## Most interesting mathematics mistake? [closed]

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:

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My mistake, not an interesting one unfortunately. – Ilya Nikokoshev Oct 18 2009 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 2010 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 2010 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 2010 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 2010 at 18:46

## closed as no longer relevant by Scott Morrison♦Mar 7 2010 at 6:40

C.N. Little listing the Perko pair as different knots in 1885 (10161 and 10162). The mistake was found almost a century later, in 1974, by Ken Perko, a NY lawyer (!)
For almost a century, when everyone thought they were different knots, people tried their best to find knot invariants to distinguish them, but of course they failed. But the effort was a major motivation to research covering linkage etc., and was surely tremendously fruitful for knot theory.

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That's a nice mistake. Do you know how it started -- presumably at some point the knots were separated by a flawed computation of some invariant? – Ryan Budney Dec 16 2009 at 3:06
Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 2<sup>10</sup> knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot, but the quest to show they are different was fruitful. – Daniel Moskovich Dec 16 2009 at 7:22
Did Conway assume they were different as well, or did the mistake persist for other reasons, like an error in computing an invariant? – Ryan Budney Dec 16 2009 at 7:56
Yes- Conway assumed they were different in his table as well, but had no invariant to prove it. I don't know of any miscalculated invariant which "showed" they were different. – Daniel Moskovich Dec 16 2009 at 11:01

All of the (in retrospect) misguided attempts to prove Euclid's Parallel Postulate, which eventually lead Gauss to develop hyperbolic geometry.

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(and/or Lobachevsky, and/or Bolyai) This gets my vote as one of the most fruitful mistakes, and one of the longest perpetuated. – Aaron Mazel-Gee Oct 17 2009 at 18:46

An error of Lebesgue. 1905 or so. Take a Borel set in the plane, project it onto a line, the result is a Borel set. Obvious: the projection of an open set is open, and the Borel sets in the plane are the least family containing the open sets, closed under countable unions and countable intersections.

But wrong. Projection doesn't commute with countable intersection.

Studying this error lead Suslin to begin the line of study now called "descriptive set theory", 1917 or so.

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Kempe's "proof" of the four-color theorem, which didn't prove the four-color theorem, but did:

1. Prove the five-color theorem
2. Somehow manage to go unnoticed for a dozen years
3. Lay the foundations for major tools in structural graph theory, and despite being fundamentally flawed, serve as the starting point for the eventual successful proof(s) of 4CT.
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A story I heard in grad school:

Once upon a time, a set theorist was writing a paper on inner models, and in it he worte, "... and we will call such models nice." When he got his manuscript back from the typist (this was back in the pre-LaTeX days of technical typists), he saw that his handwriting had been misread, and the line came out as: "... and we will call such models mice." The name stuck, and to this day if you browse almost any recent volume of the Journal of Symbolic Logic, you will find set theory articles on "mice."

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I've heard a version of this story too, but I've also heard that Jensen denied that this was the origin of "mice". I never asked Jensen himself about it, so I don't know what to believe. – Andreas Blass Oct 9 2011 at 23:55
You know what will be a great paper title? "Of mice and men" – Aleks Vlasev Oct 10 2011 at 6:50

Maybe it's not true, but there's the story of the "Grothendieck prime":

One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replies, yes, an actual prime number. Grothendieck suggested, "All right, take 57."

But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. "He doesn’t think concretely."

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An insignificant mistake, but amusing nonetheless: in Cayley's famous 1854 paper where he defines the concept of an abstract group, as an illustration he proves that there are three groups of order 6 (up to isomorphism). This is because he does not realize that the groups $Z_2\times Z_3$ and $Z_6$ are isomorphic. (See my comment for the correct Cayley reference.)

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It took me a while to track down the correct reference. It is page 51 of A. Cayley, Desiderta and suggestions: No. 1. The theory of groups, American J. Math. 1 (1878), 50-52. An interesting related paper is G. A. Miller, Contradictions in the literature of group theory, American Math. Monthly 29 (1922), 319-328. – Richard Stanley Mar 1 2010 at 15:51

I believe Kummer's failed attempt at a proof of Fermat's last theorem led to the discovery of ideals.

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I'm told that Kummer actually didn't care about Fermat's last theorem; it just happened that the techniques he developed were applicable. – Qiaochu Yuan Oct 17 2009 at 20:54
It was actually Lame who came up with that bad proof. – Ben Webster Oct 18 2009 at 1:13
Oh, ok my mistake. – Grétar Amazeen Oct 18 2009 at 14:27
Harold Edwards wrote a wonderful account of this history in his paper "The background of Kummer's proof of Fermat's last theorem for regular primes". It doesn't seem to be available online, but the mathsci net review is: ams.org/mathscinet-getitem?mr=57:12066a – Ben Linowitz Jan 6 2010 at 2:19
I don't know whether it is appropriate to say "discovery" of ideals. Maybe "recognition of the importance/relevance of ideals"? – Kevin Lin Apr 5 2010 at 6:14

Pontryagin made a famous mistake while computing the stable homotopy groups of spheres (specifically, π2) which led to the discovery of the Kervaire invariant. I won't spoil what the mistake was: watch this video of Mike Hopkins' talk (second video on the page), starting about 7 minutes in.

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Poincare defined the fundamental group and the homology groups and proved that H _1 was pi _1 abelianized. So the question came up whether there were other groups pi _n whose abelianization would give the H _n. Cech defined the higher pi _n as a proposed answer and submitted a paper on this. But Alexandroff and Hopf got the paper, proved that the higher pi _n were abelian and thus not the solution, and they persuaded Cech to withdraw the paper. Nevertheless a short note appeared and the pi _n started to be studied anyway...

Taken from http://www.intlpress.com/hha/v1/n1/a1/ ,page 17

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Frege's proposed axioms in Die Grundgesetze der Arithmetik.

Frege was trying to derive the concept of "number" from more basic concepts, and he tried to axiomatize higher-order logic (essentially, a kind of set theory), but his intuitive-seeming axioms were logically inconsistent. Russell first found the inconsistency, which we now call Russell's Paradox.

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It was "proved" in 1961 that the first right derived functor, $\lim^1_{\leftarrow}$ of the inverse limit functor is zero on Mittag-Leffler systems.

However, recently a counter-example was found by Neeman and Deligne: http://www.springerlink.com/content/aeem2yx884nnufxn/

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wait... really? this is serious, i use that a lot... dammnit! – Sean Tilson Mar 31 2010 at 6:35
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Not just a great mistake, but also a great documentation of a mistake: Stallings's How not to prove the Poincare Conjecture. (I think this paper is my answer to every community-wiki question.)

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Whitehead's similar mistake is very interesting, too, as it lead him to the construction of contractible 3-manifolds that aren't balls. – Ryan Budney Dec 15 2009 at 21:22

Hilbert's program, whose development was induced by on assumptions shattered by Goedel.

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Supposedly Stefan Bergman attended a course on orthogonal functions while an undergraduate, and misunderstood what he was hearing, believing that the functions were supposed to be analytic. This led him to the Bergman kernel and Hilbert spaces of analytic functions, which has developed into a whole field of study at the junction of complex analysis and operator theory, and also with important ramifications in the more geometric parts of SCV. If the story is true, this was certainly an extremely fruitful mistake!

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From wikipedia (http://en.wikipedia.org/wiki/Uniform_convergence), about uniform convergence:

"Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence."

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I have always loved that way Abel wrote this (in a footnote): «it appears to me that this theorem suffers exceptions»... – Mariano Suárez-Alvarez Dec 15 2009 at 23:49
Some (e.g. A. Robinson) say that this is a mis-interpretation of the situation. When Cauchy says the sequence converges at all points this includes infinitesimals and such things not recognized as real numbers nowadays. Abel's counterexample $\sum (1/n) \sin(nx)$ in fact does not converge at certain points $x$ infinitely close to $0$. We can hardly fault Cauchy if he did not use the notion of real number from Dedekind and Cantor, since that would not come until 50 years later. – Gerald Edgar Dec 16 2009 at 16:40

Steiner's count 7776 of the number of the number of plane conics tangent to 5 general plane conics certainly deserves a mention here. He gave this answer in 1848, and it wasn't fixed until 1864, when Chasles pointed out the error and came up with the correct value of 3264. You can regard this as the first recognition of needing appropriate compactifications in order to do valid calculations in enumerative geometry.

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Goodrick's "story from Grad school" is incorrect. According to Ronald Jensen, the set theorist in question, he felt that the concept was important enough that it deserved a name which had not already been used elsewhere in mathematics. And 'mice' was it. (Also, note that 'mice' is a noun, and 'nice' is an adjective --- it would not make sense.)

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But the urban legend is so funny... – Ilya Nikokoshev Oct 19 2009 at 20:02
I have heard 3 versions of the origin of the name. They all originated with Jensen, and were told at a rate of one per decade. Last I checked, he actually does not seem to remember the reason for the name. – Andres Caicedo Oct 26 2010 at 4:52

Perhaps not under this heading but I enjoy reading in Marshall Hall Group Theory book:

"Let p be any old prime."

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I find this one (it is not in the same vein as the ones that have been posted here so far, this is not a pure math mistake) to be interesting and instructive to students: patriot missile failure due to poor understanding of binary decimals

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Samuel I. Krieger made many attempts at significant contributions to the field of mathematics, unfortunately some of his efforts did not pan out.

In 1934, he claimed that the 72-digit composite number 231,584,178,474,632,390,847,141,970,017,375,815,706,593,969,331,281,128,078,915,826,259,279,871 was the largest known prime number.

He also attempted to show that the number 2^256(2^257-1) was perfect, implying that 2^257-1 is a prime number. 852,133,201 is a factor of 2^257-1.

Finally, he claimed to have a counter example to Fermat's Last Theorem x^n + y^n = z^n using the numbers x = 1324, y = 731 and z = 1961 with an undisclosed n. A reporter supposedly called Krieger to ask how the left and the right hand side could be equal, when the left hand side could only end in a 4 or a 6 plus 1, and the right hand side could only end in 1.

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A reporter who knew about modulo 10 arithmetics, now that's quite a story! – Ilya Nikokoshev Oct 22 2009 at 7:43

I don't know if this is really a mistake: Fermat's "missing proof" for Fermat's last theorem.

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

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Petrovisky-Landis solution to the second part of Hilbert 16th problem. They "proved" the existence of a bound for the number of limit cycles of planar polynomial vector fields of fixed degree. Ilyashenko pointed out the mistake.

The problem remains wide open but the basic idea of Petrovisky-Landis ( complexify ) lead to the study of holomorphic foliations.

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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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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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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 $n \equiv 2 \pmod 4$. Nearly two hundred years later, this was proved false for every $n \equiv 2 \pmod 4$ 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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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 2010 at 7:55