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.
I think there are not many mathematicians who really believe that a non-constructive proof is worthless.
One reason for this is that, generally speaking, non-constructive proofs can in fact be embedded into constructive logical systems. I think there is more controversy about what formal foundations we should endorse for mathematics. ZF(C) is good enough as a proof of concept, but type-theoretical and category-theoretical foundations seem to be better in terms of actually doing formalized mathematics in the real world.
Half-closing my eyes and looking at the recent topic of morality from a distance, I am struck by the following trend.
In mathematics, there are no substantial controversies. (I am speaking of the present era in mathematics, since around the early 20th century. There were some before then, before it had been clearly worked out what was a proof and what was not.) There are few in physics, chemistry, molecular biology, astronomy. There are some but they are not the bulk of any of these subjects. Look at biology more generally, history, psychology, sociology, and controversy is a larger and larger part of the practice, in proportion to the distance of the subject from the possibility of reasonably conclusive experiments. Finally, politics and morality consist of nothing but controversy and always have done.
Curiously, participants in discussions of all of these subjects seem equally confident, regardless of the field's distance from experimental acquisition of reliable knowledge. What correlates with distance from objective knowledge is not uncertainty, but controversy. Across these fields (not necessarily within them), opinions are firmly held, independently of how well they can be supported. They are firmly defended and attacked in inverse proportion to that support. The less information there is about actual facts, the more scope there is for continuing the fight instead of changing one's mind. (So much for the Aumann agreement of Bayesian rationalists.)
Perhaps mathematicians and hard scientists are not more rational than others, but work in fields where it is easier to be rational. When they turn into crackpots outside their discipline, they were actually that irrational already, but have wandered into an area without safety rails.