> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.

In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.

> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study).

I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.

Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you had to state which derivation of real number you were using. And I guarantee at least 90% of PhD mathematicians could do so if they really needed to.

I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure math as a hobby.

To be fair, in some fields I've seen arguments between "a widget should be defined as ABC" vs. "a widget should be defined as XYZ", to the point that I wonder how they're able to read papers about widgets at all. (If I had to guess, likely by focusing on the 'happy path' where the relevant properties hold, filling in arguments according to their favored viewpoint, and tacitly cutting out edge cases where the definitions differ.)

So if many mathematicians can go without fixed definitions, then they can certainly go without fixed foundations, and try to 'fix everything up' if something ever goes wrong.

> But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false.

I mean, mathematicians do care about the part of the foundations that affect what they do! Classical vs constructive matters, yes. But material vs structural is not something most mathematicians think about. (They don't think about classical vs constructive either, but that's because they don't really know about constructive and it's not what they're trying to do, rather than because it's irrelevant to them like material vs structural.)

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets.

You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abstraction away from what I'm doing. I pretty much only care about stuff at my layer, and maybe one layer above or below.

Try to be charitable. Remember, research mathematicians aren't HN commenters. They're forced to live within their intellectual limitations, however narrow those may be.