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." (The two disagree on what a function is.)

As for choice, try doing much functional analysis without Zorn's lemma.

There are parts of math that you can do constructively. But most of it you really can't. And choice is embedded in a lot more math than you'd guess.

> 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.