Question

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.

Answer 1

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

Answer 2

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.

Answer 3

Appel and Haken's proof of the Four Color theorem.