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What are some good papers that debunk common myths in the history of mathematics?

To give you an idea of what I'm looking for, here are some examples.

Tony Rothman, "Genius and biographers: The fictionalization of Evariste Galois," Amer. Math. Monthly 89 (1982), 84-106. Debunks various myths about Galois, in particular the idea that he furiously wrote down all the details of Galois theory for the first time the night before he died.

Jeremy Gray, "Did Poincare say 'set theory is a disease'?", Math. Intelligencer 13 (1991), 19-22. Debunks the myth that Poincare said, "Later generations will regard Mengenlehre as a disease from which one has recovered."

Colin McLarty, "Theology and its discontents: The origin myth of modern mathematics," http://people.math.jussieu.fr/~harris/theology.pdf . Debunks the myth that Gordan denounced Hilbert's proof of the basis theorem with the dismissive sentence, "This is not mathematics; this is theology!"

Some might say that my question belongs on the Historia Matematica mailing list; however, besides the fact that I don't subscribe to Historia Matematica, I think that the superior infrastructure of MathOverflow actually makes it a better home for the list I hope to create. Still, maybe someone should let the Historia Matematica mailing list know that I'm asking the question here.

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Let's hope John Stillwell checks in at some point on this question:He,of course,has a reputation as one of mathematics' most knowledgeable and prolific historical scholars. –  Andrew L May 31 '10 at 20:10
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Glad to oblige, Andrew. –  John Stillwell Jun 1 '10 at 4:23
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Historia Matematica seems to be dead for quite a long time. (Otherwise: Please give a working link.) So: Why do you flog a dead horse? –  Hans Stricker Sep 28 '10 at 17:28
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I'm not sure that the Colin McLarty paper really 'debunks' the Gordan 'myth' so much as casts a different light on it... –  Marius Kempe Jan 12 at 4:32

23 Answers 23

For over a century, books published a picture of Legendre that was not in fact a picture of Legendre. There's an AMS notices article. Does a picture count as a myth?

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That's very interesting, sigfpe. +1 –  J. H. S. May 31 '10 at 22:32
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I learnt it from the n-category cafe. –  Dan Piponi Jun 1 '10 at 4:13
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And there's a similar state of affairs about a picture of Bolyai, see article in the Notices. –  Marius Kempe yesterday

Historians like nothing better than to debunk other historians, so there are plenty of papers shooting down one myth or another. I enjoy these papers as much as the next person, but sometimes the debunking becomes the new myth. So, if I may turn the question round a bit, here is a case where I think the original "myth" has merit, and I'm not convinced by the "debunking".

"Myth". The ancient Mesopotamians who compiled Plimpton 322 were looking for Pythagorean triples and had a powerful method for finding them. For example, they found the triple (13500,12709,18541).

"Debunking". By Eleanor Robson in Historia Mathematica 28 (2001) 167-206, which may be viewed here.

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I would like to know whether there are rejoinders to Rothman's article mentioned above. That Bell's Men of Mathematics is unreliable goes without saying. However, there are several things to object to in Rothman's article. It seems to have been written with an intense dislike for romantic sensibilities - or even republican sympathies - that passes itself off for a tone of objectivity. For Rothman, Galois's troubles - including being expelled for exposing the opportunism of his school's director - were "what might have been anticipated", and Galois's objections to this are "paranoia". –  H A Helfgott Jun 7 '10 at 11:45
    
Not exactly a rejoinder to Rothman, but a nice attempt to tell the Galois story from a fresh point of view, is the novel The French Mathematician by Tom Petsinis (Walker & Company 1998). –  John Stillwell Jun 7 '10 at 12:15
    
@H A Helfgott: Such a rejoinder to Robson (not Rothman) was published by Britton, Proust, and Shnideer here. See also mathscinet review. –  katz Jul 25 '13 at 12:34

This entry is inspired by sigfpe's example of "picture as a myth".

There is an even more outrageous picture scam: for centuries, mathematics books featured

this picture of Euclid of Megara, whereas the author of the Elements was Euclid of Alexandria!

P.S. I offered a prize to any student who can find a glaring error in our linear algebra text. No one succeeded, although in fairness to them, sleazy publisher cropped the picture.

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To debunk: Behold! –  Victor Protsak Jun 1 '10 at 4:49
    
It's difficult not to get confused here. Both Euclids look the same to me. Both of them have long beards, they both are holding a compass, etc. In the Legendre case the situation was easier: the fake Legendre had a wig whereas the real one did not. –  J. H. S. Jun 1 '10 at 7:03
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It's easy: fake Euclid is wearing a bandana! Also, EVCLIDI MEGAREN should be a bit of a clue. –  Victor Protsak Jun 1 '10 at 7:32
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Great. Apparently medieval artists had a similar lack of clue about ancient Greece as the ones of today have about the Middle Ages. ;) –  darij grinberg Jun 1 '10 at 11:51

There is a nice article by Brian Hayes in the American Scientist (May-June 2006), "Gauss' Day of Reckoning," in which he looks at the history of the so-called "baby Gauss" story (that Gauss amazes his teacher by summing the first 100 positive integers).

There is a famous anecdote about Euler embarrassing Diderot in Catherine the Great's court. He claimed to have mathematical proof of the existence of God, when in fact he just stated mathematical nonsense (which Diderot did not understand): "Monseur, $(a+b^n)/n=x$ donc Dieu existe; répondez!" B. H. Brown tracked down the source of this myth in "The Euler-Diderot Anecdote" (Amer. Math. Monthly, Vol 49, 1942, reprinted in William Dunham's The genius of Euler: reflections on his life and work (2007))

Finally, there is a mathematical urban legend that I thought was surely was false, but is apparently true (according to Snopes). This is the story about the student who comes late to class and sees the homework written on the board. After a lot of effort he solves the problems. Only later did he discover that they were not homework, but open problems. It turns out that the student in the story was George Dantzig. Snopes cites a 1986 interview with Dantzig from the College Mathematics Journal.

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One of the biggest myths in number theory is that work on Fermat's last theorem played a large role in the development of ideal theory and algebraic number theory. In fact it was much loftier goals such as the quest for higher reciprocity laws that were the true sources of inspiration. For references see e.g. Lemmermeyer's book "Reciprocity Laws" p. 15 (notes on Lagrange).

Speaking of FLT, recall the legend that Kummer submitted a false proof based on the erroneous assumption that cyclotomic number rings were UFDs. Edwards argued that this may be a myth and put forth an argument that, instead, Kummer's mistake was based on a simple error not related to any UFD assumption. However R. Bolling recently discovered new evidence that seems to lend strong support to the veracity to the claim that Kummer did in fact mistakenly assume facts equivalent to unique factorization in various rings of cylotomic integers in one of his early papers (which was not an attempted proof of FLT). So only part of this legend is actually true.

The above two paragraphs don't really do justice to the complex history. For a more faithful rendition I highly recommend that the interested reader also consult Franz Lemmermeyer's recent paper [1] which, imho, is one of the most interesting historical works on number theory in quite some time.

By the way, far less known than the constructivity of Euclid's proof that there are infinitely many primes (cf. M. Hardy below) is the striking fact that Euclid's constructive proof generalizes quite widely - namely to any infinite ring having fewer units than elements. For this little-known proof see my post here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1209616#p1209616 http://google.com/groups?selm=y8zk5f3rn4e.fsf%40nestle.csail.mit.edu

[1] Franz Lemmermeyer. Jacobi and Kummer's ideal numbers.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg v. 79, 2, 2009, 165-187. http://dx.doi.org/10.1007/s12188-009-0020-5

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As for the FLT legend, Boelling's work shows clearly that Kummer's erroneous manuscript had nothing to do with FLT. The topic of Kummer's research was the "splitting of primes" in cyclotomic rings, and his error is related with unique factorization. –  Franz Lemmermeyer Jul 2 '10 at 6:18
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This is the first time I have ever heard a story in which Kummer made a mistake of assuming unique factorization. In the story I have always heard, it was Lame who made this error, and Kummer who pointed it out. See for example bookrags.com/research/fermats-last-theorem-wom –  Timothy Chow Jul 2 '10 at 15:52
    
Oops, too much cutiing and pasting in the middle of the night. Of course I didn't mean to say Kummer's false proof was on FLT (like Lame's). I corrected that. Thanks for the heads-up. –  Bill Dubuque Jul 2 '10 at 18:19
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Kummer did not assume unique factorization - he "proved" a theorem showing that the prime ideals above prime p = 1 mod q are principal. This implies, by what he would prove later (the class group is generated by ideals of degree 1) that the ring of integers in this cyclotomic field has uique factorization. As for Lame's error: this was pointed out by Liouville - Kummer explicitly constructed a counterexample (23rd roots of unity) which (my guess) he learned from Jacobi. Jacobi probably didn't think for a second that this example might make history . . . –  Franz Lemmermeyer Jul 3 '10 at 16:52
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Timothy, is it possible you never read Bell's Men Of Mathematics? Here's the Kummer story/legend/myth, from p.473: Kummer too had fallen afoul of the net which snared Cauchy, and for a time he believed that he had proved Fermat's "Last Theorem." Then Dirichlet, to whom the supposed proof was submitted for criticism, pointed out by means of an example that the fundamental theorem of arithmetic, contrary to Kummer's tacit assumption, does {\it not} hold in the field concerned. –  Gerry Myerson Jul 5 '10 at 0:45

Misconceptions about the Golden Ratio
George Markowsky
College Math Journal: Volume 23, Number 1, Pages: 2-19 1992

first paragraph...

The golden ratio, also called by different authors the golden section [Cox], golden number [Fi4], golden mean [Lin], divine proportion [Hun], and division in extreme and mean ratios [Smi], has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks such as [Ser] make many incorrect statements about the golden ratio.

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I vaguely remember reading a review of a more recent book that debunked many of the myths (e.g. by empirically measuring the most pleasing ratio of sides of rectangular objects). –  Victor Protsak Jun 3 '10 at 1:19
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You might be thinking of Mario Livio's book; there's a chapter where he does that. –  Qiaochu Yuan Jun 3 '10 at 2:13

I like this question very much - many popular history of mathematics books seem filled with legends more than history.

I would expand the question "What are some good papers that debunk common myths...?" to include books as well as papers. Maybe the best books to debunk myths are books that directly discuss and include the primary source material (in translation, perhaps). A fantastic recent book in this spirit is "The Mathematics of Egypt, China, India, and Islam: A Sourcebook," edited by Victor J. Katz. This is a great value as well! For example, an scholarly translation of the Chinese "Nine Chapters on the Mathematical Art" costs around 350 dollars on Amazon -- but one can instead find it in this sourcebook, together with a multitude of other translated texts, for around 50 or 60 dollars.

To demonstrate that this book addresses your specific question, on pages 467-477 you can find a translation of the Bijaganita of Bhaskara II. At the end is Verse 129, the end of which is translated, "Hence, for the sake of brevity, the square-root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated. And otherwise, when one has set down those parts of the figure there, [merely] seeing [it is sufficient].

The author of this section then mentions "These verses are presumably the ultimate source of the widespread legend that Bhaskara gave a proof of the Pythagorean theorem containing only the square figure shown in figure 4.19 and the word 'Behold!' "

The figure (4.19 in the book) is not among the verses in an old text, as far as I know. I think that Indian texts of the period were traditionally written on palm leaves (which degrade somewhat quickly), and copied every generation or two, so we don't have very old texts. In any case, the "Behold!" legend for the Pythagorean theorem seems to be a myth or at least a vigorous embellishment of history.

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There is a rumor (a myth?) in holomorphic dynamics about a rivalry between Fatou and Julia competing for the 1918 Grand Prize of the french science academy. In fact Fatou never submitted a memoir for the Prize.

What we now call the Julia set was first defined in a note in Comptes Rendus by Montel, and is the starting point for the work of Fatou and Julia. Fatou published a few notes on the subject in the Comptes Rendus before Julia, who started a quarrel, claiming priority for the results. Fatou didn't try to fight back, apparently because he didn't bother, and the Academia fullfilled the wish of Julia. This may be the source of the rumor.

You may also know the Parseval formula about the relationship between the square of the L2 norm of a function and its Fourier coefficients. This formula is due to Fatou. H. Lebesgue did the case of a bounded function, and Fatou extended it to any square integrable functions. This made a big impression on Lebesgue.

This is exposed in a recent book by Michele Audin entitled "Fatou, Julia, Montel (...)"

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Check out "Life on the Mathematical Frontier: Legendary Figures and Their Adventures" byr Roger Cooke in the April 2010 issue of the Notices (Volume 57, Number 4, pages 464--475), available online at http://www.ams.org/notices/201004/rtx100400464p.pdf .

I hope someone will follow up on Cooke's "Modest Proposal" (see page 473): ``I would like to invite some ambitious mathematician with time on his/her hands to write a monograph on the role played by legends in the mathematical community. Or perhaps someone would be willing to set up the mathematical equivalent of the snopes.com website, where mathematical “urban legends” can be checked out.''

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A perfect example of what I'm looking for! It's embarrassing how far behind I've gotten on reading the Notices... –  Timothy Chow Jun 1 '10 at 15:41
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Cooke's paper mentions several other papers of the type I'm interested in. I'll mention just one: "The So-Called Euler-Diderot Incident" by R. J. Gillings, Amer. Math. Monthly 61 (1954), 77-80. Gillings traces the legend that Euler confounded Diderot with a nonsensical mathematical proof of the existence of God to the memoirs of Dieudonne Thiebault. Thiebault does tell the story, but in his account it's a Russian philosopher (not Euler), and Diderot knew it was nonsense but kept quiet because he sensed they were having him on. Thiebault also says he's not sure the story was true. –  Timothy Chow Jun 2 '10 at 15:56

Here is an paper on this subject: Mathematical Myths G. A. Miller National Mathematics Magazine, Vol. 12, No. 8 (May, 1938), pp. 388-392

This is available on jstor

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"Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, pages 44--52, by me and Catherine Woodgold, debunks the widespread belief that Euclid's proof of the infinitude of primes is a proof by contradiction. The proof that Euclid actually wrote is simpler and better than the proof by contradiction often attributed to him.

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A similar mistaken belief is that Cantor's diagonal argument is a proof by contradiction. –  John Stillwell Jun 2 '10 at 23:13
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Just as Euclid proves "given finitely many primes, you can find another one," Cantor proves "given countably many real numbers, you can find another one." Also, just as Euclid's new prime can be constructed from the given primes, Cantor's new real can be constructed from the given reals. –  John Stillwell Jun 3 '10 at 2:05
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I agree that there are constructive aspects in both arguments. But the key point is what are "given" reals. Cantor's proof starts with $\textit{assuming}$ that a bijection $f:\mathbb{N}\to\mathbb{R}$ exists and specifying or fixing it. The conclusion is that there is a number not in the image of $f$, which is a contradiction. (This is quantified over all putative bijections.) This seems a quintessential proof by contradiction to me. What do you make of Liouville's proof that transcendent numbers exist? –  Victor Protsak Jun 3 '10 at 5:03
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Victor, Cantor's proof may be phrased as by contradiction, but that is not the most elegant nor even the most natural way. You can present it as follows. Given any function f:N-->R, the well-known diagonal procedure produces a real number not in the range. Therefore no function from N to R is surjective. This is not a proof by contradiction, any more than the proof that there is no largest natural number: given any n, n+1 is larger. You do not need to "assume n is the largest". –  Pietro KC Jun 3 '10 at 6:40
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I would expand the question "What are some good papers that debunk common myths...?" to include web pages as well as papers. Naturally, there is a couple of candidate web pages in the reference section of the note mentioned by sigfpe. Another candidate might be http://www.snopes.com/science/nobel.asp where the myth of the rivalry between A. Nobel and Mittag-Leffler is debunked (to some extent).

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Snopes.com is a great website and I thought carefully about whether to phrase my question to include it or not. I decided to focus on papers because on average, people take more care with their research when publishing a paper than when posting something on a website. Snopes.com may be an exception because they take great pains to be accurate. However, often when I see a website that purports to debunk a myth, I'm not sure whether to believe it. If there are exceptionally good websites then people should mention them, but be aware that I put a very high premium on reliability. –  Timothy Chow May 31 '10 at 23:27

There have been recent articles in the Mathematical Intelligencer about the well-known rivalry/animosity between Erdos and Selberg. Didier adds this link, pointing to an article of Graham and Spencer that appeared in the M.I. Emerton adds this link, pointing to an article by Goldfeld on the controversy.

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What is the myth here? –  Timothy Chow Jun 1 '10 at 1:28
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Legend has it that the two were planning a joint publication when Selberg overheard someone say something like, "Hey, have you heard that Erdos and somebody-or-other have an elementary proof of the Prime Number Theorem?" The legend continues, that Selberg, not wishing to go down in history as "somebody-or-other," rushed the proof into print without waiting for Erdos, got all the credit, the Fields medal, etc. I haven't seen the Intelligencer and don't know what, if anything, it has to say about the legend. –  Gerry Myerson Jun 1 '10 at 13:04
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cs.nyu.edu/spencer/erdosselberg.pdf –  Did Jun 1 '10 at 17:09
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One should also consider the reference math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf, written by D. Goldfeld, which contains quotes from Selberg disputing the account of Straus that is given in the reference mentioned by Didier Piau. –  Emerton Jun 2 '10 at 5:43
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I had not heard of the legend mentioned by Gerry, but it seems incredibly biased. –  H A Helfgott Jun 2 '10 at 19:44

Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking," http://arxiv.org/abs/1202.4153

The questions they address are:

  1. Were the founders of calculus working in a numerical vacuum?

  2. Was Berkeley’s criticism coherent?

  3. Were d'Alembert’s anticipations ahead of his time?

  4. Did Cauchy replace infinitesimals by rigor?

  5. Was Cauchy’s 1821 "sum theorem" false?

  6. Did Weierstrass succeed in eliminating infinitesimals?

  7. Did Dedekind discover the essence of continuity?

  8. Who invented Dirac’s delta function?

  9. Is there continuity between Leibniz and Robinson?

  10. Is Lakatos’ take on Cauchy tainted by Kuhnian relativism?

The main thread of their revisionist picture is that 17th- and 18th-century analysis had foundations that were far more rigorous and well-defined than most modern mathematicians have been led to believe.

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Myths can be such nice inventions, one should invent more of them!

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Cute, even if 61 days late. –  Victor Protsak Jun 1 '10 at 10:58
    
The link in this answer appears to be dead. –  Ricardo Andrade Nov 26 '13 at 23:07
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Donald E. Knuth's "Johann Faulhaber and sums of powers" debunks common belief (mentioned at at Mathworld, for example) that Faulhaber was first to discover Bernoulli formula and even (to some extent) Bernoulli numbers.

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David Fowler's book "The Mathematics Of Plato's Academy: A New Reconstruction" sets out to deconstruct the myth that "the early Pythagoreans based their mathematics on commensurable magnitudes, but their discovery of the phenomenon of incommensurability (the irrationality of the square root of 2) showed that this was inadequate; this provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Book V of Euclid's Elements". The arguments and conclusions are complex and interesting; here is a review with some of the main points: http://www.maa.org/publications/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction.

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The link in this answer appears to be dead. –  Ricardo Andrade Sep 25 '13 at 22:04
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@RicardoAndrade: I repaired the link. –  Will Orrick Aug 7 at 14:40

Historians of mathematics get very cross if you dare to say "Pythagoras's theorem." You have to say, "The Pythagorean theorem," to emphasize that it wasn't the brainwave of a man called Pythagoras. Indeed, that fact, as well as the irrationality of the square root of 2, were (I read) probably known long before Pythagoras, and the only Greek proofs we know of came much later. As for the story that the penalty for revealing the irrationality of the square root of 2 to outsiders was death ...

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How can you know the irrationality of the square root of 2 without having a proof? Most sources I know place the first proofs after 450 BC, long after Pythagoras had died. –  Franz Lemmermeyer Jun 1 '10 at 13:35

A recent one could be Peter Milne's "On Gödel Sentences and What They Say" Philosophia Mathematica (III) 15 (2007), 193–226. doi:10.1093/philmat/nkm015 debunking the myth that Gödel sentences are true because they say of themselves that they are unprovable.

Sorry, I realize this is not exactly what was being asked. However, there is still some analogy with the pattern "someone was believed to do or say something while in fact he/she didn't".

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You're right; this is not quite what I asked for. Misconceptions about the content of mathematical theorems are not the same as myths in the history of mathematics. As far as Goedel is concerned, my favorite expose of nonsensical interpretations of Goedel's theorem is Torkel Franzen's book "Goedel's Theorem: An Incomplete Guide to Its Use and Abuse." By the way, if you read Milne's paper carefully, you'll see that your description of its contents is not quite accurate. –  Timothy Chow Jun 1 '10 at 15:28

B. H. Brown debunks in [1] the myth that "Algebra was Hebrew to Diderot" (cf. E. T. Bell, Men of Mathematics, NY, 1937, pp. 146-147).

Gillings mentions in his note that it is D. Thiébault's account the only authority in the Euler-Diderot anecdote (even though "Thiébault himself was not convinced of the truth of it...").

It is important to add that in Thiébault's version of the story there is no explicit reference to the name of L. Euler.

References

[1] B. H. Brown. The Euler-Diderot Anecdote. Amer. Math. Monthly. Vol. 49, Issue 5, 1942, pp. 302-303.

[2] R. J. Gillings. The So-called Euler-Diderot Incident. Amer. Math. Monthly. Vol. 61, Issue 2, 1954, pp. 77-80.

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I think the myth that Kolmogorov was the first to come up with the idea of basing probability theory on measure theory is very common. This myth is debunked in

The Sources of Kolmogorov’s Grundbegriffe

G.Shafer and V.Vovk, The Sources of Kolmogorov’s Grundbegriffe, Statist. Sci. Volume 21, Number 1 (2006), 70-98. Online (free).

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Interesting paper...thanks! But does it really say what you claim it says? I quote: "Frechet thought of probability as an application of mathematics, not as a branch of pure mathematics itself, so he did not think he was axiomatizing probability theory. It was Kolmogorov who first called Frechet’s theory a foundation for probability theory." Looks pretty close to crediting Kolmogorov with the idea of using measure theory as a foundation for probability theory. –  Timothy Chow Jun 1 '10 at 15:36
    
I think a common view was that Kolmogorov provided an incomplete framework for probability. Frequentists, Subjectivists etc. would provide different axiomatizations giving rise to probabilities, based on something more empirical. –  Michael Greinecker Jun 1 '10 at 20:44

This myth is in living memory and I have not seen it debunked in print. The myth is that the Appel-Haken proof of the four colour theorem was controversial when it was published because it was a computer proof. This makes a good story but is it true?

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FWIW, R.Diestel writes in Graph Theory that the original approach of Appel-Haken "has not been immune to criticism, not only because of their use of a computer". –  Olivier Jun 1 '10 at 13:31
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This is certainly true. Nobody needs to write an article "debunking" this one because philosophers have written entire journal articles questioning the status of the theorem. See for example Thomas Tymoczko, "The four-color problem and its philosophical significance," J. Phil. 76 (1979), 57-83. –  Timothy Chow Jun 1 '10 at 15:13
    
This article says in the opening paragraphs that mathematicians accept that this is a proof of the four colour theorem. The myth is that mathematicians did not accept this proof because it was a computer proof (not for reasons that Olivier refers to). –  Bruce Westbury Jun 1 '10 at 15:59
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I certainly didn't accept the Appel-Haken proof (until Gonthier did a fully verified machine-checked version), for the same reasons I don't (yet) accept the Hales's proof of the Kepler conjecture. If you run a computer program as part of the proof, then you also have to supply a correctness proof of the actual computer program you used, before you really have a full proof. (And indeed, Hales and his collaborators are currently developing a fully machine-checked version.) –  Neel Krishnaswami Jun 1 '10 at 22:40
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In other words you want computer proofs to be peer reviewed.:) –  Bruce Westbury Jun 2 '10 at 6:45

Anellis, I.H. (1995), “Peirce Rustled, Russell Pierced : How Charles Peirce and Bertrand Russell Viewed Each Other's Work in Logic, and an Assessment of Russell's Accuracy and Role in the Historiography of Logic”, Modern Logic 5, 270–328. Online.

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