mm-1759 === Subject: Re: JSH: Sword of Damocles >> Sword of Damocles? Where's the infamous Hammer? >> Yeah, it should be The Hammer of Harris, instead of The Sword of >> Damocles. > Right now James is more concerned about the Flail of Failure. Or perhaps the Salmon of Doubt. -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: Solving the difference of squares > The surrogate factoring theorem (SFT) provides integer solutions to > sqrt(v^2 - 4A^2(A^2 - B^2)w^2) > where all are non-zero integers, and letting n be the result of taking > that square root, you have > n^2 = v^2 - 4A^2(A^2 - B^2)w^2 > and the difference of squares is seen as > v^2 - n^2 = 4A^2(A^2 - B^2)w^2 > so, simply enough, you have the factorization > (v - n)(v + n) = 4A^2(A^2 - B^2)w^2. > And the solutions the SFT provides give an infinite number of > non-trivial factorizations of A, the target integer, along with an > infinite number of trivial ones, as it shows no differentiation between > the two. You can (and should) go a little farther with this: v - n = 2g_2 (g_1 + B^2) B^2 A^2 and v + n = 2g_1 (g_1 + B^2) (g_2 + B^2)^2 where g_1 g_2 = B^2 (A^2 - B^2). The important thing to notice here is that: *** v - n is divisible by A^2 ***. What this means is that in general, GCD(v - n, A^2) = A^2, and (v - n) will therefore yield only TRIVIAL factors of A^2 (and likewise v + n). > No other theorem or mathematical result in human history can say the > same, making the significance of the SFT obvious to objective eyes. In view of the above, yes, I would agree. The significance is obvious. Nora B. > There are other methods that rely on difference of squares, like the QS > and the Number Field Sieve, but no other known theorems like the SFT. > ************************ > Full set of equations: > n = sqrt(v^2 - 4A^2(A^2 - B^2)w^2) > v/w = z - 2A^2 > where > z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) > and x is given by > x = +/- (g_1 - g_2) + 2B^2 > where > g_1 g_2 = B^2(A^2 - B^2). > James Harris === Subject: Re: Factoring problem and the SFT On 28 Apr 2005 13:18:42 -0700, Amadeus Train-Owwell Zirconium [body part remowed] ,while SFT is not verified to work cut sci.crypt from replies. It's current state belongs to other groups than sci.crypt. Juuso ps. You might consider updating two of the links in signature >--Chair Man George NAND Strep Throat @ W'gate? >http://tarpley.net/bush12.htm >http://laroucehpub.com networksolutions says 'laroucehpub.com is available. >http://members.tripod.com/~ame[CapitalEth]rican_almanac Sorry, but the page or the file that you're looking for is not here. === Subject: Re: What is the greatest error ever committed in math? > i wanna know...please help... > [...] > Or Ramanujan returning to India only to die,when so much > nourishing food was available in England, could slightly > compromise religious views ... What I heard was that the climate and food in England were what contributed to his illness. We could also mention Galois, who got involved in a duel with someone who was a much-better shot than he was ... --- Christopher Heckman === Subject: correlation coefficient rational or irrational? Does there exist such a thing as a proof that the statistic called correlation coefficient is rational or irrational? If so, could you please rho(x,y) = COV(x,y) / sigma x * sigma y; where: rho(x,y) = correlation between random variables x & y COV(x,y) = covariance between random variables x & y sigma (x) = std. deviation of random variable x sigma (y) = std. deviation of random variable y === Subject: Re: correlation coefficient rational or irrational? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Does there exist such a thing as a proof that the statistic called >correlation coefficient is rational or irrational? Since it's a continuous function of the variables, it can obviously be either. -- Richard === Subject: MuPAD bugs: 2-D summation -> Error: Recursive definition during evaluation of 'misc::maprec_main' Hello all, Hello Herr Creutzig, Herr Wehmeier, Prof Dr Oevel, Dr Kluge, Prof Dr Fuchssteiner, Yet another tiny example from the VM machine showing that the human beings, even such qualified and diligent like Herr Creutzig or me, are too budget-devastating as well as totally inadequate to achieve the SciFace GmbH's high aims successfully. The following sum is equal to 2. MuPAD 2.5.2 and MuPAD 3.0 return it unevaluated. Maple 3.1.1 invokes an error message > sum(sum(m^2*n/(2^m*(m*2^n+n*2^n)),m=1..infinity),n=1..infinity); Error: Recursive definition [See ?MAXDEPTH]; during evaluation of 'misc::maprec_main' > numeric::sum(sum(m^2*n/(2^m*(m*2^n+n*2^n)), m=1..100),n=1..100); 2.0 This bug missed by a human, is absent in the SciFace's database at http://research.mupad.de/bugs.shtml as well as other VM-discovered bugs reported by Cyber Tester recently MuPAD bugs: solve -> Error: Domain attribute _concat missing; MuPAD bugs: solve -> Error: Illegal operand [_and]; Bug in MuPAD 3.1/3/2.5 integration (Lupino+Cyber Tester) Re: MuPad: solve problem MuPAD bugs: sum(1/n!, n= 1..n) MuPAD bugs: series can return 6 terms only MuPAD bugs: A terrible flaw in indefinite integration Re: MuPAD bugs: A terrible flaw in indefinite integration MuPAD bugs: int(z^(10^10), z) -> Error: Illegal argument [_seqgen] Regression bug in MuPAD: int(1/(sin(z+PI/4) + sqrt(2)), z= 0..PI) MuPAD bugs: solve(z^a = x^a, z) keeps running after 40,000 seconds and the bugs to be shown in the immediate future. Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC 13 Dekabristov Str, Simferopol Crimea 95000, Ukraine tel: +38-(0652)-447325 tel: +38-(0652)-230243 tel: +38-(0652)-523144 fax: +38-(0652)-510700 http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ this author === Subject: MuPAD bugs: solve -> Error: Domain attribute _concat missing; Hello all, Hello Herr Creutzig, Herr Wehmeier, Prof Dr Walter Oevel, Dr Kluge, Prof Dr Fuschsteiner, The following sum is equal to 2. MuPAD 2.5.2 and MuPAD 3.0 return it unevaluated. Maple 3.1.1 invokes an error message sum(sum(m^2*n/(2^m*(m*2^n+n*2^n)), m=1..infinity), n=1..infinity); Error: Recursive definition [See ?MAXDEPTH]; during evaluation of 'misc::maprec_main' Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC 13 Dekabristov Str, Simferopol Crimea 95000, Ukraine tel: +38-(0652)-447325 tel: +38-(0652)-230243 tel: +38-(0652)-523144 fax: +38-(0652)-510700 http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ === Subject: need help solving this. hello group. let's say you have the following system. { a'[t] == -p1 a[t] - p3 c[t], b'[t] == p3 c[t] - p2 b[t] } with initial conditions {a[0]==a0, b[0]== b0, c[0]==c0} the system is underdetermined? as there are more variables then nymber of equations? so if i add the a'[t] and b'[t] equations to get { a'[t]+ b'[t] == -p1 a[t] -p2 b[t] } what type of method of solution is this? is the equation above solvable? if so, how will i go about doing this? also.... can I divide b'[t] by a'[t]? like... b'[t] p3 c[t] - p2 b[t] ----- == ------------------ a'[t] -p1 a[t] - p3 c[t] and if so... how do I get rid of c[t] ? if the notation is comfusing, then above is essentially same as... Y' p3 Z - p2 Y --- == ------------- X' -p1 X - p3 Z i need to get rid of Z. is there a method of solution for the above system? === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> It's intent is to convince the reader that their system >> can do what it claims to do. >Then why not show the proof? Wouldn't that go a long way toward that >goal? I haven't run the system on that proof, so I don't know. I expect that the output is too lengthy or too meaningless for humans, and so unilluminating for such a system report. ... >> They did not present a mathematical >> proof of UHP. >> They didn't even present the text >> of the output of their system when given a suitably symbolized >> version of UHP. >> The paper in question >> convinced [the ATP community], not by proving UHP >> or by giving the output/the proof >> (way, way too long) >No sir. Anything can be abstracted to a shorter higher level. (BTW >the proof would not be the output, but rather the logical argument that >it is mapped into.) Hmm...I'm not sure what you're saying here. The output is just a bunch of symbols. It could even be something like Yes, meaning the prover's algorithm entered the appropriate state. Or it could be a lengthy list of all the states it entered on the way to the Yes state. Anyway, whatever you're looking for, it's not in the paper. I think you're looking for the wrong thing.... OK.. just looked at the paper again. I must apologize to the authors for misrepresenting them. Sectino 5 gives you exactly what you are looking for 5. The Proof. We now prove HP. We use the following abbreviations... the proof is -from- their prover. Section 6 gives all the laborious details (Lisp code). >> Their paper does not give a >> -mathematical- proof, either in use or mention. Mea culpa. They did both. >> I vaguely recall that you claim to have written a theorem proving system >> that proves UHP. Excellent!! As cool as that is, >> it is old news. >Wouldn't actually showing such a proof be new (since they didn't)? The proof -is- in the paper. So, no, somebody else doing it would not be considered new. >> Oh.. to your original point.. >> If one presents a formalism but doesn't show that it represents a >> particular logical argument, then they have not shown that the >> formalism has any mathematical significance. >> Yes, I agree, in some very general sense. But the paper in >> question is definitely -not- self-contained. One would need >> to be acquainted with the rest of their work to be convinced >> that their formalism has mathematical significance. >So where is that particular logical argument contained in their >other papers? I'd guess the best place to start would be their book, A Computational Logic. (and download nqthm their prover). >I agree with all of the above except the final conclusion that the >proof does exist somewhere. Download the program; I bet the script that does UHP is in there. If not, then type stuff in from their paper. >I do thank you for reading the paper and verifying my contention that >Boyer and Moore do not give the proof alluded to in the title of their >paper. :) 1) If you understand my analogy, then that's not a complaint. 2) I accept the proof that they describe in sections 4 and 5 -- Mitch Harris (remove the q for email) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability :>> It's intent is to convince the reader that their system :>> can do what it claims to do. :>Then why not show the proof? Wouldn't that go a long way toward that :>goal? : I haven't run the system on that proof, so I don't know. : I expect that the output is too lengthy or too meaningless for humans, and : so unilluminating for such a system report. If you run the system on unsolv.events (the halting problem proof), it produces over 9000 lines of output, although there are a lot of blank lines. Below is a sample. I will not post the entire thing. An interested party can download the software and try themselves. Stephen Name the conjecture *1. We will try to prove it by induction. The recursive terms in the conjecture suggest three inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP FA) (p FN X FA)) (IMPLIES (AND (NOT (NLISTP FA)) (p FN X (CDR FA))) (p FN X FA))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to prove that the measure (COUNT FA) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates the following four new conjectures: Case 4. (IMPLIES (AND (NLISTP FA) (NOT (OCCUR-IN-DEFNS FN FA)) (NOT (BTMP (GET X FA)))) (NOT (OCCUR FN (CADR (GET X FA))))). This simplifies, expanding the functions NLISTP, OCCUR-IN-DEFNS, GET, and BTMP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP FA)) (OCCUR-IN-DEFNS FN (CDR FA)) (NOT (OCCUR-IN-DEFNS FN FA)) (NOT (BTMP (GET X FA)))) (NOT (OCCUR FN (CADR (GET X FA))))). This simplifies, opening up the functions NLISTP and OCCUR-IN-DEFNS, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP FA)) (BTMP (GET X (CDR FA))) (NOT (OCCUR-IN-DEFNS FN FA)) (NOT (BTMP (GET X FA)))) (NOT (OCCUR FN (CADR (GET X FA))))). This simplifies, unfolding NLISTP, OCCUR-IN-DEFNS, and GET, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP FA)) (NOT (OCCUR FN (CADR (GET X (CDR FA))))) (NOT (OCCUR-IN-DEFNS FN FA)) (NOT (BTMP (GET X FA)))) (NOT (OCCUR FN (CADR (GET X FA))))). This simplifies, unfolding the definitions of NLISTP, OCCUR-IN-DEFNS, and GET, to: T. That finishes the proof of *1. Q.E.D. === Subject: Re: Ever hear of this -- it's not homework > Someone brought this up at work and we are not math people: A squared > plus B squared equals A squared times B squared If you want rational numbers, take two integers m>n>0. Let A=(m^2+n^2)/(m^2-n^2) and B=(m^2+n^2)/(2mn). If m=5 and n=3, you get A=34/16 and B=34/30, for example. Bart === Subject: MuPAD bugs: 2-D summation -> Error: Recursive definition during evaluation of 'misc::maprec_main' Hello all, Hello Herr Creutzig, Herr Wehmeier, Prof Dr Oevel, Dr Kluge, Prof Dr Fuchssteiner, Yet another tiny example from the VM machine showing that the human beings, even such qualified and diligent like Herr Creutzig or me, are too budget-devastating as well as totally inadequate to achieve the SciFace GmbH's high aims successfully. The following sum is equal to 2. MuPAD 2.5.2 and MuPAD 3.0 return it unevaluated. MuPAD 3.1.1 invokes an error message > sum(sum(m^2*n/(2^m*(m*2^n+n*2^n)),m=1..infinity),n=1..infinity); Error: Recursive definition [See ?MAXDEPTH]; during evaluation of 'misc::maprec_main' > numeric::sum(sum(m^2*n/(2^m*(m*2^n+n*2^n)), m=1..100),n=1..100); 2.0 This bug missed by a human, is absent in the SciFace's database at http://research.mupad.de/bugs.shtml as well as other VM-discovered bugs reported by Cyber Tester recently MuPAD bugs: solve -> Error: Domain attribute _concat missing; MuPAD bugs: solve -> Error: Illegal operand [_and]; Bug in MuPAD 3.1/3/2.5 integration (Lupino+Cyber Tester) Re: MuPad: solve problem MuPAD bugs: sum(1/n!, n= 1..n) MuPAD bugs: series can return 6 terms only MuPAD bugs: A terrible flaw in indefinite integration Re: MuPAD bugs: A terrible flaw in indefinite integration MuPAD bugs: int(z^(10^10), z) -> Error: Illegal argument [_seqgen] Regression bug in MuPAD: int(1/(sin(z+PI/4) + sqrt(2)), z= 0..PI) MuPAD bugs: solve(z^a = x^a, z) keeps running after 40,000 seconds and the bugs to be shown in the immediate future. Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC 13 Dekabristov Str, Simferopol Crimea 95000, Ukraine tel: +38-(0652)-447325 tel: +38-(0652)-230243 tel: +38-(0652)-523144 fax: +38-(0652)-510700 http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> Interesting experiment: Come to Texas and see how many > people you can call yank before we have to fire up Ol' Sparky. > Interesting learning experience: Realise that some words have different > meanings in English and American (qv 'table', 'momentarily', 'pissed', > etc etc) My particular favourite is fanny. I once heard the American wife of the vicar of Wolston in the English midlands recount how she had been caught out by fanny. === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> Naturally I chose names that even a yank would have heard of. >> Interesting experiment: Come to Texas and see how many >> people you can call yank before we have to fire up Ol' Sparky. >> Interesting learning experience: Realise that some words have different >> meanings in English and American (qv 'table', 'momentarily', 'pissed', >> etc etc) >My particular favourite is fanny. I once heard the American wife of >the vicar of Wolston in the English midlands recount how she had been >caught out by fanny. That doesn't make sense. AFAIK American: Fanny=Buttocks English: Fanny=Vagina. To be caught out by either would be strange in the extreme! -- Jeremy Boden === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> <$C7V69CyBKcCFwUj@jboden.demon.co.uk> In message , Bart Goddard > huh? get your facts straight... troll. > *plonk* >Like we needed more evidence that anglos have no >>sense of humor. >> What's an anglo -is it half an Anglo-Saxon? >Anglo is an ethnic slur. > --- Christopher Heckman > Seems a pretty poor attempt at racism - I thought the > good citizens of the USA were meant to be pretty good > in this area? Some of us try to be, but cases come up continually. I learned earlier in the semster that a certain word that begins with the letter O and is used to denote people from eastern Asia is actually a racist slur. The next time Mom called me, I told her she needed to wash her mouth out with soap, because she'd been using it all while I grew up, and she didn't know it was one. Same with the five letter K-word which refers to a German. --- Christopher Heckman === Subject: Re: Crossword clue >>Anglo is an ethnic slur. >> --- Christopher Heckman >> Seems a pretty poor attempt at racism - I thought the >> good citizens of the USA were meant to be pretty good >> in this area? > Some of us try to be, but cases come up continually. I learned earlier > in the semster that a certain word that begins with the letter O and is > used to denote people from eastern Asia is actually a racist slur. The > next time Mom called me, I told her she needed to wash her mouth out > with soap, because she'd been using it all while I grew up, and she > didn't know it was one. > Same with the five letter K-word which refers to a German. Are you trying to be funnier than me? Anglo is _sometimes_ an ethnic term, but usually not. And in no case is it a slur. (I should know, I am one.) You silly occidental, you. B. === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> <$C7V69CyBKcCFwUj@jboden.demon.co.uk> In message , Bart Goddard >Anglo is an ethnic slur. > --- Christopher Heckman Seems a pretty poor attempt at racism - I thought the > good citizens of the USA were meant to be pretty good > in this area? >> Some of us try to be, but cases come up continually. I learned earlier >> in the semster that a certain word that begins with the letter O and is >> used to denote people from eastern Asia is actually a racist slur. The >> next time Mom called me, I told her she needed to wash her mouth out >> with soap, because she'd been using it all while I grew up, and she >> didn't know it was one. >> Same with the five letter K-word which refers to a German. >Are you trying to be funnier than me? Anglo is _sometimes_ >an ethnic term, but usually not. And in no case is it a slur. >(I should know, I am one.) You silly occidental, you. I think you will find that occidental is an adjective, not a noun. -- Jeremy Boden === Subject: Re: Richard Henry shows his psychopathology >> Two news groups I follow regularly, sci.math and rec.skiing.alpine, have >> similar threads going. >> In sci.math, James Harris is discussing Why would mathematicians lie? >> In rec.skiing.alpine, Scott Abraham started a thread title Why do they >> lie. > Why does Richard Henry lie? Scores of lies today, obvious and gross lies, > and not a word from Richard Henry. Just a cheap shot, a defamatory > manipulation. >> These two individuals occupy similar niches in the respective newsgroups. >> JSH is complaining about his inability to get any respect for his >> mathematical discoveries, and suspect a conspiracy to silence him. SA is >> complaining about his inability to get any respect for anything, and >> suspects a conspiracy, also. > I do not suspect a conspiracy. I know there is a proven conspiracy. > And I've never complained about a lack of respect from psychopaths like > Henry. If he respected me, I'd wonder what I was doing wrong: his respect > is saved for his fellow liars and criminals. >> Compare, contrast, discuss among yourselves. > No comparison. And the contrast is obvious: you are a pathological liar. Wow, so little Scooter makes an appearance in sci.math! But is it the calling anyone a terrorist, so it's hard to believe it's really you. -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: Question about the minesweeper consistency problem (NP-complete problems) Well, believe it or not, but I've developed an algorithm > which solves the minesweeper consistency problem and I > think it runs in polynomial time. I haven't made a formal > complexity analysis like you see in > Well, I don't believe it then --- because, as you well know, > the minesweeper problem is NP-complete. > I tested the program, it works and gives exact results, > and it takes 15 seconds in order to process a game > situation corresponding to the expert level of the > real game. It included something like 134 uncovered > cells adjacent to covered cells (which are the only > sources of information, covered cells or uncovered > cells which are adjacent to no covered cells don't add > information, they don't add conditions to be respected > by the mines distribution which could be a possible > solution). This means that my program is certainly > sub-exponential, but I can > Why does it? I've played the minesweeper game, and my > impression is that those configurations which account > for the NP-completeness are actually not that common. > So if you tested your program on random configurations, > it may have been that you just weren't likely to hit > hard situations. Also, it may be that you need to > increase the playing field further to really encounter > these difficulties. There's also a result that says that every NP-complete problem has an algorithm that solves the problem and whose EXPECTED running time is polynomial. So, for the NP-complete problems, there are a large number which can be solved quickly and a very few which are pathological. If you're interested about these pathological examples, you should NP-complete. (Math. Intelligencer 22 (2000), no. 2, 9--15.) -- or check out his website at http://web.mat.bham.ac.uk/R.W.Kaye/minesw/ordmsw.htm -- and make up a few examples based on his reduction construction. > [...] > So likely your algorithm doesn't run in polynomial time > (but ran quickly on the instances you considered), or it > isn't always right (but was right on the instances you > tested). (Or both. :P) It also sounded like you use randomization, which would also disqualify it from being in P. But this may have been a misinterpretation on my part. --- Christopher Heckman === Subject: Re: Cardinality question >I prefer to use logic universally regressible to self contradictory >alternatives. I realize that's a bit of a stretch for point whores. >>Perhaps if you would show examples of it in action to prove something I >>could figure out if it was something I could stomach. As it stands now, >>I saw a post about it that made absolutely no sense. It looked like >>there could be sense to it, but I didn't know what it was. >Well how about if I say that everything is differences because there >can be nothing different from differences? >>But that doesn't say anything. That gives me no clue what differences >>are. How is the above clearer than saying Everything is slursh because >>there can be nothing different from slursh? Yours sounds cooler, but I >>can't detect any meaningful content in it. > This is the classical approach to universal truth: Everything is > slursh because there can be nothing different from slursh. The > problem is slursh doesn't regress tautologically to self contradictory > alternatives. That is t:[slursh][not slursh] is tautologically true > but [not slursh] is not self contradictory. Whereas if I essay the > empirical observation [not] the tautologcial regression to [not not] > is self contradictory. The same with different from differences or the > contradiction of contradiction. How about explaining *why* t:[slursh][not slursh] is tautologically true. At the moment, t:[slursh][not slursh] are meaningless symbols to me. Similarly with the [not], [not slursh], [not not]. What are these supposed to represent? >Or if I say that there is a >provable Law of Contradiction because contradictions of contradiction >are self contradictory? >>I thought a contradiction was something that is self-contradictory. >>Again, this doesn't explain what you mean. It sounds good, but it >>conveys no meaning. > This is typical of what most people think but is not correct. A > contradicts C. If the proposition P not P is self contradiction, > P is the self and not the contradiction. There is no implicit > self contradiction in the fact of contradiction alone. Well, since not is generally a unary operator, I find it odd that you would apparently use it as a binary operator. Please start with some clear and precise definitions. If something is not defineable, at least offer some examples of it in action so I can get a clear idea of what's going on. >It's really difficult to know how to approach >such fundamental issues when the only response seems to be that the >issues raised are unconventional and do not conform to mathematical >and scientific conventions. >>My problem isn't that it's unconventional, but that it appears to be >>devoid of content. I'm open to being proven wrong, but I haven't seen >>anything promising yet. > Well it's really too bad you missed discussions on the earlier thread. > They ran to more than seven thousand posts and such topics formed a > large part of the discussion. I've looked at some of them. They didn't offer anything enlightening. Perhaps if I read all 7000 I would be more enlightened, but I'm not eager to commit that much time to it. Perhaps a summary with clarifications so far on a web site would help. >The problem is that science is not about arbitrary selection of axioms >or proclamations of current theory. >>No, but large chunks of mathematics may appear to be. Generally, the >>axioms are not arbitrary, but the reasons for their selection may not be >>obvious. > The problem is not that the axioms are true or not true but whether > they are universally true and demonstrably so. It's generally understood that the axioms are not necessarily true or false in any sense. In fact, some axioms are worked with in certain scenarios, and their negation in other scenarios. > The axioms of non > Euclidean geometries are presumably true given Euclidean geometry. No. Some of the axioms of a non-Euclidean geometry are simple false given Euclidean geometry. Non-Euclidean geometries are the result of *changing* or *negating* or *abandoning* axioms of Euclidean geometry. > But not without because curves can't be analyzed or even defined in > the absence of straight lines and bisection. Curves are a non-issue to the discussion. >That's the reason I forwarded the >root post to my previous thread on the subject because it covers the >most explicit treatment of what I had in mind. The difficulty goes all >the way back to Aristotle. So I'm not really surprized people find it >difficult to deal with. I just don't see any practical way to shorten >the process than by talking and explaining one point at a time. >>Have you been understood on any of your points? If not, you are failing >>in your goal of explaining. > I think there have been partial assimilations, yes. At least Tony > Orlow was able to refer to finite tautological regression to self > contradictory alternatives as a standard of universal truth with a > straight face. Albert and Allan have also indicated some degrees of > comprehension. Has anyone shouted Eureka! We are saved! from > the rooftops yet? No. Not yet. But the fact you think my regressions > sound cooler than slursh gives me hope. Note: There are many things that I think sound cool which I would never use. I think it would be cool to have long hair again, but I'm more likely to shave my head than grow it out. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Cardinality question I'd be very interested to know the branch of >>modern math which can define pi with no reference to points, lines, >>geometry, or ratios defined thereon and without the expression Let >>there be . . . that doesn't really depend directly or indirectly on >>those concepts. Calculus. Consider the series 1 - 1/3 + 1/5 - 1/7 + 1/9 + ... It is convergent and its sum is a therefore a real number S. Define >the number pi as 4*S. >>So just point out the limit of the series on a straight line and we >>can all go home. >Excuse me, you asked for a definition that didn't rely on points, >lines, geometry or ratios. Why are you putting lines back in? >> Probably because your approximation of pi relies on ratios and is only >> an approximation which was the whole problem to begin with since your >> definition is only for a series and not the limit of the series which >> is what pi is. >Ratios do not require lines. Why are you putting lines back in? >Note: what you see above are multiplicative inverses, not ratios. Strange. They seem to involve ratios, like 1, 2, 3 etc. which are cardinal ratios between straight line segments if they are anything. === Subject: Re: Cardinality question >> > I'd be very interested to know the branch of >> modern math which can define pi with no reference to points, lines, >> geometry, or ratios defined thereon and without the expression Let >> there be . . . that doesn't really depend directly or indirectly on >> those concepts. Calculus. Consider the series 1 - 1/3 + 1/5 - 1/7 + 1/9 + ... It is convergent and its sum is a therefore a real number S. Define >the number pi as 4*S. >> >> So just point out the limit of the series on a straight line and we >> can all go home. >Excuse me, you asked for a definition that didn't rely on points, >lines, geometry or ratios. Why are you putting lines back in? >> Probably because your approximation of pi relies on ratios and is only >> an approximation which was the whole problem to begin with since your >> definition is only for a series and not the limit of the series which >> is what pi is. Why? === Subject: Re: Cardinality question >> I bothered before reading your reply. The [not not] stuff is either a >> definition with no examples of application, or applications with no >> definitions. I haven't figured out which. >It is a difference of differences and a finite regression to a universal >tautology. That shold be obvious to you. You say you've been with these threads from the beginning? Hard to tell from your citations. Or maybe you're just a slow learner. === Subject: Re: Cardinality question >> I have no idea which, if any of these, correspond in any way to your >> ideas. I take it I need to read the entire Epistomology 201 thread to >> hope to understand these concepts? >Don't bother. You will come up dry if you do. I have been with this from >the beginning. Behind the facade of utter emptiness lies utter empiness. >Like the Earth before creation in Genisis. Without form and void. Which is exactly why I'm trying to give the void form. === Subject: Re: Cardinality question contradiction of space. Even though he can't explain clearly *how* >they are the fundamental contradiction in space. >>Or even what a fundemental contradiction in space is. >>Bob Kolker Dang, you beat me to it. >>Bob beats you to a lot of things like the integration of points to >>produce lines. He's uncommonly quick if not uncommonly right. >Here's an idea: why not prove us all wrong by explaining what a >fundamental contradiction in space is. It would be more useful than >the insult. >> Bob has made it long since obvious his only stock in trade is insults. >> It's less apparent what others do or don't understand. At this stage >> you've already commented you don't understand the root post to my >> preceeding thread so I'm left without any way to respond to what a >> fundamental contradiction is space is. If you understood what a >> fundamental contradiction is, we could get on from there. But I have >> no idea what you understand a contradiction to be nor a fundamental >> contradiction nor space. So what do I have to work with? What kind of >> answer would satisfy your question? >I understand a contradiction to be the result of mutually incompatible >statements or situations. A fundamental contradiction would be one >which lies at the foundation of the system it arose in. Space has a >variety of meanings depending on context, including the stuff you reach >when get beyond Earth's atmosphere and a mathematical construct with a >variety of properties. Once again, if P not P is a self contradiction, then P is the self and not the contradiction. The problem is P not P does not entail a fundamental contradiction because the tautological regression for P is not self contradictory. That is [not P] denies P but does not deny itself. On the other hand tautological regression for the empirical observation [not] is [not not] and forms a self contradiction which makes it a fundamental contradiction. (I think you really ought to reread the root post to Epistemology 201: The Science of Science a little more closely.) >I have no idea which, if any of these, correspond in any way to your >ideas. I take it I need to read the entire Epistomology 201 thread to >hope to understand these concepts? I hope not. I never specified which contradictions I had in mind because the regression is to self contradictory alternatives for all kinds of differences. People keep wanting me to explain whether I'm talking about geometric differences, neuron differences, phlogiston differences, caloric differences, or just what. I don't care what differences. The structural, geometric, and arithmetic differences that regress to self contradictory alternatives lie in the nature of differences as differences or contradiction as contradiction. For example, I once posted a thread on the origin of arithmetic symbols in western Europe which clearly illustrates the precedence of differences in the history of symbol development, just as an empirical test of the idea. === Subject: Re: Cardinality question >>contradiction of space. Even though he can't explain clearly *how* >>they are the fundamental contradiction in space. Or even what a fundemental contradiction in space is. Bob Kolker >>Dang, you beat me to it. >Bob beats you to a lot of things like the integration of points to >produce lines. He's uncommonly quick if not uncommonly right. >>Here's an idea: why not prove us all wrong by explaining what a >>fundamental contradiction in space is. It would be more useful than >>the insult. >Bob has made it long since obvious his only stock in trade is insults. >It's less apparent what others do or don't understand. At this stage >you've already commented you don't understand the root post to my >preceeding thread so I'm left without any way to respond to what a >fundamental contradiction is space is. If you understood what a >fundamental contradiction is, we could get on from there. But I have >no idea what you understand a contradiction to be nor a fundamental >contradiction nor space. So what do I have to work with? What kind of >answer would satisfy your question? >>I understand a contradiction to be the result of mutually incompatible >>statements or situations. A fundamental contradiction would be one >>which lies at the foundation of the system it arose in. Space has a >>variety of meanings depending on context, including the stuff you reach >>when get beyond Earth's atmosphere and a mathematical construct with a >>variety of properties. > Once again, if P not P is a self contradiction, then P is the self > and not the contradiction. In standard logic, P and not P is a self-contradiction. As things stand, P not P appears to be nonsense. Please explain the rules for forming sentences and deriving some kind of meaning from them. > The problem is P not P does not entail > a fundamental contradiction because the tautological regression for P > is not self contradictory. That is [not P] denies P but does not deny > itself. On the other hand tautological regression for the empirical > observation [not] is [not not] and forms a self contradiction which > makes it a fundamental contradiction. (I think you really ought to > reread the root post to Epistemology 201: The Science of Science a > little more closely.) You didn't explain what your words and symbols meant, from what I could see. What does [not] mean? What does [not not] mean? >>I have no idea which, if any of these, correspond in any way to your >>ideas. I take it I need to read the entire Epistomology 201 thread to >>hope to understand these concepts? > I hope not. I never specified which contradictions I had in mind > because the regression is to self contradictory alternatives for all > kinds of differences. People keep wanting me to explain whether I'm > talking about geometric differences, neuron differences, phlogiston > differences, caloric differences, or just what. I don't care what > differences. The structural, geometric, and arithmetic differences > that regress to self contradictory alternatives lie in the nature of > differences as differences or contradiction as contradiction. For > example, I once posted a thread on the origin of arithmetic symbols > in western Europe which clearly illustrates the precedence of > differences in the history of symbol development, just as an empirical > test of the idea. Why not just say these are different and move on? Also, some people are interested in quantifying those differences. Do you have any method to offer for doing so? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Cardinality question %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > What you consider universally true isn't the issue. What I consider > universally true isn't the issue. What you or I can prove universally > uninteresting. It shows the only was to prove something universal. > As for which branches of mathematics regress to universal truth, I > imagine Euclidean geometry and elementary arithmetic do. And I > imagine a lot of other branches do as well. The problem is we won't > know which do and which don't until we have some demonstrable way > to decide what universal truth is and what its demonstration implies. Truth is nothing more than a property of language. It has nothing to do with the universe outside the scope of language. If I say X is true, then I've defined a universal truth within the scope of my communication. Universal truth happens only by definition in language. The only universal truths are the ones we arbitrary define to be universal truths in our languages. 1+1=2 is a universal truth in the language of basic mathematics. But it's only a universal truth in that language because it was defined to be such by the definition of the language. When we use language to describe the universe, truths become dependent on time and that causes truths to be only temporary. If we say gravity sucks, that's a fact we expect to be true for eternity, but we have no way of knowing if it will or not. Gravity could start working differently 10 minutes from now. All we know is that we live our lives based on the assumption that gravity won't change. And so far, acting as if that assumption was a universal truth hasn't ever caused us a problem. Our brains are not built with universal truth technology. They work with probabilities. Everything is a fuzzy truth to them. Though they have the power to learn things which are mostly always true, they still represent this knowledge in a fuzzy logic system where nothing can be a true universal truth for all time. If we define an axiom in a formal logic system, we have created a universal truth. There really isn't anything mysterious about this. That's all there is to Universal truths. They are simply things we choose to define in our language. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cardinality question >> something better than it's worked so far? >> Also, it's not clear what problems you see with set theory in mathematics. >He doesn't like set theory and he can't stand the mathematikers. That >is because he cannot comprehend mathematics, so it must be the fault of >the mathematikers. I have refined my contempt for mathematikers in general, of which there are plenty, to point whores in set theory, of which there are even more. === Subject: Re: Cardinality question generally sufficient. >>Does that apply to Lester Zick also? He's specified the rules of a game. I'm still trying to figure out what he's talking about. His game seems >to be one of convincing mathematicians to abandon terminology that works >well for his terminology that I see no clear way to use at all, even >when I can figure his point. >>Yes, well I'm having a similar problem with the last sentence. >We have a way to communicate that works. For some reason you want us to >abandon it. In exchange, you offer something that does not seem to be a >functional replacement. Often, what you offer is something I can't even >understand enough to see that it doesn't work as a replacement. Of >course, if I can't understand it, it still doesn't work as a replacement. >> Let me see if I can explain what I see as the general problem in plain >> language so you can tell where I'm trying to get to.We have set theory >> in mathematics; we have SR and GR in cosmological physics; we have >> hermit functions in quantum physics. And we have no way to judge the >> universal truth in any proposed approaches to these kinds of problems. >> In fact we have no way to judge the universal truth of universal truth >> or the possibility of universal truth in any mechanically consistent >> terms with the problems studied by science and mathematics. That's >> what I'm after. So if the idea seems a little strange I can't really >> say I'm surprized. It's true we have a way to communicate that works >> and works better than preceeding ways to communicate such as Ptolemaic >> and Aristotelian cosmology. That doesn't mean the way we communicate >> is not dysfunctional on a very basic level that goes unrecognized >> simply because we're all well used to current ways of communication. >It sounds like you take issue with the Scientific Method. You want >something better than it's worked so far? Sure. Once again it's a shame you missed the Epistemology 101 and Epistemology 102 threads where I laid out the reasoning involved. >Also, it's not clear what problems you see with set theory in mathematics. Sets of points. I don't have problems I know of with set theory otherwise. Of course it's hard to tell for sure. A lot of these objections were covered in detail in the prior thread. === Subject: Re: Cardinality question >Let me see if I can explain what I see as the general problem in plain >language so you can tell where I'm trying to get to.We have set theory >in mathematics; we have SR and GR in cosmological physics; we have >hermit functions in quantum physics. And we have no way to judge the >universal truth in any proposed approaches to these kinds of problems. >In fact we have no way to judge the universal truth of universal truth >or the possibility of universal truth in any mechanically consistent >terms with the problems studied by science and mathematics. That's >what I'm after. So if the idea seems a little strange I can't really >say I'm surprized. It's true we have a way to communicate that works >and works better than preceeding ways to communicate such as Ptolemaic >and Aristotelian cosmology. That doesn't mean the way we communicate >is not dysfunctional on a very basic level that goes unrecognized >simply because we're all well used to current ways of communication. >>It sounds like you take issue with the Scientific Method. You want >>something better than it's worked so far? > Sure. Once again it's a shame you missed the Epistemology 101 and > Epistemology 102 threads where I laid out the reasoning involved. I browsed through the root threads. It was not the clearest exposition I've seen, but enough for me to figure out your main beef seems to be with the Scientific Method as a foundation for doing science. Alas, I don't see that you've offered anything that is in a position to replace it. >>Also, it's not clear what problems you see with set theory in mathematics. > Sets of points. I don't have problems I know of with set theory > otherwise. Of course it's hard to tell for sure. A lot of these > objections were covered in detail in the prior thread. Points are not necessarily part of set theory. If you have a problem with sets of points, it suggests to me you don't understand what set theory is. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Cardinality question >Universal truth is the set of all points which are universally true? >> You're welcome, Albert. ROTFLMAO??? >>Rolling on the floor, laughing my ass off. Behavior 12/15/3 which was quite instructive but didn't go anywhere. === Subject: Re: Cardinality question >Universal truth is the set of all points which are universally true? >> You're welcome, Albert. ROTFLMAO??? >>Rolling on the floor, laughing my ass off. Ah. Much appreciated. === Subject: Re: Cardinality question > I prefer to use logic universally regressible to self contradictory > alternatives. I realize that's a bit of a stretch for point whores. >> Perhaps if you would show examples of it in action to prove something >> I could figure out if it was something I could stomach. As it stands >> now, I saw a post about it that made absolutely no sense. It looked >> like there could be sense to it, but I didn't know what it was. > Well how about if I say that everything is differences because there > can be nothing different from differences? >> But that doesn't say anything. That gives me no clue what differences >> are. How is the above clearer than saying Everything is slursh because >> there can be nothing different from slursh? Yours sounds cooler, but I >> can't detect any meaningful content in it. >[...] >Better: >Everything is slursh because there is nothing that is slurshable from >slursh. Yes. For general reference Wolf is one of several who indicated my finite tautological regressions to self contradictory alternatives were childish nonsense and that anything could be proved that way. Many months later none of the above has vouchsafed proof of anything except their emotions, beliefs, and a childlike faith in established truth and conventional wisdom. >But I prefer Doh nastjaze doh kalma, er nastjinote doh kalamari. >That's the way the Antareans say it, anyhow, and they are fully aware of >the subtle difference between differences and differences that aren't >different from differences. Not even Lester has progressed to that >point. Yet. So there! === Subject: Re: A New (And Slightly Stupid) Set Theory > This is going to look a little silly, but Herc has given me an idea. > Probably a very bad idea, to be sure, but an idea nonetheless. > Let the empty set be a level -1 set. Any finite set is a level 0 set. > The whole numbers is a level 1 set, as it can be construected from > all possible level 1 sets. > The real numbers is a level 2 set. Meaning what by constructed? If N counts as level 1 because it can be constructed from finite sets, why doesn't R count as level 1 for the same reason? [To say The whole numbers is a level 1 set, as it can be constructed from all possible level 1 sets is circular.] === Subject: Re: SF: National security <7v92719kjtqc0seai39rpo50b87crtaij7@4ax.comSo what did they say? >James Harris > They said give up sci.crypt or I Juuso will mess up > your SFT threads beyond recognition. Good one! 8-) What are you planning on doing, including real proofs in his posts, or providing an explicit algorithm? --- Christopher Heckman === Subject: Re: SF: National security > ... > And I can guarantee some of you who may even now be waiting on RSA, That's the RSA that publishes challenge numbers, one of which has been successfully factored? When that happened, did national security (whatever that may be) suffer? What makes you think that public key cryptography has got anything to do with national security? If it does have, may I point out that there are public key cryptosystems that don't rely on the (supposed) difficulty of factoring large numbers. Do terrorists communicate with one another using public key cryptography? The IRA [*] used couriers who carried written messages wrapped in thin plastic film in their cheeks. If they were stopped by the security forces they just swallowed the evidence that was soon destroyed by stomach acid. [* That's the IRA that was financed by Americans--thank you so much.] === Subject: Re: SF: National security Randy Howard spaketh thus - > Let's see -- world's largest employer of professional mathematicians > AND employers of professional signals interception staff. Could it > somehow be possible that someone from the NSA, in their professional or > private lives, lurks here and has already seen the surrogate factoring > theorem? I would guess not, actually ... but, given JSH's record of writing to the NSA, to Congressmen, to famous mathematicians, to the Oklahoma Attorney General, to (probably) the FBI, to (maybe) the CIA, to (probably)the President, what do you think the chances are that he has made it into the Secret Service's files of suspicious kooks ? Someone they might want to keep an eye on if the Pres. happens to be visiting Atlanta ? Nora B. === Subject: Re: SF: National security > Someone they might want to keep an eye on if the Pres. happens to > be visiting Atlanta ? did JSH attend Georgia Tech? that would explain a few things. :-) m. === Subject: Re: SF: National security rleone@hotmail.com says... > Let's see -- world's largest employer of professional mathematicians > AND employers of professional signals interception staff. Could it > somehow be possible that someone from the NSA, in their professional or > private lives, lurks here and has already seen the surrogate factoring > theorem? I haven't read anything about them having problems with employees dying laughing, but they're probably good at covering that sort of thing up. -- Randy Howard (2reply remove FOOBAR) Making it hard to do stupid things often makes it hard to do smart ones too. -- Andrew Koenig === Subject: Re: SF: National security >impossible to follow. The only thing you'll accomplish by adding mindlessly >repetitive replies is increase his threads' message counts -- which is My standard replies will be done in a minute. He is the one taking a risk; perhaps someone else will also start to repeat with standard replies, then the SFT thread mess becomes quickly complete. Standard replies are terrible boring. D:>JSH 'JSH' is not recognized as an internal or external command, operable program or batch file. D:>SFT 'SFT' is not recognized as an internal or external command, operable program or batch file. Ok, this is an experimental trick I wonder for how long posters will want to try to have an intellect conversation with two weirdoes - one who repeats SFT like a parrot - and an another who gives standard error messages like a parrot. >seems to get off on just that there _are_ replies. Well, then I guess he will soon be in ecstasy. Or at least he will have to learn to become happy with a larger amount of boring syntax error messages: D:>JSH 'JSH' is not recognized as an internal or external command, operable program or batch file. D:>SFT 'SFT' is not recognized as an internal or external command, operable program or batch file. The trick is that constant off topic announcements reduce anyone's interest to seriously talk about SFT in favor for replying to off topic thing. But there are other benefits too. likely he will soon have to choose between SFT development and sci.crypt. I am sorry for the annoyance my posts are / will be causing. Btw. can that you said be interpreted that if no one would reply to his threads , then he would be non-happy - and people here reply to his threads just for crating him some weird satisfaction. That's weird - perhaps. Juuso campaign in the history of advertisement. === Subject: JSH: Brainstorming over, for now Ok, I've finished brainstorming on the SFT and how to best present it. It's been a VERY useful few days as most importantly I've managed to air out my paranoia about the dangers of this research, and communicate loud and clear I hope, so that the people who are supposed to pay attention to problem areas assuredly noticed! Which makes me feel a little silly for being worried--as nothing has happened--but then again, how do you know if you don't check? My fears about working on the factoring problem go back for YEARS and have affected a lot of things for me, so it's a tremendous relief to be here now with what I feel is a major result, and it looks like everything is fine. No worries. Things get a lot more boring from here for the rest of you, as there's less need for me to talk anything out, as I think I've learned what I needed to know, and a lot less interest on my part in this subject area anyway, as I'm getting that been there done that feeling. Extreme mathematics is about the extreme--pushing limits and the envelope. Maybe I burned myself out ahead of time on factoring, worrying about it so much, but now it just seems like so much old hat. Of course, papers to be sent off, but I have a backlog now. I've been sitting on papers versus working them out to be sent off, as it's all just kind of tedious and annoying--the social crap. In any event, the world is still here. The economy didn't crash, and I'm feeling stupid but giddy. Sometimes fears are just uncalled for, and unnecessary, but you have them anyway. James Harris === Subject: Re: JSH: Brainstorming over, for now > In any event, the world is still here. The economy didn't crash, > and I'm feeling stupid but giddy. This is a good sign. You're more in touch with reality than you usually seem. === Subject: Re: Why do we have the natural numbers? >What's so good about them? What are they for? Historically, their use was for counting and accounting -- sheep, people, bushels of grain, taxes paid, etc. etc. More recently, they have been shown capable of formally capturing our intuitive notions of mathematical proveability. So if something can't be done via the natural numbers than it can't be done at all. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Why do we have the natural numbers? > What's so good about them? What are they for? Do you mean for in mathematics, or for in extra-mathematical applications? === Subject: Re: Epistemology 202: Advanced Topics > But when _I_ talked about > space, I meant empty space. And it seems odd to me to claim that a region > of space that has a line drawn on it is still (empty) space. Oho, now we have this new empty qualifier jumping in ... Would you mind defining empty space now? -- Giuseppe Oblomov Bilotta I'm never quite so stupid as when I'm being smart --Linus van Pelt === Subject: Re: Epistemology 202: Advanced Topics algebra, calculus, real analysis, and modern geometry. Are you familiar >>with these topics? Zick is a total mathematical incompetent. He prefers to be so. >>Of course he prefers to be so. He just prefers not to remain so >>incompetent as Bob prefers to remain. Go figure. >But do you know what it is you are attempting to replace? Are you sure >you will have an impact on the majority of mathematics, even if your >ideas are accepted? >> Mathematics is not universally demonstrated as it stands. I don't >> expect a lot of mathematics to change once universally demonstrated. >> But I do expect certain conceptual problems to be resolved such that >> some parts will be reinterpreted. Among them the treatment of angular >> momentum and transcendentals. >Angular momentum is a physics concept. The application of angular momentum may be a physics concept but its derivation is strictly mathematical apart from mass and time constants. >I'm looking for a clear definition of what you mean by >transcendentals, but I still don't know what you mean by a curve. The question is do you know what you mean by a curve. I think you'll agree that curves are straight lines and cannot be pointed out exactly on straight line segments. That's what I mean by transcendentals.