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