how quantum uncertainty gives rise to macrocosmic certainty. >> For now, such approach can result only in confusion. >You seem to have strong opinions about things you obviously don't >know. Bohm did a good job of explaining maro-certaintanty as a >was decades ago. Teaching is always behind the cutting edge. > I was taught at school that Newton's prism experiment proved that > light was composed of seven colours, and we were made to memorise the > seven colours and conduct the experiment ourselves and see the seven > colours ourselves. > Then we were given coloured filters to play with and were taught that > white light was actually composed of three distinct colours, and we > were shown how to conduct expeirments to prove /that/. > No wonder the poor kids were a bit bewildered. > I survived those classes by being arrogant enough to assume that if > something sounded wrong it was probably bull. Good, most things are oversimplifications to the point where they are wrong. > Hopefully anyone going through that syllabus with a real talent for > physcs woudl have realised that thge problem was not with them but > with the teaching, but I do sometimes wonder how many adepts leave > physcs because they think,wrongly, that their misgivings about certain > subjects are becuase they arenlt good at the subject. > If the people taking up the subject are predominantly people who are > more prepared to suspend disbelief than the norm, then that might not > be good for the subject. I suppose the people who already have a > grounding in physcs before they take the class (eg family background), > or those who are pig-headed enough to believe that they are right and > the system is wrong, may still get through. You don't have to be simply pig-headed, you can just call your teachers on anything that doesn't make sense and then disregard their assertion if they don't back them up well enough. > I do notice that a strangely high percentage of physics people seem to > have physicists or teachers as parents or as elder siblings etc, > Perhaps a support network makes it easier for one to survive > introductory physics classes without having one's brain scrambled. I took introductory physics by not caring about the subject. >> But remember, you still have a problem. SR+GR are >> axiomatic systems. No different from Euclidean geometry in that >> respect. You gain understanding of the theories early on but also you >> raise the doubt in their foundations. > Huh? You want to avoid doubt? Science has doubt, predictions are >> made, the predictions can be compared to data, they might pass or >> fail. > Eventually, someone will lose. > Only theories can lose, not people, please explain more what you mean. > I meant that popularization of SR, GR will turn out more questioning >> and eventually abandoment. This is the fate of every theory that >> becomes pupolarized. >I seriously doubt that. I think that when the correct SR theory is >finally popularized the incorrect alleged SR theories will FINALLY >be adandoned, and not a moment too soon, yeak! > I think that perhaps part of why SR is such a bad subject is that > there are probably a lot of pro-SR people strenuously insisting that > what they were taught is right, even when it isn't. > I think the Penrose/Terrell case illustrates this nicely, we had a > situation where professional physcists had supposedly been pushing a > wrong result for decades, even though it disagreed with the math, > because they had been /taught/ that moving objects are seen to be > contracted under SR. > Social conditionaing overrode mathematics and geometry. > Penrose was a mathematician who snuck his result out as a > non-peer-revirewed letter in an obscure local journal, Terrell was an > undergraduate who struggled for years to get his paper through peer > review. > So the matter was eventually tackled by a newbie and a mathematican, > not by mature physics people. > For some reason, these things always seem to end up being corrected by > outsiders, the highly-trained mainstream don't seem to be willing or > able to do it themselves. I've considered that there might be large numbers of physicists that think clearly about SR, but just assume that everyone else does too and hence don't get invovled in debates about it. The irony is that all of these physicists could compute what people SEE correctly, if only they CARED enough to do so. >> Those that are afraid of ending up with a loser keep SR+GR as advance >> subjects. > I'd love to have GR and SR as grade-school subjects, does that mean >> I'm not afraid of losing? I don't know what you mean by afraid of >> losing or lose, I do hope my predictions match the data, that's >> because if it's easy to be wrong, so why bother learning a theory if >> it makes wrong predictions? > Remember what happened to Euclidean geometry? > What happened? It's still around! > As soos as >> they started teaching it, thousands attempted to disprove the Playfair >> axiom (parallel lines never meet). The result was a >> relative-consistency with spherical and hypoerbolic geometries. > Elemetary euclidean geometry was proved consistent, you can drop >> relative-consistency and just say consistent if what you mean is >> that elementary hyperbolic geometry is as consistent as elementary >> Euclidean geometry. And if you mean something other than elementary, >> then that useally means bringing in sets and once you've done that >> then all become equiconsistent with set theory. > I agree. But I hope you recall that the consistency of Euclidean >> geometry was under question for some time. >Well, the same grounding to elementary euclidean geometry applies to >elementary hyperbolic geometry, and I can do SR predictions with >ELEMENTARY hyperbolic geometry. If you want not instantaneous >accelerations, then I need FULL hyperbolic geometry, which is as >consistent as FULL euclidean geometry. > Mmmm, but the GR experience has hopefully taught everyone that > mathematical consistency is not necessarily enough -- in physics, a > structure also has to be appropriate. > If basic Euclidean geometry is consistent, it doesn't neccessarily > mean that it is appropriate or sufficient for describing lightbeam > geometry in the presence of gravitational fields or accelerating > masses (until you start adding dimensions), or even relativiely moving > masses. What was originally being debated was the internal consistency of SR predictions. Therefore a constructed a slim version of SR without frames or curves, just points and straight lines, which is KNOWN to be consistent, and hence the internal consistency of the SR predictions in my slim model should safe and garanteed. And accelerating bodies can be handled to arbitrary precision. People forget that the accelerating results of SR are KNOWN, and that THAT is HOW we get the gravitational results in GR. > So a theory or model can be completely consistent in its own > (artificial) context, but physcally wrong when it comes to attempts to > use it to model the behaviour of the real world. > Mathematical theorems can appear to be rock-solid, but still be > hopelessly wrong (or misunderstood) in the context of attempts to > construct physical theory. > Nuances of language can be incredibly important. That's exactly what I'm doing, I'm contrasting a 4-d Newtonian model that uses the manhattan metric with a minkowksi metric. Each is internally consistent, but I'm asserting that the minkowski one would match experiments, which I have GOTTEN to doing yet, because eleaticus and Androcles are acting afraid of the model and I don't think you guys are helping at all. > BTW, This is one of the reasons why I'm still an SR sceptic > (even though I count myself as a harcore relativist): apart from the > fact that many or most of the experimental SR proofs seem to have been > badly compromised, and that there still seems to me to be at least one > major theoretical loophole that hasn't yet been dealt with (and which > IMO should have been tackled decades ago), it's the sheer badness of > most of the analysis. > I don't honestly believe that physics people are usually this bad, > without good reason. How many people want to test SR for a living? Most good people do that USES the results of SR as a necissary component rather than testing SR as an end to itself. > If SR was wrong, and the community was heroically struggling along > with a reference theory that didn't work properly, then perhaps those > analyses might be the best that could be achieved without exposing > apparent conflicts with SR, and perhaps then we'd have a logical > reason why the analyses seem to be so consistently compromised. It's just not consistent, and there aren't problems with SR theory, but there *might* be problems with that *some* people think is SR theory, which as I'm saying is a problem for physics education to fix. > If SR is /not/ a correct physcal theory, then perhaps these repeated > screwups make some sort of sense and tell a story. > Otherwise, if SR really is correct, I suppose the explanation would > have to be that the whole community is just hopeless at these sorts of > calculations, period. These are fallacies that everyone screws up. Physicis education literature discusses screw ups and how to avoid them in teachers as well as students. For instance look at me, I'm not special in any way, just a normal educator, but I don't make these erros you accuse other people of. And there are tons of people like me. I do tend to find frames bad news, for instance if one person hops frames, then the distant events that were called *simultaneous* change in mid hop, so clearly the notion of simultaneous is rather limited in it's use. === Subject: Re: SR consistency is crap. <309c03F2sn568U1@uni-berlin.de>: >> How stupid are you going to get? Now you claim Einstein's theories are >> physical theories? >Any theory that leads to testable quantitative predictions about what >happens in the world, under specified circumstance is a physical theory. >The word physics in a physics theory pertains to the domain of >application, not the methodology. >GTR has not yet been falsified by empirical means. It's difficult to say for sure, I think that depends on how the dark matter thing pans out. Dark matter was only postulated to explain why current GTR predictions don't match the available data, so if suitable dark matter is found to /not/ be out there, we might decide, retrospectively, that GTR was discovered to be empirically incorrect years ago. History might decide that dark matter was a brilliant prediction, or was a desperate attempt to salvage a theory that had been experimentally falsified. In a few decades time we might have a better perspective on this. ----------- I guess it's also difficult to say when a theory is really falsified experimentally, except with hindsight -- if an experimental result goes against a theory, does subsequent history then say that the theory was officially falsified on the date when that result got through peer review? What if it wasn't taken seriously at the time, or wasn't considered good enough? Or is the theory only falsified on the date when most physicists /believe/ it to have been falsified? This definition is probably more practical in that it allows people at the time to come up with a solid definition of whether a theory is falsified or not, but then we are talking about a value judgment based on a spread of evidence and the prevailing social belief systems, rather than on just hard experimental results. Perhaps we could refer to the theory being discredited than falsified (a theory can be discredited, due to human error, and still be physically correct). Although its nice to think that current experiments support our current theories well, history tells us that this sort of confidence can sometimes be ill-founded. We can /now/ say with the luxury of hindsight that the Michelson/Morley experiment falsified simple crude fixed-aether theory, but contemporary physicists might have felt entitled to say that nobody really took the M&M experiment seriously, and that if you only considered proper independently-verified results, the M&M result didn't count. Nobody else at the time was able to replicate Michelson's figures - Einstein attributed this to Michelson's exceptionally keen eyesight, which (if true) effectively meant that M&M's extreme precision was based partly on Michelson's being able to see things that his colleagues couldn't. (!) Couple this with the fact that M&M was carried out partly in response to Lorentz pointing out that Michelson had screwed up the math in his earlier experiment in a way that made the experiment look better than it actually had been, and the perception in Europe that any serious work on the subject would be done by people working at the major European institutions, and its easy to see why Michelson's result failed to set the world alight -- it may have looked like an inexplicable and irreproducible result by an incompetent researcher in some backwater institution, and the M&M experimental falsification of simple absolute aether theory may not have been obvious until much later. Considering the avalanche of papers being published on SR&GR nowadays, if someone finds an experiment crucis that blows one or both theories away, and they are in Michelson's situation, I'm not sure that we would notice, it might just stay buried in the pile. It's tricky. :( >What is a physics theory. It is a verbal artifact which is used to grind >out testable predictions of what will happen in the world, under >specified condition. It is a predictor. Nothing more, nothing less. >Bob Kolker Hmm. Not a bad description, IMO But I think physical theories also have a use as things that let us understand and appreciate (to some extent) how our universe operates around us. It's easier to get a grasp of how chemistry operates if one has a theory of the structure of the atom that explains how properties and valancies change from element to element. Even if the theory is wrong or artificial in some respects, it has mnemonic value and makes it easier to collate and assemble facts and patterns of data in our minds, which in turn makes it easier to devise newer or more complete theories. Most people in the general public don't need to know the precise atomic weight of oxygen to six decimal places, and don't need to know how many shells etc it has, but knowing the theory that the reason why things burn is because they react with oxygen in the air ... that's useful knowledge in day-to-day life. If you have a house fire, its easier to remember the instructions someone gave you years ago on how to put it out if you understood those instructions in the context of a theory, rather than as a completely random-sounding set of instructions with no obvious context. Theories reduce the number of isolated special-case facts that we need to carry around in our heads, as well as being able to let us predict new things, they can also be useful for the data-compression (and error-correction) of things that we already know. =Erk= (Eric Baird) : Who is Number One? : You are [,] Number Six. : -- The Prisoner === Subject: Re: SR consistency is crap. > <309c03F2sn568U1@uni-berlin.de>: >> How stupid are you going to get? Now you claim Einstein's theories are >> physical theories? >Any theory that leads to testable quantitative predictions about what >happens in the world, under specified circumstance is a physical theory. >The word physics in a physics theory pertains to the domain of >application, not the methodology. >GTR has not yet been falsified by empirical means. > It's difficult to say for sure, I think that depends on how the dark > matter thing pans out. > Dark matter was only postulated to explain why current GTR predictions > don't match the available data, so if suitable dark matter is found to > /not/ be out there, we might decide, retrospectively, that GTR was > discovered to be empirically incorrect years ago. > History might decide that dark matter was a brilliant prediction, or > was a desperate attempt to salvage a theory that had been > experimentally falsified. > In a few decades time we might have a better perspective on this. In a course on extra-galactic astronomy given by Paul Hodge at the University of Washington, I had the obvious pointed out to me. The universe isn't an experiment! You can measure things pertaining to it, but you can't control some parameters to isolate the effects of others. In other words, you can't really falsify a theory using astronomical observations unless the theory doesn't give itself any leeway to deal with how it differs from observation. The predictions of GR, or any other theory of cosmology, are obviously dependent on the observational capabilities of current astronomy. Indeed, astronomy is full of discoveries that involved finding things that explain anomalous observations such as the planet Neptune. John Anderson === Subject: Re: SR consistency is crap. Steve Harris sbharris@ROMAN9.netcom.com: >This is easily proved. The Shroedinger equation is the >non-relativistic limit of the Klein-Gordon *scalar* wave equation, >which explicitly does not include room for spin. However, the pauli equation (which is the schroedinger equation with the electron spin coupled to a magnetic field) _is_ the non-relativistic reduction of the dirac equation. The dirac hamiltonian is, HPsi = (alpha.p + beta m)Psi Using the canonical momentum, p -> p - (e/c)A, one obtains, H = alpha.(p - (e/c)A) + beta m (Where I've left out the scalar potential, which is irreevant here). The solutions can be decomposed into a two-component spinor, [Phi] Psi = [ ] [Chi] If you plug this in to the hamiltonian and uncouple the two spinors, you'll get one equation of the form: HPhi = (1/2m)(delta.p delta.p)Phi where delta are the pauli matrices. You then have the identity, delta.a delta.b = a.b + idelta . (a x b) and since p == p - (e/c)A, you get, HPsi = (1/2m) [ (p - (e/c)A)^2 - (ehbar/c)delta . B ] As you can see, the term containing delta.B contains the spin. This gives the correct magnetic moment (not including the anomalous part). === Subject: Re: SR consistency is crap. > Steve Harris sbharris@ROMAN9.netcom.com: > >This is easily proved. The Shroedinger equation is the > >non-relativistic limit of the Klein-Gordon *scalar* wave equation, > >which explicitly does not include room for spin. > However, the pauli equation (which is the schroedinger equation with > the electron spin coupled to a magnetic field) _is_ the non-relativistic > reduction of the dirac equation. COMMENT: Yes, I never said it wasn't. I missed your point, I suppose. There's this discussion which goes on periodically (ahem) in physics about the electron's spin. Somehow, since Dirac came out with an equation in which the electron's spin is taken into account and the g-factor doesn't have to be added in by hand, there was for quite some time in physics the myth that the electron's spin is somehow a quantum property, or at least a relativistic one. Neither is true. The electron's spin (spin angular momentum) is quantized in large multiples in a given direction, but only because it's so small. There's nothing there new, or that isn't true of anything else's angular momentum, or any other kind of angular momentum (for example orbital angular momentum). And just because Dirac's equation happens to be relativistic, doesn't mean the relativistic treatment does anything special about spin. The fact that the non-relativistic limit of Dirac is Pauli, proves that quite well. The spin energy states of the electron (as non-degenerately appear as separate wave functions in external fields) are described if the equation leaves room for them. Which Pauli's does. Which even the Shroedinger equation can do if written as a pair of coupled equations in two unknowns (one for each spin wave function, up or down), with external field term included. This thread contains the idea that *spin* is somehow tied up in, or linked with, the i which appears in the Schroedinger equation. Of course that's not true. There's an i in the usual way of writing the because it's a wave equation and it's first-order in time, and any equation first order in time which is going to spit out a wave solution (something like e^i(kx-wt)), is going to need an explicit i in there somewhere, because the i comes out in front when you differentiate with regard to time, once. The classic wave equation (for violin strings or EM waves or whatever) is second-order in time, so it has an i^2, or negative sign. It's the square root of that -1 which ends up on the time-dependent exponents of the solution, and thus gives wave solutions. Take it away and you don't have a diff equation which has waves as solutions. So there's your wave behavior, if it's anywhere. The Klein-Gordon equation is a classic wave equation in this sense of having a -1 and a second order time derivative, even though it's of course a relativistic equation. Its solutions are matter waves, but they still show up as waves and not other exponential functions, basically due to that -1. The quantum mechanics shows up in the factors you stick in to require that the matter waves wave with the proper frequency w. There has been some discussion of whether or not i is necessary to quantum mechanics, and that Schroedinger's genius lay in putting the i in to stand for the time-phase of matter waves. However, I think this is basically a wrong idea, also. Wave functions as written in QM are usually complex, but they don't always need to be in every single instance. The reason psi functions are *usually* complex, as I understand it imperfectly, is because they *usually* describe charged which can be time-reversed with a complex conjugate, because otherwise bad things happen in describing standing-waves (they show up with permanent currents or dipole moments or something). So Shroedinger was stuck needing to use a complex psi for electrons, because otherwise his standing wave solutions came out badly. And working backwards, his equation, as noted, had to have i in it because of his complex solutions and his use of a first order time derivative. But also, in relativistic QM you need a mechanism for relativistic charged matter waves to gain in energy and momentum without gaining in charge, and the complex notation allows a second time-reversed a high-energy matter wave, which otherwise it would gain along with energy and momentum. That keeps an electron which goes by with (say) the energy of 3 electrons, to keep from looking like it has a charge of -3 as though it really was 3 electrons. Basically, if you require that an electron which has the energy of 3 electrons look like it has the charge of just one electron, you need a positron wave solution which adds mass (energy) but subtracts charge. You can see all this in the solutions to the Klein-Gordon equation for pions of +1, -1, or 0 charge. The solutions for charged pions need to be complex, because otherwise there's no way to describe the appear under relativistic energies. However, for uncharged pions, there's no need for such solutions (or rather, conjugate solutinos look the same and are simply additive). So the wave equation for the relativistic neutral pion can be written as a simple real-valued function, with cosines and such, and that's it. So neither relativity, nor QM, nor relativistic QM, really requires complex psi functions in all circumstances. Just most of them. So, what does the i do in quantum mechanics? So far as I can tell, it sort of stands in as an extra dimention (something like polar coordinates) that allows one to specify and add up as a vector, the time directionality of charge transport in a wave, if you need to. But if you have an uncharged wave, like our uncharged pion wave, it's not strictly necessary. For an uncharged pion, with no need to correct use a real function that simply describes a matter wave that gains in matter. Thoughts from you physicists on this point are solicited. SBH === Subject: analysis question Let 0 be a point of Lebesgue density of E subset mathbb{R} (i.e. lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is taken over all balls about 0). Prove that there exists an infinite sequence of points x_n in E, with x_n != 0, and also x_n -> 0 as n -> infinity, such that the sequence also satisfies -x_n in E and 2x_n in E, for all n. any hints appreciated === Subject: Re: analysis question > Let 0 be a point of Lebesgue density of E subset mathbb{R} (i.e. > lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is taken over all > balls about 0). > Prove that there exists an infinite sequence of points x_n in E, with x_n > != 0, and also x_n -> 0 as n -> infinity, > such that the sequence also satisfies -x_n in E and 2x_n in E, for all n. Please, no TEX when you post here; this is a plain text newsgroup. === Subject: Re: analysis question > such that the sequence also satisfies -x_n in E and 2x_n in E, for all n. > Please, no TEX when you post here; this is a plain text newsgroup. On the contrary, this is a mathematics newsgroup, where TeX is quite common. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: The joy of plain text > Please, no TEX when you post here; this is a plain text newsgroup. > On the contrary, this is a mathematics newsgroup, where TeX is quite > common. I'm not sure what common is supposed to imply: Poorly stated questions are much more common on sci.math than is TeX; that is hardly an argument for them. I was under the impression that sci.math is a plain text ng. Isn't this forum for the worldwide mathematics community at large? This would include many who are unfamiliar with TeX: kids, high-school teachers, all sorts of amateurs and hobbyists, engineers, math Ph.D.s who got their degrees decades ago and are in other fields now, etc. I once knew TeX well enough to write a few papers in it, but I left academia years ago and today I much prefer plain old text to TeX on sci.math. === Subject: Re: The joy of plain text >> Please, no TEX when you post here; this is a plain text newsgroup. >> On the contrary, this is a mathematics newsgroup, where TeX is quite >> common. >I'm not sure what common is supposed to imply: Poorly stated questions >are much more common on sci.math than is TeX; that is hardly an argument >for them. >I was under the impression that sci.math is a plain text ng. Isn't this >forum for the worldwide mathematics community at large? This would include >many who are unfamiliar with TeX: kids, high-school teachers, all sorts of >amateurs and hobbyists, engineers, math Ph.D.s who got their degrees >decades ago and are in other fields now, etc. I once knew TeX well enough >to write a few papers in it, but I left academia years ago and today I much >prefer plain old text to TeX on sci.math. That is if the plain old text can intelligently state the problem. Too often, it cannot. -- 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: The joy of plain text > Please, no TEX when you post here; this is a plain text newsgroup. > On the contrary, this is a mathematics newsgroup, where TeX is quite > common. > I'm not sure what common is supposed to imply: Poorly stated questions > are much more common on sci.math than is TeX; that is hardly an argument > for them. > I was under the impression that sci.math is a plain text ng. Isn't this > forum for the worldwide mathematics community at large? This would include > many who are unfamiliar with TeX: kids, high-school teachers, all sorts of > amateurs and hobbyists, engineers, math Ph.D.s who got their degrees > decades ago and are in other fields now, etc. I once knew TeX well enough > to write a few papers in it, but I left academia years ago and today I much > prefer plain old text to TeX on sci.math. Plain text to me means no binaries; no .jpg files, or Word files, etc. TeX is plain text, at least the way I understand the term. Occasionally I post something here that I've cut 'n' pasted from another source, where it was in TeX. Going through the TeX, especially if it's at all long, and manually editing it into a form you'd approve is not an option; your options are, I post the TeX, or I don't post at all. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: The joy of plain text > Occasionally I post something here that I've cut 'n' pasted from > another source, where it was in TeX. Going through the TeX, especially > if it's at all long, and manually editing it into a form you'd approve > is not an option; your options are, I post the TeX, or I don't post > at all. Then don't post it. Unless it's simple Tex, likely I may never read it. Of the many many problems in the 7 groups I read, I've little time to translate hard to read stuff. So if you've not time to make your post readable, why should I take the time to make it readable and then even more time to solve the problem and yet more time to present an answer? You are asking too much of free tutoring. Those who post readable problems are those I read and (when able) answer first. === Subject: Re: The joy of plain text > Occasionally I post something here that I've cut 'n' pasted from > another source, where it was in TeX. Going through the TeX, especially > if it's at all long, and manually editing it into a form you'd approve > is not an option; your options are, I post the TeX, or I don't post > at all. > Then don't post it. Unless it's simple Tex, likely I may never read it. > Of the many many problems in the 7 groups I read, I've little time to > translate hard to read stuff. So if you've not time to make your post > readable, why should I take the time to make it readable and then even > more time to solve the problem and yet more time to present an answer? > You are asking too much of free tutoring. Those who post readable > problems are those I read and (when able) answer first. I see that I didn't make myself clear. Most of what I post here is not problems but answers to other people's problems. I've been providing, not asking for, the free tutoring for over a decade now (for the most part), though I guess it's too much to expect that even regulars contributors such as yourself would have noticed. The typical situation in which I'll post in TeX is when someone asks a research-level question, and I happen to know that there's a paper on the topic already in the literature, and I'll go cut 'n' paste the review from Math Reviews, which review will be written in TeX. If the person who asked the question can't decode the TeX, too bad - how much can she ask of free tutoring? You are hereby excused from reading anything I post in TeX. I think I'll survive. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: The joy of plain text >The typical situation in which I'll post in TeX is when someone >asks a research-level question, and I happen to know that there's >a paper on the topic already in the literature, and I'll go cut >'n' paste the review from Math Reviews, which review will be written >in TeX. For William's sake, perhaps you could use Zentralblatt instead. Of course, then you'd have to schneid'n'klebb. Lee Rudolph === Subject: Re: The joy of plain text >> Occasionally I post something here that I've cut 'n' pasted from >> another source, where it was in TeX. Going through the TeX, especially >> if it's at all long, and manually editing it into a form you'd approve >> is not an option; your options are, I post the TeX, or I don't post >> at all. >Then don't post it. [...] No, please do. Derek Holt's post says it all, so I won't labour the point. I just wanted to add another vote for sanity (not that I'm claiming to be sane). -- Angus Rodgers Contains mild peril === Subject: Re: The joy of plain text >> Occasionally I post something here that I've cut 'n' pasted from >> another source, where it was in TeX. Going through the TeX, especially >> if it's at all long, and manually editing it into a form you'd approve >> is not an option; your options are, I post the TeX, or I don't post >> at all. >Then don't post it. Unless it's simple Tex, likely I may never read it. The fact that you may never read it is a poor argument for not posting it. There may be others who do want to read it. My impression is that the most generally preferred method of writing mathematics on this newsgroup is to use a simplified version of TeX, removing all symbols (such as dollars) that are unnecessary for comprehension - for example: alpha^2 + beta^{5/2} = sum_{i=0} ^infty gamma_i ^{-3}. is not difficult to read. How would you prefer that to be written? With a longer chunk of material, cutting and pasting from a TeX document is not ideal, but as long as a handful of people are interested in reading it, it is worth posting it. There is so much total junk on this newsgroup already, that it seems strange to attmept to outlaw potentially interesting material on account of minor shortcomings. Derek Holt. >Of the many many problems in the 7 groups I read, I've little time to >translate hard to read stuff. So if you've not time to make your post >readable, why should I take the time to make it readable and then even >more time to solve the problem and yet more time to present an answer? >You are asking too much of free tutoring. Those who post readable >problems are those I read and (when able) answer first. === Subject: Re: The joy of plain text > My impression is that the most generally preferred method of writing > mathematics on this newsgroup is to use a simplified version of TeX, > removing all symbols (such as dollars) that are unnecessary for > comprehension - for example: > alpha^2 + beta^{5/2} = sum_{i=0} ^infty gamma_i ^{-3}. > is not difficult to read. How would you prefer that to be written? My sentiment more or less exactly! Everybody understands TeX (or can guess the meaning). My practice is to also leave out the backslashes (''), if it looks like that won't lead to any misunderstandings, so I might write: alpha^2 + beta^{5/2} = sum_{i=0}^infty gamma_i^{-3}. I think that this is a reasonable compromise and IMVHO slightly more readable. I also prefer not to use 'frac' or 'over' (me the plainTeX-fan:) at all. I think that {daadaa}/{doobedoo} is better than the alternative ways of writing a quotient:) I do feel that cutting and pasting from TeX-source has certain other drawbacks. E.g. if I were to cut and paste the above from a TeX-file I had written, it probably wouldn't have that extra space surrounding 'plus' and 'equal to' signs. I feel that this extra space does enhance legibility a bit, so I would normally do it that way. Ok. Sometimes I relax on that rule, if I'm in a hurry and don't have the time to edit or proofread my postings. But to summarize: Degustibus non est disputandum. Jyrki Lahtonen, Turku, Finland === Subject: Re: The joy of plain text is not difficult to read. How would you prefer that to be written? > I might write: > alpha^2 + beta^{5/2} = sum_{i=0}^infty gamma_i^{-3}. Much easier to read. Other hard to read stuff is ax^2+bx+c=(x-r)(x-s)=x^2-(r+s)x+rs=hardtoread === Subject: Re: analysis question >Let 0 be a point of Lebesgue density of E subset mathbb{R} (i.e. >lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is taken over all >balls about 0). >Prove that there exists an infinite sequence of points x_n in E, with x_n >!= 0, and also x_n -> 0 as n -> infinity, >such that the sequence also satisfies -x_n in E and 2x_n in E, for all n. Hint: 0 will also be a point of density of -E and of 1/2 E. Make sure that m(B cap E)/m(B) and m(B cap (-E))/m(B) and m(B cap (1/2 E)/m(B) are large enough, and you'll be able to ... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: analysis question >Let 0 be a point of Lebesgue density of E subset mathbb{R} (i.e. >lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is taken over all >balls about 0). >Prove that there exists an infinite sequence of points x_n in E, with x_n >!= 0, and also x_n -> 0 as n -> infinity, >such that the sequence also satisfies -x_n in E and 2x_n in E, for all n. > Hint: 0 will also be a point of density of -E and of 1/2 E. Make sure > that m(B cap E)/m(B) and m(B cap (-E))/m(B) and m(B cap (1/2 E)/m(B) > are large enough, and you'll be able to ... I'd only add that if x is a point of Lebesgue density of both E and F, then x is a point of Lebesgue density of E * F, where * denotes intersection. So 0 is a point of Lebesgue density of E * (-E) * (E/2). === Subject: Convergence Question (just for fun) I was doing some work the other day, and had to keep taking the ceiling of half of a value. Eventually, I found that I was taking the ceiling of half of the ceiling of half of a value. And so on. This led me to think about the following sequence of functions, and whether it converges or not. f_1(x) = ceil(x/2) f_2(x) = ceil(ceil(x/2)/2) = ceil(f_1(x)/2) f_3(x) = ceil(ceil(ceil(x/2)/2)/2) = ceil(ceil(f_1(x)/2)) = ceil(f_2(x)/2) ... f_n(x) = ceil(f_n-1(x)/2) ... It is not quite the same as g_n(x) = x/(2^n) I was just wondering if it converged. - Tim -- Timothy M. Brauch NSF Fellow Department of Mathematics University of Louisville email is: news (dot) post (at) tbrauch (dot) com === Subject: Re: Convergence Question (just for fun) > I was doing some work the other day, and had to keep taking the ceiling > of half of a value. Eventually, I found that I was taking the ceiling of > half of the ceiling of half of a value. And so on. > This led me to think about the following sequence of functions, and > whether it converges or not. > f_1(x) = ceil(x/2) > f_2(x) = ceil(ceil(x/2)/2) = ceil(f_1(x)/2) > f_3(x) = ceil(ceil(ceil(x/2)/2)/2) = ceil(ceil(f_1(x)/2)) = > ceil(f_2(x)/2) ... > f_n(x) = ceil(f_n-1(x)/2) > ... > It is not quite the same as > g_n(x) = x/(2^n) > I was just wondering if it converged. It converges to 0 if x <= 0, 1 if x > 0 (which is similar to the Heaviside unit step function). David === Subject: Re: Convergence Question (just for fun) > It converges to > 0 if x <= 0, > 1 if x > 0 > (which is similar to the Heaviside unit step function). > David Sorry, I realized that if it converged, it must converge to above. I guess, I was really wondering what kind of convergence does it have? Given any fixed n, I can always find an x such that f_n(x) > 1. Thus it is not uniform convergence. Other than that, I am stumped as to what kind of convergence it does exhibit. - Tim -- Timothy M. Brauch NSF Fellow Department of Mathematics University of Louisville email is: news (dot) post (at) tbrauch (dot) com === Subject: Re: Convergence Question (just for fun) > It converges to > 0 if x <= 0, > 1 if x > 0 > (which is similar to the Heaviside unit step function). > Sorry, I realized that if it converged, it must converge to above. I > guess, I was really wondering what kind of convergence does it have? > Given any fixed n, I can always find an x such that f_n(x) > 1. Thus it > is not uniform convergence. Other than that, I am stumped as to what > kind of convergence it does exhibit. This doesn't answer your question, but FWIW: You'd said before that f_n(x) is not quite the same as g_n(x) = x/(2^n). But f_n(x) is precisely the same as ceiling(x/(2^n)). David === Subject: Proposed definition for comparing the sizes of two sets Proposed Definition: A set Y is said to be larger than a set X iff there exists no function mapping X onto all of Y. (i.e. there is no surjection from X to Y) Is this a workable definition that covers all cases? If so, is it widely used? Dan === Subject: Re: Proposed definition for comparing the sizes of two sets at 08:57 PM, dchris@netcom.ca (Dan Christensen) said: >Proposed Definition: A set Y is said to be larger than a set X iff >there exists no function mapping X onto all of Y. (i.e. there is no >surjection from X to Y) >Is this a workable definition that covers all cases? If so, is it >widely used? Are you assuming the Axiom of Choice, or some equivalent? If not, you have to deal with incomparable pairs of sets. -- 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 === Subject: Re: Proposed definition for comparing the sizes of two sets > at 08:57 PM, dchris@netcom.ca (Dan Christensen) said: >Proposed Definition: A set Y is said to be larger than a set X iff >there exists no function mapping X onto all of Y. (i.e. there is no >surjection from X to Y) >Is this a workable definition that covers all cases? If so, is it >widely used? > Are you assuming the Axiom of Choice, or some equivalent? If not, you > have to deal with incomparable pairs of sets. I don't currently have AC built into my program, but you can introduce it as a premise at the beginning of any proof. I am planning to make it a true axiom, as well as building in some defintions for cardinal numbers in a future release. Some details need to be worked out. Dan Download DC Proof 1.0 at http://www.dcproof.com === Subject: Re: Proposed definition for comparing the sizes of two sets >Proposed Definition: A set Y is said to be larger than a set X iff there >exists no function mapping X onto all of Y. (i.e. there is no surjection >from X to Y) >Is this a workable definition that covers all cases? If so, is it widely >used? This is not a workable definition. Without at least some form of AC, one can have too sets, each of which comes out as larger than the other. Consider a non-wellorderable set X and a well-orderable set Y which is not smaller than or equal to (in the usual sense) P(X). If there is a non-wellorderable set, this must exist. Now a surjection of X onto Y is equivalent to an injection of Y into P(X), which we have assumed does not hold. And a surjection of a well-orderable set onto any set gives a well-ordering of that set. -- 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: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) Whoops forgot to include William's Metatheorem in my other post. A malady which I now cure. There is no know proof of WMT nor has anyone, not even the mighty James Harris, been as yet able to produce a counter example. > Is this a workable definition that covers all cases? If so, is it widely > used? Yes and since it's already well know, it also fails to be a counter example to: Whatever math I dream up is already old hat. -- William's Metatheorem. === Subject: Re: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) > Whoops forgot to include William's Metatheorem in my other post. A malady > which I now cure. There is no know proof of WMT nor has anyone, not even > the mighty James Harris, been as yet able to produce a counter example. > Is this a workable definition that covers all cases? If so, is it widely > used? > Yes and since it's already well know, it also fails to be a counter > example to: > Whatever math I dream up is already old hat. > -- William's Metatheorem. I have a version of Cantor's Theorem included with my DC Proof software that satisfied with my explanation, but I began to wonder if perhaps my definition might need some work. So, I am actually relieved to hear that is Dan Download DC Proof 1.0 at http://www.dcproof.com === Subject: Re: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) > Whoops forgot to include William's Metatheorem in my other post. A malady > which I now cure. There is no know proof of WMT nor has anyone, not even > the mighty James Harris, been as yet able to produce a counter example. > Is this a workable definition that covers all cases? If so, is it widely > used? > Yes and since it's already well know, it also fails to be a counter > example to: > Whatever math I dream up is already old hat. > -- William's Metatheorem. I have a version of Cantor's Theorem included with my DC Proof questioning the proof. He was satisfied with my explanation, but I began to wonder if perhaps my definition might need some work. It is not exactly the same as I have found elsewhere. So, I am actually Dan Download DC Proof 1.0 at http://www.dcproof.com === Subject: Re: Proposed definition for comparing the sizes of two sets >[...] > Whatever math I dream up is already old hat. > -- William's Metatheorem. Actually that includes William's Metatheorem. Sorry... ************************ David C. Ullrich === Subject: Re: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) > Is this a workable definition that covers all cases? If so, is it widely > used? > Dan It is essentially the definition Cantor used, and I believe it covers all cases provided one has the axiom of choice. === Subject: Re: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) > Is this a workable definition that covers all cases? If so, is it widely > used? > Dan > It is essentially the definition Cantor used, and I believe it covers > all cases provided one has the axiom of choice. There are definitions of relative set sizing besides cardinality, for example, there is a definition of set sizing that a proper superset of a set is larger than the set, for finite and infinite sets, as verified by Katz. As well, in the consideration of number theory, the set of even integers, as an example, has half the asymptotic density of the integers, within the integers. Another notion of larger is X > Y when X has infinitely many proper subsets that are proper supersets of Y, essentially a stronger condition than the proper superset size relation. The asymptotic density is rather useful, for example, half of the integers are even integers, and twice as many integers are not multiples of three as are multiples of three. For example, consider the power set of the naturals where half of the subsets of the naturals have as an element a given element of the naturals, and a quarter have both of two elements. The set of all subsets of the naturals containing as an element a given element of the naturals is half the size of the entire set. One way to reinforce that intuitive concept is as so: compare 1/2 to 1/4. Pick a subset of the set of all natural numbers at random, the chance that it contains as an element zero is one half, and that it contains 18 zillion even is one half, and that it contains both is one fourth. For disjoint sets, the above methods can work by comparing each to their union, or some other superset of their union, and then comparing those relative sizes. There are probably yet other ways to meaningfully compare two sets' sizes. Using cardinality will probably get you a passing grade. There is considerable argument about whether infinite sets automatically have bijections. Ross F. === Subject: Re: Proposed definition for comparing the sizes of two sets > There are definitions of relative set sizing besides cardinality, for > example, there is a definition of set sizing that a proper superset > of a set is larger than the set Are you referring to the partial order you get by using the subset predicate? I.e. the question is whether S1 is a subset of S2? Unfortunately that method leaves almost all pairs of sets uncomparable. For example consider two sets {a, b} and {a, c}. Neither is a subset of the other, so the two sets are uncomparable by the subset method, even though almost anyone can see that each has two elements so they ought to be considered the same size. The whole idea of cardinality is to clarify what the common definition means for finite sets in a way that directly applies in the same way to non-finite sets. Your proposal of using the subset property doesn't even give the right answer for finite sets, so it fails at the desired task. > in the consideration of number theory, the set of even integers, as > an example, has half the asymptotic density of the integers, within > the integers. Do you know the difference between a set, which has no properties other than what are elements and what aren't elements, and a more complicated structure that has a set plus some additional operators defined on the elements of the set? The set of integers, and the set of even integers, have no such thing as asymptotic density when all you can consider is what elements are in the sets and what elements aren't. Asymptotic density can be defined only when you have some kind of metric defined between elements of the sets. If you use the real metric, whereby the distance between any two integers is the absolute value of their arithmetic difference, and as with any metric space you can define neighborhoods of various radii, and you can define various kids of limits with respect to that metric, such as the limit as the size of the ball from some fixed point grows larger without limit, then you can define asymptotic density in terms of those metric-space-related terms. But if you use a different metric on the very same set, such as the 2-adic metric, you get a completely different result. So obviously asymptotic density isn't a property of the set itself, but only a property of one or another specific metric space using the set as elements in the space. > The asymptotic density is rather useful Only in a metric space. In just set theory it's not even defineable (unless you use set theory to define a metric space of course, like the way set theory is used to construct the natural numbers, from which you can construct the integers, from which you can construct the rationals, from which you can define a metric, etc.). === Subject: Re: Proposed definition for comparing the sizes of two sets > There are definitions of relative set sizing besides cardinality, for > example, there is a definition of set sizing that a proper superset > of a set is larger than the set > Are you referring to the partial order you get by using the subset predicate? > I.e. the question is whether S1 is a subset of S2? > Unfortunately that method leaves almost all pairs of sets uncomparable. > For example consider two sets {a, b} and {a, c}. Neither is a subset of > the other, so the two sets are uncomparable by the subset method, even > though almost anyone can see that each has two elements so they ought > to be considered the same size. The whole idea of cardinality is to > clarify what the common definition means for finite sets in a way that > directly applies in the same way to non-finite sets. Your proposal of > using the subset property doesn't even give the right answer for > finite sets, so it fails at the desired task. > in the consideration of number theory, the set of even integers, as > an example, has half the asymptotic density of the integers, within > the integers. > Do you know the difference between a set, which has no properties other > than what are elements and what aren't elements, and a more complicated > structure that has a set plus some additional operators defined on the > elements of the set? The set of integers, and the set of even integers, > have no such thing as asymptotic density when all you can consider is > what elements are in the sets and what elements aren't. Asymptotic > density can be defined only when you have some kind of metric defined > between elements of the sets. If you use the real metric, whereby the > distance between any two integers is the absolute value of their > arithmetic difference, and as with any metric space you can define > neighborhoods of various radii, and you can define various kids of > limits with respect to that metric, such as the limit as the size of > the ball from some fixed point grows larger without limit, then you can > define asymptotic density in terms of those metric-space-related terms. > But if you use a different metric on the very same set, such as the > 2-adic metric, you get a completely different result. So obviously > asymptotic density isn't a property of the set itself, but only a > property of one or another specific metric space using the set as > elements in the space. > The asymptotic density is rather useful > Only in a metric space. In just set theory it's not even defineable > (unless you use set theory to define a metric space of course, like the > way set theory is used to construct the natural numbers, from which you > can construct the integers, from which you can construct the rationals, > from which you can define a metric, etc.). Do you have any infinite sets? Please name an infinite set besides the integers or reals. The counting numbers, the natural integers, are very useful for counting these other sets of things. As each set contains only unique elements and there is an ordering relation on each of those sets of things, it is simple to see why each is numbered individually. When the sets are disjoint, disjoint, or partially disjoint, then there exists a proper superset containing each element of both. You'll notice that the proper superset is larger than the set. (You'll notice that was ignored.) You might see why you could apply that to any pair of sets, thus that it is universally applicable. About density and metrics, the metric is a great thing. I'm interested in this measure theory. For example, I consider the sigma algebra in relation to my rootsets and decorated ordinals, in the theory with ubiquitous ordinals where the ordinals are Z, the integers, in the complete, concrete, and consistent theory. I browse the MathWorld definition of the sigma algebra and wonder what he means when Eric has sequences in the statement. In terms of the sets where they are defined to be containing the, for example, integers, they are then those integers. The notion of counting in any form, enumerating each, for example via a choice function or well ordering and induction, reflects back upon the completely intuitive natural counting numbers. That's especially so in the finite case. Besides the metrics there is also the intertwined notion of probability distributions. Not about that, the set of even integers within the integers, is defined by the integral modulus. Cardinality has nothing to say about asymptotic density except no opinion. Are half the infinite binary sequences normal or through restricted sequence element interchange convertible to the canonical sequence with equal zero and one density, one third or two thirds? The infinitesimal predates the cardinal, in a sense being the classical. The cardinal is in a sense a modern red herring. The set of all sets is its own powerset. The infinite is not necessarily simple nor intuitive except that it is: half of the integers are even. Bijections exist between the integers and even integers, and the particular one f(x)=2x, a plain straight line, shows that there are twice as many integers as even integers, in the integers or superset of the integers. You have some good points there, but in comparing the relative sizes or infinite sets there is sometimes a reason that has to do with solving a real world problem instead of flights of fancy about meaningless escapisms from foundational foundations. Asymptotic density is a useful notion that runs right back to one plus one equals two. Ken, 2 + 2 = 4. To Scandinavians herring is a way of life, to Americans it's a Monty Python sketch. I saw this the other day, it's haunting, yet funny: http://www.khaaan.com/ . It gets more haunting and less funny. There is an implied point set topology and metric space, from nothing, wherefrom all is implied. Ross Finlayson === Subject: Re: Proposed definition for comparing the sizes of two sets > Do you have any infinite sets? Please state what you mean when you use the word have. The ordinary meaning, synonym with own, isn't applicable, because sets are abstract mathematical ideas which can't be owned by any one person. > Please name an infinite set ... Please state what you mean when you use the word name. The ordinary meaning, meaning to assign a name to an object, for example naming a baby, doesn't seem useful in this context. So there's this infinite set, and you want me to name it Fred?? > The counting numbers, the natural integers, are very useful for > counting these other sets of things. Are you talking about using natural numbers (positive integers) as context-sensitive names for individual elements of sets, for example in the context of the set of rational numbers you can call 1/3 #1 and you can call 4/7 #2 and you can call 0 #3 and you can call -5000 #4 etc., assigning a different natural-number label to each element in the set? Or are you talking about cardinality of finite subsets of some context-establishing set, for example the rationals, whereby the subset consisting of {1/3, 4/7} would be counted as size 2, and the subset consisting of {-5000, 0, 22/7, 1/2, 1/3} would be counted as size 5? > As each set contains only unique elements I have no idea what you mean by a unique element. Every thing is unique, so of course every element of any set is a unique thing hence a unique element. Do you actually mean anything by what you said? > and there is an ordering relation on each of those sets of things, That is such a gross understatment that it's grossly misleading hence basically a lie. Not only is there one ordering relation on a set, but for any set containing at least two elements there are at least two different ordering relation on the set, and for any set containing at least three elements there are at least six different ordering relations on the set, and for the integers there are an uncountable infinity (aleph_null factorial, which equals aleph_one) of different ordering relations. > completely intuitive natural counting numbers. The natural numbers aren't completely intuitive. Only the numbers from one up to about four or five are completely intuitive for humans, where they can just glance at a visual image showing that many similar objects in any random orientation and immediately know intuitively, without needing to count them, how many there are. Some birds can intuitively recognize cardinality up to about seven, beating humans by a couple, which is very useful for detecting if any eggs have been removed from the nest or added to the nest. If objects are in standard partterns, such as pips on a die-face or half-domino, then we can recognize them up to nine, but put those same pips in random pattern and suddenly the problem gets much more difficult. > the set of even integers within the integers, is defined by the > integral modulus. Wrong! If you treat the integers as *nothing* except a set, no order relation, no arithmetic properties, and the individual integers aren't defined in terms of something else such as cardinality of sets whereby you can use that definition to generate arithmetic properties, there is no way whatsoever to define which are even and which aren't even except by an infinitely long list enumerating each and every even integer (or alternately by enumerating the complement set which are odd). Here are four examples of sets of integers: (1) Recursive definition: The empty set, and any set containing exactly one element which is an integer. Thus {} {{}} {{{}}} etc. are the consecutive integers. Even can be defined recursively like this: N is even iff N = {M} and M is not even. This works because integers aren't just abstract elements, but are actually constructed via set theory in a way such that even can be defined from that. (2) Arithmetic definition: Start with Peano's postulates, in particular the successor function S, with natural numbers defined recursively as 1, and any S(N) where N is an integer. Even can be defined recursively like this: N is even iff N = S(M) where M is not even. (3) Explicit listing of just a few integers because each one is listed separately and I don't have an infinite amount of time to type them all: {apat, isa, lima, delawa, tatlo}. Unless you know that I've used Tagalog names for those five integers, and unless you actually know what those five Tagalog words mean, and unless I'm actually using the corect Tagalog names instead of shuffling them, you can't figure out which of those are odd and which are even in my set of integers. If I tell you that apat and delawa are even, and the other three aren't even, would you believe me? If I told you something else instead, would you believe me? On what basis could you decide for yourself which are even and which aren't per my definition unless I simply tell you the answer and promise not to change my definition to pull a trick on you? (4) In all the above, I had some sort of name for each integer. But suppose I don't have any names at all. Here are a bunch of integers, each displayed as an asterisk: * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * Now suppose that I claimed that was a viewport into a small section of my integers, that really there are an infinite number of them, and they are not displayed in any particular order that would make sense to you, but I promise I do have a grand design whereby that pattern you see above is one 3-by-62 portion of the overall design. Now how would you decide which of those asterisks are even numbers and which aren't? No matter how you guess which are even and which aren't, I could legitimately claim you are 100% wrong. If you get two guesses, I could legitimately claim you are 50% wrong, or worse, in each case. Suppose I told you an algorithm for deciding in each position whether one of the integers is there or that position is empty, not just for that 3-by-62 viewport but for the entire grand pattern. How would you define even vs. not-even for all the asterisks? Well because they are located in a lattice, you could perhaps make up a definition that is based on the location within the pattern. It wouldn't match my definition, but at least you *could* define what is meant by even on that set of integers. But that would not be defining even on the elements themselves, rather you'd be definining even based on their position within some sort of structue, in this case a regular lattice. But suppose they weren't in any lattice structure, but just abstract objects that you could get somehow, not in any particular order, not in any structure, no way to examine them internally, the only predicates you have are: (a) For any item, either that item is in the set (of integers) or not; (b) For any two items in the set, they are either the same element or not the same. With only those two predicates, there's no way to define even rigorously. Suppose in some object-oriented language, for example Java, I provided for you a playpen whereby you could execute commands interactively. I provide for you the following functions/methods: (1) static getInteger() ==> an integer object (2) integerObject.toString() ==> SomeInteger (every integer prints the same) (3) integerObject.hashCode() ==> 0 (every integer has the same hashcode too!) (4) integerObject.equals(integer) ==> true if same object, false otherwise By calling getInteger() from time time, getting several such integerObjects, can you decide just from calls to the above API which of them are even and which aren't? No, the best you can do is make arbitrary decisions, which wouldn't be consistent from one run of the demo to another. Can you define a predicate: boolean isEven() which is well defined, using *only* calls to the API I've specified above? No, you can't. The best you can do is randomly decide for each new integerObject whether it's even or odd, and keep a cache of such decisions so you don't contradict yourself later, but that's not a definition, that's a random sampler, and again, just like the manual demo, you'd get different results with each run of the program, so your function/method doesn't *define* a function, it merely generates a new random sample each time it's run. By the way, with *all* the integers, not just the positive integers, even if you know the total ordering, that's still not enough to define what is even and what isn't even. The best you can do is divide the ordered set into two subsets, alternating, and then pick one at random to be even. And it's even worse: If you have a playpen where you can call an API which gives you random integers and tells for any two integers whether they are equal and if not which is the smaller of the two, since you don't know whether two integers are adjacent or not you can't in a finite number of steps determine the two alternating sets. > The set of all sets ... There's no such thing. If you understood proof-by-contradiction I could present you a very simple proof to that fact, but you don't seem to understand even that simple aspect of mathematical logic so it would be a waste of my time to present it to you. > half of the integers are even. As just a set, 99% of them are even too. > Bijections exist between the integers and even integers, and the > particular one f(x)=2x, a plain straight line, shows that there are > twice as many integers as even integers, in the integers or superset of > the integers. It shows no such thing!! The bijection shows there are exactly the same number of integers as even integers. For every integer there's a corresponding even integer, and vice versa. What you said is equivalent to saying there's a bijection between the fingers on my left hand and the fingers on my right hahd, which shows there are twice as many fingers on my left hand as on my right hand. === Subject: Re: Proposed definition for comparing the sizes of two sets > Do you have any infinite sets? > Please state what you mean when you use the word have. The ordinary > meaning, synonym with own, isn't applicable, because sets are > abstract mathematical ideas which can't be owned by any one person. > Please name an infinite set ... > Please state what you mean when you use the word name. The ordinary > meaning, meaning to assign a name to an object, for example naming a > baby, doesn't seem useful in this context. So there's this infinite > set, and you want me to name it Fred?? > The counting numbers, the natural integers, are very useful for > counting these other sets of things. > Are you talking about using natural numbers (positive integers) as > context-sensitive names for individual elements of sets, for example in > the context of the set of rational numbers you can call 1/3 #1 and you > can call 4/7 #2 and you can call 0 #3 and you can call -5000 #4 etc., > assigning a different natural-number label to each element in the set? > Or are you talking about cardinality of finite subsets of some > context-establishing set, for example the rationals, whereby the subset > consisting of {1/3, 4/7} would be counted as size 2, and the subset > consisting of {-5000, 0, 22/7, 1/2, 1/3} would be counted as size 5? Identify an infinite set. > As each set contains only unique elements > I have no idea what you mean by a unique element. Every thing is > unique, so of course every element of any set is a unique thing hence a > unique element. Do you actually mean anything by what you said? Yes, I tend to be sufficiently exact. > and there is an ordering relation on each of those sets of things, > That is such a gross understatment that it's grossly misleading hence > basically a lie. Not only is there one ordering relation on a set, but > for any set containing at least two elements there are at least two > different ordering relation on the set, and for any set containing at > least three elements there are at least six different ordering > relations on the set, and for the integers there are an uncountable > infinity (aleph_null factorial, which equals aleph_one) of different > ordering relations. It's definitely not a lie. There are obviously many ordering relations, pick one. > completely intuitive natural counting numbers. > The natural numbers aren't completely intuitive. Only the numbers from > one up to about four or five are completely intuitive for humans, where > they can just glance at a visual image showing that many similar > objects in any random orientation and immediately know intuitively, > without needing to count them, how many there are. Some birds can > intuitively recognize cardinality up to about seven, beating humans by > a couple, which is very useful for detecting if any eggs have been > removed from the nest or added to the nest. If objects are in standard > partterns, such as pips on a die-face or half-domino, then we can > recognize them up to nine, but put those same pips in random pattern > and suddenly the problem gets much more difficult. How many states are in the union? How many continents are on the planet? How many stars are in the sky? > the set of even integers within the integers, is defined by the > integral modulus. > Wrong! If you treat the integers as *nothing* except a set, no order > relation, no arithmetic properties, and the individual integers aren't > defined in terms of something else such as cardinality of sets whereby > you can use that definition to generate arithmetic properties, there is > no way whatsoever to define which are even and which aren't even except > by an infinitely long list enumerating each and every even integer (or > alternately by enumerating the complement set which are odd). > Here are four examples of sets of integers: > (1) Recursive definition: The empty set, and any set containing exactly > one element which is an integer. Thus {} {{}} {{{}}} etc. are the > consecutive integers. Even can be defined recursively like this: N is > even iff N = {M} and M is not even. This works because integers aren't > just abstract elements, but are actually constructed via set theory in > a way such that even can be defined from that. > (2) Arithmetic definition: Start with Peano's postulates, in particular > the successor function S, with natural numbers defined recursively as > 1, and any S(N) where N is an integer. Even can be defined recursively > like this: N is even iff N = S(M) where M is not even. OK. The powerset is the successor is the order type. > (3) Explicit listing of just a few integers because each one is listed > separately and I don't have an infinite amount of time to type them > all: {apat, isa, lima, delawa, tatlo}. Unless you know that I've used > Tagalog names for those five integers, and unless you actually know > what those five Tagalog words mean, and unless I'm actually using the > corect Tagalog names instead of shuffling them, you can't figure out > which of those are odd and which are even in my set of integers. > If I tell you that apat and delawa are even, and the other three aren't > even, would you believe me? If I told you something else instead, would > you believe me? On what basis could you decide for yourself which are > even and which aren't per my definition unless I simply tell you the > answer and promise not to change my definition to pull a trick on you? Do they mean the same thing as {1, 2, 3, 4, 5}? > (4) In all the above, I had some sort of name for each integer. But > suppose I don't have any names at all. Here are a bunch of integers, > each displayed as an asterisk: > * * * * * * * * * * > * * * * * * * * * * * * > * * ** * * * * * * * * > Now suppose that I claimed that was a viewport into a small section of > my integers, that really there are an infinite number of them, and they > are not displayed in any particular order that would make sense to you, > but I promise I do have a grand design whereby that pattern you see > above is one 3-by-62 portion of the overall design. Now how would you > decide which of those asterisks are even numbers and which aren't? I don't care. I wouldn't. > No matter how you guess which are even and which aren't, I could > legitimately claim you are 100% wrong. If you get two guesses, I could > legitimately claim you are 50% wrong, or worse, in each case. Suppose I > told you an algorithm for deciding in each position whether one of the > integers is there or that position is empty, not just for that 3-by-62 > viewport but for the entire grand pattern. How would you define even > vs. not-even for all the asterisks? Well because they are located in a > lattice, you could perhaps make up a definition that is based on the > location within the pattern. It wouldn't match my definition, but at > least you *could* define what is meant by even on that set of > integers. But that would not be defining even on the elements > themselves, rather you'd be definining even based on their position > within some sort of structue, in this case a regular lattice. But > suppose they weren't in any lattice structure, but just abstract > objects that you could get somehow, not in any particular order, not in > any structure, no way to examine them internally, the only predicates > you have are: (a) For any item, either that item is in the set (of > integers) or not; (b) For any two items in the set, they are either the > same element or not the same. With only those two predicates, there's > no way to define even rigorously. > Suppose in some object-oriented language, for example Java, I provided > for you a playpen whereby you could execute commands interactively. I > provide for you the following functions/methods: > (1) static getInteger() ==> an integer object > (2) integerObject.toString() ==> SomeInteger (every integer prints the same) > (3) integerObject.hashCode() ==> 0 (every integer has the same hashcode too!) > (4) integerObject.equals(integer) ==> true if same object, false otherwise > By calling getInteger() from time time, getting several such > integerObjects, can you decide just from calls to the above API which > of them are even and which aren't? No, the best you can do is make > arbitrary decisions, which wouldn't be consistent from one run of the > demo to another. An integer has an integer value. If you have integerObject.add(IntegerObject i) returning the sum, then you should be able to tell which are even. > Can you define a predicate: > boolean isEven() > which is well defined, using *only* calls to the API I've specified above? > No, you can't. The best you can do is randomly decide for each new > integerObject whether it's even or odd, and keep a cache of such > decisions so you don't contradict yourself later, but that's not a > definition, that's a random sampler, and again, just like the manual > demo, you'd get different results with each run of the program, so your > function/method doesn't *define* a function, it merely generates a new > random sample each time it's run. That's irrelevant. Half of the integers are even. > By the way, with *all* the integers, not just the positive integers, > even if you know the total ordering, that's still not enough to define > what is even and what isn't even. The best you can do is divide the > ordered set into two subsets, alternating, and then pick one at random > to be even. And it's even worse: If you have a playpen where you can > call an API which gives you random integers and tells for any two > integers whether they are equal and if not which is the smaller of the > two, since you don't know whether two integers are adjacent or not you > can't in a finite number of steps determine the two alternating sets. > The set of all sets ... > There's no such thing. No, there is. Obviously enough there is not in ZF. > If you understood proof-by-contradiction I could > present you a very simple proof to that fact, but you don't seem to > understand even that simple aspect of mathematical logic so it would be > a waste of my time to present it to you. No, you're wrong. Several theories including my own have sets of all sets. > half of the integers are even. > As just a set, 99% of them are even too. No, the integers have integer values. > Bijections exist between the integers and even integers, and the > particular one f(x)=2x, a plain straight line, shows that there are > twice as many integers as even integers, in the integers or superset of > the integers. > It shows no such thing!! The bijection shows there are exactly the same > number of integers as even integers. For every integer there's a > corresponding even integer, and vice versa. You're wrong, it shows exactly that thing. > What you said is equivalent to saying there's a bijection between the > fingers on my left hand and the fingers on my right hahd, which shows > there are twice as many fingers on my left hand as on my right hand. No, it doesn't. Look at any function from the integers to the integers of the form y=mx+b, a straight line, for integer m. The range has asymptotic density of 1/m in the integers. As it is so for any straight line function, except for arguably m=0, where that bijection exists for the domain of the integers, it shows that the range has an asyptotic density, which is a useful comparison of set's sizes, in this case the sets of the domain and range, comparing the integers to a subset of the integers and illustrating why the subset comprises half of the integers, or generally 1/m. Do you not see how terribly, horribly wrong you've been about all this? Half of the integers are even, true or false? It's true. If it's false then through contradiction the integer is neither even nor odd. Besides your caffeinated integers, an integer is even or odd. Anyways, the key point to consider is that the proper subset definition of sizing is universally applicable. Also, when you talk about sets of only integers, not labels but integers, then all structural aspects of the integers hold true in the comparison of collections of them. Ross === Subject: Re: Proposed definition for comparing the sizes of two sets > Proposed Definition: A set Y is said to be larger than a set X iff there > exists no function mapping X onto all of Y. (i.e. there is no surjection > from X to Y) This is no different that what is already being used. Because magnitudes of sets are linear ordered X < Y iff not Y <= X (you may prefer to read X < Y as |X| < |Y|) You've presented X < Y when for all f:X -> Y, f not a surjection Thus not X < Y iff some f:X -> Y, f is a surjection which by above is equivalent to Y <= X and the definition or theorem Y <= X when some surjection f:X -> Y is well known along with it's intimate associate AxC. > Is this a workable definition that covers all cases? > If so, is it widely used? Yes and Yes. === Subject: Re: Proposed definition for comparing the sizes of two sets >> Proposed Definition: A set Y is said to be larger than a set X iff there >> exists no function mapping X onto all of Y. (i.e. there is no surjection >> from X to Y) >This is no different that what is already being used. >Because magnitudes of sets are linear ordered > X < Y iff not Y <= X >(you may prefer to read X < Y as |X| < |Y|) Linear ordering of the magnitudes is equivalent to the axiom of choice. There are lots of models without it, and the definition is never a good one without it. -- 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: So you want to count an infinite power set ? > To construct a powerset of an infinite set I use an > infinite cartesian grid populated with 1s and 0s, When you say infinite cartesian grid, do you mean a two-dimensional grid, consisting of elements indexed by the positive integers along each axis? If so, that's not enough elements to hold the powerset. > then use logical AND operation with those values from every row > applied to the original sequence. What original sequence?? You need to start at the beginning. Define what you have to start with. Define all the personal jargon you use. Define what you construct at each step of the way. If you jump in the middle where you somehow already have some original sequence, but don't tell us where that original sequence came from or what it is, we can't possibly be expected to understand what you're talking about. > SET = { <0,3>, <1,1>, <2,4>, <4,1>, <5,5> ... } I see no pattern there, so I have no idea what ... means at the end. Please define clearly what SET is supposed to contain. I don't even know what you mean by each individual element, such as <0,3>. Is that supposed to be an ordered pair, or what? > P_1(SET) = { > {1000000000000.. AND SET}, > {0100000000000.. AND SET}, > {1100000000000.. AND SET}...} I assume each of the things that looks like 1000000000000.. etc. is supposed to be a bitvector that is infinitely long, right? I have no idea how to perform an AND operation between a bit vector and a set of ordered pairs. Please define what AND does, or tell how SET is supposed to be treated as if it were a bit vector so that AND would apply in the usual way as the bitwise-AND operation. === Subject: Re: So you want to count an infinite power set ? > P_1(SET) = { > {1000000000000.. AND SET}, > {0100000000000.. AND SET}, > {1100000000000.. AND SET}...} That is my only introduced notation, but I manually calculate the step straight after. P_1(SET) = { {1 AND <0,3>, 0 AND <1,1>, 0 AND <2,4> ...}, {0 AND <0,3>, 1 AND <1,1>, 0 AND <2,4> ...}, {1 AND <0,3>, 1 AND <1,1>, 0 AND <2,4> ...}, ...} 1 = true, 0 = false. TRUE AND X = X Look up 'bit masking The original SET is { <0,3>, <1,1>, <2,4>, ...} i.e. pi 314159.. put into <0 > <1 > <2 > <3 > ... Its the quickest way I could think of to make infinite distinct members that werent trivial N. I use { } for set and < > for sequence, fairly standard for programmers atleast. Herc === Subject: Mathematical Ideas I am trying to find an idea in the area of number theory or combinatorics that I can do my undergraduate senior research paper/presentation on. I've seen how people on here have harrassed others asking for idea telling them to go see their advisor. However I have a good case for asking here: the two math profs at my school only have interest in statistics, calculus, and geometry. And these are not my strong fields, as I am double majoring in math and computer science. I have met with my professors once a week on average for the entire semester and together have still to come up with a topic they deem viable. I have to do more than just research the idea, but in some way I have to do something with it or contribute to the topic (nothing major mind you). So if you have any constructive === Subject: connecting opposite edges of the cube If the 3-cube is a finite union {A_i} of open subsets, any 3 of them having empty intersection, is it true that some A_i is intersecting two opposite edges of the cube? Opposite here means symmetric with respect to the cubocenter. [I proved that if the 3-cube is a finite union {A_i} of open subsets, any 4 of them with empty intersection, then some A_i is intersecting two opposite FACES of the cube] vedran === Subject: Philosopher/marketer is stumped by this Probability Problem. Plz help. Hi. I hope someone can at least point me in the right direction. I'm looking for a general solution to the following problem: Suppose you have two urns, each containing an equivalent VERY LARGE (maybe not infinite, but very large) number of balls. Each urn has only red and green balls in it. Suppose you draw from urn A 11 times, with replacement, and you get 7 green balls. So the ratio of green balls to total draws in the sample is 7/11. Suppose you draw from urn B 15 times, with replacement, and you get 8 green balls. So the ratio, for urn B, of green balls to total draws in the sample is 8/15. Assuming random sampling, and given these samples, what is the probability that urn A contains more green balls than urn B? I am actually looking for the general solution. Given a sample from urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and Ys are integers), what is the probability that there are more greens in urn A than in urn B. I'm kind of stumped. Giblar === Subject: Re: Philosopher/marketer is stumped by this Probability Problem. Plz help. >Hi. I hope someone can at least point me in the right direction. >I'm looking for a general solution to the following problem: >Suppose you have two urns, each containing an equivalent VERY LARGE >(maybe not infinite, but very large) number of balls. Each urn has >only red and green balls in it. >Suppose you draw from urn A 11 times, with replacement, and you get 7 >green balls. So the ratio of green balls to total draws in the sample >is 7/11. >Suppose you draw from urn B 15 times, with replacement, and you get 8 >green balls. So the ratio, for urn B, of green balls to total draws >in the sample is 8/15. >Assuming random sampling, and given these samples, what is the >probability that urn A contains more green balls than urn B? >I am actually looking for the general solution. Given a sample from >urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and Ys are >integers), what is the probability that there are more greens in urn A >than in urn B. >I'm kind of stumped. There is not much information on the problem. What does equivalent mean? If the total number of balls is equal, the is the standard problem of comparing proportions. One still needs prior assumptions to compute the probability. -- 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: Philosopher/marketer is stumped by this Probability Problem. Plz help. >I'm looking for a general solution to the following problem: >Suppose you have two urns, each containing an equivalent VERY LARGE >(maybe not infinite, but very large) number of balls. Each urn has >only red and green balls in it. >Suppose you draw from urn A 11 times, with replacement, and you get 7 >green balls. So the ratio of green balls to total draws in the sample >is 7/11. >Suppose you draw from urn B 15 times, with replacement, and you get 8 >green balls. So the ratio, for urn B, of green balls to total draws >in the sample is 8/15. >Assuming random sampling, and given these samples, what is the >probability that urn A contains more green balls than urn B? >I am actually looking for the general solution. Given a sample from >urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and Ys are >integers), what is the probability that there are more greens in urn A >than in urn B. If you actually want a *probability*, you need to specify a prior joint distribution on the fractions of green balls in the two urns. Use Bayesian updating. It is also possible to do a frequentist hypothesis test: Y1 and Y2 should be slightly larger (but not necessarily too much so), and use the test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)), which has approximately a standard normal distribution. Here, Qi = Xi / Yi, and Qc = (X1+X2) / (Y1+Y2) -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Philosopher/marketer is stumped by this Probability Problem. Plz help. [bcc'd to OP, who e-mailed me] You asked what is the *probability* that one method is better than the other, but I suspect that that is not what you really want. In the frequentist paradigm, one poses hypotheses in terms of unknown parameters which are not random variables, hence it makes no sense to speak of the probability of a hypothesis. Rather, one decides whether an observed event is excessively improbable in the case that a hypothesis is true; if so, one rejects the hypothesis and accepts the alternative. Suppose you want to ask the question, Is there evidence that two methods differ in their efficacy? The frequentist method is as follows. Let p1 and p2 be the yields of the two methods. You pose the null hypothesis and its alternative, H0: p1 = p2 H1: p1 != p2 Under the null hypothesis, the test statistic Z = (Q1-Q2) / sqrt(Qc (1-Qc) / n) has approximately a standard normal distribution, where the Q's are defined as in my post and n is the combined sample size (your Y1+Y2). You want Y1 and Y2 to be sufficiently large; if each are at least 15, you are probably O.K. Thus, if the observed value of the test statistic is too extreme (e.g., if P{|Z| > z_obs} < 0.10; 0.10 is called ths significance level), you reject the null hypothesis and find evidence of a difference. Ohterwise, you find no evidence of a difference. This is all rather standard stuff; also, it makes several assumptions, in particular, independent samples. In Bayesian statistics, one models the unknown parameters as random variables, hence one can come up with a coherent statement of the probabiliy of a difference given the data. This requires the specification of a prior joint distribution on the parameters, and one computes the posterior conditional probability given the observed data. Standard elementary texts on statistics should discuss all of this. >> I'm looking for a general solution to the following problem: >> Suppose you have two urns, each containing an equivalent VERY LARGE >> (maybe not infinite, but very large) number of balls. Each urn has >> only red and green balls in it. >> Suppose you draw from urn A 11 times, with replacement, and you get 7 >> green balls. So the ratio of green balls to total draws in the sample >> is 7/11. >> Suppose you draw from urn B 15 times, with replacement, and you get 8 >> green balls. So the ratio, for urn B, of green balls to total draws >> in the sample is 8/15. >> Assuming random sampling, and given these samples, what is the >> probability that urn A contains more green balls than urn B? >> I am actually looking for the general solution. Given a sample from >> urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and Ys are >> integers), what is the probability that there are more greens in urn A >> than in urn B. >> > If you actually want a *probability*, you need to specify a prior > joint distribution on the fractions of green balls in the two urns. > Use Bayesian updating. > It is also possible to do a frequentist hypothesis test: Y1 and Y2 > should be slightly larger (but not necessarily too much so), and use > the test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)), which has > approximately a standard normal distribution. Here, Qi = Xi / Yi, > and Qc = (X1+X2) / (Y1+Y2) -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Philosopher/marketer is stumped by this Probability Problem. Plz help. Hi Stephen (Keith and Herman, too!) might be able to use much of your present answers, but I worry now that my Urn analogy isn't apt for my real problem. Please, if you'll indulge me, let me state my real-world application, then maybe you can help me set up the problem correctly. (In other words, I'm not really dealing with urns filled with red and green balls :-) ) I'm developing a marketing tool that analyzes split run tests. So, say, on a sales letter, copies that run headline A might have a sales conversion ratio of 5 sales out of 347 exposures, and copies that run headline B might have a sales conversion ratio of 6 sales out of 334 exposures. I want to provide an estimate of how likely it is that ultimately headline B will convert better than headline A. This will help determine whether to continue running the test or to terminate it. I thought the urn problem was a good analogy, but it seems that I wasn't explicit about several assumptions, and those assumptions were important for answering the question. If you have a straightforward solution, please ignore the rest of this message and proceed to enlighten me. I would be much grateful. But to show I'm not just being lazy, and that I'm trying to figure it out, here is an approach I came up with, but have minimal confidence in. One approach I thought might work would be to figure out some intermediate value X between the two conversion ratios (i.e., in this case 5/347 < X < 6/334)where Prob(CRA>=X | (EA=347) & (SA<=5)) = Prob( CRB<=X |(EB=334) & (SB>=6)) Where CRA = the true conversion rate of letter A CRB = the true conversion rate of letter B SA = Sales with letter A SB = Sales with letter B EA = Exposures for letter A EB = Exposures for letter B Then, it seems that the probability that letter B does NOT ultimately have a higher conversion rate than letter A is the probability of a conjunction, namely, Prob[(SA<=5 | (EA=347) & (CRA=X)) & (SB>=6 | (EB=334) & (CRB=X))] Call this Z. And then the probability that letter B has a greater conversion ratio than letter A is 1-Z. But this seems a lot more complicated than I'm hoping it has to be. And I'm not sure it's right anyway. And, of course, again, I'm after the general solution. I'm sorry if you wind up repeating yourself, but I wanted to make sure you had more information before I tried to apply the advice already given. Giblar. (By the way, I abandoned mathematics in college to pursue philosophy. So far it seems the mathematics would have been more useful :-) >I'm looking for a general solution to the following problem: >Suppose you have two urns, each containing an equivalent VERY LARGE >(maybe not infinite, but very large) number of balls. Each urn has >only red and green balls in it. >Suppose you draw from urn A 11 times, with replacement, and you get 7 >green balls. So the ratio of green balls to total draws in the sample >is 7/11. >Suppose you draw from urn B 15 times, with replacement, and you get 8 >green balls. So the ratio, for urn B, of green balls to total draws >in the sample is 8/15. >Assuming random sampling, and given these samples, what is the >probability that urn A contains more green balls than urn B? >I am actually looking for the general solution. Given a sample from >urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and Ys are >integers), what is the probability that there are more greens in urn A >than in urn B. > > If you actually want a *probability*, you need to specify a prior joint > distribution on the fractions of green balls in the two urns. Use > Bayesian updating. > It is also possible to do a frequentist hypothesis test: Y1 and Y2 > should be slightly larger (but not necessarily too much so), and use the > test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)), which has > approximately a standard normal distribution. Here, Qi = Xi / Yi, > and Qc = (X1+X2) / (Y1+Y2) === Subject: Re: Philosopher/marketer is stumped by this Probability Problem. Plz help. >Suppose you have two urns, each containing an equivalent VERY LARGE >(maybe not infinite, but very large) number of balls. Each urn has >only red and green balls in it. >Suppose you draw from urn A 11 times, with replacement, and you get 7 >green balls. So the ratio of green balls to total draws in the sample >is 7/11. >Suppose you draw from urn B 15 times, with replacement, and you get 8 >green balls. So the ratio, for urn B, of green balls to total draws >in the sample is 8/15. >Assuming random sampling, and given these samples, what is the >probability that urn A contains more green balls than urn B? Since nobody else has said anything, I'll start on this problem and see where it goes. No promises... I think for your purposes you can simplify this from the discrete problem of ball ratios to a more abstract one involving real number probabilities. Let green_A be the ratio of green balls to total balls in urn A, and green_B be the same ratio in urn B. Your samples are cases of binomial distribution. P(7 of 11 from A) = choose(11,7) * green_A^7 * (1-green_A)^4 P(8 of 15 from B) = choose(15,8) * green_B^8 * (1-green_B)^7 choose(n,m) = n! / (m! * (n-m)!) P(7 of 11 from A) = 330 * green_A^7 * (1-green_A)^4 P(8 of 15 from B) = 5435 * green_B^8 * (1-green_B)^7 To continue, we need to assume a probability distribution for green_A and green_B. Let's assume a uniform distribution; that is, 1/1000 green is just as likely as 500/1000. We are going to work in the samples next, so don't worry about that. I assume you haven't been given any other info about what the distribution might be, such as at least 25% of the balls in each urn are green. S = sample event R = ratio event P(S&R) = P(R) * P(S|R) P(S1&R1&S2&R2) = P(R1) * P(S1|R1) * P(R2) * P(S2|R2) We know both P(S|R) from the binomial distributions, and we're assuming P(R) to be uniform on [0,1] so I think that: P(green_A > green_B) = int(green_A=0,1)[int(green_B=0,green_A)[ 330 * green_A^7 * (1-green_A)^4 * 5435 * green_B^8 * (1-green_B)^7 * 1 * 1]] A (double) polynomial integral -- not too hard but the more samples you have the more terms you get. A good check would be to reverse the samples, and the two probabilities should add to 1. If I made a mistake they probably won't. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Continued Fraction of n*x with n-th Partial Quotient = 1 Consider a real number x that satisfies the condition: the n-th partial quotient of the continued fraction expansion of n*x equals 1 for all positive integer n where the constant term is assigned index 1. Example. A possible solution to x such that CF(n*x)[n]=1 may begin: x=1.8661698591292536563675165915962758366350388262... and the continued fraction expansions of n*x are CF(x)=[_1; 1, 6, 2, 8, 2, 11, 1, 1, 1, 1, 22,...] CF(2x)=[3;_1, 2, 1, 2, 1, 3, 1, 2, 1, 5, ...] CF(3x)=[5; 1,_1, 2, 26, 2, 3, 1, 1, 6, 1,...] CF(4x)=[7; 2, 6,_1, 1, 2, 1, 2, 2, 1, 1,...] CF(5x)=[9; 3, 44, 2,_1, 1, 11, 1, 1, 4,...] CF(6x)=[11; 5, 13, 4, 1,_1, 3, 3, 2, 3, ...] CF(7x)=[13; 15, 1, 4, 1, 2,_1, 2, 1, 2,...] CF(8x)=[14; 1, 13, 6, 2, 2, 4,_1, 2, 1,...] CF(9x)=[16; 1, 3, 1, 8, 6, 1, 5,_1, 1, ...] CF(10x)=[18; 1, 1, 1, 21, 1, 2, 3, 5,_1, ...] where the partial quotients along the main diagonal are all to equal 1. It seems that there may be an infinite number of attractors that satisfy this condition. What I am interested in are the extremes (assuming that at least one solution exists). The minimum value for such an x is greater than 1.25, and the maximum value of such x is less than 1.88. What are the aproximate minimum and maximum values that satisfy the condition? Paul === Subject: Re: 4400 > The reason I posted that was to inform other people exactly what the heck > John was talking about. Well, they don't know the Usenet Rules*: Usenet Rule #37 (Faisal Nameer Jawdat): Read the thread from the beginning, or else. (* For you newbies, these rules can be found at http://www.faqs.org/faqs/usenet/legends/godwin/ at the bottom of the page.) -- Christopher Heckman, Usenet surfer since 1994. > Similarly to his post, I should have given more information. > > American Zoetrope, comes a haunting new limited series: THE 4400. > Over the last century, thousands of people have gone missing. > Suddenly > and inexplicably, 4400 missing people are returned all at once, as they > were > on the day they vanished. Unclear what this world altering-event means, > the > government investigates the 4400 to piece together where they've been > and > why they've been returned. It quickly becomes apparent that their > presence > will change the human race in ways no one could have ever foreseen. post: Why 4400, instead of, say, 5280? > -- Christopher Heckman > Can anyone tell me what is special about the number 4400? As in the TV show the the 4400 > what does it stand for? > Its not a prime number nor does it have a even square root. any ideas??? === Subject: Re: 4400 Sorry my friend, but the information I posted is no where in this thread - hence the reason why I posted it. But I'm sure many other people will listen to your wisdom for future posts. > The reason I posted that was to inform other people exactly what the heck > John was talking about. > Well, they don't know the Usenet Rules*: > Usenet Rule #37 (Faisal Nameer Jawdat): Read the thread from > the beginning, or else. > (* For you newbies, these rules can be found at > http://www.faqs.org/faqs/usenet/legends/godwin/ at the bottom of the page.) > -- Christopher Heckman, Usenet surfer since 1994. > Similarly to his post, I should have given more information. > company, > American Zoetrope, comes a haunting new limited series: THE 4400. > Over the last century, thousands of people have gone missing. > Suddenly > and inexplicably, 4400 missing people are returned all at once, as they > were > on the day they vanished. Unclear what this world altering-event means, > the > government investigates the 4400 to piece together where they've been > and > why they've been returned. It quickly becomes apparent that their > presence > will change the human race in ways no one could have ever foreseen. > post: Why 4400, instead of, say, 5280? > -- Christopher Heckman Can anyone tell me what is special about the number 4400? As in the TV show the the 4400 > what does it stand for? > Its not a prime number nor does it have a even square root. any ideas??? === Subject: Re: 4400 | Can anyone tell me what is special about the number 4400? | | As in the TV show the the 4400 | what does it stand for? | Its not a prime number nor does it have a even square root. | | any ideas??? It has a square root which I can express precisely in the form of a repeating pattern in the Stern-Brocot number system. Start with a Stern-Brocot tree. Descend 66 steps to the right, followed by 3 steps to the left, followed by 66 more steps to the right. Repeat the 66R, 3L, 66R sequence over and over and it converges on the square root of 4400. That square root can be approximated by the following fraction of whole numbers: 835119505470355596931039942986758467 / 12589900248874196950608348266938385 Also: If you take an ASCII code value for a character used to represent a digit in the hexadecimal number system, including either upper or lower case for those digits which are normally letters (e.g. a..f and A..F), an do a bitwise logical-OR of those code with the bits representing the value 4400, then what you get is a number that when divided by 55 results in the value that ASCII character represents in the remainder of that division. -- ---------------------------------------------------------------------------- - | Phil Howard KA9WGN | http://linuxhomepage.com/ http://ham.org/ | | (first name) at ipal.net | http://phil.ipal.org/ http://ka9wgn.ham.org/ | ---------------------------------------------------------------------------- - === Subject: Re: logic of the Cantorian followers mind > I've been told Cantor's proof has nothing to do with computability. Diagonalization is a powerful technique with many applications. It can be used to show, among other things, that there are uncomputable functions from N -> N. > I don't find it strange that I was able to lead you here though. The concept of computability relates to the existence of unidentifiable real numbers insofar as there is to be an effective method for distinguishing between identifiers. === Subject: Re: 2-manifold metric spaces with many symmetries > A property of euclidean 2-D space is that: > Given two congruent triangles with distinct edge-lengths, say > triangle(A, B, C) and triangle(A', B', C') , then there is just one > isometry that maps the first triangle to the second. ( A, B, C, A', > B', C' are points in euclidean 2-D space.) [ Property 1 ] > I think the same is true for a torus, viewed as a finite-height > cylinder with vertical axis of rotation and where the upper boundary > of the cylinder is glued to the lower boundary. >> The result does not hold for the torus T^2. T^2 has all the >> translation symmetries of R^2, but rotations in T^2 (i.e., >> isometries of T^2 that fix a point) are severely limited. > [...] > That came as a surprise to me. But now I see that viewing T^2 [...] >> Presumably, you want the surface to be a smooth submanifold >> of R^3. > I want the surface to be a smooth submanifold of R^n, > for some positive n. > ( I had a look at Riemannian manifold > here: > http://en.wikipedia.org/wiki/Riemannian_manifold ) For more on the geometry of spacetime assuming spatial homogeneity and isotropy: (perhaps this is physics?) List of links: David Eppstein's Geometry Junkyard page: http://www.ics.uci.edu/~eppstein/junkyard/topic.html has a Hyperbolic Geometry link which has this link: Visualization of a hyperbolic universe, (by Martin Bucher) which takes one here: (Martin Bucher's homepage): http://www.damtp.cam.ac.uk/user/mab43/ There one finds images for a hyperbolic (3-D) universe, a flat or euclidean universe and a spherical universe. Also, the following text: The above three images illustrate the difference in perspective of the three types spacetime geometries possible if the requirements of spatial homogeneity and isotropy are imposed. Perhaps this is a result in the domain of physics (General Relativity). Anyway, the images are interesting. David Bernier === Subject: Re: 2-manifold metric spaces with many symmetries > >> A property of euclidean 2-D space is that: >> >> Given two congruent triangles with distinct edge-lengths, say >> triangle(A, B, C) and triangle(A', B', C') , then there is just >> one isometry that maps the first triangle to the second. ( A, >> B, C, A', B', C' are points in euclidean 2-D space.) >> [ Property 1 ] >> >> I think the same is true for a torus, viewed as a finite-height >> cylinder with vertical axis of rotation and where the upper >> boundary of the cylinder is glued to the lower boundary. The result does not hold for the torus T^2. T^2 has all the > translation symmetries of R^2, but rotations in T^2 (i.e., > isometries of T^2 that fix a point) are severely limited. >> [...] >> That came as a surprise to me. But now I see that viewing T^2 > [...] > Presumably, you want the surface to be a smooth submanifold of > R^3. >> I want the surface to be a smooth submanifold of R^n, for some >> positive n. >> ( I had a look at Riemannian manifold here: >> http://en.wikipedia.org/wiki/Riemannian_manifold ) > For more on the geometry of spacetime assuming spatial homogeneity > and isotropy: (perhaps this is physics?) > List of links: > David Eppstein's Geometry Junkyard page: > http://www.ics.uci.edu/~eppstein/junkyard/topic.html > has a Hyperbolic Geometry link which has this link: Visualization > of a hyperbolic universe, (by Martin Bucher) > which takes one here: (Martin Bucher's homepage): > http://www.damtp.cam.ac.uk/user/mab43/ > There one finds images for a hyperbolic (3-D) universe, a flat or > euclidean universe and a spherical universe. > Also, the following text: > The above three images illustrate the difference in perspective of > the three types spacetime geometries possible if the requirements of > spatial homogeneity and isotropy are imposed. > Perhaps this is a result in the domain of physics (General > Relativity). Anyway, the images are interesting. Probably not. Physics is not geometry. It's more like differential geometry. > David Bernier The result you want requires the manifold to have a transitive action of isometries by some Lie group G; in addition, this action must contain the rotation group SO(n) in the isotropy subgroup of G at x for each element x. That means that your n-manifold has a symmetry group of dimension at least n + n(n-1)/2 = n(n+1)/2. I note that this is the dimension of the group SO(n+1), the group of isometries of the n-sphere. I'll note that among the various differential structures on S^n (for n >= 7, these are non-diffeomorphic smooth structures on the same topological space S^n), the only one that has a group of isometries of this dimension is the standard, round sphere. I suspect that this condition poses a severe constraint on the manifold, since it's as symmetric as the n-sphere. Dale === Subject: Re: 2-manifold metric spaces with many symmetries Yet another correction. I should have proofread this stuff better. ... stuff deleted ... > I want the surface to be a smooth submanifold of R^n, for some > positive n. ... more stuff deleted ... Here, I hadn't scanned up to see the use of n as the dimension of the ambient space, and used it incorrectly as the dimension of the submanifold; I've replaced n by k in this paragraph, and note that it refers to the dimension of the submanifold: > The result you want requires the manifold to have a transitive > action of isometries by some Lie group G; in addition, this > action must contain the rotation group SO(k) in the isotropy > subgroup of G at x for each element x. That means that your > k-manifold has a symmetry group of dimension at least > k + k(k-1)/2 = k(k+1)/2. I note that this is the dimension > of the group SO(k+1), the group of isometries of the k-sphere. > I'll note that among the various differential structures on > S^k (for k >= 7, these are non-diffeomorphic smooth structures > on the same topological space S^k), the only one that has a > group of isometries of this dimension is the standard, round > sphere. > I suspect that this condition poses a severe constraint on > the manifold, since it's as symmetric as the k-sphere. > Dale There it is. Now get back to work. Dale. === Subject: Re: 2-manifold metric spaces with many symmetries Just fixing a poorly-worded sentence. ... stuff deleted ... >> Perhaps this is a result in the domain of physics (General >> Relativity). Anyway, the images are interesting. > Probably not. Physics is not geometry. It's more like differential > geometry. What I meant to have said was that the OP's problem is more like differential geometry than it is like physics. >> David Bernier ... the rest deleted ... > Dale === Subject: Re: mean value of the roots of a stochastic polynom a is fixed, and the hypothesis b << c won't be too false, so that the series expansion migth be an interesting solution. I will try it for the case where b is follows a Weibull distribution and where the pdf of c is empirically known. Manu === Subject: What's this function called? I have defined the following functions, which I find intriguing and beautiful. I vaguely remember seeing some of it before, so I doubt its that original. Can anyone tell me what these functions called? For all integers n, we define s(n) as the sum of all the prime factors of n, counting repeated factors multiple times. For example s(10) = 7, since 10=2*5; 8=2*2*2, thus s(8) = 6. And s(p) = p for all primes p. s(mn) = s(m) + s(n). We can extend it to negative integers, so that s(-m) = s(m) + 1/2 Argument: s(mn) = s(m) + s(n), s(-1 * -1) = s(-1) + s(-1) = s(1), s(1) = 1, hence 2*s(-1) = 1, hence s(-1) = 1/2. s(-m) = s(-1 * m) = s(-1) + s(m), thus s(-m) = s(m) + 1/2. Also, s(m^n) = n*s(m), so arguably s(m^-1) = -s(m). Thus s(m/n) = s(m) - s(n). Which provides an extension to the rationals. [Could it sensibly be extended further, say to the reals?] Can we give a sensible value to s(0)? Say, s(0) = 0? But, ignoring this extension to negative integers and rationals: Now, we can iterate this function s, by applying its result to itself. For example s(8) = 6. s(6) = 5. Let us define another function t(n), to be the result of iterating s(n) until we keep on getting the same value (which will be a prime, or the number 4). Finally, we can define another function u(n), which is the number of times we must iterate s(n) before the process terminates. For example u(4) = 0, since s(4) = 4. u(any prime) = 0, since s(any prime) = that prime. u(6) = 1, since s(6) = 5, and s(5) = 5. u(8) = 2, since s(8) = 6, s(6) = 5, and s(5) = 5. So basically, are there names for these functions I have termed s, t, u? Simon Kissane === Subject: Re: What's this function called? > For all integers n, we define s(n) as the sum of all the prime factors > of n, counting repeated factors multiple times. For example s(10) = 7, s(0) = 0 = s(1) ? > s(mn) = s(m) + s(n). You're messing around with a homomorphism from a multiplicative group to an additive group. That's all you're doing. s(0) = s(0*7) = s(0) + s(7) s(0) = s(0*5) = s(0) + s(5) s(0) cannot be defined when requiring s(mn) = s(m) + s(n) > We can extend it to negative integers, so that s(-m) = s(m) + 1/2 > Argument: s(mn) = s(m) + s(n), s(-1 * -1) = s(-1) + s(-1) = s(1), s(1) > = 1, hence 2*s(-1) = 1, hence s(-1) = 1/2. s(-m) = s(-1 * m) = s(-1) + > s(m), > thus s(-m) = s(m) + 1/2. s(1) = s(1*1) = s(1) + s(1); s(1) = 1 ??? No! s(1) = 0 So explain this wonderful mess s(1) = s(-1 * -1 * -1 * -1) = s(-1) + s(-1) + s(-1) + s(-1) = 4(s(1) + 1/2) = 4s(1) + 2 Or this one s(2) = s(--2) = s(-2) + 1/2 = s(2) + 1 > Also, s(m^n) = n*s(m), so arguably s(m^-1) = -s(m). Thus s(m/n) = s(m) > - s(n). Which provides an extension to the rationals. [Could it s(1) = s((-1)^2) = 2s(-1) s(1) = s((-1)^4) = 4s(-1) s(-1) = 0, you've no other choice > sensibly be extended further, say to the reals?] log ab = log a + log b for a,b > 0 log 1 = 0 log a^n = n log a log a/b = log a - log b log |ab| = log |a| + log |b| for all the reals /= 0 > Can we give a sensible value to s(0)? Say, s(0) = 0? No as explain above. Also s(1) = s(-1) = 0, so s(-n) = s(-1*n) = s(-1) + s(n) = s(n) > But, ignoring this extension to negative integers and rationals: Now, > we can iterate this function s, by applying its result to itself. For > example s(8) = 6. s(6) = 5. Let us define another function t(n), to be > the result of iterating s(n) until we keep on getting the same value > (which will be a prime, or the number 4). So now you're talking about orbits of a group action or something like that. s(1) = 0, s(0) undefinable, t(0), t(1) undefinable > Finally, we can define another function u(n), which is the number of > times we must iterate s(n) before the process terminates. > For example u(4) = 0, since s(4) = 4. > u(any prime) = 0, since s(any prime) = that prime. > u(6) = 1, since s(6) = 5, and s(5) = 5. > u(8) = 2, since s(8) = 6, s(6) = 5, and s(5) = 5. As before u(1), u(0) not definable. > So basically, are there names for these functions I have termed s, t, u? tus === Subject: Re: What's this function called? S(1) = 0 because 1 doesn't have any prime factors. > s(0) = s(0*7) = s(0) + s(7) The usual trick when dealing with this problem is to realize that 0 is divisible by an infinite number of copies of any prime(s) you want, so s(0) is the sum of them all, i.e. infinity. Then good old Cantor cardinal number arithmethic shows that infinity plus any finite number is infinity again, satisfying the above equation. The same sort of thing comes up when dealing with ideals generated by 1 or 0 and asking questions such as unique factorization. The OP's attempt to define S(-1) = 1/2 did no good at all, as you pointed out, because S(-1 * -1) doesn't equal S(-1) + (-1). Perhaps units as factors should be ignored, as they usually are when dealing with factorizations. So S(-1) = 0. As for the OP's question whether it can be extended to rationals: NO, because in a field *all* nonzero numbers are units, so S(n) would have to be zero for *all* nonzero values, contradicting the original definition involving prime factors as integers. === Subject: Re: Ozkural was Re: Platonism >> > That was a pedagogical question (e.g. how do you tell what Z is to > somebody who's never studied formal set theory, who is *not* me if you > will try another sore joke), and it was a real explanatory issue that > I faced trouble with over a coffee table. >> >> Right. You were wondering if there are integers with an infinite >> number of digits. >> Indeed. His question was not >> How do I explain to my interlocutors that each integer has finitely >> many digits in its base ten representation? >> it was >> are there integers with an infinite number of digits?. > Doh! Not interlocutors, friends. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Need help with inverse laplace transform!! Hi! I could use some help with inversing a Laplace Transform. The function looks as follows: exp(-v*z)*s*(1-v) where v=sqrt(1-2/s) and I want to perform an inverse laplace transform (s ->u) When I performed the inversion myself, I ended up with a modified Besselfunction*exp(u)/u, but this does not seem to be right.. I am very grateful for any ideas!!! /Malin *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Re: Painful but inevitable resignation >> Of course, I too can envision probable continuation of the trail >> into future, extrapolated from the existing trail. But I see such >> continuations more as the ability to predict the logical end or >> continuation of coherent processes that we understand. >> The point is, real life is never completely predictible but embedded in >> unseen external influence. > Of course, but I don't see why this should be a problem, since you > agree tht the future is not written yet. I see the problem in lacking public awarenes. Even John L. Bell http://publish.uwo.ca/~jbell/ shares the widespread deterministic notion of causality. He is otherwise expressing my mathematical understanding quite accurately. > Sure. I never treated future time as if it was observable. Was there any objection? === Subject: Re: Painful but inevitable resignation >> Of course, I too can envision probable continuation of the trail >> into future, extrapolated from the existing trail. But I see such >> continuations more as the ability to predict the logical end or >> continuation of coherent processes that we understand. >> The point is, real life is never completely predictible but embedded in >> unseen external influence. > Of course, but I don't see why this should be a problem, since you > agree tht the future is not written yet. > I see the problem in lacking public awarenes. Even John L. Bell > http://publish.uwo.ca/~jbell/ > shares the widespread deterministic notion of causality. He is > otherwise expressing my mathematical understanding quite accurately. I agree. I observed no awareness at all, even in the scientific community. > Sure. I never treated future time as if it was observable. > Was there any objection? If there were, they kept their objections to themselves. Besides, physicists tend to dislike discussing with me. Causality is not very popular these days. Andr.8e Michaud === Subject: High school factorization of algebraic fraction Hi Group, I was wondering how I could go about writing this algebraic fraction in the simplest factored form: ((6aî+2a)/(4a+8))*((6a+12)/(a)) What steps should I take? James Midolo === Subject: Re: High school factorization of algebraic fraction > I was wondering how I could go about writing this algebraic fraction > in the simplest factored form: > ((6a2+2a)/(4a+8))*((6a+12)/(a)) > What steps should I take? When you say 6a2, do you mean 6 * a^2 ? I'll assume the answer is yes. Well, before posting you should at least *try* to do the obvious stuff. For example, do you see that 4a+8 has a common factor of 4? So you can factor that as 4 * (a+2), right? So why didn't you at least do that tiny bit of the work before posting? There are two other places where you should have seen common factors that are obvious and easy to pull out. So why didn't you do those simple operations before posting? Are you so terribly shy and afraid of making the slighest mistake and being embarrassed that you would rather have somebody else do your homework for you then even try to do it yourself?? Or are you just plain too lazy to do your own work? Unfortunately Jeroen Boschma already posted a response with most of that easy obvious stuff already done for you, and I presume you looked at and copied his work, so you have forever lost the chance to figure that sort of stuff out yourself. On your final exam you won't have him to do your work for you, so I suggest you find another example from your book that looks about as difficult as this example, and try that example without any help, get as far along with the easy stuff as you can, then post the question and your partial work here and ask if there's anything further you can do. That's the only way you'll learn how to do these sorts of factorizations and simplifications of polymomial fractions. === Subject: Re: High school factorization of algebraic fraction > I was wondering how I could go about writing this algebraic fraction > in the simplest factored form: > ((6a2+2a)/(4a+8))*((6a+12)/(a)) > What steps should I take? > When you say 6a2, do you mean 6 * a^2 ? > I'll assume the answer is yes. > Well, before posting you should at least *try* to do the obvious stuff. > For example, do you see that 4a+8 has a common factor of 4? So you can > factor that as 4 * (a+2), right? So why didn't you at least do that > tiny bit of the work before posting? There are two other places where > you should have seen common factors that are obvious and easy to pull > out. So why didn't you do those simple operations before posting? Are > you so terribly shy and afraid of making the slighest mistake and being > embarrassed that you would rather have somebody else do your homework > for you then even try to do it yourself?? Or are you just plain too > lazy to do your own work? > Unfortunately Jeroen Boschma already posted a response with most of > that easy obvious stuff already done for you, and I presume you looked You are very right. I have a habit of posting a response to questions I think I can handle adequatly (as a non-mathematician), but in this case I should not have worked towards the final solution as far as in my response. An equivalent example should be enough to show the 'trick'. Something to keep in mind for the next time :) > at and copied his work, so you have forever lost the chance to figure > that sort of stuff out yourself. On your final exam you won't have him > to do your work for you, so I suggest you find another example from > your book that looks about as difficult as this example, and try that > example without any help, get as far along with the easy stuff as you > can, then post the question and your partial work here and ask if > there's anything further you can do. That's the only way you'll learn > how to do these sorts of factorizations and simplifications of > polymomial fractions. === Subject: Re: High school factorization of algebraic fraction >> I was wondering how I could go about writing this algebraic fraction >> in the simplest factored form: >> ((6a2+2a)/(4a+8))*((6a+12)/(a)) >> What steps should I take? > When you say 6a2, do you mean 6 * a^2 ? > I'll assume the answer is yes. get the little 2 for squared. I wasn't aware that it wouldn't work. > Well, before posting you should at least *try* to do the obvious stuff. > For example, do you see that 4a+8 has a common factor of 4? So you can > factor that as 4 * (a+2), right? So why didn't you at least do that > tiny bit of the work before posting? There are two other places where > you should have seen common factors that are obvious and easy to pull > out. So why didn't you do those simple operations before posting? Are > you so terribly shy and afraid of making the slighest mistake and being > embarrassed that you would rather have somebody else do your homework > for you then even try to do it yourself?? Or are you just plain too > lazy to do your own work? I have maths revision sheets here that our teacher gave us to use to study for our upcoming mathematics exam. I tried this question for about 4 hours over the weekend. Every time I tried it, I came up with the 3(3a + 1). I tried it several different ways, like expanding the top and bottom lines and attempting to factorize, etc. The reason that I kept trying was that the answer in the answer section on the sheets was: (a(3a+1))/(3). I was attempting to get this answer for a long time. I thought that there must be something harder about this question, because it was the last question in the exercise. I should have substituted values in for the answer and the question to make sure that it was actually a correct answer in the answers section. I decided that the people here might be able to explain or whatever how to get to the (a(3a+1))/(3). I wasn't aware of the alt.math.undergrad newsgroup that Stan Brown has told me, and I will be using that group in future. So yeah, sorry about wasting peoples' time and asking questions that I wasn't sure if the answer shown was correct anyway. > Unfortunately Jeroen Boschma already posted a response with most of > that easy obvious stuff already done for you, and I presume you looked > at and copied his work, so you have forever lost the chance to figure > that sort of stuff out yourself. On your final exam you won't have him > to do your work for you, so I suggest you find another example from > your book that looks about as difficult as this example, and try that > example without any help, get as far along with the easy stuff as you > can, then post the question and your partial work here and ask if > there's anything further you can do. That's the only way you'll learn > how to do these sorts of factorizations and simplifications of > polymomial fractions. Well I thought that I had the grasp of the subject, and this question came out and stumped me when the given answer didn't match mine. I have now gone and tried all the questions in the algebraic exercise and I am confident that I know how to do them. Sorry that I didn't show my partial working, because I thought that it was wrong, but I guess now looking back I should have shown you my working so that you could point out where I had made the error if I had made an error. Sorry for wasting everyones' time... === Subject: Re: High school factorization of algebraic fraction >I was wondering how I could go about writing this algebraic fraction in the >simplest factored form: >((6aî+2a)/(4a+8))*((6a+12)/(a)) The tops and bottoms can be multiplied: [ (6a^2+2a) (6a+12) ] / [ (4a+8) a ] Then just factor top and bottom normally, and look for common factors to remove. Start with common monomials: [ 2a(3a+1) * 6(a+2) ] / [ 4(a+2) a ] You see that (a+2) is a factor of the whole top and the whole bottom, so it's gone: [ 2a(3a+1) * 6 ] / [ 4a ] And a is also a common factor, so it's gone: [ 2(3a+1) * 6 ] / 4 Now on the top you have 2*6 = 12 = 4*3: 4*3(3a+1) / 4 and the common factor of 4 is divided out of top and bottom: 3(3a+1) For really basic questions you might want to post to alt.math.undergrad or even one of the k12 groups in future -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com A: Maybe because some people are too annoyed by top-posting. Q: Why do I not get an answer to my question(s)? A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? === Subject: Re: High school factorization of algebraic fraction > Hi Group, > I was wondering how I could go about writing this algebraic fraction in the > simplest factored form: > ((6aî+2a)/(4a+8))*((6a+12)/(a)) > What steps should I take? What steps have you taken so far? === Subject: Re: High school factorization of algebraic fraction > Hi Group, > I was wondering how I could go about writing this algebraic fraction in the > simplest factored form: > ((6aî+2a)/(4a+8))*((6a+12)/(a)) > What steps should I take? > James Midolo You have: (6a^2 + 2a)*(6a+12) ------------------- (4a+8)*a Try to move terms out of the ()-brackets, like: (4a+8) = 4*(a+2) (6a^2+2a) = 2a*(3a + 1) You'll get: 2a*(3a + 1) * 6(a+2) ---------------------- 4*(a+2) * a Then eliminate equal terms up and below the division line, recalling that 2*6/4=3 you get your answer. Jeroen New job listings at http://jobs.phds.org - Jobs for PhDs List your job at no cost! http://jobs.phds.org/jobs/post * Research on Algorithms and Architectures for Computational Biochemistry: D.E. Shaw, New York, NY. Extraordinarily gifted computer scientists, systems architects, electrical engineers and systems software professionals are sought to join a rapidly growing New Yorkbased research group pursuing an ambitious, long-term project aimed at achieving major scientific advances in... * Systems Architects and ASIC Engineers: Specialized Supercomputer for Computational Drug Design: D.E. Shaw, New York, NY. Extraordinarily gifted systems architects and ASIC design and verification engineers are sought to participate in the development of a special-purpose supercomputer designed to fundamentally transform the process of drug discovery within the pharmaceutical industry. This... synthesized by nonaqueous solgel chemistry: Martin-Luther-Universitíót Halle-Wittenberg - Fachbereich Chemie - Institut fí.b9r Anorganische Chemie, Halle (Saale) - Germany. Two PhD positions are opened at the Martin Luther University of Halle-Wittenberg (Germany) in the department of chemistry. In this project we are going to extend the known... * Mathematics Instructor, HR233-05: Truckee Meadows Community College, Reno, Nevada. Truckee Meadows Community College, located in Reno Nevada, seeks a full-time, tenure-track, Mathematics instructor for the Mathematics, Science, Engineering and Technology division. The primary teaching assignment will be pre-calculus mathematics.... === Subject: FLTMA: Fermat's Last Theorem and Modular Arithmetic topic. I have no excuse. I will get to it. You can always use Google Groups with search terms dgoncz@ and Fermat for a rough scan. I am using the AOL proportionally spaced newsreader to post this list of computer output. I will read this tonight in Google. How does it look in your reader? pa=phi(a) etc. n N a ta pa (c^n-b^n) mod a b tb pb (c^n-a^n ) mod b c tc pc (a^n+b^n) mod c 1 2 3 4 5 6 7 8 9 10 11 2 3 2 2 0 1 1 4 2 2 0 2 5 4 1 2 6 7 6 6 0 1 5 5 2 1 2 9 6 3 0 3 6 10 4 4 2 6 0 2 6 7 6 6 0 4 4 5 6 2 2 9 6 3 0 6 3 13 12 12 2 3 0 6 5 1 0 2 6 9 0 5 6 8 4 2 0 4 2 9 6 6 0 5 6 2 3 8 13 12 12 2 4 2 6 10 11 0 1 4 9 8 8 6 7 6 6 0 3 6 0 4 1 2 10 4 4 0 6 0 4 13 12 12 2 4 6 4 12 2 0 3 12 7 7 6 10 3 2 2 0 2 6 11 10 10 0 10 6 3 1 9 6 9 9 2 13 12 12 2 1 0 6 6 3 0 5 5 9 0 2 6 9 6 6 0 2 3 2 6 2 10 11 10 10 0 4 0 5 0 9 0 3 0 1 13 12 12 2 7 7 6 12 10 0 11 12 9 6 9 2 5 4 4 0 1 0 4 2 12 4 2 0 8 13 12 4 2 4 0 7 6 7 6 6 0 1 4 0 6 3 2 12 4 2 0 6 13 12 12 2 6 11 4 10 10 0 5 4 7 5 1 30 11 10 10 0 1 3 7 4 9 8 6 2 5 2 12 4 2 0 2 13 12 12 2 10 5 4 4 6 0 1 10 7 11 5 OK, that's a little rough. The two smaller sequences should be repeated to extend to the alignment with the longer, and all three should end with zeroes. (Zeros?) It's a start. The output from Mathcad is a matrix. It gets copied to the Works spreadsheet where zeroes get blanked. That gets output as text and tabs and converted to four space tabs with CLR Text to Spaces. That gets read into the AOL text editor in Courier font to check spacing and then pasted into the chatty proportional newsreader. Working with Outlook Express and Outlook is a little frustrating, but I am on it. I pasted a few zeroes back in after zeroes were blanked. You see, it appears N is always even, intersecting with n prime to give n = 2. N is the location of the triple coincidence of 0, if it occurs. What I do is generate vectors A, B, and C, mod a, b, and c, as above, listing the dual exponential congruential sequences to limits phi(a), phi(b), and phi(c), then determine the actual periods ta, tb, and tc. I compute lcm(ta,tb,tc). If the triple coincidence occurs, it must occur in this range. I recompute A, B, and C in this new, larger range, and look for the coincidence A.n+B.n+C.n=0, and if found, N=n, so then I add with Mathcad's stack function to the output matrix. I compute lcm(ta,tb,tc)/N to give Q, the number below N, and forget to include that in the header. Sorry. I don't dare tinker with it now that I am in the proportional editor. In tests to c=25, N ranged as high as 330. Printed and faxed output of this list, larger lists, and the Mathcad source is available for those interested in FLTMA. I tolerance everything and tolerate everyone. I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri, alphabetically. I drive: A double-step Thunderbolt with 657% range. I fight terrorism by: Using less gasoline. === Subject: Re: Skolem's Paradox and why is math the way it is? >>I mostly agree, but at least one of my physics professors in college >>considered treating the wave-function as real as wrong, wrong, wrong. >>It applies only to a statistical ensemble, not to just a single >>system. (!) > Doesn't the Aspect experiment make that view untenable? Quoting from [1]: However, the low efficiency of the detectors used in the experiments means that additional assumptions (essentially that those photons detected are a fair sample of the total flux) have to be made to test the Bell inequality. If these assumptions are made, the results are found to rule out local realist theories, and to be in good agreement with the quantum predictions. Most physicists now accept that quantum theory is correct, and that local realism has to be abandoned. However, other physicists, often known as the realists, strongly disagree. They question the additional [...] Reference [1] presents an interesting biographical sketch of John Bell. [1] Andrew Whitaker: John Bell and the most profound discovery of science, Physics World, December 1998 URL: David Bernier === Subject: Re: Skolem's Paradox and why is math the way it is? > Axioms for FOL and ZFC are unambiguously stated at: > (C => B) (A2) (B => (C => D)) => ((B => C) => (B => D)) [...] Compare to and . === Subject: Re: Skolem's Paradox and why is math the way it is? [...] |> I think it will be mighty difficult for you to order your development |> unless you admit the necessity of starting informally, for at least a |> brief time, or you decide to go formal all the way and not care about |> such things as whether the theory has a model. | |Which theory are you talking about? Any one that one intends to treat as foundational. | Quine is discussing statements, |and Hintikka has games, either of those seems fine to treat |informally, they are both just abstractions of VERBS, things I can do |personally. You can perform uncomputable strategies and communicate them to your allies, can you? I don't think so. My sets are abstractions of ADJECTIVES. In order for an adjective to make sense, I don't necessarily have to be able to do anything besides understand what it means. Isn't that less problematic? | And so it is disprovable and subject to experimental |observation ultimately. How? Disproof requires a shared notion of what counts as proven from the premises in question. This ordinarily is done in an informal way, but if we want to resolve all disputes in a rigorous way, we ultimately want to formalize the theory in question. IF logic hasn't been formalize as far as I know (and any formalization would necessarily be partial). [...] |> It seems, then, that we have found a dictionary between whatever I |> might have to say mathematically that concerns this large submodel |> of the cumulative hierarchy (the sets having rank less than the |> smallest inaccessible cardinal) can be translated into a language |> that you can understand! | |I'm still not sure that there is such a submodel that fits the |standard interpretation. Fine, lots of people doubt that too. The issue is whether the properties that it would have, were it to exist, can be discussed in an unambiguous way. Since you believe in IF logic, you agree that they do; don't you have to? | If I look at the set theory validities, |i.e. the valid formulas N or T where T is an oFOL closed formula and |N is the IF-FOL translation of the SO negations of the SO set theory |axioms SO alternated together, then it can be true in all models, but |that doesn't mean that there is a model where N is actually false. It |just means that N or T is a validity. I understand. The same could be said by a person who believes in the coherence of FOL as applied to models, but who doubts the existence of any infinite sets. |> |This is FINE for physics because in physics we ALSO play verification |> |and falsification games in the laboratory, so I can make them match |> But you don't play uncomputable strategies against the universe, |> so far as I know. IF-logic only works the way that it does because |> one is implicitly assuming the possibility of using uncomputable |> strategies. | |Where did computable come from? As far as I know, you are unable to play uncomputable strategies. Second, IF logic develops a completely different semantics if we only consider computable strategies. For example, suppose we play with rational numbers the game corresponding to the sentence (*) (A)(B) {[(EN) (x)(y) x>N -> y^2<>x^3+Ax+B] v [(N) (Ex)(Ey) x>N & y^2=x^3+Ax+B]. This is just a sentence of first-order logic. It's regarded as true because the two subformulas in [] are negations of each other, but this is a nonconstructive fact. In order actually to be able to win every time, you would need an algorithm for determining whether there are arbitrarily large solutions in x and y to y^2=x^3+Ax+B. I don't know offhand whether such an algorithm exists. I believe it to be a reasonably nontrivial question. Now take Joe Random mathematician and ask him whether (*) is true. Yes, of course, he says. Can he always play the game and win? He will inform you that believing in (*) is not based on a belief that we can *actually* play the game associated with it and win. Some would say that (*) is true because a hypothetical omniscient being can play it and win every time, but that's a completely different story. | The universe itself plays the part of |the initial falsifier, the existance of the winning strategy means you |can defeat the universe at the game, every time. If you can evaluate the winning strategy. | If you can proof the |validity, then you can show that you could win the game N or T in |any model. So I'm not making the assumption that the proofs construct |all the validities, so why should I? No, that's beside the point. | And you are totally forgeting |that the game is usually a premise as well as a conclusion, so to say |that something is continuous, you might be more precise and say that |the game Ax Ay (~(0 (Az ~(a{conclusion} can be won by the verifier only if he can first determine either that he can win the game of {premise} playing the part of the falsifier, or that he can win the game of {conclusion} playing the part of the verifier. That makes it terribly hard to play the part of verifier successfully. The function in question might have a very small discontinuity in it hidden somewhere.... I said that it would be more consistent with what people ordinarily mean by -> to allow the verifier access to the strategy for the premise, but this is not part of IF logic. |Validities is based partly on the fact that an oFOL statement T always |HAS a strategy for one side, so then you can assume a strategy for one |side and try to prove that a strategy exists so that the E-team wins |the game N or T and then you assume a strategy exists for the other |side and try to prove that a strategy exists for N or T for the |E-team. The word computable (whatever it means) never comes up, the |fact that T is oFOL and has a strategy for either side is useful. Now you are not learning how to play the game; you are learning how to deduce that a strategy would exist. That's just a dodge. The fact is, these strategies you refer to as abstractly existing are highly idealized, and have relatively little to do with any strategies you can actually follow. |Maybe you are bothered that there are two subjects, logic, the study |of formulas, and set theory the study of sentances T such for a fixed |N (unkown to me) the formulas N or T are validities. Set theory |needs logic, it really does, I didn't think this was a problem. If you look in the mathematical reviews subject categorization, there's a field known as logic that is broad enough to include set theory as a subfield. People do sometimes define logic per se to be just the study of logical validities. Your guess that this would bother me is quite a wild guess! |> How else can it be true that either for every x, ~P(x), or there |> exists an x such that P(x)? The only way to win that game is to |> be able to search the whole domain of discourse to check whether |> any of the elements satisfies P! | |I don't know what P(x) is, but saying Ax ~P(x) is true MEANS that if |you searched the universe that ~P(x) is true for every x, Ex P(x) is |true means that some choice of x makes P(x) true, whether [Ax ~P(x)] |or [Ey P(y)] is a validity or not is just a matter of whether P(z) is |true or false for every z, if it is, that's enough, then you know it's |a validity. The point of a validity is that since it is true for any |play of the game in any model, it is subject to disproof, that's the |best you can hope for. Proof in science is impossible, so having a |verb that serves as capable of disproof is the best you can hope for. |You can't perform every experiment in every place at every time, but |since the claims are universal, you can attempt disproof to your |heart's content. I don't think this is correct. How do such sentences falsify? Suppose Dr. Smith has a theory that turns out to be equivalent to the winnability of some game played against the universe. But poor Dr. Smith keeps losing this game. Has his theory been falsified? Maybe Smith is just a poor player of this game. If the only way he can win every time is by being able to solve problems without computable algorithms for solving them, I can hardly blame him for failing, can I? I think integers are either prime or composite. If we play the game (n) {[(Es)(Et) 1 st<>n]} on the natural numbers, the falsifier can probably ensure that the verifier will go to his grave before finding the necessary s and t to win the game. Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? [...] |I've asked many times what the axioms are. You know, I haven't seen it. I looked through a lot of the messages from you on this thread and didn't see any such question. I have seen lots of messages that suggest to me you think you already know enough about them to serve as a good critic. One of your original questions: Why do mathematicians like a nondescriptively complete axiom system as a basis? You asked various times why we liked the axioms, hinting that you knew of some problem. [...] how do we know everyone is doing the same math if the axioms don't describe real numbers uniquely? Why all the blather about uncountability then? It tends to be a bad sign when someone casually describes standard results in a field as blather. [Addressed to Torkel Franzen] What axioms do YOU use? He soon explained that ordinary mathematics is perfectly adequate for physics, and presumably that's what he practices. The axioms of ZF assert the existence of a set CALLED the power set of the naturals, but I've seen no proofthat this set has no logical correspondance to the naturals, in fact Skolem seems do give a convincing case that it does and the the models of ZF don't allow a set to be created from this correspondance. So you thought you understood the power set axiom well enough? What I am saying is that the first order ZF set theory axioms don't prove the existence of uncountably many subsets of the naturals. I.e., J.E. knows his ZF well enough already to say what they do and do not prove. I'm not trying to do mathematics. I've said that all along. What I'm doing is asking mathematicians about what the axioms THEY use mean to THEM so that I can decide if I was to use them in my PHYSICAL models. At this point it seemed pretty clear that what you were after was the MEANING of the axioms, i.e. a philosophical account, not just a statement of what they were. There are only countably many of them, but the others appear to be useless since for MANY MANY years NO ONE has proven theorems about them (since I haven't seen new axioms added to set theory to prove more subsets of the naturals exist than formerly could be proven to exist). So J.E. has been keeping track of which axioms have been added! I HAVE written out definitions of countable models of ZF(C), and they are well-founded and pure, AND obviously incomplete in that there are things that should be sets that aren't in it. J.E. has done model theory! Honestly, you claimed to have written out definitions of countable models of ZF(C), and now you tell us you never actually knew what the axioms were. |> has been scarcely any strictly mathematical (as opposed to |> philosophy of math) question in this discussion. You ask things |> like How do theorists know that the SEQUENCE to generate h |> [Planck's constant] exists in ZF? that doesn't have any clear |> meaning. | |Now you can't hold me responsible for whatever other people bring to |the discussion. Of course. But for some reason their discussion of Planck's constant prompted you to wonder whether it exists in ZF, a meaningless phrase. | I would wonder if ZF is the right theory in which to |create a model that PREDICTS the value of h [Planck's constant], Well that's a heap of a lot better question. The way to answer it is definitely not to worry about model theory. It's to examine the kind of mathematics actually being used. | but |as far as I'm considered the value we measure in the lab is a rational |number. The point is that people assert that everything is in set |theory, but they go OUTSIDE set theory to assert that, and I don't |know WHERE that brazen confidence comes from. ANYONE can just ASSUME |that all sets are in their theory. There is no such concept as set in a theory. Completely undefined! You keep tossing it about as if it meant something, but it doesn't. The brazen confidence is the same chutzpah as leads us to think that when we say the bananas of the world, we mean, the bananas of the world, and not some subset of them. Does it take a special argument to show that when we say banana, we mean banana? No. Is this a bizarre assertion? No. [...] |In physics if they had a wrong theory, they'd state at the very very |beginning that it was wrong, that's very different than saying that |theorems are true for a whole semester. They knew that we knew already. The professor made a blanket statement at the beginning of the course that every theory in physics was an approximation. I do not remember him reiterating this in reference to Newtonian mechanics. |> There were a number of places where we had to fudge. Infinities |> appear in certain places that one just has to accept as being |> not quite right. The energy in the electric field of a charged |> can't be correct; the product of the charge and the electric |> field vector at it is undefined, because the electric field goes | |WHAT! Those can all be fixed by being more careful. There have been threads in sci.physics.research discussing this-- it's not entirely trivial. I think calling it being careful is to understate the case quite a bit. Professional physicists have wondered aloud whether classical E&M is consistent at all. And 99% of your problems with mathematics could be cured by being half as careful as all that. |NO ONE cares |electrodynamics and I kept asking why don't we compute the trajectory You mean j? | And the answer is that |it's harder and no one cares. This is assuming that your charges are small enough and evenly enough distributed that you can treat them like they form a continuous current density. | But it's up to the people who care |about something to make it work, the purpose of most physics classes |is not to teach you how to find analytic solutions, it's to develope |physical intuition so that you can recognize correct results and fix |problems caused by doing numerical approximations sloppily and to get |approximate answers so that you don't have to get the true answer for |every situation. Aha! And what do you suppose your mathematics courses are for, huh? Oh, wait, sorry, we're supposed to be mindless drudges with nothing else to do but be as precise as possible. I keep forgetting that. [...] |> What exactly is the Dirac delta function? Well, it's not really |> a function, and telling you what it precisely is, is beyond the |> scope of this course. :-) This willingness to work with an ill-defined |> concept is not just accepted; actual pride is taken by physicists |> in their willingness to dispense with precise definitions and | |Distribution theory, if we want to have a pissing contest about who |has better teachers, then I can concede that many many bad physics |teachers exist, in fact that's what I'd like to change, that's my goal |to improve physics teaching. Terrific. My point was not that these were bad teachers-- they were good. My point is that it was never a game of but you keep refusing to tell me the precise version of the story; I'm going to whine until you do, and if I had taken that attitude, I'd have about the same relationship to physics as you have to mathematics. Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? |> | I consider |> |math to be science. |> Then try treating it like one. |> The key objections to be made to a physical theory are that its |> predictions are observed to be incorrect, it's been superceded by |> a more comprehensive theory, and that it's needlessly complex. |> The corresponding objections can be made of a mathematical theory |> too, in principle, but I see you making an awful lot of objections |> that are of a completely different kind. | |In IF-FOL I can imagine an observational refutation of a proposed |validity (by demostrating game play inconsistent with that |description), but I don't see how you can observe other things to |invalidate a theory, the theorems are treated formally if you object |to the English, and then someone will continue using the English, so |for most people it's just a waste of time to observe anything in |math. Not at all. We can compute, for one thing. If a theory says a computation will work out one way and it works out another way (and we haven't made a mistake), then the theory must be wrong. |> Allow it to be just as sloppy and messy as physics is and nearly |> all of those remaining troubles are gone! If you are still of the |> impression that you can't tell what ZF is, try naming some statement |> which you either have trouble knowing how to formalize, or don't know |> whether it's an axiom, or a proof which you don't know whether it's |> valid, and we can surely clear it up for you. | |Is Ax ~Ef (Ea aex & {0,{0,a}}ef) & (An Ab (bex => ({n,{n,b}}ef => (Ec |cex & {nU{n},{nU{n},c}}ef & ceb)))) the foundation axiom? Is it |considered part of set theory? I have more questions like this, but |for all I know I'm already kill-filed by everyone. No, that is not how the foundation axiom is usually written. It's an odd formulation; where did you get it? For one thing this representation of an ordered pair (a,b) as {a,{a,b}} is a bit unforunate, since the foundation axiom is needed to prove that if {a,{a,b}}={c,{c,d}} then a=c and b=d! I can't see offhand whether this creates any real troubles to have this *inside* the foundation axiom itself. Also the role played by x seems a little obscure. For each f, there exists an x which contains the image of f (as a function), i.e. the set of b such that for some n, (n,b) is in f. So unless I'm missing something, x is superfluous, and we can just say that there does not exist an f satisfying (Ea (0,a)ef) & (An Ab ((n,b)ef => (Ec (n+1,c)ef & ceb)). Foundation is usually expressed so as to say that each nonempty set has a member disjoint from it. I don't know what the original form it was written in. It's always possible that it's been rewritten somewhat since it was first stated as an axiom. The way to go from the axiom of foundation to what you've got is to supose there were such an f, and then consider the set S={f(n) : n is a natural number}. Each member f(n) would share a member, namely f(n+1), with S. So by the usual axiom of foundation no such S exists, hence no such S. [...] |> We don't depend upon axiom systems in the way that you imagine. |> When someone claims that a result is a theorem, they mean that it |> has been proven, period. End of sentence. Not, proven in... but |> proven. | |What? You don't seriously mean proven without assuming any standards |of proof or axioms, do you? I didn't say anything about standards of proof, but since you ask, we do of course have standards. They are just informal, not set by a fixed axiom system. I don't know how it is that people like you get so deeply indoctrinated into the idea that everything in mathematics depends upon a choice of axioms. Without getting a heavy dose of very consistently worked out formalist philosophy of mathematics along with it. Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? > [...] > |> We don't depend upon axiom systems in the way that you imagine. > |> When someone claims that a result is a theorem, they mean that it > |> has been proven, period. End of sentence. Not, proven in... but > |> proven. > |What? You don't seriously mean proven without assuming any standards > |of proof or axioms, do you? > I didn't say anything about standards of proof, but since you ask, we > do of course have standards. They are just informal, not set by a fixed > axiom system. But different axioms systems are different. For instance you can have a non-well founded set theory, then you usually have collection instead of repalcement, but in a normal set theory class a proof of collection is required. > I don't know how it is that people like you get so deeply indoctrinated > into the idea that everything in mathematics depends upon a choice of > axioms. Without getting a heavy dose of very consistently worked out > formalist philosophy of mathematics along with it. I took many classes where people said theorem and truth a lot but always turned out to really mean proof in ZFC. Personally, I'm actually surprsied to hear that most people don't turn out like I did. === Subject: Re: Skolem's Paradox and why is math the way it is? J.E.: [...] |I've never studied intuitionist logic, and I don't even know what |inconsistent is defined to be in intuitionist logic. A set of statements S is inconsistent if for some statement P, one can intuitionistically deduce both P and ~P from the statements. |Assuming y is |in f(y) is true leads to ~f(y) is in f(y), which is a |contradiction. Assuming ~y is in f(y) is true leads to ~~f(y) is |in f(y), which is a contradiction. So we are lead to conlude that y |is in f(y) is neither true nor false. This is a problem if you have |an excluded middle, but otherwise, what's the big deal. I suspect |from talking to you that intuitionist logic DOES have an excluded |middle, but that it has a limited power to discuss that fact and a |limited ability to infer from that fact. Quite the opposite-- three-valued logic is essentially classical logic, but with a limited ability to use it. The distinctions between a statement being true or not, and between it being false or not are still essentially being treated as classical distinctions, just with less of a chance to use them. Three-valued logic is classical logic without negation as applied to pairs of propositions that contradict each other. Given such a pair (P, Q) where P->Q, we can decide to call it true when P is true, and false when Q is true, but indeterminate when both P and Q are false. We define the connectives like this: (P1,Q1)&(P2,Q2) = (P1&P2,Q1vQ2), (P1,Q1)v(P2,Q2) = (P1vP2,Q1&Q2), ~(P,Q)=(Q,P), (P1,Q1)->(P2,Q2) = (Q1vP2, Q2&P1). The vanilla quantifiers are similar to conjunction and disjunction: Ex (P(x), Q(x)) = (ExP(x), AxQ(x)) Ax (P(x), Q(x)) = (AxP(x), ExQ(x)). The real effect of this is to put a restriction on the kind of proposition that you can express in it. The propositions you get are essentially pairs of ones that can be defined without using negation starting from the original propositions. The weakening effect of this is not so serious if you have an actual negation on atomic formulas, like Hintikka does. Each statement in classical first-order logic can be converted into an equivalent one where the negations are all on atomic sentences. This is how he ends up having classical first-order logic embedded in his logic. It's well-known that the usual proof of the undefinability of truth relies upon the language doing the defining being rich enough to include a negation. The usual conclusion drawn from the undefinability of truth argument is that for each fixed, well-defined language there is another language that expresses more than it does. I think that's correct. But the richer language as usually defined applies negations to statements of the original language. There's a form of the diagonal argument for predicates on natural numbers-- if P_0(n), P_1(n),... is an enumeration of the predicates on natural numbers in some language, then there is another one, Q(n) = ~P_n(n), that can't be expressed in the language. There are two ways to describe how Hintikka circumvents the undefinability of truth. One is to say that he realizes it's appropriate to allow a generalization of first-order logic where paradoxical sentences satisfying p<->~p can exist. The other is what I've been saying here, that the essential reason for his success is that he redefines negation to be less expressive than it usually is. The negation of this graph has at least four connected components should be this graph has at most three connected components as usual; it's just that Hintikka ham-strings negation to keep us from expressing propositions like that. This second explanation seems more illuminating to me than Hintikka's. Intuitionist logic represents a more fundamental departure from classical logic than three-valued logic does. Instead of just pasting together two distinctions, each of which looks just like an ordinary classical proposition, to get a trichotomy, intuitionist logic permits one to make an array of more subtle distinctions. Classical logic has the connective & and ~ and the quantifier for all that are pretty close to intuitionist ones. However, given those three, the remaining connectives and quantifiers of classical logic are redundant: P v Q can be defined as ~(~P&~Q). P -> Q can be defined as ~(P&~Q). (Ex) P(x) can be defined as ~(Ax)~P(x). In intuitionist logic, however, none of these three is redundant. They all are needed to express additional shades of meaning that classical logic and 3-valued logic wash out. The Ex and v are taken in a constructive sense. Ex means that one can effectively get an x. AvB is treated similarly, and means that one can effectively get a choice of side, left or right of the disjunction that holds true. Intuitionist implication is more subtle still. In classical logic, starting from n propositions P1,...,Pn, we get 2^n possible assignments of truth-values. For any propositional expression involving the P1,...,Pn, we get an assignment of a truth value to each of these assignments (i.e., the truth-table for the expression). That gives us 2^(2^n) possibilities. Every formula of classical propositional calculus in n variables is equivalent to one of those 2^(2^n). In three-valued logic, this argument no longer applies because one doesn't have excluded middle. But instead one simply has three truth-values. The situation is similar, except that not all of the 3^(3^n) possibilities for assignments of truth-values are allowed. Intuitionist logic, however, can't be exhausted by any finite set of possible truth-values to assign to propositions. It's a famous metatheorem about intuitionist logic there given a proposition p, there is an infinite sequence of propositions that can be expressed in terms of p, none of them necessarily equivalent to each other. It's true that there isn't a special distinction between ~P is true and P is not true or P is false, because in intuitionist logic there wasn't a need to redefine ~P to mean something unusual. So the usual proof of the undefinability of truth still works, and ~(P and ~P) is valid, as is ~(~P and ~~P). But neither of them entails P or ~P, which means that we can in principle determine the truth or falsity of P, which we can't. |> I think you're avoiding this conclusion by considering it possible |> for some statement to be true if and only if it is not true. I |> understand how IF-logic permits such a thing to occur, but it's |> not convincing to me that this makes good sense. | |And I don't understand what IT MEANS to not have an exluded middle and |disallow that. The law of excluded middle says that p or ~p holds for every p. The absence of a p for which p is true if and only if it is false means that ~(p<->~p) holds for every p. A system having the one rule and a system having the other rule are not the same thing. I just happen to know of a very well-known system where the second rule holds but not the first. |We are coming from different worlds and I don't know |the basis for your ideas, while you know that my meanings are derived |from the semantics of games. So it's a bit unfair for me to be |explaining things to you. I don't agree. It's rare for a person to be familiar with IF logic, and since we've started the discussion, I've learned more about it than I knew to begin with. This is more than you're entitled to expect. I haven't asked you to explain things that I already know. If you want to make claims about how the law of excluded middle affects the nature of a logical system, I think it's entirely reasonable of me to mention facts about the most well-known and most heavily studied logical system lacking the law of excluded middle. You've done an unusual amount of complaining about people failing to explain things to you. I can appreciate your complaining if you have had professors who were paid to explain things to you, but were stingy with their time in office hours, or something like that, but nobody here is being paid to do anything, and in fact we've used a bunch of our own spare time to try to explain things. As far as leaving the basis for my ideas obscure, I think you're expecting a different kind of explanation from me that is reasonable to expect. I'm more willing than most people are to approach issues either from a formalist or a realist perspective. To put it crudely, there is a kind of chicken-and-egg problem: either you start with sentences or with objects. The formalist treats as fundamental the sentences, but in doing so he is stuck having to treat them as uninterpreted sentences, because he lacks any real objects for them to refer to. The realist coming before any formalization, he has to discuss them to begin with in informal terms. The realist approach makes people uneasy, because they don't feel like there has been enough of an explanation of the nature of the objects referred to by the fundamental, undefined terms. In our case, these would be the sets. I suspect your perception that I haven't explained to you where my ideas come from is largely due to your discomfort with the undefined term set. You appear to be craving a kind explanation of what set means that would no longer treat it as a fundamental term. But if one wants to treat one's mathematical statements as having a definite meaning, one needs some terms whose meanings are only informally explained; it's just how it is. I think it's no better to regard IF logic as a (realistic) foundation. The way most people understand the meanings of the terms involves a belief in the existence of these domains on which the games are played, as well as the strategies of the players which might or might not be winning strategies. Domains are essentially classes, and the strategies are essentially functions on them. The strategies are not necessarily actually playable in practice (they often are uncomputable), so we're not gaining extra support from some understanding of our actual game-playing abilities. It's possible to be consistently formalistic, but it means that a lot of the questions you ask no longer make any sense. It's just incoherent to ask whether your model has all the sets it should have, as a formalist, because you are not basing your theory on the idea that there exists a model of it. If you believe in the existence of a model of your theory, then you are being somewhat of a realist. I don't think it's very sensible to be partially a realist but to brush the fact under the rug by pretending that all of your terms are defined inside of one or the other theory. If your notion of model is defined inside a theory, then use that theory, because it's more fundamental than the theory assumed to have a model. |And there is going to be huge problems |because we define implication differently, I use a stronger version |than use, it is more expressive and says more, but therefore is has |fewer rules of inference. Your implication is stronger and satisfies fewer rules of inference, but it is not more expressive. Just as in classical logic, it's redundant; one could use ~AvB instead. Intuitionist implication is not redundant, and allows us to express the existence of a connection between the premise and conclusion. A classical implication is supposedly true always by virtue of some property holding of one of the sides; either A is false, which makes the implication true regardless of what B is, or B is true, which makes the implication true regardless of what A is. Intuitionist implication can hold without either side necessarily making it true by itself. | So my biconditional says A <-> ~A means that |(~A or ~A) & (A or A) which is logically equivalent to ~A & A which |means that A cannot be true or false, full stop. Why not, in addition to this implication, include the real one? Allow me to say that, given a winning strategy for A, I can give you a winning strategy for B. |> Once we have a logic with three truth-values, true, false and |> indeterminate, I don't see how it can be invalid for me to start |> talking about which of the three bins a sentence falls into. The |> claim that a certain sentence simply fails to be true, i.e., is |> either false or indeterminate, appears to make logical sense. It's |> just not a claim that can be expressed in IF-logic. | |Huh? Truth is about a winning stratgey in all models, same with |falsity. I don't think this is standard terminology. Certainly in classical first-order logic, one defines the notion of the sentence being true in a model. It can be true in some models and not in others. I'm saying that the sentences of IF logic intentionally exclude such model-relative claims as in this model, the verifier has no winning strategy for the game associated with S. That's what the negation of S should mean. The language is weakened by our not being allowed to say such a thing. | Being neither can mean totally different things. It could |mean that it has a winning strategy for one team in one model one for |the other team in another model, or it could mean that there is a |model where the sentance has no winning strategy for either side. You're providing good examples here of the kind of thing that one wants to say when discussing IF logic, that doesn't really fit inside of it. The statement in this model, such-and-such sentence has no winning strategy for either side isn't expressible in IF logic. Everything we say about winning strategies is part of the metalanguage, since winning strategy doesn't have a translation into the logic itself. This is why I don't see it as a fundamental theory; almost anyone who believes it is a coherent theory also believes in the existence of things like winning strategies that don't belong inside it. | Why |does it make sense to lump these cases together? I don't. | What we care about |is truth, which means winning in all models. I thought winning was everything? But seriously, suppose I claim to care about whether a sentence lacks a winning strategy for either player in all models. What's wrong with that, aside from not belonging to the formalism? | For instance if N is the |negation of an axiom and T is a theorem of the axioms, then N or T |is a validity, true, true in all models, in all worlds. And there are |TWO ways to negate that CONCEPT, to say false in all worlds or to |say not the case that it is true in all worlds. IF-logic does the |FIRST, because the point is that you write SENTANCES and then you |assert that their truth means something about THE DOMAIN OF DISCOURSE, |this is what we do in physics, we state the world is such that T is |true, it's what philosophers do. To discuss anything else invovles |actually quantizying over possible worlds, which I didn't think people |were still serious about doing. But you've been quantifying over all possible models throughout the discussion, here. How does it even make sense for you to divide up the possible cases the way you just did, if it's inappropriate to quantify over possible domains of discourse. Quantifying over possible worlds has a different connotation from quantifying over all possible models. The former is an extension of modal logic; the latter is set theory. |> This is really what I want Hintikka to tell me: why am I mistaken |> when I think I've made an assertion such as the domain of discourse |> is finite that is true *exactly* when a certain sentence of IF-logic |> fails to be true. In what way is this an incoherent sentence? It |> seems as though he simply ham-strings his theory to make it impossible |> to say certain things in it. | |It's an funny difference of opinions because you think he ham-stringed |his theory, and it seems to me that you want him to ham-string his |theory when he hasn't. Yes, he has. I've given more than one example. It's not just that there are things that can't be expressed in the language, since that's always true. It's that the things that can't be expressed are so much like the ones that can be, aside from being prevented. Even just to allow games where to win a player needs to make a finite number of moves of a certain kind, without a fixed upper limit, extends the language enourmously at essentially no cost. |Validities are what we care about, things true |in all worlds, Not everybody. Many of us care about other things as well. If you want to keep saying what we care about is... you should say why we shouldn't care about other things as well. [...] |(~A or B) is stronger than A=>B |~A is stronger than it is not the case that A is true, so we can |say things that you can't otherwise say in FOL. Preventing yourself from saying it is not the case that A is true is not an advantage. The only reason ~A fails to be expressible is that it's been reduced to the status of a second proposition that happens to contradict A. There can be different A's, where there exists a winning strategy in the same domains, but whose negations are different, just because. The reason you can say things in IF logic that can't be said in ordinary FOL is the use of these independence requirements on variables. You could just as well include the ordinary ~ and => of FOL, and it would give you a stronger theory, equivalent to second-order logic. |It's is a WEAKER claim about the universe of discourse, it merely says |consider the worlds where the theorems are true not consider the |worlds where the (second order) axioms are true. You want nonsesnse? | It IS nonsense to say consider a FO universe of discourse where the |SO axioms are true if you just want the FO axioms and oFOL theorems, |then IF-FOL is useless. Which second-order axioms? Why not consider those domains on which a collection of second-order axioms are true? You haven't presented anything like a cogent argument against the good sense of this. The natural numbers N={0,1,2,...} are definable up to isomorphism as the model of a set of second order axioms. Where's the big problem with this? Why should I limit myself to some less expressive language? |If you want to translate SO axioms into FO |language without assuming a universe of discourse that INCLUDES SO |entities. A strategy for one of these games is already a second-order entity, since it is made up of functions on the domain of discourse. How are you really hoping to get away from that? | Then instead of considering the universe where all the |axioms are true you should instead consider the universe of discourse |where all the theorems are true, do you really not get it? I don't think you're explaining your point very well here. You write as if you think you have a really strong argument, but I don't see that you do. | The IF |logic claim is more powerful because when you actually translate SO |axioms into IF-FOL you get statements that do NOT have contradictory |negations in second-order logic, that is because they simply do NOT |have winning strategies for either side, so INSTEAD of considering |universes where you can VERIFY the truth of the axioms, you consider |the ones where it is impossible to VERIFY the negation of the axioms. I'm not sure of which translation you have in mind. Overall, it's especially unclear what the actual benefit is supposed to be. You accept the coherence of IF logic, which is fine, but seems to require accepting the meaningfulness of terms like strategy and model or domain. Having accepted those, the reason for having special qualms about second-order logic is obscure. |It is simply astonding that the ordinary proofs of theorems carry over |to IF-FOL validities because it is a much much stronger claim to say |that the theorems are all true in models where the axioms are not |false, then to MERELY say they are true when the axioms are true, |because the INTENDED axioms are NOT true in any model. That's what YOU think, but do you have any argument for it? |I'm trying to |explain why what you are asking for is nonsense, but I don't know if |you can see it. The problem isn't that I can't see it; the problem is that it isn't nonsense. |> Wittgenstein gave an example of an incoherent description: when it's |> five o'clock on the Sun. He imagines someone asking, what does that |> mean? and the answer is, Just like what it means to be five o'clock |> here-- but on the Sun. I can imagine that some of the things I think |> and talk about are confused in sort of this way: it only seems to me |> that I'm considering a well-defined proposition. The only way I can |> see how it would make sense to consider a system like IF-logic to be |> ultimate is if I were confused in such a manner about the things that |> I say that appear not to be expressible in IF-logic. | |The claims you want to make (contradictory negations of IF-FOL |formulas) are actually either IF-FOL formulas (in the special case |where the original formula was actually logically equivalent to an |oFOL formula) or the negations are ACTUALLY SO statements and they |require a CHANGE of the universe of discourse to INCLUDE SO entities. |If you want to stay FO, then I'd be hard pressed to come up with some |stronger and more expressive logic than IF-FOL, but maybe there is |one. I think Hintikka had an extended IF-logic. I think IF-logic is first order in only a phony way. Its semantics appears to require having an intuitive understanding of what a strategy is, which is already a second-order notion. The fact that one does not allow direct references to strategies (although they clearly do exist) doesn't strike me as an advantage of IF-logic. Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? |I should also point out that the Aspect experiment's |results were published in December of 1982, and I don't remember |for sure whether the remark was made before or after the |paper was published. So maybe he changed his mind |afterward. Now that I think about it, the Aspect results must have been published within a year or two before this professor made his remark. Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? Sorry about the lag; I think I've been trying to do too many things all together. I'll probably have to taper off my contribution soon. |> Most mathematics, incidentally, uses only a relatively uncontroversial |> portion of set theory. People deal with things like sets of real |> numbers all the time, but not so often the parts that depend on |> (say) the axiom of replacement. | |Since we haven't proven that ALL the set theory axioms taken together |are consistent, then you'd expect that if a smaller subset works for |physics, that someone would have tried to prove that that smaller set |of axioms was consistent. Is there such a proof? There is indeed a proof that the axioms of set theory, without replacement, are consistent. However! The proof uses the axiom of replacement. By Goedel's second incompleteness theorem, a system like Z (the system without the axiom of replacement) can't prove its own consistency, so some additional axiom is needed, of course. The most familiar proof works by showing the existence of a model of Z. The most obvious model of Z is V_{omega+omega}, which is the first part of the cumulative hierarchy: Let V_0 be the empty set, and by induction let V_{n+1} be the set of all subsets of V_n for n=0,1,2,.... These are all finite sets, which can be written as finite strings if need be. They're known as the hereditarily finite sets, since not only are they finite, but their members are finite, the members of their members are finite, and so on. Then let V_{omega} be the set of all hereditarily finite sets. This is a countably infinite set. It requires the axiom of infinity to prove that it exists, but its elements can be put into correspondence with the natural numbers. Incidentally V_{omega} is a model of all the axioms of ZFC except for the axiom of infinity. Define V_{omega+(n+1)} to be the set of all subsets of V_{omega+n} inductively. Then finally let V_{omega+omega} be the union of all of the V_{omega+n} for natural numbers n. That last step is the only step in the proof where the axiom of replacement is needed. [...] |> I don't think there is a be-all theory. | |Isn't being a be all theory part and parcel of the standard |interpretation that everything that could exist for anyone that is |small enough to be a set, is a actually a set? No. It seems as though you're confusing an interpretation with a theory. One can believe that one has a coherent interpretation in which when one says set, it really means any kind of set. That doesn't mean that one has a be all theory of what they're like. It's like the difference between believing that you know what an apple is, and knowing everything that there is to know about apples. |The only reason to |insist on this rather than just that enough sets exist to satisfy the |axioms is because one wants to pretend like one can have an |everything, where the universe of discourse of standard |interpretation set theory is superior to all other universe of |discourses. No. There are larger domains of discourse. Set theory is merely a relatively large one. |Otherwise why is that interpretation consider necissary or even |standard? It seems to me that you're putting the cart before the horse again. I consider the primary purpose of the axioms to be to investigate a domain of inquiry systematically, *not* to define the limits of the domain of inquiry. Unless you have a good reason to think that it's impossible to talk about all real numbers (and I don't think you do), it's a very strange suggestion that we should talk only about some of them. |> |But IF |> |logic avoids having infinite regresses into higher-order logics so |> |that we CAN sit down and discuss how you make theories, so isn't that |> |worth considering? |> What infinite regress into higher-order logic is there for anyone? | |Like you say in your previous post, you need SO set theory to define |the strongly inaccessible cardinal, in order to get a faithful model |of set theory, but once you introduce SO set theory, people will want |the other sets too, because the whole POINT of introducing that |cardinal was to get all the sets that were missing in previous |models. You aren't succeeding at getting all the sets. I don't think I said you needed second-order set theory to define a strongly inaccessible cardinal. It only needs a first-order definition in terms of the epsilon relation. As usual, of course, to say that an element of a model satisfies this definition as relativized to the model doesn't mean the same thing as saying that its a cardinal satisfying the definition. I think you need to distinguish between various senses of get, here, pertaining to the scope of variables, the language as a whole, and the axioms. If I say that all real numbers are either <0, >0 or =0, then my statement implicitly contains a quantifier for a variable ranging over all the real numbers. The statement succeeds in getting all the reals in the sense that it quantifies over them. If I say that I can define any arbitrary algebraic number, then I've gotten them all in second sense. This sense is relative to my language, since what definitions I can provide depend on how rich my language is. If all I have are the elementary school operations of +,-,*,/ and maybe simple roots x^(1/n), then my language is too weak to define all of them. There is no language (with finite expressions) strong enough to get all the real numbers in this sense. If I say that I can prove the existence of a weak inaccessible cardinal, then I have gotten it in a third sense, which depends on what can be proven (which depends on which axioms are accepted). These three senses are in order of increasing narrowness. In order to prove that something exists, I need to be able to describe it. In order to describe it, I need to have (implicitly at least) variables that range over a domain that includes it. When you say get, you often seem to be sliding between these senses. You seem often to be trying to treat them as if they were the same thing. You don't seem to see any problem with treating the narrowest sense (what we can prove to exist) as if it should somehow be the same as the range of our quantifiers. I don't see any point in doing that. If we have been talking about real numbers, I don't see any point in deciding to assume that we are *always* talking about some subset of definable or provably existing real numbers instead. The only reason I can think of for wanting to do that, not just sometimes but generally, would be if there was somehow a serious problem with the concept of real number, some genuine ambiguity or incoherence. [...] |> You can't say that a graph has three connected components, in it. | |Do you have a citation for that result, or better yet can you state |your definition of graph and connected component? I don't have a citation for it offhand. There's nothing special about three, by the way. I used that because I was thinking I could go on to point out that This graph has at least three connected components as well as this graph has at least four connected components were both expressible in IF logic, but not this graph has (exactly) three connected components. Connectedness is a familiar example of a non-first-order property of a structure. It's not expressible in IF logic because IF logic extends first-order logic by permitting an existential quantifier over subsets of the structure (in effect). Those are the sigma-1-1 properties. But connectedness is a pi-1-1 property, which is how it escapes being first-order definable. It's usually defined as meaning that any two vertices are joined by a path. That's equivalent to the nonexistence of a way to divide the graph into two nonempty disjoint subsets, with no edges connecting any vertex in the one with any vertex in the other. Lemme see if I can sketch a proof that it's not also sigma-1-1. Suppose there is a game (associated to a sentence in IF-logic) with a winning strategy for the verifier on an infinite connected graph where the degree of the vertices is bounded above by some natural number n. For simplicity, we can take a set of vertices indexed by the integers (including negative integers) where the edges connect adjacent vertices. The winning strategy consists of a collection of functions f(a1,...,a_m) for different values of m. I claim that there exist disconnected graphs where the verifier also has a winning strategy. First, a simple example. I'm pretty sure that the graph consisting of two disjoint copies of the original graph is an example. Take the points (x,y) in the plane where y=0,1 and x is an integer, and join the points (x,y) and (x+1,y) by edges to form the graph. I'm suffering a little writer's block on the proof, though. Second, pulling out the big guns. The original graph has just two relations on it, xEy meaning that x and y are joined by an edge, and x=y. Now augment the structure by adding the functions f that correspond to the verifier's winning strategy. That's now a model of the first-order sentence saying that the verifier wins the game regardless of what the falsifier plays. The upward Lowenheim-Skolem theorem says (as a special case) that if a first-order sentence has an infinite model, it also has an uncountable model. So the sentence saying the functions f are a winning strategy for the verifier, and that all of the vertices have degree 2, also holds true for some functions f' on a graph with uncountably many vertices. Since a connected components of a graph whose vertices have degree 2 is always countable, this graph with uncountably many vertices is disconnected. The same proof works just as well for the sentence, this graph has three connected components. Any game associated with a sentence of IF logic that can be won on a graph with three components can always be won on a graph with more components. To me this just reveals something missing in IF logic. We can understand nearly as easily what it means to be able to win the following game: the refuter picks two vertices, and to win the verifier has to present vertices one at a time, each connected to the previous one, and starting with the first vertex given by the refuter get to the other one. It's true that this involves the notion of a finite sequence of moves, but I don't see how that can be much worse than the kind of arbitrary strategy allowed the players in a game associated with an IF logic sentence. IF logic just is so limited that we can't say it. We can even say what I would call the REAL negation of the claim that the graph is connected: that it can be divided into two nonempty parts that aren't connected to each other. [...] |> |Every model of set theory lacks a set that should exist as much as the |> |alleged uncounted real should exist. |> If by model you mean a set with an epsilon relation on it, then |> this is correct, but people often mean by model either a set *or* |> a proper class with an epsilon relation on it. The cumulative |> hierarchy does not lack a set that should exist-- it consists |> by definition in all the well-founded pure sets. | |This is really hard to discuss non-circularly. The words structure, |class, function, set, collection, relation all have definitions in |the theory and to use the same words outside of the theory is begging |for confusion. I think trying to force them to be theory-dependent in an inconsistent way is begging for deeper confusion. I don't think there is such a thing as a different definition of function, for example, in the theory. It's possible that you are alluding to relativizing some of these concepts to models, but you need to distinguish theories from models. |What do you want to take as given? I'm pretty flexible about what to take as given, so long as we're consistent with it. We can start the formal development by taking set as an undefined term, either believing that we know a definite meaning for it, or by merely proceeding as though it does. To remain consistent with such a starting point, however, makes a lot of statements nonsensical, like saying that this domain of sets lacks a set that should exist. If by model, you mean a set having an epsilon relation and so on, then it's circular to try to define set relative to model, because this sense of model depends on the concept of set already. Set needs to have a meaning that doesn't depend on models. We do not need to start out with a model in this sense of some theory-- it's circular to try to start that way. It's true of this kind of model that there are sets that are not members of it; it does not contain itself, for instance, and it is by definition a set. If by model you mean something broader, like what philosophers sometimes call a domain of discourse, then we can, if you like, call the domain of sets that we start out with a model, but then there is no longer any sense in saying that our starting model is missing any sets. We have just defined the domain of discourse to consist of all such objects that we are going to be calling sets. When you asked questions about deciding to use the minimal model, since the minimal model is defined in terms of the concept of set, whatever you meant by minimal model was dependent on some prior concept of set. Certainly if you want to use such a model for physics there isn't an inconsistency, but you are still stuck with the fact that you started out with one brand of set theory, and then created a second kind that depends on the original kind. Most people figure that they are better off just sticking to whatever kind of set they started out with. | We could have a |third person in the game, and have the third person start talking, |saying a in M, aea in A, a' in M, a'ea' in A, aea' in E, a'ea in A, |a'' in M, a''ea'' in A, a''ea' in A, a''ea in A, a'ea'' in A, aea'' in |A, a''' in M, a'''ea''' in A, a'''ea in A, a'''ea'' in A, a'''ea'' in |A, aea''' in A, a'ea''' in A, a''ea''' in A, ... and but where the |third person chooses freely whether to say xey in A or xey in E, These little scenarios where you describe some outside source generating a structure incrementally strike me as having so little about them that is analogous to the way mathematics actually works or how we talk about it, that I can hardly think of anything to say about them. [...] |I don't understand your claim about the cumulative hierarchy, once you |finish the model, someone can take the standard interpretation and |say that some sets are missing, Why? |isn't V=L considered restrictive by |mathematicians? The best guess I can come up with here is that you're confusing two definitions of hierarchies here, the definition of the cumulative hierarchy (whose members constitute V) and the definition of the constructive hierarchy (whose members constitute L). Keith Ramsay === Subject: Re: Skolem's Paradox and why is math the way it is? > Sorry about the lag; I think I've been trying to do too many things > all together. I'll probably have to taper off my contribution soon. > |> Most mathematics, incidentally, uses only a relatively uncontroversial > |> portion of set theory. People deal with things like sets of real > |> numbers all the time, but not so often the parts that depend on > |> (say) the axiom of replacement. > |Since we haven't proven that ALL the set theory axioms taken together > |are consistent, then you'd expect that if a smaller subset works for > |physics, that someone would have tried to prove that that smaller set > |of axioms was consistent. Is there such a proof? > There is indeed a proof that the axioms of set theory, without > replacement, are consistent. However! The proof uses the axiom > of replacement. By Goedel's second incompleteness theorem, a system > like Z (the system without the axiom of replacement) can't prove > its own consistency, so some additional axiom is needed, of course. > The most familiar proof works by showing the existence of a model > of Z. The most obvious model of Z is V_{omega+omega}, which is the > first part of the cumulative hierarchy: Let V_0 be the empty set, > and by induction let V_{n+1} be the set of all subsets of V_n for > n=0,1,2,.... These are all finite sets, which can be written as > finite strings if need be. They're known as the hereditarily finite > sets, since not only are they finite, but their members are finite, > the members of their members are finite, and so on. > Then let V_{omega} be the set of all hereditarily finite sets. This > is a countably infinite set. It requires the axiom of infinity to > prove that it exists, but its elements can be put into correspondence > with the natural numbers. Incidentally V_{omega} is a model of all the > axioms of ZFC except for the axiom of infinity. > Define V_{omega+(n+1)} to be the set of all subsets of V_{omega+n} > inductively. Then finally let V_{omega+omega} be the union of all > of the V_{omega+n} for natural numbers n. That last step is the only > step in the proof where the axiom of replacement is needed. > [...] > |> I don't think there is a be-all theory. > |Isn't being a be all theory part and parcel of the standard > |interpretation that everything that could exist for anyone that is > |small enough to be a set, is a actually a set? > No. It seems as though you're confusing an interpretation with a > theory. One can believe that one has a coherent interpretation in > which when one says set, it really means any kind of set. That > doesn't mean that one has a be all theory of what they're like. > It's like the difference between believing that you know what an > apple is, and knowing everything that there is to know about apples. > |The only reason to > |insist on this rather than just that enough sets exist to satisfy the > |axioms is because one wants to pretend like one can have an > |everything, where the universe of discourse of standard > |interpretation set theory is superior to all other universe of > |discourses. > No. There are larger domains of discourse. Set theory is merely a > relatively large one. > |Otherwise why is that interpretation consider necissary or even > |standard? > It seems to me that you're putting the cart before the horse again. > I consider the primary purpose of the axioms to be to investigate a > domain of inquiry systematically, *not* to define the limits of the > domain of inquiry. > Unless you have a good reason to think that it's impossible to talk > about all real numbers (and I don't think you do), it's a very > strange suggestion that we should talk only about some of them. > |> |But IF > |> |logic avoids having infinite regresses into higher-order logics so > |> |that we CAN sit down and discuss how you make theories, so isn't that > |> |worth considering? > | |> What infinite regress into higher-order logic is there for anyone? > |Like you say in your previous post, you need SO set theory to define > |the strongly inaccessible cardinal, in order to get a faithful model > |of set theory, but once you introduce SO set theory, people will want > |the other sets too, because the whole POINT of introducing that > |cardinal was to get all the sets that were missing in previous > |models. You aren't succeeding at getting all the sets. > I don't think I said you needed second-order set theory to define a > strongly inaccessible cardinal. It only needs a first-order definition > in terms of the epsilon relation. As usual, of course, to say that > an element of a model satisfies this definition as relativized to the > model doesn't mean the same thing as saying that its a cardinal > satisfying the definition. > I think you need to distinguish between various senses of get, here, > pertaining to the scope of variables, the language as a whole, and the > axioms. > If I say that all real numbers are either <0, >0 or =0, then my > statement implicitly contains a quantifier for a variable ranging > over all the real numbers. The statement succeeds in getting all > the reals in the sense that it quantifies over them. > If I say that I can define any arbitrary algebraic number, then I've > gotten them all in second sense. This sense is relative to my language, > since what definitions I can provide depend on how rich my language is. > If all I have are the elementary school operations of +,-,*,/ and maybe > simple roots x^(1/n), then my language is too weak to define all of them. > There is no language (with finite expressions) strong enough to get > all the real numbers in this sense. > If I say that I can prove the existence of a weak inaccessible cardinal, > then I have gotten it in a third sense, which depends on what can be > proven (which depends on which axioms are accepted). > These three senses are in order of increasing narrowness. In order to > prove that something exists, I need to be able to describe it. In order > to describe it, I need to have (implicitly at least) variables that > range over a domain that includes it. > When you say get, you often seem to be sliding between these senses. > You seem often to be trying to treat them as if they were the same > thing. You don't seem to see any problem with treating the narrowest > sense (what we can prove to exist) as if it should somehow be the same > as the range of our quantifiers. I don't see any point in doing that. > If we have been talking about real numbers, I don't see any point in > deciding to assume that we are *always* talking about some subset of > definable or provably existing real numbers instead. The only reason I > can think of for wanting to do that, not just sometimes but generally, > would be if there was somehow a serious problem with the concept of > real number, some genuine ambiguity or incoherence. > [...] > |> You can't say that a graph has three connected components, in it. > |Do you have a citation for that result, or better yet can you state > |your definition of graph and connected component? > I don't have a citation for it offhand. There's nothing special > about three, by the way. I used that because I was thinking I could > go on to point out that This graph has at least three connected > components as well as this graph has at least four connected > components were both expressible in IF logic, but not this graph > has (exactly) three connected components. > Connectedness is a familiar example of a non-first-order property of > a structure. It's not expressible in IF logic because IF logic > extends first-order logic by permitting an existential quantifier > over subsets of the structure (in effect). > Those are the sigma-1-1 properties. But connectedness is a pi-1-1 > property, which is how it escapes being first-order definable. It's > usually defined as meaning that any two vertices are joined by a path. > That's equivalent to the nonexistence of a way to divide the graph > into two nonempty disjoint subsets, with no edges connecting any > vertex in the one with any vertex in the other. > Lemme see if I can sketch a proof that it's not also sigma-1-1. > Suppose there is a game (associated to a sentence in IF-logic) > with a winning strategy for the verifier on an infinite connected > graph where the degree of the vertices is bounded above by some > natural number n. For simplicity, we can take a set of vertices > indexed by the integers (including negative integers) where the > edges connect adjacent vertices. The winning strategy consists of > a collection of functions f(a1,...,a_m) for different values of m. > I claim that there exist disconnected graphs where the verifier > also has a winning strategy. > First, a simple example. I'm pretty sure that the graph consisting > of two disjoint copies of the original graph is an example. Take > the points (x,y) in the plane where y=0,1 and x is an integer, and > join the points (x,y) and (x+1,y) by edges to form the graph. I'm > suffering a little writer's block on the proof, though. > Second, pulling out the big guns. The original graph has just two > relations on it, xEy meaning that x and y are joined by an edge, > and x=y. Now augment the structure by adding the functions f that > correspond to the verifier's winning strategy. That's now a model > of the first-order sentence saying that the verifier wins the game > regardless of what the falsifier plays. > The upward Lowenheim-Skolem theorem says (as a special case) that if > a first-order sentence has an infinite model, it also has an uncountable > model. So the sentence saying the functions f are a winning strategy > for the verifier, and that all of the vertices have degree 2, also holds > true for some functions f' on a graph with uncountably many vertices. > Since a connected components of a graph whose vertices have degree 2 > is always countable, this graph with uncountably many vertices is > disconnected. > The same proof works just as well for the sentence, this graph has > three connected components. Any game associated with a sentence of > IF logic that can be won on a graph with three components can always > be won on a graph with more components. > To me this just reveals something missing in IF logic. We can understand > nearly as easily what it means to be able to win the following game: > the refuter picks two vertices, and to win the verifier has to present > vertices one at a time, each connected to the previous one, and starting > with the first vertex given by the refuter get to the other one. It's true > that this involves the notion of a finite sequence of moves, but I don't > see how that can be much worse than the kind of arbitrary strategy > allowed the players in a game associated with an IF logic sentence. > IF logic just is so limited that we can't say it. We can even say what I > would call the REAL negation of the claim that the graph is connected: > that it can be divided into two nonempty parts that aren't connected to > each other. > [...] > |> |Every model of set theory lacks a set that should exist as much as the > |> |alleged uncounted real should exist. > | |> If by model you mean a set with an epsilon relation on it, then > |> this is correct, but people often mean by model either a set *or* > |> a proper class with an epsilon relation on it. The cumulative > |> hierarchy does not lack a set that should exist-- it consists > |> by definition in all the well-founded pure sets. > |This is really hard to discuss non-circularly. The words structure, > |class, function, set, collection, relation all have definitions in > |the theory and to use the same words outside of the theory is begging > |for confusion. > I think trying to force them to be theory-dependent in an inconsistent > way is begging for deeper confusion. > I don't think there is such a thing as a different definition of > function, for example, in the theory. It's possible that you are > alluding to relativizing some of these concepts to models, but you > need to distinguish theories from models. > |What do you want to take as given? > I'm pretty flexible about what to take as given, so long as we're > consistent with it. > We can start the formal development by taking set as an undefined > term, either believing that we know a definite meaning for it, or by > merely proceeding as though it does. To remain consistent with such > a starting point, however, makes a lot of statements nonsensical, like > saying that this domain of sets lacks a set that should exist. > If by model, you mean a set having an epsilon relation and so on, then > it's circular to try to define set relative to model, because this > sense of model depends on the concept of set already. Set needs > to have a meaning that doesn't depend on models. We do not need to start > out with a model in this sense of some theory-- it's circular to try > to start that way. It's true of this kind of model that there are sets > that are not members of it; it does not contain itself, for instance, and > it is by definition a set. > If by model you mean something broader, like what philosophers sometimes > call a domain of discourse, then we can, if you like, call the domain of > sets that we start out with a model, but then there is no longer any > sense in saying that our starting model is missing any sets. We have just > defined the domain of discourse to consist of all such objects that we > are going to be calling sets. > When you asked questions about deciding to use the minimal model, > since the minimal model is defined in terms of the concept of set, > whatever you meant by minimal model was dependent on some prior > concept of set. Certainly if you want to use such a model for physics > there isn't an inconsistency, but you are still stuck with the fact that > you started out with one brand of set theory, and then created a second > kind that depends on the original kind. Most people figure that they > are better off just sticking to whatever kind of set they started out > with. > | We could have a > |third person in the game, and have the third person start talking, > |saying a in M, aea in A, a' in M, a'ea' in A, aea' in E, a'ea in A, > |a'' in M, a''ea'' in A, a''ea' in A, a''ea in A, a'ea'' in A, aea'' in > |A, a''' in M, a'''ea''' in A, a'''ea in A, a'''ea'' in A, a'''ea'' in > |A, aea''' in A, a'ea''' in A, a''ea''' in A, ... and but where the > |third person chooses freely whether to say xey in A or xey in E, > These little scenarios where you describe some outside source generating > a structure incrementally strike me as having so little about them that > is analogous to the way mathematics actually works or how we talk about > it, that I can hardly think of anything to say about them. > [...] > |I don't understand your claim about the cumulative hierarchy, once you > |finish the model, someone can take the standard interpretation and > |say that some sets are missing, > Why? > |isn't V=L considered restrictive by > |mathematicians? > The best guess I can come up with here is that you're confusing two > definitions of hierarchies here, the definition of the cumulative > hierarchy (whose members constitute V) and the definition of the > constructive hierarchy (whose members constitute L). > Keith Ramsay I want to compliment you on your recent posts, I think they have been really excellent. I've learned some things from them, mostly interfaces to specific terms and words that then offer ready venues into relatively more well developed mathematical topics than mine. I'm talking about ready interfaces for my theory, of course. I have a question about the universe of discourse, could you please expand what you mean when you say that there are larger realms of discourse than set theory? Yeah, V=L. Consider your cumulative hierarchy in the theory with ubiquitous ordinals, V{n+1} = P(V{n}) = succ(V{n}) = n+1. Particularly, consider them with their description as Z, the set of integers. Surely, that gets into N-1, etcetera, omega - 1, because it's talk about negative one. The idea is to find the mechanistic operation upon the set representing the positive and the set representing the negative integer to sum them. I think they might be from fully decorating the ordinals, instead of using the naked ordinals, the more and less ornate ordinals in the ubiquitous ordinals. As well, in the theory of ubiquitous naturals, then I promote splitting infinity in half. Half of the integers are positive. About the graph problem there, just embed them in matroids. I must consider Replacement vis-a-vis the confirmator, and U's mask. Consider Russell. Skolem: why is math the way it is? Ross F. === Subject: anti-cantorian probability theory and dart throwing Suppose the reals are countable. :) Then R = Q / Irr, where Irr is RQ. If somebody throws a dart at a dartboard, what is the probability that the x-coordinate of the point where the dart lands is in Q (the rationals), and why? David Bernier === Subject: Re: anti-cantorian probability theory and dart throwing > Suppose the reals are countable. :) > Then R = Q / Irr, where Irr is RQ. > If somebody throws a dart at a dartboard, > what is the probability that the x-coordinate > of the point where the dart lands is in Q (the rationals), > and why? > David Bernier I think the probability depends on the velocity and distance of the dart. On a massless spaceship moving at an acceleration of 1g and no air, then the dart would move in a parabola, so if you threw it with a rational velocity in the right direction from the right distand, then it would hit a rational point. In other circumstance it would be irrational. So the question is begged back towards you providing a PD of the phase space of the center of mass of the dart, and from that, we can kinematically transform the PD of the dart's position at one time into a PD on the x coordinate of the dart later, and from that answer your probability question. === Subject: Re: anti-cantorian probability theory and dart throwing > Suppose the reals are countable. :) Starting from a false premise, anything that follows is worthless. If you want anyone to read the rest of your message, I suggest you not start off with a blatantly false premise. I suggest you avoid all mention of the reals. I suggest you devise some model that doesn't involve real numbers. === Subject: Re: anti-cantorian probability theory and dart throwing Suppose the reals are countable. :) > Then R = Q / Irr, where Irr is RQ. So Irr = empty set , which means that pie (=3.14159...), an irrational number, does not exist. This implies that the area of your darts board, which equals (pie*r^2) (where r is the radius) does not exist either. The probability that you would hit your dartsboard would therefore be virtually zero, and so the answer to your question is straigtforward: Only intelligent questions can have a meaningfull answer. thomas *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Re: anti-cantorian probability theory and dart throwing > Suppose the reals are countable. :) >> Then R = Q / Irr, where Irr is RQ. > So Irr = empty set , the mere fact that R is a countable set containing Q it does not follow that RQ is empty, but from the fact that R is both countable and uncountable you can conclude that Bertrand Russell is the Pope. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: anti-cantorian probability theory and dart throwing > Suppose the reals are countable. :) >>Then R = Q / Irr, where Irr is RQ. > So Irr = empty set , which means that pie (=3.14159...), an irrational > number, does not exist. This implies that the area of your darts > board, which equals (pie*r^2) (where r is the radius) does not exist > either. The probability that you would hit your dartsboard would > therefore be virtually zero, and so the answer to your question is > straigtforward: Only intelligent questions can have a meaningfull > answer. > thomas I'm not saying that the reals are countable. (hence the :) ). For those in this newsgroup (a vocal few) who don't accept the uncountability of the real numbers, and declare that the reals are countable, all I can say is that it would be interesting to see how they would answer the question... I accept the validity of Cantor's proof that R is uncountable. David Bernier === Subject: Re: anti-cantorian probability theory and dart throwing The probability is 0. The number of irrationals in any given interval is uncountable, while the number of rationals is countable. The probability is 1. A dart has finite width and thus its point will span infinitely many rationals no matter where it lands. Take your pick. === Subject: Re: anti-cantorian probability theory and dart throwing >Suppose the reals are countable. :) >Then R = Q / Irr, where Irr is RQ. >If somebody throws a dart at a dartboard, >what is the probability that the x-coordinate >of the point where the dart lands is in Q (the rationals), >and why? This probability is 17. (Simple proof by contradiction snipped.) >David Bernier ************************ David C. Ullrich === Subject: Higher degree congruences it's easy to find congruences like x = a (mod N) and there could also be solutions for quadratic congruences like x^2 = a (mod N) but is there something to solve congruences like x^2 + ax = b (mod N) or x^a + x^b + cx = d (mod N) where a, b, c, d are integers That is, what about degrees higher than 2 congruences? Only a curiosity. === Subject: Re: Higher degree congruences > it's easy to find congruences like > x = a (mod N) > and there could also be solutions for quadratic congruences like > x^2 = a (mod N) > but is there something to solve congruences like > x^2 + ax = b (mod N) > or x^a + x^b + cx = d (mod N) > where a, b, c, d are integers > That is, what about degrees higher than 2 congruences? > Only a curiosity. Well, for x^2 + ax = b (mod N) first, factorise N into its prime factors. Then we have to solve x^2 + ax = b (mod p) for every prime p dividing N. for every p^t dividing N. Now use the Chinese remainder theorem to piece together the solution for N. To solve x^2 + ax = b (mod p) simply complete the square to reduce it to y^2 = A(mod p). I don't think there is an algorithm to solve congruences of higher degree than 2. For example, how does one solve x^3+ax+b= 0(mod p), other than by trial? Ray Steiner === Subject: Re: Higher degree congruences > x^2 + ax = b (mod N) Well, you can allways use normal procedure of solving quadratic equation - make a quadrat. It will give You solution of particular equation, but won't rather tell You much in general... sirix. === Subject: Re: Higher degree congruences >> x^2 + ax = b (mod N) > Well, you can allways use normal procedure of solving quadratic equation - > make a quadrat. Not allways the normal procedure... for instance how do you manage to make a quadrat with x^2 + 7x + 2 = 0 (mod 8) This is easy, but not with the usual make a quadrat... Because 2 has no inverse modulo 8, you can't write as (x + 7/2)^2 +... You need a different method. -- philippe (chephip at free dot fr) === Subject: Re: What can we expect by taking Fourier Transform of noise or random samples? > Hi all, > I am trying to understand using Fourier Transform to do denoising. [ ... ] > Any thoughts? It all depends upon what your real problem is. But ... You could consider denoising _without_ using a Fourier transform. Basically, what you do then is taking the convolution of your signal with a filter function, for example a bell shaped (Gauss) function: Z(t) = norm . integral_(-oo)^(+oo) exp(-(x-t)^2/sigma^2/2) z(x) dx Where norm = 1/(sigma*sqrt(2.pi)) The bandwidth sigma must be chosen properly. Due to properties of the 'exp', the (numerical) domain of integration is rather limited, say (t-2.pi.sigma) < x < (t+2.pi.sigma), which makes the method feasible. BTW. It is noted that the above convolution integral becomes a product of functions in the Fourier domain: exp(-(omega*sigma)^2/2) * Y(omega) Where Y(omega) = Fourier transform of z(t) . IF your decision is in favour of the Fourier approach, nevertheless, be well aware then that there exist subtle, but nasty, differences between the so-called FFT (discrete version) and the continuous FT. Han de Bruijn === Subject: Re: What can we expect by taking Fourier Transform of noise or random samples? > Hi all, > I am trying to understand using Fourier Transform to do denoising. > I heard many times that people say suppose I have z(t)=y(t)+n(t), where z is > the observations of the useful signal y and n is the noise... then we can > look at the spectrum of z(t) and then if we know our useful signal is in > some certain frequency range then we can do some filtering to recover the > useful signal and get rid of the noise. > I want to understand this approach. > I feel n(t) is a random process, y(t) is a deterministic signal, then is > z(t) wide sense stationary? My guess is that z(t) is a non-WSS random > process... what is the meaning of taking Fourier Transform of a random > process? > If we take FT of z(t)=y(t)+n(t), what is the meaning of taking FT of a > random process n(t)? Commonly people say noise n(t) is white... but this > white is talking about the power spectral density of the random process, > it has nothing to do with taking the FT of n(t) and consequently z(t)... Am > I right? > I did some experiments in Matlab: >>plot(abs(fftshift(fft(rand(1, 20000))))) > gives a huge peak/impulse at DC frequency f=0... what does this mean? It means that the sample mean of the data is non-zero. That's what the FFT is estimating at f=0. Many people preprocess the data by subtracting the average from each observation to center at zero. By the way, to have a true spectrum you should be squaring the transform results, not abs'ing. >>plot(abs(fftshift(fft(rand(1, 20000))))) > gives a uniformly chaotic noisy spectrum, looks like white... Uniform spectrum indicates no correlation. This follows directly from the Weiner-Khintchine theorem, which says that the spectrum can be found as the Fourier transform of the autocovariance function. If your noise terms are independent, then the autocovariance is zero for all lags other than zero, yielding a flat spectrum. (A common misconception is that the noise needs to be Gaussian for white noise, but uncorrelated is sufficient.) >>plot(abs(fftshift(fft(random('rayleigh', 10, 1, 20000))))) > also gives a huge peak/impulse at DC frequency f=0... what does this mean? > -------------------- > Maybe I should ask what shall we expect to see if we take FT of random > samples? > Any thoughts? Chatfield has written a very nice text called Time Series Analysis. It's not as comprehensive or theoretical as many others, but it's quite accessible. === Subject: Re: definition of definition >>What makes it any better than some other definition like my friend's >>alternative definition? That was my concern, and *you* have no answer >>to that. (And neither do anybody else on this group) I don't think you >>even understand the above questions. ... > 2) if you're not really talking about a definition (which is purely > prescriptive (there's some other polysyllabic word to describe this > better: a definition in math just tells you exactly what the word should > be understood to mean, regardless of any preconceptions you might have for > the word)) Is it stipulative? I recall recently there being a book published (in mathematics or philosphy) titled something like Definitions which mentions this concept. Any confirmation? (my own web searching did not). -- Mitch Harris (remove q to reply) === Subject: Re: definition of definition > Is it stipulative? > I recall recently there being a book published (in mathematics or > philosphy) titled something like Definitions which mentions this > concept. Any confirmation? (my own web searching did not). Mathematical Monthly has some interesting comments about the role of stipulative definitions in mathematics: Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions by Barbara S. Edwards and Michael B. Ward -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: definition of definition Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> Is it stipulative? ... >Mathematical Monthly has some interesting comments about the >role of stipulative definitions in mathematics: >Surprises from Mathematics Education Research: Student (Mis)use of >Mathematical Definitions >by Barbara S. Edwards and Michael B. Ward Mitch === Subject: Re: definition of definition <30s9o0F2k7h10U1@uni-berlin.de> <310l73F3629qhU1@uni-berlin.de> Discussion, linux) >> Is it stipulative? >> I recall recently there being a book published (in mathematics or >> philosphy) titled something like Definitions which mentions this >> concept. Any confirmation? (my own web searching did not). > Mathematical Monthly has some interesting comments about the > role of stipulative definitions in mathematics: > Surprises from Mathematics Education Research: Student (Mis)use of > Mathematical Definitions > by Barbara S. Edwards and Michael B. Ward Available online at -- The papers are currently at journals. [When published,] make no mistake, there will be no place on this planet where you can hide. Remember, I'm not talking about something vague here. I'm talking about publication in journals. James S. Harris. Wow. Journals. === Subject: Help!! Please help me to sove the Question︰ S（ω）= 26+006ω/ω, ω>=5 Graph S and state the process (domain , s', s) Please re-mail: hanhaly@yahoo.com.tw 3Q very, very, very much!!!!!! === Subject: Re: Help!! >Please help me to sove the Question︰ >S（ω）= 26+006ω/ω, ω>=5 >Graph S and state the process (domain , s', s) >Please re-mail: hanhaly@yahoo.com.tw >3Q very, very, very much!!!!!! I count five times now that you've posted this. If you were present in the room, would you just walk up and start yanking at a person's sleeve until they dropped what they were doing and answered you RIGHT NOW? Hmm .. since you demand an e-mail response, I guess a better analogy is until they left the room and answered you in private. Hint: Usenet is like God: it helps those who help themselves. Post _once_, and show what you yourself did to solve the problem. And since you post here, have the courtesy to read answers here. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com A: Maybe because some people are too annoyed by top-posting. Q: Why do I not get an answer to my question(s)? A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? === Subject: SOS! Help!!! 各位仁慈善心的大哥 823;姊高手高高手們，救 & #25937;命吧！火燒屁股，ą 2 6;天要交作業中，這題&# 2 5105;不會寫，SOS！Help！！！ & #24863;激& 19981;盡！ Please help me!! One Question︰ S(w)=26+0.06w/w w>=5 Graph S and state the process (domain , s', s) Please re-mail: hanhaly@yahoo.com.tw 3Q very, very, very much!!!!!! === Subject: SOS!!! Please help me!! S(w)=26+0.06w/w w>=5 Graph S and state the process (domain , s', s) Please re-mail: hanhaly@yahoo.com.tw 3Q very, very, very much!!!!!! === Subject: Simulation Question Does anyone know how to get random draws uniformly from ellipsoid: {x = (x1, ... , xn): x1^2/a1^2 + x2^2 / a2^2 + ... + xn^2 / an^2 = 1}? Mike === Subject: Re: Simulation Question >Does anyone know how to get random draws uniformly from ellipsoid: >{x = (x1, ... , xn): x1^2/a1^2 + x2^2 / a2^2 + ... + xn^2 / an^2 = 1}? >Mike There was a thread about this not too long ago: --Lynn === Subject: SOS! One question in Calculus.Please help me!! SOS! One question in Calculus.Please help me!! S(w)=26+0.06w/w w>=5 Graph S and state the process (domain , s', s) Please re-mail: hanhaly@yahoo.com.tw 3Q very, very, very much!!!!!! === Subject: SOS! One question in Calculus.Please help me!! SOS! One question in Calculus.Please help me!! S(w)=26+0.06w/w w>=5 Graph S and state the process (domain , s', s) Please re-mail: hanhaly@yahoo.com.tw 3Q very, very, very much!!!!!! === Subject: Re: SOS! One question in Calculus.Please help me!! han scribbled the following: > SOS! One question in Calculus.Please help me!! > S(w)=26+0.06w/w w>=5 > Graph S and state the process (domain , s', s) > Please re-mail: hanhaly@yahoo.com.tw > 3Q very, very, very much!!!!!! You know, one post would have been enough. -- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland -------- -------------------------------------------------------- rules! --------/ The trouble with the French is they don't have a word for entrepreneur. - George Bush === Subject: Re: logic is innate? >>Wow. Do you know of any scientific support for this? > Just about all the accumulated data for psychological testing over the > past century support this. Maybe you could be more explicit just to help me out. >>There's quite a bit of support for the innateness of language >>ability, but I have never heard of such a claim for innateness >>of -deductive logic- ability. > Since it's difficult to survive without it, I'm surprised that anyone > questions it. Well, that's not a particularly satisfying reason to not question something. Humans seem to need reasoning to survive, but jellyfish don't seem to need much. Do rats and crows reason (successfully use abstract logical inferences)? Where and how does it start (both in the individual and in the species)? > In any case, one certainly doesn't need a math class to learn to reason. > We all are born with the ability to reason, both deductively and > inductively. Sure. Informally, I'm with you. But just not convinced. Maybe people do learn how to reason. Or maybe there's an innate capacity for reason (a logic module, analogous to a face recognition area of the brain) which, through exposure, we develop well. Is there a brain lesion/freezing study that shows where modus ponens lies?. -- Mitch Harris (remove q to reply) === Subject: Re: logic is innate? >>Wow. Do you know of any scientific support for this? > Just about all the accumulated data for psychological testing over the > past century support this. > Maybe you could be more explicit just to help me out. >>There's quite a bit of support for the innateness of language >>ability, but I have never heard of such a claim for innateness >>of -deductive logic- ability. > Since it's difficult to survive without it, I'm surprised that anyone > questions it. > Well, that's not a particularly satisfying reason to not question > something. Humans seem to need reasoning to survive, but jellyfish > don't seem to need much. Do rats and crows reason (successfully use > abstract logical inferences)? Where and how does it start (both in > the individual and in the species)? Logical reasoning is probably something learnt, through trial and error, and creative thinking. It's a way of making sense of the world. I believe it to be a very misleading picture to think of a human brain as a simple inference engine. Besides, there is no compelling evidence for the innateness of language ability. I'm surprised at how people believe Chomsky's outdated science so easily. Learning the basics of perception is a much harder problem than drawing simple inferences. What's so fantastic about logic, except that it's the simplest thing we can think? In addition, logical reasoning is not the only kind of reasoning you can learn. Analogy making, etc., in general common sense reasoning is more than logic. It's only one of the things in your toolkit, it's not at the root, or even hardwired. I thought it was obvious to everybody that logic is a generalization we learn from observations of causal processes in the world. It's just an abstract computational model, it's not the law of reasoning or anything like that. It's an idealization, that sometimes works, and sometimes doesn't. Didn't you read The Name of The Rose? In general, when we say logic in commonsense, we do not mean predicate logic or anything as naive, instead we refer to intelligent decision making that's so much more complicated than simple linear chains. -- Eray === Subject: Re: logic is innate? >>Wow. Do you know of any scientific support for this? > Just about all the accumulated data for psychological testing over the > past century support this. > Maybe you could be more explicit just to help me out. >>There's quite a bit of support for the innateness of language >>ability, but I have never heard of such a claim for innateness >>of -deductive logic- ability. > Since it's difficult to survive without it, I'm surprised that anyone > questions it. > Well, that's not a particularly satisfying reason to not question > something. Humans seem to need reasoning to survive, but jellyfish > don't seem to need much. Do rats and crows reason (successfully use > abstract logical inferences)? Where and how does it start (both in > the individual and in the species)? > In any case, one certainly doesn't need a math class to learn to reason. > We all are born with the ability to reason, both deductively and > inductively. > Sure. Informally, I'm with you. But just not convinced. Maybe people > do learn how to reason. Or maybe there's an innate capacity for > reason (a logic module, analogous to a face recognition area of the > brain) which, through exposure, we develop well. Is there a brain > lesion/freezing study that shows where modus ponens lies?. I think that most people have strong intuitions about the meanings of the words 'truth' and 'falsity' and rely on those to do the intuitive equivalent of a model-theoretic argument. However, their notions of truth and falsity are often not sophisticated enough, since things like the Paradox of Material Implication, are, well, paradoxical to many people until they shift their point of view. I have *some* empirical evidence for this, since I've tutored both math and logic -- but obviously not enough to make sweeping claims. 'cid 'ooh === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> Discussion, linux) > I think that most people have strong intuitions about the meanings of > the words 'truth' and 'falsity' and rely on those to do the intuitive > equivalent of a model-theoretic argument. However, their notions of > truth and falsity are often not sophisticated enough, since things > like the Paradox of Material Implication, are, well, paradoxical to > many people until they shift their point of view. What paradox of material implication? -- Jesse F. Hughes How lucky we are to be able to hear how miserable Willie Nelson could imagine himself to be. -- Ken Tucker on Fresh Air === Subject: Re: logic is innate? > I think that most people have strong intuitions about the meanings of > the words 'truth' and 'falsity' and rely on those to do the intuitive > equivalent of a model-theoretic argument. However, their notions of > truth and falsity are often not sophisticated enough, since things > like the Paradox of Material Implication, are, well, paradoxical to > many people until they shift their point of view. > What paradox of material implication? The term Paradox of Material Implication refers to the fact that if P is false, then P -> Q for any value of Q. This is unintuitive, since many natural English if... then... statements are parsed as biconditionals. 'cid 'ooh === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> Discussion, linux) >> What paradox of material implication? > The term Paradox of Material Implication refers to the fact that if > P is false, then P -> Q for any value of Q. This is unintuitive, > since many natural English if... then... statements are parsed as > biconditionals. Unintuitive, perhaps, but not what I'd call a paradox. -- Jesse F. Hughes Even I, who know beyond doubt that my death will be caused by a silly girl, will not hesitate when that girl passes by. -- Merlin, as reported by John Steinbeck. === Subject: Re: logic is innate? >What paradox of material implication? >>The term Paradox of Material Implication refers to the fact that if >>P is false, then P -> Q for any value of Q. This is unintuitive, >>since many natural English if... then... statements are parsed as >>biconditionals. > Unintuitive, perhaps, but not what I'd call a paradox. it is not uncommon to call these classic difficulties paradoxes. -- Mitch Harris (remove q to reply) === Subject: Re: logic is innate? > The term Paradox of Material Implication refers to the fact that if > P is false, then P -> Q for any value of Q. This is unintuitive, > since many natural English if... then... statements are parsed as > biconditionals. I think in these cases it's usually more accurate to say that the sentences are understood as relevant implications. They're not really if and only if; they're just not truth functional at all. === Subject: Re: logic is innate? > The term Paradox of Material Implication refers to the fact that if > P is false, then P -> Q for any value of Q. This is unintuitive, > since many natural English if... then... statements are parsed as > biconditionals. > I think in these cases it's usually more accurate to say that the > sentences are understood as relevant implications. They're not > really if and only if; they're just not truth functional at all. Sure, but that's on the natural language side. Suppose a person, during normal discourse, tells his son: If you eat your vegetables, I'll give you some ice cream. Conversationally implied is the sentence If you don't eat your vegetables, I won't give you ice cream. To the uninitiated, seeing an If... then... and considering the conversational implicature leads them to treat that the arrow like a biconditional and leads to such zaniness as denying the antecedent. 'cid 'ooh === Subject: Re: logic is innate? >The term Paradox of Material Implication refers to the fact that if >P is false, then P -> Q for any value of Q. This is unintuitive, >since many natural English if... then... statements are parsed as >biconditionals. >>I think in these cases it's usually more accurate to say that the >>sentences are understood as relevant implications. They're not >>really if and only if; they're just not truth functional at all. > Sure, but that's on the natural language side. Suppose a person, > during normal discourse, tells his son: If you eat your vegetables, > I'll give you some ice cream. Conversationally implied is the > sentence If you don't eat your vegetables, I won't give you ice > cream. No, I don't really think it is. I think that's an oversimplification. What's understood conversationally is that there's a relationship between the *meaning* of the hypothesis and the conclusion, not just between their *truth*values*. That leads to a reasonable inference that, if the father were planning to fork over (spoon over?) the ice cream anyway, he wouldn't have put it as he did. But if he caves in at the end and allows ice cream with no vegetables, it's not generally considered that he's been inaccurate. These intensional relationships are much more difficult to model mathematically than truth-functional relations such as material implication. === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> <313slaF34ah79U1@uni-berlin.de> <314djvF377aqnU1@individual.net> Discussion, linux) >>The term Paradox of Material Implication refers to the fact that if >>P is false, then P -> Q for any value of Q. This is unintuitive, >>since many natural English if... then... statements are parsed as >>biconditionals. >I think in these cases it's usually more accurate to say that the >sentences are understood as relevant implications. They're not >really if and only if; they're just not truth functional at all. >> Sure, but that's on the natural language side. Suppose a person, >> during normal discourse, tells his son: If you eat your vegetables, >> I'll give you some ice cream. Conversationally implied is the >> sentence If you don't eat your vegetables, I won't give you ice >> cream. > No, I don't really think it is. I think that's an oversimplification. > What's understood conversationally is that there's a relationship > between the *meaning* of the hypothesis and the conclusion, not > just between their *truth*values*. That leads to a reasonable > inference that, if the father were planning to fork over (spoon over?) > the ice cream anyway, he wouldn't have put it as he did. But > if he caves in at the end and allows ice cream with no vegetables, > it's not generally considered that he's been inaccurate. > These intensional relationships are much more difficult to > model mathematically than truth-functional relations such > as material implication. I think this sums it up very well. After all, the father doesn't even mean that if the son eats his vegetables, he will be guaranteed ice cream. The conditional operator of natural language is non-monotonic. Eat veggies ----> get ice cream but presumably *not* Eat veggies & kill sister ----> get ice cream Even though eat veggies & kill sister implies eat veggies, one shouldn't conclude that the kid still gets the ice cream after eating his veggies and killing his sister. Depending, of course, on Dad's opinions of sis. These aren't paradoxes of material implication. They are problems of natural language, relevance, and so on. If they're relevant at all in the current discussion, then they cast doubt on the claim that logic is innate. How does one claim that logic is part of the fundamental features of the human brain, when natural language conditionals are so obviously not truth-functional? -- Jesse F. Hughes That's the base tautological space where by tautological space I mean a region of truth. -- James S. Harris does philosophy of mathematics. JSH is a renaissance man. === Subject: Re: logic is innate? > >>The term Paradox of Material Implication refers to the fact that if >>P is false, then P -> Q for any value of Q. This is unintuitive, >>since many natural English if... then... statements are parsed as >>biconditionals. I think in these cases it's usually more accurate to say that the >sentences are understood as relevant implications. They're not >really if and only if; they're just not truth functional at all. >> Sure, but that's on the natural language side. Suppose a person, >> during normal discourse, tells his son: If you eat your vegetables, >> I'll give you some ice cream. Conversationally implied is the >> sentence If you don't eat your vegetables, I won't give you ice >> cream. > No, I don't really think it is. I think that's an oversimplification. > What's understood conversationally is that there's a relationship > between the *meaning* of the hypothesis and the conclusion, not > just between their *truth*values*. That leads to a reasonable > inference that, if the father were planning to fork over (spoon over?) > the ice cream anyway, he wouldn't have put it as he did. But > if he caves in at the end and allows ice cream with no vegetables, > it's not generally considered that he's been inaccurate. > These intensional relationships are much more difficult to > model mathematically than truth-functional relations such > as material implication. > I think this sums it up very well. > After all, the father doesn't even mean that if the son eats his > vegetables, he will be guaranteed ice cream. The conditional operator > of natural language is non-monotonic. > Eat veggies ----> get ice cream > but presumably *not* > Eat veggies & kill sister ----> get ice cream > Even though eat veggies & kill sister implies eat veggies, one > shouldn't conclude that the kid still gets the ice cream after eating > his veggies and killing his sister. Depending, of course, on Dad's > opinions of sis. > These aren't paradoxes of material implication. They are problems of > natural language, relevance, and so on. They are called Paradoxes of Material Implication for historical reasons. Note that these aren't problems for natural language *speakers*, just people who wish to translate a natural language into the FOL. >If they're relevant at all in > the current discussion, then they cast doubt on the claim that logic > is innate. How does one claim that logic is part of the fundamental > features of the human brain, when natural language conditionals are so > obviously not truth-functional? Because there are different natural language conditionals, some of which are truth functional. I still stand by my Eat veggies, get ice cream example, since in my experience, if the father gives the kid ice cream without him having eaten his vegetables, people would say that the caved. Of course, your experience could be different. Assuming that this reaction is universal is too far out. Granting this, your example of the non-monotonicity of this conversational conditional doesn't really show what you think it does. It shows that the context generated (this may not be the best phrase for what I mean, but...) by the sentences Eat veggies, get ice cream and Eat veggies and kill sis, get ice cream are different -- very different. In short, you've shown the existence of two classes of conversational conditionals. This isn't new -- we should not forget that logical english, like what a mathematician would use when stating a theorem, is a fragment of the whole of english. For what it's worth, my view is that our natural language abilities combined with rough, though strong intuitions of truth and falsity constitute most people's logic abilities. Conversational implicatures can lead people to stray from classical logic because they cannot differentiate between a truth functional and a non-truth functional conditional. 'cid 'ooh === Subject: Re: logic is innate? > Granting this, your example of the non-monotonicity of this > conversational conditional doesn't really show what you think it does. > It shows that the context generated (this may not be the best phrase > for what I mean, but...) by the sentences Eat veggies, get ice cream > and Eat veggies and kill sis, get ice cream are different -- very > different. In short, you've shown the existence of two classes of > conversational conditionals. Can you explain why you think this example involves two different kinds of conditional? 'Different contexts generated' is almost always what is going on in this kind of examples of non-monotony. -- Herman Jurjus === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> <313slaF34ah79U1@uni-berlin.de> <314djvF377aqnU1@individual.net> <871xeasayt.fsf@phiwumbda.org> Discussion, linux) >>If they're relevant at all in >> the current discussion, then they cast doubt on the claim that logic >> is innate. How does one claim that logic is part of the fundamental >> features of the human brain, when natural language conditionals are so >> obviously not truth-functional? > Because there are different natural language conditionals, some of > which are truth functional. Can you give a simple example of a natural language conditional that is best interpreted truth-functionally? Let's leave informal mathematical discussions out, if we may. > I still stand by my Eat veggies, get ice cream example, since in my > experience, if the father gives the kid ice cream without him having > eaten his vegetables, people would say that the caved. Of course, > your experience could be different. Assuming that this reaction is > universal is too far out. I didn't disagree with your claim that this example is better understood as a biconditional, but even then, I'd say that each of the two conditionals is probably non-monotonic, rather than material implication. > Granting this, your example of the non-monotonicity of this > conversational conditional doesn't really show what you think it > does. I'm not sure what you're correcting here. I didn't want to claim that the fact natural language conditionals aren't truth-functional *proves* logic is not innate. But if natural language gives us any hints on that question, I'd guess it pushes us towards the negative at least as strongly as to the positive. To be honest, though, it's not the kind of philosophical question I spend much thought on. Matter of taste, I suppose. > It shows that the context generated (this may not be the best phrase > for what I mean, but...) by the sentences Eat veggies, get ice cream > and Eat veggies and kill sis, get ice cream are different -- very > different. In short, you've shown the existence of two classes of > conversational conditionals. This isn't new -- we should not forget > that logical english, like what a mathematician would use when > stating a theorem, is a fragment of the whole of english. I certainly didn't claim that anything I mentioned was new. -- Sure, [my Usenet presence is] like Shaq playing against you in your backyard, but that has its perks, as I find ways to have my fun *and* I can send messages to certain people in the United States Government without concern that the rest of you understand them. -- James Harris === Subject: Re: logic is innate? >>If they're relevant at all in >> the current discussion, then they cast doubt on the claim that logic >> is innate. How does one claim that logic is part of the fundamental >> features of the human brain, when natural language conditionals are so >> obviously not truth-functional? > Because there are different natural language conditionals, some of > which are truth functional. > Can you give a simple example of a natural language conditional that > is best interpreted truth-functionally? Let's leave informal > mathematical discussions out, if we may. How about informal philosophical discussions? This might seem like cheating, since this sort of discourse shares many of the features of mathematical discussions (and includes more than a small amount of equivocation). I'll ask my linguistics buddies if they can think of any. > I still stand by my Eat veggies, get ice cream example, since in my > experience, if the father gives the kid ice cream without him having > eaten his vegetables, people would say that the caved. Of course, > your experience could be different. Assuming that this reaction is > universal is too far out. > I didn't disagree with your claim that this example is better > understood as a biconditional, but even then, I'd say that each of the > two conditionals is probably non-monotonic, rather than material > implication. > Granting this, your example of the non-monotonicity of this > conversational conditional doesn't really show what you think it > does. > I'm not sure what you're correcting here. Well, you claimed that _The_ conditional operator of natural language is non-monotonic, whereas your example shows that it isn't unique. (emph mine) > I didn't want to claim that > the fact natural language conditionals aren't truth-functional > *proves* logic is not innate. But if natural language gives us any > hints on that question, I'd guess it pushes us towards the negative at > least as strongly as to the positive. I don't have any strong opinions (that would settle the issue, anyway). Like I've said before, in my experience -- say, during freshman year analysis, I would reason by internally thinking things along the lines of So 'X' is true, and 'X->Y' is true and using my intuitions about what truth meant. My abilities have grown in sophistication since then (not to toot my own horn, but the fact that we can have intelligent discourse on abstract topics is testament). But one can mean many things by logic. Mentally applying an algorithm to solve an NP-complete game would be equivalent to applying an algorithm to solve the satisfiability problem -- it would be difficult to say that this person wasn't using logic, even though all they've done is raw, likely brute calculation. This sort of data crunching ability might seem to be innate, but it seems like a different skill than *finding* the algorithm. The best I think we can say about this issue is sort of a compromise -- we each start with certain mental faculties (the raw data crunching stuff) that we can develop into higher level skills with intense critical thinking. Maybe we can even improve our raw data crunching skills with lots of practice (yuck). Developmentally, we might find people's rough, intuitive notions of how logic works somewhere between the two. But this isn't philosophical or empirical. It's downright flakey. > To be honest, though, it's not the kind of philosophical question I > spend much thought on. Matter of taste, I suppose. Me neither, usually. But you and Mitch brought up some interesting points regarding my posts. 'cid 'ooh === Subject: Re: logic is innate? <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> <313slaF34ah79U1@uni-berlin.de> <314djvF377aqnU1@individual.net> <871xeasayt.fsf@phiwumbda.org> <87vfblyqx5.fsf@phiwumbda.org> Discussion, linux) >> Granting this, your example of the non-monotonicity of this >> conversational conditional doesn't really show what you think it >> does. >> I'm not sure what you're correcting here. > Well, you claimed that _The_ conditional operator of natural language > is non-monotonic, whereas your example shows that it isn't unique. > (emph mine) -- Jesse F. Hughes LOL. How arrogant you are. Now when you realize that I DID prove Goldbach's conjecture and that I proved Fermat's Last Theorem as well, how are you going to feel then? -- James Harris === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> <313slaF34ah79U1@uni-berlin.de> <314djvF377aqnU1@individual.net> <871xeasayt.fsf@phiwumbda.org >The term Paradox of Material Implication refers to the fact that if >>P is false, then P -> Q for any value of Q. This is unintuitive, >>since many natural English if... then... statements are parsed as >>biconditionals. I think in these cases it's usually more accurate to say that the >sentences are understood as relevant implications. They're not >really if and only if; they're just not truth functional at all. >> Sure, but that's on the natural language side. Suppose a person, >> during normal discourse, tells his son: If you eat your vegetables, >> I'll give you some ice cream. Conversationally implied is the >> sentence If you don't eat your vegetables, I won't give you ice >> cream. > No, I don't really think it is. I think that's an oversimplification. > What's understood conversationally is that there's a relationship > between the *meaning* of the hypothesis and the conclusion, not > just between their *truth*values*. That leads to a reasonable > inference that, if the father were planning to fork over (spoon over?) > the ice cream anyway, he wouldn't have put it as he did. But > if he caves in at the end and allows ice cream with no vegetables, > it's not generally considered that he's been inaccurate. > These intensional relationships are much more difficult to > model mathematically than truth-functional relations such > as material implication. > I think this sums it up very well. > After all, the father doesn't even mean that if the son eats his > vegetables, he will be guaranteed ice cream. The conditional operator > of natural language is non-monotonic. > Eat veggies ----> get ice cream > but presumably *not* > Eat veggies & kill sister ----> get ice cream > Even though eat veggies & kill sister implies eat veggies, one > shouldn't conclude that the kid still gets the ice cream after eating > his veggies and killing his sister. Depending, of course, on Dad's > opinions of sis. > These aren't paradoxes of material implication. They are problems of > natural language, relevance, and so on. If they're relevant at all in > the current discussion, then they cast doubt on the claim that logic > is innate. How does one claim that logic is part of the fundamental > features of the human brain, when natural language conditionals are so > obviously not truth-functional? > -- > Jesse F. Hughes > That's the base tautological space where by tautological space I mean > a region of truth. -- James S. Harris does philosophy of mathematics. > JSH is a renaissance man. It's not random alien stranger A and random alien stranger B, besides the fact that if the kid does finish his vegetables he will get some ice cream, the kid already knows not to kill his sister. The kid presumably knows that kill sister or murder dog or other excessive and unusual circumstances would make void the pledge of ice cream. If two aliens meet and one confers to the other, eat your vegetables, and then you can have this ice cream, then there are no strings attached, because they don't have a relationship outside of that. The two close relatives have a rich context of expectations already in place, that vegetables lead to ice cream is presumably a new agreement in their relationship, or ice cream is a special event, where presumably the kid has vegetables more often than ice cream, unless it's some round little ice cream addict. I think it's generally implied that no vegetables means no ice cream, else there would be no reason for the statement, meant to induce action, unless it was some mantra, and alien dinner manners would probably, bereft of the familiar surroundings of the dinner that may be so, the exact same conversation would probably take place. Logic is definitely a part of the human nervous system, and in most cases cognition. Higher function is quite dependent on being able to derive logical meaning. So, it's Son, eat vegetables and get ice cream. There are no paradoxes, calling it a paradox is stupid. I see that Ross has succeeded in altering the language of sci.math. It's enough to make one weep. - Randy Ross === Subject: Re: logic is innate? > These aren't paradoxes of material implication. They are problems of > natural language, relevance, and so on. If they're relevant at all in > the current discussion, then they cast doubt on the claim that logic > is innate. How does one claim that logic is part of the fundamental > features of the human brain, when natural language conditionals are so > obviously not truth-functional? Good point. The currently accepted propositional logic rules are a social construction from the past 2 centuries (whether they were always there or not). But maybe we shouldn't be so strict as to what logic is. In analogy with language, we (humans) certainly have biological adaptations to allow us to speak and hear (breathing control, trachea connected with mouth, threshold neurons for phonological features (yes, chinchillas have some too but that's irrelevant), brain areas with specific language function). Any particular language is not forced on us by our parents but language capability certainly is. To tell if some specific logic capability is innate, we'd have to be pretty clever in defining our notions. And as usual, even though there is a question as to how the individual gets the skill (through sheer conditioning, triggering, or built in), you still need the explanation of how/when/why the development of a species gained whatever mechanisms necessary to do logic or math or language. -- Mitch Harris (remove q to reply) === Subject: Re: logic is innate? > .... Is there a brain > lesion/freezing study that shows where modus ponens lies?. Given the number of people who think that If P then Q, Q, therefore P is a valid inference, probably not. I used to think that logic, being (as I thought) necessary for survival in the world, might have evolved. But given the number of stupid people who seem to do very well, I now have my doubts. === Subject: Re: logic is innate?