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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. Poincaré's 3d sphere characterization 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?

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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  • $\begingroup$ Yeah, I've also been thinking there should be a tag like "not-math-related" -- perhaps "meta", if that's doesn't suggest 'related to the operation of MO' too narrowly. $\endgroup$
    – Alex Fink
    Commented Oct 18, 2009 at 20:30
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    $\begingroup$ 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... $\endgroup$
    – vonjd
    Commented Mar 12, 2010 at 18:28
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    $\begingroup$ 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.... $\endgroup$
    – Cam McLeman
    Commented Mar 12, 2010 at 18:40
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    $\begingroup$ 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 $\endgroup$
    – vonjd
    Commented Mar 12, 2010 at 18:46
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    $\begingroup$ A discussion thread was started on meta partly inspired by this: tea.mathoverflow.net/discussion/284/…. $\endgroup$
    – Jonas Meyer
    Commented Mar 13, 2010 at 0:19

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For surfaces of constant mean curvature, it is alleged that Hopf thought that all compact CMC surfaces in $\mathbb{R}^3$ were round spheres. CMC surfaces are what you get if you have a soap film bounding a fixed volume, so after a childhood full of blowing bubbles this is a pretty reasonable thing to think. And it even happens to be mostly true: Hopf proved that immersed CMC spheres are round, and Alexandrov proved with a nice reflection argument that embedded CMC surfaces of any genus must actually be round spheres.

But a bit later, Wente discovered a collection of CMC tori. Ivan Sterling has some nice pictures of these on his website, as does MSRI. There are many very pretty connections between these surfaces and algebraic geometry, so to me they sort of mark the start of the modern "integrable systems" era of CMC research.

I should probably add that nobody actually seems sure if Hopf believed that compact CMC surfaces are spheres, but it makes a good creation story for the subfield!

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Euler conjectured that there were no pairs of orthogonal Latin squares for orders $n \equiv 2 (\text{mod}~ 4)$. Nearly two hundred years later, this was proved false for every $n \equiv 2 (\text{mod}~ 4)$ except $ 2 $ and $ 6 $. Here's the link to Euler's paper. Regardless, Euler's work certainly helped spur research into Latin squares.

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    $\begingroup$ It seems odd to call a conjecture a mistake. $\endgroup$
    – Jim Conant
    Commented Sep 29, 2015 at 13:23
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Something I came across a long time ago during my years in Oxford. A bit off a tangent, but still worth a quick read:

http://eprints.maths.ox.ac.uk/104/1/balls.pdf (Wayback Machine)

"If I remember rightly, cos(pi/2) = 1"

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A mistake that isn't a mistake: in an early edition of Number Theory by Borevich and Shafarevich (corrected in later editions), there is a typographical error which doesn't change the meaning or correctness of the intended text, yet it is obvious that it is a typographical error. Namely, there is a table which is divided into four cases: $n\equiv 0\ (\mathrm{mod}\ 4)$, $n\equiv 1\ (\mathrm{mod}\ 4)$, $n\equiv 22\ (\mathrm{mod}\ 4)$, and $n\equiv 3\ (\mathrm{mod}\ 4)$. (I don't have a copy available, so I cannot give the precise reference.)

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  • $\begingroup$ Do you mean an early Russian edition? I believe the book in English only ever had one edition. Back in the 1990s I sent the translator Newcomb Greenleaf an errata list for the English version of the book and he thanked me but said he did not expect Academic Press would ever come out with a second edition. A genuinely bad typographical error in the English translation of that book was mentioned by me an in answer to the MO question mathoverflow.net/questions/25263/…. $\endgroup$
    – KConrad
    Commented Oct 16, 2021 at 2:28
  • $\begingroup$ @KConrad: I checked that the error is on page 422 (English edition). Perhaps there is a later printing without this error; otherwise I was mistaken about it being later corrected. $\endgroup$
    – Richard Stanley
    Commented Oct 20, 2021 at 21:43
  • $\begingroup$ I see that error in my copy, where I had just crossed out one of the 2’s in 22 to make it 2. I would not be surprised if such typos were never corrected in any later printing. $\endgroup$
    – KConrad
    Commented Oct 21, 2021 at 4:23
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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 $n\le3$ and constructed counter-examples for every $n\ge4.$

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

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If Hilbert's program was a "mistake", then surely so was Russell-Whitehead's Principia Mathematica.

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  • $\begingroup$ Why? Did they claim any form of completeness? $\endgroup$
    – Joël
    Commented Mar 26, 2018 at 20:20
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A posthumous book on number theory by Dirichlet appeared in 1859. It stated that Euclid's proof of the infinitude of primes was by contradiction, starting with an assumption that only finitely many primes exist and then deducing a contradiction.

Euclid's actual proof, recast in modern language, was that if $S$ is any finite set of primes (with no assumption that it contains the smallest $n$ primes nor that it contains all primes), then the prime factors of $1+\prod S$ are not members of $S;$ hence there are always more primes than what one already has.

For example, if $S=\{5,7\}$ then $1+\prod S=36=2\times2\times3\times3$ and the new primes are $2$ and $3.$

This requires no assumption that $S$ contains all primes.

Only the assumption that $S$ contains all primes could justify the conclusion that $1+\prod S$ has no prime factors, and so "is therefore itself prime", to quote G. H. Hardy (no relation to me, as far as I know) on pages 122–123 of the 1908 edition of A Course of Pure Mathematics (but not in the posthumous 10th edition).

The erroroneous belief that $1+\prod S$ is prime whenever $S$ is the set of the smallest $n$ primes for some $n$ has been held by some conscientious persons. The smallest (but not the only) counterexample is $1+(2\times3\times5\times7\times11\times13) = 59\times509.$

Catherine Woodgold and I examined in some detail the error of thinking that this proof is by contradiction in "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, Fall 2009, pages 44–52. 

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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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Karl Pearson's contributions in the development of statistics are so ubiquitous that most users take his assumptions for granted. One key contribution and mistake of his was to claim that all distributions are parametric. Such models are still predominantly used in social and behavioral sciences, but his insistence led to a lot of interesting and very useful developments in mathematical statistics and its applications by people who published refutations of his work (like R.A. Fisher).

As a non-math mistake, Karl Pearson avidly advocated eugenics towards racial purity. Big mistake.

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  • $\begingroup$ Do you have a source? I have trouble picturing why he would think that. $\endgroup$
    – arsmath
    Commented Nov 7, 2013 at 14:05
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    $\begingroup$ On your final point, Pearson was indeed part of the Eugenics movement, as were many leading statisticians and human biologists at the time, including Pearson's predecessor Galton (creator of the poorly named regression and much more) as well as Fisher and Haldane. But I have not seen evidence that they advocated racial purity. $\endgroup$
    – Henry
    Commented Sep 20, 2020 at 18:25
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The Grunwald-Wang theorem: https://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem on the injectivity of $K^\times/n \to \prod K_\mathfrak{p}^\times/n$ for a global field $K$. (Proof with mistake by Grunwald in 1933, corrected by Wang in 1948, who found a counterexample, but showed that it is correct if one is not in a "special case") See also https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html (Cohomology of Number Fields) Chapter IX, especially Theorem 9.1.11.

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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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    $\begingroup$ But was this a fruitful mistake? $\endgroup$
    – Jim Conant
    Commented Sep 29, 2015 at 13:27
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The recent paper by Richard Brent “Some instructive mathematical errors” certainly deserves a mention in this thread:

We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the author's own published papers.

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

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    $\begingroup$ Any examples of such errors? Any interesting ones? $\endgroup$
    – Ilya Grigoriev
    Commented Mar 13, 2010 at 7:55
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    $\begingroup$ But "Contains errors from start to finish", seems quite an exaggeration. $\endgroup$
    – Pietro Majer
    Commented Nov 7, 2013 at 13:36
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    $\begingroup$ Kevin, at the risk of asking something stupid: what's the problem with circles intersecting in 0, 1, or 2 points? $\endgroup$
    – Todd Trimble
    Commented Nov 7, 2013 at 22:11
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    $\begingroup$ @GerryMyerson Yes, sure, but the way my pedantic mind works: I think it's still true that over any field, two distinct circles intersect in less than three points. So I guess what you're suggesting is that there was something buggy about the description of when each of those three cases occurs (and that Kevin's description was a shorthand). $\endgroup$
    – Todd Trimble
    Commented Nov 7, 2013 at 22:47
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    $\begingroup$ That's what I had in mind, sorry for the confusion. But in my defense, can't circles on the surface of a torus have 4 intersections? And $p$-adic circles have infinitely many? $\endgroup$
    – Kevin O'Bryant
    Commented Nov 10, 2013 at 23:21
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The question above says:

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems,

This will be a very minor example, included because the person who made the mistake is very smart and very famous, but not because it led to anything.

Take three regular pentagons sharing a vertex and three edges, as in a dodecahedron. Set it down on a horizontal mirror, so that the three edges opposite the common vertex make contact with their reflections.

These three pentagons and their mirror-images are (but are they??) six faces of a polyhedron that also has three faces that are rhombuses or rhombi or rhomboi or whatever they're called.

So Donald Knuth once thought, according to what he said in a seminar that I attended.

But the four edges of the putative rhomboi are not coplanar, so instead of those three faces you have six triangular faces.

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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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    $\begingroup$ COuld you elaborate more on this? $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 26, 2009 at 22:05
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    $\begingroup$ 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). $\endgroup$
    – Ilya Grigoriev
    Commented Mar 13, 2010 at 7:57
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Cantor's set theory - had he known the related paradoxes, he would probably not have started developing set theory.

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    $\begingroup$ 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. $\endgroup$
    – John Goodrick
    Commented Oct 17, 2009 at 15:12
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    $\begingroup$ 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>). ) $\endgroup$
    – Thomas Riepe
    Commented Oct 17, 2009 at 15:45
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    $\begingroup$ Well, I think maybe he was aware: en.wikipedia.org/wiki/Cantor%27s_paradox But is there actually an identifiable mistake he made in his writings? Frege on the other hand went further, and made a mistake. $\endgroup$
    – Todd Trimble
    Commented Apr 11, 2015 at 19:57
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