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It is widely believed that mathematicians have a uniform standard of what constitutes a correct proof. However, this standard has, at minimum, changed over time. What are some striking examples where controversies have arisen over what constitutes a correct proof?

Examples of this include:

  1. The acceptability of the use of the axiom of choice
  2. The acceptability of proofs that rely on assuming that a computer has performed a certain computation correctly
  3. The debate over intuitionistic logic versus classical logic
  4. Hilbert's re-examination of Euclid's axioms and his discovery of unstated assumptions therein
  5. Debates over the use of infinitesimals in calculus, culminating in Weierstrass's epsilons and deltas. There are of course many others.

Edit:

The above re-formulation of the question was provided by Timothy Chow and copied directly from the meta.mathoverflow thread about this question.

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    $\begingroup$ mathoverflow.net/questions/13896/… $\endgroup$
    – Steve Huntsman
    Commented Jun 6, 2010 at 23:17
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    $\begingroup$ Re: first sentence: let me introduce you to a mathematician named Euler... $\endgroup$
    – Qiaochu Yuan
    Commented Jun 6, 2010 at 23:47
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    $\begingroup$ I've voted to close as I think this question is too controversial and will generate more heat than light. $\endgroup$
    – Noah Snyder
    Commented Jun 7, 2010 at 5:35
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    $\begingroup$ well, the only possibly controversial issue is the idea that a question about controversial be controversial. $\endgroup$
    – Pietro Majer
    Commented Jun 7, 2010 at 10:53
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    $\begingroup$ I don't particularly dislike controversial issues, but I second the comment on "more heat than light". I'm not even sure that I agree with the statement that Cantor's theory was controversial. Sure, Kronecker didn't like it, but he felt the same about a lot of Dedekind's work. Which mathematical community didn't accept Cantor's work? Many people perhaps thought it wasn't particularly important, but that's not the same thing as being controversial. $\endgroup$
    – Franz Lemmermeyer
    Commented Jun 7, 2010 at 17:55

3 Answers 3

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Paolo Ruffini's work on the impossibility of solving the quintic by radicals did meet a strong passive resistance. Around 1800 he proved the theorem up to a minor gap, that himself or somebody else could have fixed soon, had his book met the attention that deserved. But times were not ready for a such a revolutionary idea as proving the impossibility; 20-30 years later this idea had slowly spread and become more natural, and Abel and Galois got more lucky (so to speak).

This is in my opinion a major example of a particular theorem that was met with resistance before being accepted, and in fact it also shows that resistance is not necessarily associated with controversials, but sometimes even with indifference (which may be even worse).

A short and well written account of the story is in J.J.O'Connor and E.F.Robertson's article for the History of Mathematics archive: http://www-history.mcs.st-and.ac.uk/Biographies/Ruffini.html

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    $\begingroup$ By the way, Ruffini's 1799 proof can be downloaded in full off Google books. It is a two volume work Teoria generale delle equazioni totaling 548 pages! A proof that long was unheard-of in those days (and pretty rare today, outside of group theory) which is surely one reason that the book was not studied carefully. $\endgroup$
    – John Stillwell
    Commented Jun 7, 2010 at 6:01
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    $\begingroup$ Anyone who today is willing to read an illegible 550pp manuscript from an otherwise unknown mathematician is allowed to call Ruffini's proof controversial. Who is going to cast the first stone on the mathematical community (= Lagrange and Cauchy; no one else probably could have understood the work anyway) in 1800? $\endgroup$
    – Franz Lemmermeyer
    Commented Jun 7, 2010 at 17:59
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    $\begingroup$ the pages are smaller than a modern page of maths, at a guess i'd say 50 characters per line, 20 lines per page versus maybe 80 characters by 40 lines for a trans ams article -- so a back of envelope calcuation says maybe it's more like 170 pages in modern typesetting? $\endgroup$
    – Max Flander
    Commented Jun 9, 2010 at 4:25
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    $\begingroup$ On a plus side (for us and Lagrange, but maybe not most 1800s mathematicians), it was written in Italian. I find it rather remarkable! $\endgroup$
    – Victor Protsak
    Commented Jun 9, 2010 at 5:04
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    $\begingroup$ Well, at that time, reading Italian was quite natural for educated people. $\endgroup$
    – Pietro Majer
    Commented Jan 29, 2013 at 9:54
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Appel and Haken's proof of the Four Color theorem.

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Hsiang's proof of the Kepler conjecture has never been accepted by the mathematical community. The proof by Hales and Ferguson has fared better but there is still some resistance to it, which is one reason Hales is pursuing the Flyspeck project.

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    $\begingroup$ Proof --> "proof". Gaps in Hsiang's approach have been well documented. $\endgroup$
    – Victor Protsak
    Commented Jun 7, 2010 at 4:53
  • $\begingroup$ @VictorProtsak: Hasn't Hsiang already responded to them in the article A Rejoinder to Hale's Article? $\endgroup$
    – user57432
    Commented Dec 2, 2018 at 14:33
  • $\begingroup$ @user170039 : Hsiang has indeed responded but the mathematical community has judged Hsiang's response to be insufficient. In a nutshell, there are many gaps in Hsiang's proof; Hsiang says that these gaps are trivial for any competent mathematician to fill, but nobody else seems to know how to fill most of them. $\endgroup$
    – Timothy Chow
    Commented Dec 2, 2018 at 19:41
  • $\begingroup$ @TimothyChow: Very interesting. Thanks for the response. $\endgroup$
    – user57432
    Commented Dec 3, 2018 at 3:06

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