What do you think about debates about which axioms or rules of inference to endorse? I'm thinking here about disputes between classical mathematicians and varieties of constructivist mathematicians), which sometime show themselves in which proofs are counted as legitimate.

First-, or possibly second-order predicate logic has swept the board. Constructivism is just a branch of mathematics. Everyone understands the difference between constructive and non-constructive proofs, and while building logical systems in which only constructive proofs can be expressed is a useful activity, I think there are not many mathematicians who really believe that a non-constructive proof is worthless. There is some ambivalence toward such things as the continuum hypothesis and the axiom of choice, but those issues never seem to have any practical import outside of their own domain.