I'll admit that calling all constructivists crackpots is unfair, as there are legitimate mathematicians studying homotopy type theory and whatnot that goes way over my head. What I refer to as crackpottery is people who argue that a random proof by contradiction is invalid or who reject things like Cantor's diagonalization argument with no justification besides their personal "intuition" or "philosophy" that math should be constructivist. And I think the latter group vastly outnumbers the former, at least on random internet forums. But I'll apologize if I offend any of those mathematicians with my flippant generalizations.
I'll not take offense, nor will I spout off about metaphysics here, but I can't let it pass without noting that Cantor's diagonal argument is in fact constructively valid --- given a function from a set S to its power set 2^S, Cantor constructs an element of 2^S which can't be in the image of f. The subtlety here is that there are statements equivalent to Cantor's theorem in classical settings that are not intuitionistically valid, for instance that there's no injection from 2^S -> S.
