# Most interesting mathematics mistake?

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

• 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
• A discussion thread was started on meta partly inspired by this: tea.mathoverflow.net/discussion/284/…. – Jonas Meyer Mar 13 '10 at 0:19

## 42 Answers

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.

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

• Why? Did they claim any form of completeness? – Joël Mar 26 '18 at 20:20

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"

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.

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!

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 Nov 7 '13 at 14:05

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.

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.

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

• 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
• @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 Nov 7 '13 at 22:47
• 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? – Kevin O'Bryant Nov 10 '13 at 23:21

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

• But was this a fruitful mistake? – Jim Conant Sep 29 '15 at 13:27

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 Oct 26 '09 at 22:05
• 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.

• 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
• 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. – Todd Trimble Apr 11 '15 at 19:57

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

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