a tangential response on mathematics

Today there is little disagreement over inference, but a century ago there was a well-known conflict over the axiom of choice and a less known conflict over propositional logic. I've never been clear on the philosophy of intuitionism, but it was the driving force behind constructive mathematics. And it is pretty clear that Platonism demands proof by contradiction.

As for axioms of set theory, Platonists debate which axioms to add, while formalists say that undecidability is the end of the story. Platonists pretty consistently approve higher cardinal axioms, but I don't know that there's a good reason for their agreement. They certainly disagree about the continuum hypothesis. That's just Platonic set theorists. Mainstream mathematicians tend to (1) have less pronounced philosophy and (2) not care about higher cardinals, even if they are Platonists (but perhaps only because they haven't studied set theory). Bourbaki and Grothendieck used higher cardinals in mainstream work, but lately there has been a turn to standardizing on ZFC.

Going back to the more fundamental issue of constructive math: many years ago, I heard a talk by a mathematician who looked into formal proof checkers. They came out of CS departments and he was surprised to find that they were all constructivist. I'm not sure this reflects a philosophical difference between math and CS, rather than minimalism or planned application to the Curry-Howard correspondence.