## Most interesting mathematics mistake?

150

124

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

'meta' is kind of reserved keyword which indeed refers to things like meta.stackoverflow.com. I'm thinking about soft-questions. Is it a good idea? – Ilya Nikokoshev – 2009-10-19T20:01:24.323

6Closed: big-list questions don't need to keep cycling back to the front page, after some point. – Scott Morrison – 2010-03-07T06:41:11.880

25doesn'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 – 2010-03-12T18:28:14.943

12Well, 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 – 2010-03-12T18:40:35.617

11I 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 – 2010-03-12T18:46:04.270

3

A discussion thread was started on meta partly inspired by this: http://tea.mathoverflow.net/discussion/284/why-was-my-question-closed-section-in-the-faq/#Item_1.

– Jonas Meyer – 2010-03-13T00:19:21.537

Cassini expected his ovals could describe planetary orbits around two big masses ... later found incorrect. – Narasimham – 2017-06-05T11:10:59.707

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. – Alex Fink – 2009-10-18T20:30:23.713

177

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.
alt text http://mathworld.wolfram.com/images/eps-gif/PerkoPair_1000.gif

Update (2013):
This morning I received a letter from Ken Perko himself, revealing the true history of the Perko pair, which is so much more interesting! Perko writes:

The duplicate knot in tables compiled by Tait-Little [3], Conway [1], and Rolfsen-Bailey-Roth [4], is not just a bookkeeping error. It is a counterexample to an 1899 "Theorem" of C.N. Little (Yale PhD, 1885), accepted as true by P.G. Tait [3], and incorporated by Dehn and Heegaard in their important survey article on "Analysis situs" in the German Encyclopedia of Mathematics [2].

Little's Theorem' was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little's "Theorem", if true, would prove them to be distinct!

Perko continues:

Yet still, after 40 years, learned scholars do not speak of Little's false theorem, describing instead its decapitated remnants as a Tait Conjecture- and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thistlethwaite.

I had no idea! Perko concludes (boldface is my own):

I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.

And the final nail in the coffin is that the image above isn't of the Perko pair!!! It's the Weisstein pair' $10_{161}$ and mirror $10_{163}$, described by Perko as "those magenta colored, almost matching non-twins that add beauty and confusion to the Perko Pair page of Wolfram Web’s Math World website. In a way, it’s an honor to have my name attached to such a well-crafted likeness of a couple of Bhuddist prayer wheels, but it certainly must be treated with the caution that its color suggests by anyone seriously interested in mathematics."

The real Perko pair is this:

Well, I'll be jiggered! The most interesting mathematics mistake that I know turns out to be more interesting than I had ever imagined!

1. J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, Proc. Conf. Oxford, 1967, p. 329-358 (Pergamon Press, 1970). 2. M. Dehn and P. Heegaard, Enzyk. der Math. Wiss. III AB 3 (1907), p. 212: "Die algebraische Zahl der Ueberkreuzungen ist fuer die reduzierte Form jedes Knotens bestimmt." 3. C.N. Little, Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774 and plate III. This paper describes itself at p. 771 as "Communicated by Prof. Tait." 4. D. Rolfsen, Knots and links (Publish or Perish, 1976).

2That'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 – 2009-12-16T03:06:27.567

12Little (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 – 2009-12-16T07:22:57.330

2Did Conway assume they were different as well, or did the mistake persist for other reasons, like an error in computing an invariant? – Ryan Budney – 2009-12-16T07:56:32.647

2Yes- 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 – 2009-12-16T11:01:31.397

Ryan- you were right. See the update! – Daniel Moskovich – 2013-11-07T14:59:21.540

7+1 for the update that shows that there was a mistake wrapped in a mistake hidden in a mistake. :) – Michael – 2013-11-07T21:45:04.410

10Ken Perko attempted to make another edit, by adding the following to the citation of Conway's paper: CONWAY WAS NOT MISLED BY THIS FALSE THEOREM OF C.N.LITTLE. HE FOUND THREE COUNTEREXAMPLES AMONG HIS 11-CROSSING NON-ALTERNATING KNOTS AND CORRECTLY WEEDED OUT THE DUPLICATE KNOT TYPES. Cf. Hoste-Thistlethwaite-Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998) FOOTNOTE 8 and Jablan-Radovic-Saxdanovic, Adequacy of link fanilies, Publictiones de L'Institute Mathematique, Nouvelle Serie, Tome 88(102) (2010), 21-52. – S. Carnahan – 2013-11-23T05:07:40.240

9(Perko's comment continues...) FOR CONWAY, KNOT THEORY WAS A HIGH SCHOOL HOBBY AND HIS CHECKING AND EXTENSION OF THE NINETEENTH CENTURY TABLES "AN AFTERNOON'S WORK." He just didn't look very closely at the 10-crossing knots. – S. Carnahan – 2013-11-23T05:08:22.247

11Image for the real Perko pair is gone :( – მამუკა ჯიბლაძე – 2015-09-29T17:09:28.307

2This answer is in need of some editing. The proposed edit by Perko should definitely be included, and the picture of the Perko pair should be restored. – Danu – 2015-11-08T07:53:19.707

1@მამუკაჯიბლაძე FTFY (meaning: "Fixed that for you" -- for civilians like myself -- Todd Trimble). – David Roberts – 2016-02-06T12:55:56.993

95

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 decreasing intersections.

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

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

15Projection doesn't even commute with finite intersection. – domotorp – 2014-11-02T12:33:58.733

4To generate Borel sets, it is enough to use countable decreasing intersections. Will correct. – Gerald Edgar – 2014-11-02T12:57:59.140

3The story was recounted by Kazimierz Kuratowski in his A half century of Polish mathematics." Of course neither Suslin nor his advisor Lusin were Polish, but Waclaw Sierpinski, a Polish mathematician who was interned in Russia at that time, witnessed the conversation between Suslin and Lusin in which the student communicated the discovery to the professor. – Margaret Friedland – 2015-04-10T17:55:03.687

84

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

18(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 – 2009-10-17T18:46:02.573

73

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.

The news of that "proof" was even given in an Iranian public newspaper at the time when a few Iranian (could) read anything but religious texts! – Amir Asghari – 2017-12-07T22:40:12.390

57

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 wrote, "... 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."

10I'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 – 2011-10-09T23:55:32.640

26You know what will be a great paper title? "Of mice and men" – Aleks Vlasev – 2011-10-10T06:50:57.073

5

Note that this story is not true.

– Danu – 2015-11-08T08:52:56.237

54

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."

12But does this qualify as an interesting mistake? – Todd Trimble – 2013-11-07T14:17:46.923

4@Todd, yes, in the sense that it is a fun mistake. – Joël – 2013-11-07T14:49:18.820

8@Joël I agree that it's an amusing mistake, but I was reading the question more in terms of mistakes that led to interesting developments (and I think the highest voted answers went with that reading). – Todd Trimble – 2013-11-07T15:39:40.523

3I was told this story by my lecturer in a graduate algebra course back in 2015, except the lecturer ended with the punchline "All right, take 59". There was a dead silence for about three seconds until we all realized what had happened and everyone started laughing. – Improve – 2016-03-28T02:47:40.447

8Grothendieck was not the first eminent mathematician to give 57 as an example of a prime. Hermann Weyl (American Mathematical Monthly 1951, p.532) mentioned Goldbach's conjecture about "primes of the smallest possible difference 2, like 57 and 59." – John Stillwell – 2016-10-29T18:47:49.180

1

@JohnStillwell: this seems to attribute the twin prime conjecture to Goldbach. This would seem to also be a mistake - see e.g. http://www.math.sjsu.edu/~goldston/twinprimes.pdf , which credits the conjecture to de Polignac.

– Michael Lugo – 2016-12-06T19:01:30.097

50

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/

56wait... really? this is serious, i use that a lot... dammnit! – Sean Tilson – 2010-03-31T06:35:01.050

7

Cf. http://mathoverflow.net/a/35864/27465 and jlms.oxfordjournals.org/content/73/1/65. A sufficient extra structure in an abelian category for this to hold is: Grothendieck's axioms AB3, AB4* and having a set of generators. In particular, it is true in module categories (and even categories of "almost modules").

– Torsten Schoeneberg – 2014-01-19T00:28:44.537

2Apparently all you need is that the abelian category have a set of generators, so for example it is true in the category of abelian groups. Also, my advisor, and i believe some others, use it differently then it is stated on wikipedia and in the paper... i think – Sean Tilson – 2010-04-06T18:15:01.707

46

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."

26I have always loved that way Abel wrote this (in a footnote): «it appears to me that this theorem suffers exceptions»... – Mariano Suárez-Álvarez – 2009-12-15T23:49:33.243

42Some (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 – 2009-12-16T16:40:17.677

46

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.

This is found on page 51 of A. Cayley, Desiderata 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.

Very nice one and very much in spirit of computer bugs! – Ilya Nikokoshev – 2009-12-24T18:51:01.153

Iǘe had a couple of students who are now going to be proud! :P – Mariano Suárez-Álvarez – 2009-12-25T23:09:33.323

1

Can you provide a reference? I just checked his paper "On the theory of groups, as depending on the symbolic equation θ^n = 1" (1854) which Wikipedia gives as the first definition of an abstract group. He says, "And we have thus two, and only two, essentially distinct forms of a group of six", and then gives their Cayley tables. The paper I was reading is http://pdfserve.informaworld.com/26005_751318052_910849049.pdf linked from http://www.informaworld.com/smpp/content~content=a910849049&db=all

– aorq – 2010-02-03T01:05:46.313

16It 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 – 2010-03-01T15:51:56.437

45

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.

+1, although I don't see this as a "mistake" in the sense of the other examples. He didn't do what he intended, but he wasn't wrong at any point. An inconsistent formal system is a perfectly fine mathematical setup, just one that people are mostly uninterested in. – Nikolaj-K – 2015-02-23T18:17:50.837

40

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

2

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: http://www.ams.org/mathscinet-getitem?mr=57:12066a

– Ben Linowitz – 2010-01-06T02:19:32.570

2I don't know whether it is appropriate to say "discovery" of ideals. Maybe "recognition of the importance/relevance of ideals"? – Kevin H. Lin – 2010-04-05T06:14:37.087

1I'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 – 2009-10-17T20:54:20.680

14It was actually Lame who came up with that bad proof. – Ben Webster – 2009-10-18T01:13:54.067

3Oh, ok my mistake. – Grétar Amazeen – 2009-10-18T14:27:28.907

39

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

39

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.

1A mistake in a close field (I can't do another answer): before Milnor, everybody thought it obvious that two topological n-spheres could have different structures as differentiable manifolds. I'm pretty sure Milnor himself thought he made a mistake when some invariant turned out to be different for two topological spheres. For details, see the third volume of his collected papers. – Ilya Grigoriev – 2010-03-13T08:02:56.850

8@IlyaGrigoriev: Do you mean "before Milnor, everybody thought it obvious that two topological n-spheres could NOT have different structures as differentiable manifolds"? – Jim Conant – 2015-09-29T13:30:50.943

2@JimConant: Yes, of course. – Ilya Grigoriev – 2015-10-01T23:09:05.760

I am glad this mistake had some good consequence. i had only heard that it delayed Bott's proof of his periodicity theorem, since it seemed to contradict that result. – roy smith – 2016-02-07T00:13:04.507

37

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!

2Paul Cohen told a version this story in his Complex Analysis class at Stanford when I was a graduate student. – Dan Ramras – 2013-11-07T22:02:49.630

34

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.)

3Whitehead's similar mistake is very interesting, too, as it lead him to the construction of contractible 3-manifolds that aren't balls. – Ryan Budney – 2009-12-15T21:22:48.083

2There is also a paper by Cartier with a similar title: "Comment l'hypothèse de Riemann ne fut pas prouvée." – Joël – 2013-11-07T14:50:59.767

4

Stallings's page went down. The paper is at https://web.archive.org/web/20140630032821/http://math.berkeley.edu/~stall/notPC.pdf

– Samuel Lelièvre – 2016-02-22T15:41:36.970

26

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.

I must add to this that this story is even linked with modern developments, in the following direction: if the 5 conics are all real, how many among the 3264 conics tangents to all 5 of them are real? In 1997, Ronga, Tognoli and Vust found an example where all 3264 tangents conics are real. In 2005, Welschinger proved that if the 5 conics are ellipses, no two nested one inside the other, then at least 32 of the 3264 tangent conics must be real. This is related to a lot of deep modern tools. – Benoît Kloeckner – 2015-04-11T08:41:18.277

Eisenbud and Harris consider this example so important that they named a book after the number 3264: D. Eisenbud, J. Harris: 3264 and all that. Cambridge University Press (2016). In a nutshell, what is is to be appreciated in this story is that mathematicians a hundred years later succeeded in completing a formal framework which guarantees that "number of plane conics tangent to 5 general plane conics" is well-posed. (The deeply problematic word to make sense of in the latter is "general".) – Peter Heinig – 2017-09-23T08:43:27.160

26

Poincaré's discovery of homoclinic points grew out of a extremely serious mistake he made in his original submission for a prize essay contest sponsored by Acta Mathematica in 1888. His original 200 page manuscript, on the restricted three-body problem, was evaluated by Weierstrass, Mittag-Leffler, and Phragmén, who had great difficulty following his arguments. Poincaré responded with a dozen further explanations, totaling 100 pages. After many further exchanges, the editors finally decided to accept the manuscript (this was, after all, Poincaré, and he must know what he's doing) and awarded him the prize.

But around the time of publication, Phragmén was still puzzled by some points and Mittag-Leffler wrote to Poincaré. They received back a telegram from Poincaré asking that publication be stopped immediately! Poincaré realized that his belief that the stable and unstable manifolds could not intersect transversally was wrong, and that such intersection points, which he later called homoclinic points, immediately forced very complicated dynamically behavior, invalidating much of his work. He wrote to Mittag-Leffler:

"I have written this morning to Mr. Phragmén to tell him about an error which I have committed and he has undoubtedly informed you of my letter. But the consequences of this error are more serious than I first thought. It is not true that the asymptotic surfaces are closed, at least not in the sense that I meant before. What is true, is that if one considers the two parts of that surface (which I yesterday still believed coincided with each other) they intersect along infinitely many asymptotic trajectories and furthermore their distance is an infinitesimal of higher order than $\mu^p$ however big p is.

I don't conceal from you the trouble this discovery gives me."

Mittag-Leffler immediately halted the presses and recalled all copies of this issue he could get, destroying them all (except for a few, one of which remains in the library of the Mittag-Leffler Institute). They asked Poincare to pay for the suppression of this issue, which he did.

Poincare then wrote a new essay, incorporating many of the added notes from the original, and this was the version that Acta Mathematica published (with no mention of the earlier one). Eventually Poincaré used this as the basis of his three volume classic Les méthodes nouvelles de la mécanique céleste.

A riveting account of this story is contained in Poincaré's discovery of homoclinic points by K. G. Anderson, Archive History of Exact Sciences, 48(2) (1994), 133–147.

And, apparently, Poincaré had to spend (all or most of) his prize money to pay for the destruction and reprinting of that Acta issue. Would be good to get an authoritative source for this. – David Roberts – 2016-02-06T12:57:58.650

The article by K. G. Anderson cited above does give authoritative account of these issues. It's not clear how much of Poincare'd prize was spent on the recall however. – Douglas Lind – 2016-02-08T06:11:22.480

24

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

1Just stumbled over this answer, and would like to point out that this sentence 'does not parse', as they say. I will not touch it, out of respect to any writing which is not technically-wrong, yet it might be good if you reformulated it. The 'shattered' is a bit extreme, by the way. And in case you don't have umlauts ready, here is an ö to copy and paste. @Thomas Riepe – Peter Heinig – 2017-09-23T08:48:20.790

1I think "induced by" could be replaced by "founded" and then it would read sensibly. – Todd Trimble – 2017-09-23T17:06:15.850

19

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.)

22I 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. – Andrés E. Caicedo – 2010-10-26T04:52:59.980

12But the urban legend is so funny... – Ilya Nikokoshev – 2009-10-19T20:02:58.533

15

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

"Let p be any old prime."

I can't find this in my copy of Hall's book :-( Google led me to a suggestion that it was on "p419" but the only mention of primes on p419 of my copy is in the middle of Theorem 20.9.13 where it's an odd prime. I have the second printing (published 1976). Maybe this is the problem? – eric – 2015-09-11T12:24:28.490

1@eric the mistake is indeed "old" instead if "odd" in the first edition! – Andrea Di Biagio – 2015-12-30T14:43:05.333

14

In chapter 3 of What Is Mathematics, Really? (pages 43-45), Prof. Hersh writes:

How is it possible that mistakes occur in mathematics?

René Descartes's Method was so clear, he said, a mistake could only happen by inadvertence. Yet, ... his Géométrie contains conceptual mistakes about three-dimensional space.

Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse—there are only so many things we can keep in mind at once.

Wittgenstein said that mathematics could be characterized as the subject where it's possible to make mistakes. (Actually, it's not just possible, it's inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.

Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren't uncommon. It shows that mathematical knowledge is fallible, like other knowledge.

...

Some mistakes come from keeping old assumptions in a new context.

Infinite dimensionl space is just like finite dimensional space—except for one or two properties, which are entirely different.

...

Riemann stated and used what he called "Dirichlet's principle" incorrectly [when trying to prove his mapping theorem].

Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)

Sometimes mathematicians try to give a complete classification of an object of interest. It's a mistake to claim a complete classification while leaving out several cases. That's what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [cf. this annotation by Peter Shor.]

Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer... Mistake or not?

...

"The mistakes of a great mathematician are worth more than the correctness of a mediocrity." I've heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable—rigorous deductions from premises. If you made a mistake, your deduction wasn't rigorous, By definition, then, it wasn't mathematics!

So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren't mathematics, and needn't be considered by the philosopher of mathematics.

Riemann's incorrect statement of Dirichlet's principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere.

A famous oversight of Euclid and his students (don't call it a mistake) was neglecting the relation of "between-ness" of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.

Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can't. Our mathematics can't be certain.

The reference to the said article by Philip J. Davis is:

Fidelity in mathematical discourse: Is one and one really two? Amer. Math. Monthly 79 (1972), 252–263.

From that article, I find particularly amusing the following couple of paragraphs from page 262:

There is a book entitled Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900.There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in "On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors", Philos. Mag., (1851) pp. 295-305. This error was corrected by H. E. Baker in the Collected Papers of Sylvester, Vol. I, pp. 647-650.

...

A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in Time magazine.

2IIRC, one reason this is interesting is that this got people looking at calculus of variations in more detail/anxiety than had been done previously (i.e. in arguments where one assumed a minimizing function existed, and then used properties of said function, it wasn't always clear that a minimizer existed) – Yemon Choi – 2009-12-16T06:22:41.960

2Re Poincaré's explanation: IMHO, it's frequently not just memory lapse, but just plain laziness. Whenever you see "it is obvious that", take a moment to see if it's really so obvious. I find this a useful rule of thumb when refereeing papers. – Todd Trimble – 2015-09-29T11:52:32.073

13

Bringing in sort of tragic flavor to this question, - the following came to my mind:

He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I've realized that it's very difficult to make good mistakes.

(Shimura on Taniyama, seen it in the "BBC Horizon Season 1996 Episode 2 - Fermat's Last Theorem" available on Youtube)

13

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

13

Petrovskiĭ-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 Petrovskiĭ-Landis ( complexification of real differential equations ) lead to the study of holomorphic foliations.

12

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. 2^257-1 is actually a composite number: its smallest prime factor is 535,006,138,814,359.

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.

65A reporter who knew about modulo 10 arithmetics, now that's quite a story! – Ilya Nikokoshev – 2009-10-22T07:43:50.073

There is a little bit more about Krieger at https://outofthenormmaths.wordpress.com/2011/11/09/blitz-krieger/ Amazon.com claims to have a book, Krieger's Mathematical Formulae, that he published in 1934, but the page also says, Out of Print--Limited Availability.

– Gerry Myerson – 2016-12-06T22:50:42.343

12

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

8

8

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.

10It seems odd to call a conjecture a mistake. – Jim Conant – 2015-09-29T13:23:51.750

7

Lakatos' work "Proof and refutation" contains many examples of mistakes concerning the development of Euler's polyhedron formula, along with an extensive treatment of what mistakes are and how they can crucially contribute to the development of mathematics.

7

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!

5

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

5

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

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

4

Foias constant also comes from a mistake. I quote directly from An interesting serendipitous real number by John Ewing and Ciprias Foias:

This is the story of a remarkable real number, the discovery of which was due to a misprint. Namely, in the midseventies, while Ciprian was at the University of Bucharest, one of his former students approached him with the following question:

(Q) If $x_1>0$ and $x_{n+1}=\left(1+\frac1{x_n}\right)^n$ $(n= 1,2, ... )$, can $x_n\rightarrow\infty$?

This was listed in a fall issue of the "Gazeta Matematica" as one of the problems given at the previous summer admission examination for prospective freshmen in the Department of Mathematics at the University of Bucharest. Ciprian found the answer in about one day, but considered that the problem was even above the sophomore level. He also found that (Q) is a misprinted version of the following question (given by Professor N. Boboc):

(Q') If $x_1>0$ and $x_{n+1}=\left(1+\frac1{x_n}\right)^{x_n}$ $(n= 1,2, ... )$, can $x_n\rightarrow\infty$?

This was an appropriate exam question since the answer is clearly "No." Years later, in the 1980s, Ciprian told the story to Professor P. Halmos, who in turn told the story to John, but mischieveously did not mention at all Ciprian's answer to (Q), so a day later John also found the answer. This answer is given by the following

Theorem 1.1 There exists exactly one real number $a\sim 1.187$ such that if $x_1=a$ then $x_n\rightarrow\infty$. Moreover in this case $$x_n\frac{\ln n}n\rightarrow 1 \text{ for } n\rightarrow\infty \ \ (1)$$ Relation (1) can be rewritten as $$\lim_{n\rightarrow\infty} \frac{x_n}{\pi(n)}=1 \ \ (2)$$ where $\pi(n)$ is the number of primes less than $n$. However after many attempts to establish a deeper connection with the Prime Number Theorem, we came to believe that relation (2) is fortuitous. A strong argument for this opinion is provided by Theorem 3.1 below in which we show that the estimate for the error $$x_n\frac{\ln n}n-1$$ differs from its analog in the Prime Number Theorem.

An interesting serendipitous real number. J. Ewing, C. Foias. Finite versus infinite. Contributions to an eternal dilemma. Springer (2000), pages 119-126.

4

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.

3

A wonderful mistake, which paved the way to singular cardinals, was done by Felix Bernstein in his dissertation. I learnt this from Menachem Kojman. Berstein thought he had proved that for every ordinal $\alpha$, $\aleph_\alpha^\omega=\aleph_\alpha \cdot 2^{\aleph_0}$. This is true for every $\alpha < \omega$ but already fails for $\alpha=\omega$. Berstein's mistake was to assume that every cardinal has an immediate predecessor.

Kőnig later used Berstein's result to prove that the continuum is not an aleph, thus disproving at once two of Cantor's main beliefs: 1) every set can be well-ordered and 2) the continuum hypothesis! He presented his result at the third International Congress of Mathematicians in Heidelberg in 1904 and the organizers cancelled all parallel session to allow all participants (which included Cantor and Hilbert) to attend Kőnig's lecture. And his discovery was even reported in the local news!

Here is Kőnig's reasoning:

First he proves the correct result that for every ordinal $\beta$, $\aleph_{\beta+\omega}^\omega>\aleph_{\beta+\omega}$ (a special case of what is now known as Kőnig's Theorem). He then reasons that if the continuum were an aleph, say $2^{\aleph_0}=\aleph_\beta$, then substituting $\alpha=\beta+\omega$ into Berstein's result one obtains that $\aleph_{\beta+\omega}^\omega=\aleph_{\beta+\omega} \cdot 2^{\aleph_0}=\aleph_{\beta+\omega}$, which is a contradiction!

1

For some additional details, see MR3409864. Kojman, Menachem. Singular cardinals: from Hausdorff's gaps to Shelah's PCF theory. In Sets and extensions in the twentieth century, 509–558, Handb. Hist. Log., 6, Elsevier/North-Holland, Amsterdam, 2012.

– Andrés E. Caicedo – 2017-12-22T22:32:43.597

3

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.

Do you have a source? I have trouble picturing why he would think that. – arsmath – 2013-11-07T14:05:20.177

2

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.

2

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

1Wh.. what ?? – Qfwfq – 2011-06-26T13:25:22.973

1

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). http://en.wikipedia.org/wiki/Hilbert%27s_axioms

– Kevin O'Bryant – 2011-06-26T19:38:55.447

4Any examples of such errors? Any interesting ones? – Ilya Grigoriev – 2010-03-13T07:55:54.573

1

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 http://www.gap-system.org/~history/Printonly/Theon.html corrected some errors), and from what I've read Hilbert did also.

– Kevin O'Bryant – 2010-03-14T19:36:01.043

But "Contains errors from start to finish", seems quite an exaggeration. – Pietro Majer – 2013-11-07T13:36:58.603

1Kevin, at the risk of asking something stupid: what's the problem with circles intersecting in 0, 1, or 2 points? – Todd Trimble – 2013-11-07T22:11:30.767

@Todd, perhaps what Kevin was referring to was the assumption that, for example, the circles centered at $A$ and $B$ with radius $AB$ intersect at all. This assumes a completeness which Euclid never made explicit. – Gerry Myerson – 2013-11-07T22:37:35.210

2@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). – Todd Trimble – 2013-11-07T22:47:25.767

@Todd, yes, unless Kevin is thinking of something that has escaped both of us, then I think it is as you say. – Gerry Myerson – 2013-11-07T22:55:48.707

2That'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? – Kevin O'Bryant – 2013-11-10T23:21:17.620

1

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

1

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.

-3

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

COuld you elaborate more on this? – Ilya Nikokoshev – 2009-10-26T22:05:38.500

9I 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 – 2010-03-13T07:57:24.210

-6

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

2

Well, I think maybe he was aware: http://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.

– Todd Trimble – 2015-04-11T19:57:13.673

6Seems 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 – 2009-10-17T15:12:19.457

1

Yes - fortunately their intuitions had enough force to made them jump over (or blind towards) the problems (and infinitesimals may <a href="http://www.maths.nott.ac.uk/personal/ibf/rem.pdf" title="Fesenko's essay">come back</a>). )

– Thomas Riepe – 2009-10-17T15:45:09.187