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Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove that Euclid's parallel axiom is really necessary unnecessary.

But I also think there are less famous mistakes worth hearing about. So, here's a question:

What's the most interesting mathematics mistake that you know of?

This question is community wiki, meaning neither the question nor the answers receive points (which are reserved for "hard" questions). So please post as much as you like (indeed please post one answer per post so that others can upvote the ones easier), vote a lot and vote freely.

(should there be a tag 'not-math-related' or similar?)

EDIT: There is a similar question which has been closed as a duplicate to this one, but which also garnered some new answers. It can be found here:

Failures that lead eventually to new mathematics

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My mistake, not an interesting one unfortunately. –  Ilya Nikokoshev Oct 18 '09 at 7:13
Closed: big-list questions don't need to keep cycling back to the front page, after some point. –  Scott Morrison Mar 7 '10 at 6:41
doesn't "cycling back to the front page" could also mean that it is still of interest? e.g. this one has been just been edited and therefore got to the front page again. Therefore it gets closed??? I don't get the logic behind that... –  vonjd Mar 12 '10 at 18:28
Well, cycling well-viewed topics back to the front comes at the cost of pushing newer questions out of immediate visibility faster, so I understand the motivation. On the other hand, as the site grows, we get new perspectives on old questions which, and as vonjd points out, are apparently still of interest. We shouldn't close things just because the site old-timers are tired of seeing them. This discussion is probably on meta somewhere already.... –  Cam McLeman Mar 12 '10 at 18:40
I agree with Cam - and in this case additionally: the big-list-tag means it is a big list and it can only become a big-list because many people make it a big list - so to close big-lists because they became big-lists is kind of absurd. Perhaps the underlying mechanism of bringing things to the front page should be changed in the software then. Just closing it is no solution –  vonjd Mar 12 '10 at 18:46

36 Answers 36

Certainly not the most interesting mistake in math, but it deserves to be mentioned.

Hesse claimed that homogeneous polynomials in n variables with vanishing Hessian are, after a linear change of coordinates, polynomials in at most n-1 variables.

Gordan and M. Noether verified Hesse's claim for n<=3 and constructed counter-examples for every n>=4.

It is ironic that there is no hesitation today to call the hessian hessian.

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William Shanks (1812-1882), who calculated pi to the 707th place, by hand, but it was only correct for the first 527 places.

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Cantor's been mentioned, but I think the lessons there should be different. First, the really big mistake was that of highly-reputed academics (including, I believe, Poincare, Kronecker and even Wittgenstein) who rejected his ideas. And (related) second, even in a wiki devoted to mistakes it seems somewhat carping to fault Cantor for failing to spot a subtlety without at the same time adequately crediting his genius.

Somewhat along the same lines, one might mention Fourier's difficulties in getting his ideas accepted.

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The mother of all examples: Euclid's Elements contains errors from start to finish.

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Any examples of such errors? Any interesting ones? –  Ilya Grigoriev Mar 13 '10 at 7:55
The most famous is the assumption that two circles intersect in 0, 1, or 2 points. It was already seen to be buggy in ancient times (Theon or Hypatia gap-system.org/~history/Printonly/Theon.html corrected some errors), and from what I've read Hilbert did also. –  Kevin O'Bryant Mar 14 '10 at 19:36
Wh.. what ?? –  Qfwfq Jun 26 '11 at 13:25
Some poor grammar there on my part. Hilbert tried to correct Euclid's work, and needed around 20 axioms (not 5) to do so. These are seen today as being primarily topological in nature (like that the points on a line are ordered meaningfully, or that circles have insides and outsides). en.wikipedia.org/wiki/Hilbert%27s_axioms –  Kevin O'Bryant Jun 26 '11 at 19:38
Kevin, at the risk of asking something stupid: what's the problem with circles intersecting in 0, 1, or 2 points? –  Todd Trimble Nov 7 '13 at 22:11

I think The Feynman path integral may be regarded as a great mathematical mistake, as once remarked by Richard Borcherds in a conversation.

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I don't think it was a mistake. Feynman's argument was not mathematically rigorous, but I don't think he ever claimed it was, or wanted it to be. The important thing to him, I think, was just that it gave the right answers (verifiable by experiments). –  Ilya Grigoriev Mar 13 '10 at 7:57

Cantor's set theory - had he known the related paradoxes, he would probably not have started developing set theory.

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Seems fairly controversial to call the development of set theory a "mistake" :) I guess you mean that Cantor's mistake was not being careful and rigorous enough? But then you could probably say the same thing about 18th-century analysts who played around with infinitessimals. –  John Goodrick Oct 17 '09 at 15:12
Yes - fortunately their intuitions had enough force to made them jump over (or blind towards) the problems (and infinitesimals may <a href="maths.nott.ac.uk/personal/ibf/rem.pdf"; title="Fesenko's essay">come back</a>). ) –  Thomas Riepe Oct 17 '09 at 15:45

protected by François G. Dorais Nov 7 '13 at 13:17

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