> In theory, maybe. In practice you have to work to reconcile the constructive "all functions are continuous" with classical results like "a strictly increasing function can be discontinuous on a dense set." [T]he two disagree on what a function is.

Well yes, it's still a different kind of math. What's not true however is this common notion that a constructivist can only ever view all "classical" math as pure nonsense. In many ways, it ought to be a lot easier for someone committed to constructivist semantics to grok a classical development than the converse - because decision procedures, computations etc. are way more of an afterthought to mainstream mathematicians.

That's why for me, constructivism - or in general mathematics with different axioms - are additions to classical mathematics, not replacements. And in that sense, they're fine for me. You'll definitely find people online defending the position that all classical maths is bullshit - I'm not saying that actual researchers behave like that, though.

But I'm also a strict formalist, or more accurately, I think whichever axioms happen to be useful should be used, without much need for there to be a epistemological commitment.