[black_knight]

This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game.

I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. 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. So, unless they are just playing a game, the mathematicians should care!

[steppi]

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.

[black_knight]

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

[johngossman]

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.

[black_knight]

We are not talking about having to return to foundational axioms in every argument! Just that what axioms one chooses has an impact on which arguments are valid, and thus in turn what truths there are.

[AnimalMuppet]

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.

[black_knight]

Very little of mathematics, like analysis? I am sure the analyst will care about all functions on the reals suddenly turning continuous. (Or rather losing the discontinuous ones)

Or what of commutative algebra and their beloved existence of maximal ideals!

[romangarnett]

Do you not care if your vector space has a basis?

[LegionMammal978]

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.

[soVeryTired]

In my experience those debates are usually between experts who deeply understand the difference between ABC and XYZ widgets (the example I'm thinking of in my head is whether manifolds should be paracompact). The decision between the two is usually an aesthetic one. For example, certain theorems might be streamlined if you use the ABC definition instead of the XYZ one, at the cost of generality.

But the key is that proponents of both definitions can convert freely between the two in their understandings.

[Sniffnoy]

> 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.)

[oh_my_goodness]

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.