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.

I am tempted to back up a level and say that there is little or no dispute about conditional claims: if you give me these axioms and these rules of inference, then these are the provable claims. The constructivist might say, "Yes, that's a perfectly good non-constructive proof, but only a constructive proof is worth having!" But then, in a lot of moral philosophy, you have the same sort of agreement. Given a normative moral theory and the relevant empirical facts, moral philosophers are unlikely to disagree about what actions are recommended. The controversy is at the level of which moral theory to endorse. At least, that's the way it looks to me.