* Gerald Edgar According to the experts, there is a "small hole" in Wiles' proof I hear the same thing. But every report (inluding yours) is third- or fourth-hand. So some skepticism might be wise. * Gerald Edgar to Lou Talman (281193) Here is a report from the Internet newsgroup: From R.G.E.Pinch@pmms.cam.ac.uk Mon Nov 22 15:19:28 1993 Date: Mon, 22 Nov 93 20:16 GMT From: Richard Pinch Subject: Re: Fermat hole : Coates said in a lecture at the Newton Institue here last week that in his opinion there is a gap in the 'geometric Euler systems' part of the proof which "might take a week, or might take two years" to fill. I have spoken to him several times, but am still not sure on what basis he makes the claim: he does not have a copy of the MS. As far as I know the only copy in Cambridge is with Richard Taylor as one of the five referees of the paper for Inventiones, and he has consistently declined to comment until all the referees reach a common conclusion. So the situation is confused. Myself I don't see how Coates's view can be taken as authoritative at this stage: I plan to wait for word from Richard Taylor. Richard Pinch * Steven Conrad to Bob Ewell (291193) I am *not* more knowledgeable, but both my sons are doing doctoral work in number theory -- one is a student of Andrew Wiles. Both told me about what they call "Marilyn Idiot Savant" and had nothing by sarcasm for her remarks. They gave me a good analogy about her very poorly chosen example, but they hit it on the head when they said that, for her, an untrained person in matters of advanced number theory, to make commentary on an advanced academic field is far out in left field. They then said "What do you expect, I guess, from someone who makes her living writing an advice column for a Sunday supplement?" I don't criticize her her employment -- but she is out of her league and should apologize. She's shown herself a fool if she does not recant. Steve * Steven Conrad to Bruce Bowyer (291193) The claim that there is an error was made by a man from Cambridge who does not get along with Wiles and who may (in fact) never have seen the paper. The reviewers are neither numerous nor scarce -- but if anything appeared to be a difficulty, it would wot be made know, first, to someone with whom Wiles is not a close buddy. Steve * Steven Conrad to Gerald Edgar (291193) Real! Coates, of Cambridge, does not have a copy of the paper and is not on good terms with Wiles. His motivation is suspect, as is everything he has said. No one on the committee is likley to have shared a formal opinion with Cotes -- who was Wiles' dissertation advisor when Wiles got his PhD at Cambridge. Steve * Allan Trojan to Steven Conrad (291193) There certainly is something exceedingly strange about all this! The fact that someone would give a SEMINAR on the falsehood of a proof when he does not even possess the official manuscript and the referees have not finished their work!! I had not heard before that Coates was Wiles thesis advisor. Where did you learn this?? If this is the case, it also must be a first in mathematical history, a thesis advisor giving a seminar to condemn the work of a former student!! No doubt it all will be written up some day in the official biography. So far, there seems to be nothing in the way of news from authoritative voices, aside from the brief statement of Ribet that I read. Yours truly, Allan Trojan. * Kevin Delaney to all (291193) to all, I am happy to hear that there may be a hole in the Andrew Wile's proof. This hole gives me an excuse to play with one of my favorite math exercises for one last time. I would be interested in hearing your opinions on this proposed solution for Fermat's Last Theorem. Three minor theorems Theorem 1: if a and b,are mutually prime and are greater than 1; then a + b is divisible by neither a nor b. Theorem 2: if a and b are not divisible by p, and n < m then ap^n + bp^m will be divisible by p^n but not by p^(n+1). Theorem 3: Fermat's Little theorem states that if n is prime then n!/(n-k)!k! is divisible by n, where k > 0. This says that all terms in the binomial expansion, except the first and last, of (a+b)^n will be divisible by n. Main Proof: I would like to find the first counter-example to FLT. a^n + b^n = c^n is the FCE if n is the smallest integer for which there is a counter-example, and a is the smallest integer under n for which there is a counter-example. In the FCE a, b and c are mutually prime. If they shared a common factor, then (a/k, b/k, c/k) would be a solution. a/k < a; contrary to the definition of FCE. In the FCE n is prime or equal to 4. If n = jk then a^jk + b^jk = c^jk = (a^j)^k + (b^j)^k = (c^j)^k. Since k < n, this implies that there is a smaller counter-example to the problem. Fermat took care of the solution where to n = 4. I will skip this for now. I define d as the difference between a and c. That is a + d = c. Since a and c are mutually prime, a & d will be mutually prime. Since b is between a and c; d must be > 1. I can do the binomial expansion of (a + d)^n: (a + d)^n = c^n a^n + na^(n-1)d + ... + nad^(n-1) + d^n = = c^n na^(n-1)d + ... + nad^(n-1) + d^n = c^n - a^n = b^n d(na^(n-1) + ... + nad^(n-2) + d^(n-1)) = = b^n Interesting, d must be a divisor of b^n. I will define a new variable e = b/d or b = de. na^(n-1)d + ... + nad^(n-1) + d^n = b^n = (de)^n na^(n-1)d + ... + nad^(n-1) = = e^nd^n - d^n na^(n-1)d + ... + nad^(n-1) = (e^n - 1)d^n d(na^(n-1) + ... + nad^(n-2)) = = (e^n - 1)d^n This is now extremely interesting. We have a situation where xd = yd^n, where x and y are integers. The only way this can be true is if x is a multiple of d^(n-1). The sum of (na^(n-1) + ... + nad^(n-2)) must be divisible by d^(n-1). All the terms in the series except the first are divisible by d. For this series to add up to a factor of d^(n-1), na^(n-1) must be divisible by d. As stated a & d are mutually prime, a^(n-1) contributes nothing. The variable n is prime. If n <> d, then the proof is over by theorem 1. If n = d; Then we could rewrite the series as (da^(n-1) + ... + ad^(n-1)). From theorem 3 and the binomial expansion, all of the terms except the first will be divisible by d^2. Using theorem 2 the terms will add up to a figure which is divisible by d but not by d^2. We now have the situation where if n = 2, the problem is solvable. If n is greater than 2, then there is no solution. Basically we have x*d^2 = y*d^n, where x and y are not divisible by d, and n > 2. This cannot be true; therefore, there can be no first counter example. Q.E.D. It's not 210 pages, but works for me. I would not be surprised if this is along the same line of reasoning as used by Pierre himself used. * Bob Ewell to Steven Conrad Thanks for your clear and strong response, Steven. I have sent her a letter with the previous comments from this forum. That, coupled with the hundreds of other letters she's no doubt received, should communicate. Maybe she's testing the market to see how many of us read her column! Bob * Jean Bausch to Kevin Delaney (301193) Kevin, Brilliant! To complete your proof you only need a small lemma. Am I asking too much when I claim it should be called "Jean's Great Lemma" (JGL)? Here it is: > If a,b,c,n are defined by the FCE and if (c-a) divides b^n, > then (c-a) divides b. I hope that JGL's proof will take less than 200 pages and 300 years. Jean * Kevin Delaney to Jean Bausch (301193) Jean, Thank you for catching that glaring error. I really have to watch out about those 3:00 Am messages. The proof shows that (c - a) and (c - b), for that matter, must be either an integer to the nth power, or a n^k(n-1), where k is an integer. Of course, (c - a) and (c - b) cannot both be divisible by n; so that leaves only three possiblities: (c-a) is of the form n^k(n-1), and (c-b) is of the form i^n (c-b) is of the form n^k(n-1), and (c-a) is of the form i^n or both (c-a) and (c-b) are an integer to the nth power. kd * Josef EschgfŠller to Steven Conrad How can I retrieve Marilyn's remarks? Thank you. * Steven Conrad to me I think you'd need to go to the library and read her column from last Sunday. Steve * Marilyn's article Here is the Marilyn Article I retrieved from Internet news. Yours truly, Allan Trojan, Toronto 72072,1656 From: wgd@zurich.ai.mit.edu (William G. Dubuque) Newsgroups: sci.math Subject: Full Text of Savant FLT Parade Article Date: 29 Nov 93 05:50:15 Organization: M.I.T. Artificial Intelligence Lab. Lines: 170 Distribution: sci Message-ID: NNTP-Posting-Host: martigny.ai.mit.edu Following is the full text of Marilyn vos Savant's Parade article on Fermat's Last Theorem, as scanned by a HPIIcx scanner and OCRed with Xerox's Textbridge. Hopefully this should make the article available to the international audience. [Parade Magazine, November 21, 1993, page 10] Ask Marilyn by Marilyn vos Savant When a British professor solved a problem that had been puzzling mathematicians for centuries, newspapers reported the news on their front pages. But for some of us, the mystery had just begun. Who was this fellow Fermat, anyway, and why the headlines over some notes scribbled in a margin? Marilyn vos Savant sorts it all out in this response to a reader's question and in a book now out, titled "The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries" (St. Martin's Press). ---- Andrew Wiles, a Princeton mathematics pro- fessor, claims to have solved the most famous problem in mathematics. The theorem--actu- ally a conjecture until proved--was stated by the French mathematician Pierre de Fermat in the 17th century. It went this way: Can it be proved that in the equation x^n + y^n = z^n there is no solution if "n", represents any whole number larger than 2? There are plen- ty of solutions if "n" represents 2. For ex- ample, 3^2 + 4^2 = 5^2. But if "n" represents 3 or more, according to Fermat, there are no solutions. Have you ever tried to prove it, and if so, did you succeed?. -Joseph McGriff, Odessa, Tex. No, I've never tried, and I don't think I would have succeeded even if I had. Moreover, I don't think the current work succeeds in proving "Fermat's last theorem" either--even if no mathematical errors are discovered in it. Here's why: More than 350 years ago, Pierre de Fermat wrote down his apparently simple little "theorem" in the margins of a mathematical book he was reading, adding that he had discovered a remarkable proof for it but that there was no room to include the proof in the margins too. He died without ever presenting the step-by-step logic to substantiate this tantalizing claim, confounding the best of mathematicians ever since in their efforts to do so. Since the arrival of computers, it has been shown that the theorem clearly holds true, even for ex- tremely high numbers. That might seem proof enough for the general public, but for mathemati- cians it's no proof at all. Finally, many of them came to the reluctant conclusion that Fermat had made a mistake and didn't have a proof after all. But on June 23, 1993--at the end of a three-day lecture series at the Isaac Newton Institute for Math- ematical Sciences at Cambridge University in Eng- land--Dr. Andrew Wiles, a British mathematician who teaches at Princeton University, made a sur- prise announcement that he had proved Fermat's last theorem (also known as F.L.T.). Almost at once, telephones began to ring, faxes churned out copy, electronic mail zapped into computers all over the world, and the communications satellites went into overdrive. Very few people knew what Wiles had been do- ing for seven years in his little third-floor attic of- fice at home, where he worked away in secret at the problem--and he wanted it that way, for good rea- son. After all, what would people think? Worse, what would they think if he had worked on it for a lifetime and failed? The assessments would proba- bly not be charitable, especially for a man with a wife and children and a house with an average as- sortment of squeaky screen doors, leaf-filled gut- ters and dandelions in the backyard. It would surprise no one if, after seven years of this, every window in the house eventually had become stuck shut. But there's more to the story than what appeared in the news. This is where I think it all goes astray. The mathematics of today is a far cry from the mathematics of Fermat's time. In brief, here's what has happened in the field and how this relates to the current work on Fermat's last theorem. The Euclid- ean geometry of Fermat's day is a set of principles derived by rigorous logical steps from the axioms detailed by Euclid, the Greek mathematician of the third century B.C. The fifth axiom is known as "Eu- clid's parallel postulate," and it can be rephrased this way: If a point lies outside a straight line, one (and only one) straight line can be drawn through that point that will be parallel to the first line. Some mathematicians in the 19th century began to disagree with the "parallel postulate," and a few of them even invented their own geometries, called non-Euclidean geometries, of which there are two important forms--both of which replace the fifth postulate with alternatives. One of the two main alternatives allows an infinite number of parallels through any outside point, from which was devel- oped "hyperbolic" geometry. The other main alter- native allows no parallels through any outside point, from which was developed "elliptic" geometry. Superficially, this seems ridiculous to the non- mathematician, but the new systems of geometry have their own definitions and systems of logic. The best-known example of a non-Euclidean idea is Einstein's general theory of relativity, which has little valldity outside elliptic geometry. Most people are unaware of this. If elliptic geometry is in error, so is Einstein's world. Wiles' proof is also non-Euclidean. The chain of proof is based in hyperbolic geometry, which one of its founders himself named "imaginary geome- try." Here's the crux of the matter. Three of the oldest problems in mathematics- all more than 2000 years old-are known as "Dou- bling the Cube," "Trisecting the Angle" and "Squar- ing the Circle." (All constructions were to be accomplished using only a ruler--as a straight edge, not as a measuring device--and a compass.) The problem of doubling the cube is to construct a cube with twice the volume of a given cube; the prob- lem of trisecting an arbitrary angle is to construct a method to divide any given angle into three equal parts (it must work for every angle); and the prob- lem of squaring the circle is to construct a square with an area equal to that of a given circle. It wasn't until the 19th century that the problems were all proved impossible to solve, and they are now considered "famous impossibilities." Scientif- ic American noted that "Fermat's last theorem dif- fers from circle-squaring and angle-trisecting in that those tasks are known to be impossible, and so any purported solutions can be rejected out of hand." Bearing all this in mind, what would we think if it were discovered that Janos Bolyai, one of the three founders of hyperbolic geometry, managed to "square the circle"--but only by using his own hy- perbolic geometry? Well, that's exactly what hap- pened. And Bolyai himself said that his hyperbolic proof would not work in Euclidean geometry. So one of the founders of hyperbolic geometry (the geometry used in the current proof of Fermat's last theorem) managed to square the circle?! Then why is it known as such a famous impossibility? Has the circle been squared, or has it not? Has Fermat's last theorem been proved, or has it not? I would say it has not; if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat's last theorem. This is not a matter of merely changing the rules (for ex- ample, using a ruler as a measuring device instead of a straight edge). It's much more significant than that. Instead, it's a matter of changing whole defi- nitions. And, regardless, it is logically inconsistent to reject a hyperbolic method of squaring the circle and accept a hyperbolic method of proving F.L.T.! ---- If you have a question for Marilyn vos Savant, who is listed in 'The Guinness Book of World Records" Hall of Fame for "Highest IQ," send it to: Ask Marilyn, PA- RADE, 750 Third Ave., New York, N.Y. 10017. Because of volume of mail, personal replies are not possible. $ directory * To Allan Trojan Allan, Thank you very much for mailing me Marilyn Vos Savant's article on Fermat's last theorem. It seems to me that this is a good example of how difficult it is to explain to non-mathematicians that in mathematics one cannot mix inside (mathematical) concepts and truth with outside ("world") concepts and truth, i.e., that mathematical statements are in some sense always contentless, so the only things what count in mathematics are the logical relations between the statements, not the concrete content of statements or names (e.g. "hyperbolic"). Usually in normal conversation and in many other sciences one communicates by showing up some group of facts, and the relationships between these facts are loose or automatic. I try often in my lectures to explain it by drawing a graph with circles and arrows, the circles being the contents and the arrows the logical relations, and saying that in normal communication the circles are the important thing, but in mathematics only the arrows are important. If I explain this to mathematics students, they usully apperceive that difference, if I explain it to biologists, they often do not grasp it. On the other side, I don't feel this should be a reason of triumph for mathematicians, because seldom we do enough for bridging this gap to other scientists and to other people, although it would be fruitful to pursue to some extent such ways of thought. And what Marilyn did, using technical terms improperly to deduce an assessment of something not well understood, is this not even common among mathematicians, which may be are prudent and experienced enough not to commit technical errors, but rather often use similar superficial and formal assertions in the assessment of other people's work or as an excuse for not bothering with things outside of sometimes narrow interests? J.E. * Sumedha Sengupta to me Being a statistician ( who had to learn math) it is very irritating sometimes to see, how people can make something very big out of nothing while ignoring the most important fact that might lead to something important. There is no such thing like psuedo-mathematics. * To Sumedha Sengupta What do you mean? I think, one should discuss sometimes the importance of things. The scientist should not let go the world events as they may happen. But if he tries to have an interactive relationship with reality (human and non-human), he finds himself struggling against multiple obstacles and uncertainties. * Sumedha Sengupta to me Could you please explain. I think everyone to some extent communicates with Human and Non-Human. There are different methods of communications. That is why I am so much interested in different languages, and also belong to that forum. What I was trying to say, is that most of the people do not go by facts. Mathematics is an exact science. But there is also the concept of uncertainities. * Mike Christie to all (Marilyn's book) Information for anyone who's interested: Marilyn vos Savant now has a book out called "The World's Most Famous Math Problem" in which she repeats her assertions. It's published by St. Martin's and costs $7.95 in the US. The book is not too bad (apart from her mistaken critique), but it contains a couple of mis-statements; for example she says that Fermat claimed he had a proof that all Fermat numbers were prime (which was later shown by Euler to be false). In fact Fermat knew perfectly well he had no proof. * Mike Christie to all Here's the text of an e-mail that Allan Trojan just forwarded to me: Jay H. Beder writes: > Someone just showed me her book. The article contains excerpts of > it. Martin Gardner read the ms of the book and is quoted on the > back cover. vos Savant thanks him and also Barry Mazur, Kenneth > Ribet, and Karl Rubin, with whom she was in touch by fax, although > evidently they did not have anything to do with the book. She > also has read extensively, although she suffers from the common > problems of the autodidact. Just in case there's any doubt about this, I want to report that I had nothing to do with the making of her book. I haven't seen it yet, and can't remember ever speaking to her. It is remotely possible that someone from her office phoned to request a copy of my Notices article about Wiles; perhaps a copy was faxed to her from the Math Department here. But even that sounds unlikely to me. -ken ribet (That's the end of the forwarded e-mail.) Martin Gardner is cited in the acknowledgments in vos Savant's books as follows: "I want to thank the incomparable Martin Gardner for reading my manuscript, for being my dear long-distance friend, and for brightening the world for all thinking people." Gardner has not contributed an introduction, but he did write a blurb for the back cover as follows: "How Marilyn vos Savant managed to write such a delightful, informative and accurate book about the probable proof of Fermat's Last Theorem beats me! [This book is] highly recommended even to readers who think they hate math." A little later, she says "a personal thank you to Barry Mazur, Kenneth Ribet, and Karl Rubin for being such good sports and for putting up with my faxes . . Mazur generously provided me with a text that evolved from his talk at the Symposium on Number Theory held in Washington, D.C. . . ." An appendix includes a copy of a two page sketch of the proof posted by Karl Rubin to a maths bulletin board. If anyone is in contact with Rubin, Mazur or Gardner it would be interesting to know if their reaction is the same as Ribet's. * Steven Conrad to all I have just uploaded to the mathematics library (library 5) the complete text of the message Wiles placed on the INTERNET as a reply to the rumors that have been circulating about the state of his proof. You may be interested in reading his statement. Steve * Internet message by Andrew Wiles Date: 04-Dec-93 01:13 EST From: "Brian Conrad" >INTERNET:conrad@math.Princeton.EDU Subj: wiles posts to sci.math! Sender: conrad@math.Princeton.EDU Received: from Princeton.EDU by arl-img-1.compuserve.com (8.6.4/5.930129sam) id BAA08214; Sat, 4 Dec 1993 01:12:27 -0500 Received: from math.Princeton.EDU by Princeton.EDU (5.65b/2.103/princeton) id AA11740; Sat, 4 Dec 93 01:12:21 -0500 Received: by math.Princeton.EDU (4.1/1.113) id AA07145; Sat, 4 Dec 93 01:12:19 EST From: "Brian Conrad" Message-Id: <9312040612.AA07145@math.Princeton.EDU> Subject: wiles posts to sci.math! To: 76116.2272@compuserve.com (Steven R. Conrad) X-Mailer: ELM [version 2.4 PL22] Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 1145 Wiles posted the following to sci.math: > In view of the speculation on the status of my work on the > Taniyama-Shimura conjecture and Fermat's Last Theorem I will give a > brief account of the situation. During the review process a number > of problems emerged, most of which have been resolved, but one in > particular I have not yet settled. The key reduction of (most cases > of ) the Taniyama-Shimura conjecture to the calculation of the > Selmer group is correct. However the final calculation of a precise > upper bound for the Selmer group in the semistable case (of the > symmetric square representation associated to a modular form) is not > yet complete as it stands. I believe that I will be able to finish > this in the near future using the ideas explained in my Cambridge > lectures. The fact that a lot of work remains to be done on the > manuscript makes it still unsuitable for release as a preprint . In > my course in Princeton beginning in February I will give a full > account of this work. Andrew Wiles. Brian * Bob Ewell to Mike Christie (Marilyn made another mistake) (Poor Marilyn. Many dogs ... ) Fascinating info, Mike. Will Jay provide us a definition of autodidact? Sounds impressive, but I'm not familiar with it. Does it mean "self-proclaimed teacher or expert"? Anyway, speaking of Marilyn, did she blow it again yesterday? Here's the problem and her solution. A boy has to study for a test, but his favorite movie is playing at the theater. He decides to go to the movies and leaves home. Halfway there, he feels guilty about not studying so he decides to go back home. Halfway home, he tells himself that he knows his materials for the test, so why bother studying? Halfway back to the movies, he stops again and heads back home, stopping halfway again. Going on in this manner, where will he end up? Marilyn's resposne: After an infinite amount of time, he will end up quivering like a tuning fork 1/3 of the way from his house to the movies. The sum of his forward motion to the movies totals 2/3...but the sum of his backward motion toward home totals 1/3 of the way from the theater. 2/3 - 1/3 = 1/3. My comment: it seems to me that the first two legs he will go out to 1/2 and back to 1/4. Then he will cover the half the remaining distance to the theater getting as far as 5/8 before returning half way ending up at 5/16. This progression will continue until coming from the movies, halfway is 1/3 from home. From this point he will cover half the distance to the theater, turning around at 2/3 and returning half the distance to 1/3. Thus 1/3 is the limit of the progression, but he doesn't there; rather he oscillates between 1/3 and 2/3. This can also be set up and solved algebraically where x, the ultimate starting distance from home is set equal to ((1+x)/2)/2 half the distance from which he turns around while going to the theater. Is Marilyn losing it or am I confused about something? Bob * To Sumedha Sengupta Probably we mean the same thing, that in order to apply mathematics to reality one has to know (and to appreciate) at the same time the mathematics and the reality. * To Bob Ewell (on Marilyn's walk problem) I agree with your idea, but I think that the distance 1/3 shall never occur precisely. However, if pos[n] is the position at the n-th iteration (between 0 and 1) with pos[0]=0, then, if a[n]=pos[n+1]-pos[n], we have a[n+1]+a[n]=1/(2^(n+1)). Solving this difference equation one obtains a[n]=(1/3)((-1)^(n+1)+1/(2^n)), hence pos[n], which is the sum a[0]+...+a[n], becomes 1/2 - (1/3)(1/2)^n - ((-1)^n)/6, and this does not converge, but, since the middle term goes to 0, oscillates (at the limit) between 1/2+1/6 = 2/3 and 1/2-1/6 = 1/3, as you predicted. * Sumedha Sengupta to me yes , I can not imagine doing imaginary math in the real world and vice versa. * Bob Ewell to me Confirms my suspicion. What I can't figure out is what Marilyn was thinking about. Bob * Bob Ewell to me I've figured out the problem Marilyn worked. Instead of having the boy go halfway to each destination (as the problem is written), she had him returning halfway to where he turned around the last time. In that case, he would have gone a total distance of 2/3 in one direction and 1/3 in the other. However, this is definitely NOT what the problem stated. Bob * Bruce Frech to Bob Ewell She was probably thinking he went half way to the last place he stoped at. then her figures are correct. I think your interpretation of the phrase 'goes half way home' is the correct one. So: she blew it. * Mike Christie to Bob Ewell (how Marilyn's book has been received) Here's another net posting forwarded to me by Allan Trojan. I've added a few comments of my own at the end. From: amandel@husc9.harvard.edu (Aaron Mandel) Date: 5 Dec 1993 10:34:39 GMT Organization: Harvard University, Cambridge, Massachusetts For he who asked: The consensus on sci.math about vos Savant's book has been negative. Assessments range from 'pointless' to 'sensationalist garbage'. Major problems mentioned [tell me if I miss any]: 1. The squaring-the-circle business. Vos Savant used a tenuous chain of connections to equate FLT with Euclidean geometry and then implied that Wiles was therefore 'cheating' by using hyperbolic geometry. (I believe her actual phrase was "changing the rules".) It went something like this: Fermat's equation, for n=2, is the Pythagorean Theorem, which can be interpreted in terms of right triangles in the Euclidean plane. She appears to not understand that squares in Euclidean and Lobachevskian geometries BEHAVE differently, while the integral exponents in FLT do not experience the same shifts. 2. She makes an oblique reference to Wiles's reclusive tendencies during the years preceding the lecture: "... especially a man with a wife and children and a house with an average assortment of squeaky screen doors, leaf-filled gutters and dandelions in the backyard" (p.10). Here and elsewhere, there is an implication that Wiles valued proving FLT above family or academic reputation or even validity. 3. Most damning, in my view, for its complete ignorance, is the mistake made near book's end. Vos Savant points out (correctly?) that Wiles used what is known as mathematical induction in his proof. The one sentence cited from Ribet makes it clear to any sensible reader that that type of induction (the 'knocking-over-dominoes' strategy, if you like) is what is referred to. Marilyn then blithely tells her (presumably otherwise uninformed) audience about the distinction between inductive and deductive logic, naming conjectures which were true up to very large test cases but ultimately false. She apparently does not know that mathematical "induction" is a perfectly valid _deductive_ technique, nor, we infer, does she care. 4. In places far, far, far too numerous to point out, she makes statements and observations which to me seem equivalent to "If you can't draw me a pretty picture than a layman understands, then you don't have a valid proof." Emphasis here is on the layman... vS repeats ad nauseum the fact that most mathematical fields contain only a few experts, and are incomprehensible or partially so to outsiders even within the mathematical community. She rarely states anything at all outright (perhaps planning for the future necessity of defense against critics), preferring to build an insufficient or false line of reasoning and let the imagination of the reader do the rest; _however_, the implication here is that if most people don't even have the chance to disprove something, we can't accept it. I concluded therefrom that, for instance, calculus was not guaranteed to produce true statements until Newton or Leibniz demonstrated it to enough people. Though it is a noble (and quite, quite worthwhile) cause to make math interesting to that segment of the public which is usually uninterested in it, vos Savant does so, it seems, with an agenda. The book includes the usual complement of cute math conundrums (Zeno's paradoxes, the Goldbach conjecture, spurious proofs dating back to Lewis Carroll) as if to say that anyone who can understand these things is a full-fledged amateur mathematician capable of turning out insights, and it is nothing short of elitist on the part of Wiles to write about things that cannot be explained from scratch in an 80-page monograph. Hopefully, the thrill of being 'smarter' than a Princeton mathemtician and perhaps of Einstein (he used non-Euclidean geometry too, remember) won't promote too many of the book's readers to become devotees of Ms. vS, but I have to admit I'm worried... 5. Generally speaking, she tosses her own conjectures around as if she expects somebody to come back and prove _them_ in a few years. A sample: "A possible fatal flaw in Wiles's proof is whether the same arguments could be constructed to hold true for ({ital.} all) exponents... if it could, the same proof would 'prove' the Pythagorean theorem to be false" (p.62). Does she have any reason to THINK this actually MIGHT be a problem in the proof? If so, she doesn't deign to share it with us. Having only read the book yesterday, I'm still in the throes of rage, and as such do not yet have constructive suggestions as to what (if anything) could or should be done about this woman. But we shouldn't let this die without a hiccup. Does anyone know whether Martin Gardner, quoted on the back cover, thinks MvS is on the right track? Aaron Mandel amandel@husc.harvard.edu "Everywhere you go... clever people..." -Oscar Wilde _The Importance Of Being Earnest_ (End of forwarded info) My feeling is that Mandel is over-reacting. She wrote the book in three weeks from scratch: she is dead wrong on her criticism, and doesn't know much about maths or maths history. But some of the points made above sound to me like Mandel was just angry: the stuff about Wiles being more into FLT than his family. I didn't get the impression that there was anything pejorative about her description. I think she is just annoying. Besides, if the book gets people reading about maths, what harm can it do? I also wonder if a man, writing what she has done, would have drawn a response like "what (if anything) could or should be done about this woman[?]" An autodidact is someone who has learnt primarily by teaching themselves, Bruce. * Bob to Mike (on Marilyn's walking problem) Great stuff, Mike. Appreciate your taking time to share it with us. I enjoy Marilyn's column for the most part although every now and then I get the impression she deliberately clouds her explanations to encourage more people to write that she can "correct." e.g., the 3 doors problem which I think is easier to understand than she explains. I am a little concerned that not two weeks after she published bogus criticism of FLT she had a problem that she analyzed completely incorrectly. Thanks also for the new word. Bob * Donald Burke to Mike (a high IQ) <<> I agree strongly with A & disagree strongly with B. I think her criticisms are excellent for a layman. Her recounting of history is fine. I fit in her niche (high IQ, interest in math, career in another field so only college level formal education in math). My IQ might well be as high as her (measurement above 170 or so is guesswork) so I can relate to her POV. And her book was fun & informative for a VERY niche audience likely to buy it (a few math majors & Mensans). Note that she never says FLT proof/Wiles is false. Also note the proof is possibly flawed/false & certainly not proven. All of her caveats may in fact be actual proof problems. And they may not. And she says so. Graduate math work or at least majoring is required for good familiarity with non- Euclidean geometry so its a nice intro. The biggest problem for me with the book (and it was a small one) was that people unfamiliar with her were not explicitly told where she was coming from (ie a smart layman learning details about FLT for the first time, not a career number theorist or even mathematician). The whole I-Net post sounded like a nonsocial grumpy career math wierdo with no lay friends-- the book after all never remotely claims to be an actual academic challenge to the proof, nor MOST IMPORTANTLY does she ever say she feels FLT is not yet proven. It is a thought provoker, & an attempted boon to her syndication claim (highest IQ). All IMO. Don * Steven to Mike I find the comments rather right on target. Anyone who supposes to meddle in a field in which they have a high level of ignorance should expect to come out bruised and, perhaps, worse for the experience. Don't you think so? Steve * Mike to Steven >> Anyone who supposes to meddle in a field in which they have a high level of ignorance should expect to come out bruised and, perhaps, worse for the experience. Don't you think so? << Are you referring to my comments or Mandel's, Steven? Assuming the former, then yes. She's bright, alright, and has outwitted maths professors before, but I think she took on a challenge bigger than she realized. However, Donald's comments make me wonder whether the flaws in the book are even more minor than I thought. To a reader without a fairly substantial academic background in maths maybe her book has very few flaws. It did seem warmed over to me, though, and I could tell she wasn't writing about it with the same insight that Keith Devlin, or Martin Gardner, or Ivars Petersen, or Ian Stewart (some of my favourite maths popularizers) would have brought to the topic. Perhaps Mandel is offended that an "outsider" should think they can just dive in and write something about his field. * Allan to Mike Dear Mike, There is no mathematician who wouldn't be DELIGHTED if some outsider came up with something significant pertaining to FLT. (There are many examples where 'outsiders' have had brilliant ideas. Fermat, himself, was an 'outsider', since he was a lawyer, not a professional scientist.) Unfortunately, mathematics is a hard subject, and number theory is the hardest branch of all. Modern number theory requires tremendous background knowledge. Furthermore, FLT has been worked over by countless thousands of mathematicians, both insiders and outsiders. The probability of an 'outsider' having something useful to say is nil. Yours truly, Allan Trojan * Mike to Allan I'm in two minds about your comments, Allan. I agree that number theory is probably the hardest branch of maths. (I want to specialize in analytic number theory myself.) But it's also the branch that has historically had the most successful amateurs, starting with Fermat himself, who was only a part-time mathematician. Of course, it's no longer possible for someone to have anything remotely like the impact on number theory that Fermat or Gauss had, but new questions are appallingly easy to ask in number theory. Many of them are interesting, too; the 4-number problem; the "3n+1" problem; and so on. In fact, my own involvement with the University of Texas came about because I asked a number theory professor there about a question I had come up with. It turned out to be a new question, and he sent one of his Ph.D. students off hunting for the answer. (He didn't find it.) So I take issue with you if you're saying no outsider can say anything interesting about number theory. However, if (as I suspect) you're actually saying that nobody is likely to say anything interesting about FLT, then I do agree with you. What I was objecting to in Mandel's letter is the sense I got that Marilyn had no *right* to stick her damn interfering nose in, and she should just shut up about things she doesn't understand. He himself said he was enraged by the book; I wonder if piqued might be a better description. Here's an example. I live in Williamson County, Texas, which has been in the national news recently for denying Apple Computers a tax break because it gives medical benefits to same sex partners of employees. When journalists wrote about the hicks who live here, I was similarly annoyed: local polls show a strong majority supporting Apple against the county, but the journalists unanimously sought out fundamentalists and homophobics to interview. I felt misrepresented, and invaded, and I think Mandel does too. But that's what publicity does, both to me and Mandel, and I don't see the point in reacting like he did. * Mike to all This satire on Marilyn's FLT article was forwarded to me by Allan Trojan. I'm posting it on his behalf: X-News: sol sci.math:41769 From: beals@gargoyle.uchicago.edu (Robert Beals) Subject:FLAW FOUND in FAMOUS PROOF Date: 12 Dec 93 00:36:24 GMT Message-ID: I am posting this for a friend who unfortunately has lost his net access: -------------------------------------------------------------------------- It SEEMS to be a WELL-KNOWN RESULT that if A is an (n x n) INTEGER matrix of FINITE order then the trace of A is BOUNDED in absolute value by n. The short ``PROOF'' goes AS FOLLOWS: SUPPOSE A^k = I. Then ANY EIGENVALUE x of A satisfies x^k = 1, from which |x| = 1. Since the trace of A is the sum, WITH MULTIPLICITIES, of the eigenvalues, we get |Tr(A)| <= n by the TRIANGLE INEQUALITY. However, this SO-CALLED proof makes use of COMPLEX NUMBERS to prove a RESULT about INTEGER MATRICES. Complex numbers are numbers of the form (a + bi), where i is the SQUARE ROOT OF MINUS ONE. For THOUSANDS of years, mathematicians TRIED to find the square root of minus one WITHOUT SUCCESS, until GAUSS and others THREW COMMON SENSE OUT THE WINDOW and POSTULATED that i^2 = -1. WHAT would people THINK if they knew that GAUSS had FACTORED the integer 17 into SMALLER INTEGERS (one of the ``FAMOUS IMPOSSIBILITIES'')? Well, USING COMPLEX NUMBERS, that is EXACTLY what HAPPENED! Observe: (4 + i) (4 - i) = 4^2 - i^2 = 16 - (-1) = 17 GAUSS himself ADMITTED that his FACTORIZATION would NOT work in the ORDINARY INTEGERS. So has 17 been FACTORED, or has it NOT? Has a BOUND on the TRACES of (n x n) MATRICES of FINITE ORDER been PROVED, or has it NOT? I say that IT HAS NOT; if we REJECT a COMPLEX factorization of 17, then we should also REJECT a COMPLEX proof that the TRACES of such MATRICES are BOUNDED by n. UN-altered REPRODUCTION and DISSEMINATION of this IMPORTANT Information is ENCOURAGED. * Kevin to Allan Allan, Many of the most insightful advances in math came from mathematician's in their youth. Gauss, Abel, Galois were under 20 when they came up with many of their most interesting observations. I fear today that the language used in math is so difficult that students will not get to the frontiers until they are in their late twenties and thirties. After they have lost the vigor of youth. Hardy said "No mathematician should ever allow himself to forget tha mathematics,..., is a young man's game." Today the language is so difficult to learn that the "young man," or woman for that matter, is excluded from the debate. * Tom to Mike Mike, >>>Of course, it's no longer possible for someone to have anything remotely like the impact on number theory that Fermat or Gauss had, but new questions are appallingly easy to ask in number theory.<<< I work with teachers and would be delighted if you could mention some of the easily-asked number theory problems. Possible to do? * Steven to Mike At least someone like MArtin Gardner ASKED for help -- and never claimed anything that wasn't supported by professional oopinion. What would give MArilyn Vos Savant the chutzpah to write about a field in which she is untrained? Let's put it this way -- getting A's in high school math does not make her a valid commentator on the field. Steve