> Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians?
I would hope so! The short answer is the philosophy of Math will help you determine whether what you're researching is true! Surely it would be very bizarre to research something with complete apathy regarding its truth value. A few examples:
The famous Peano axioms [1] are widely used to prove such things as the commutative property of multiplication (ab=ba). But as the name "axiom" suggests, you just have to accept them as true or the whole thing crumbles. So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism? If so, what are its limits? When is mathematical research warranted, and when can we simply regard mathematical beliefs as properly basic?
Also, consider the realist/anti-realist debate [2, 3] which seeks to answer the question "are numbers, sets, functions, etc. actual features of the real world, or are they all just in our heads?" (or some refined variation thereof). If they are real entities, how is it that these non-causal things (like 5) lie at the very heart of the laws governing the physical, causal universe? But if they aren't real, then what possible explanation can one give for the perfect harmony of the physical world and these functions, that are ultimately all in my head? Moreover, why is belief in these unreal entities so widespread (I know of no "amathists")?
>So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism?
Math major here. To me, this seems like a non-nonsensical question. 0 is defined to be a natural number. You could say (as Platonism does), that there is some "real" natural numbers out there in the universe, and ask if the Peano axioms define a system that is isomorphic to the real "natural numbers". Or you could ask if the Peano axioms acuratly define the artificial system that we informally think of as the natural numbers.
You could also ask if any system (real or abstract) can satisfy the Peano axioms. That is to say, are they internally consistent (and is there a constructive proof of such). Assuming this is the case, then (by definition) there will be an object that we call "0" that is a natural number. You can also ask if there is a unique (up to isomorphism) instantiation of the Peano axioms (there is).
> ...and ask if the Peano axioms define a system that is isomorphic to the real "natural numbers". Or you could ask if the Peano axioms accurately define the artificial system...
This is exactly what I meant. You can define anything you want. You can even do so in a sophisticated way such that your defined system is internally coherent. But internal coherence alone can't be the standard of measure of truth -- we need to ask "does our internally coherent system correspond with reality?" To the questioner on MathOverflow, math will answer the coherence question, philosophy of math will answer the correspondence question.
> To me, this seems like a non-nonsensical question. 0 is defined to be a natural number.
Is it merely definitional though? They're called "natural" numbers for a reason! These are the set of numbers that seem most obvious to us, the kind that (I don't think) we can teach. For example, you can tell a child "this is 1 apple", "these are 2 apples", etc., but other than the name, you really can't teach a person what numbers are. They just know. Why is that? It's clearly is more than merely definitional. And I think 0 falls into this category unlike, say, complex numbers. But notice, whatever your response -- even if you disagree -- we're deep into philosophy territory here with the simple axiom "0 is a natural number".
> For example, you can tell a child "this is 1 apple", "these are 2 apples", etc., but other than the name, you really can't teach a person what numbers are. They just know. Why is that?
I would argue that this is a question for psychology, with input from neuroscience, biology, and likely numerous other fields of science. What you are asking is not a question about the universe, but rather a question about the human mind: why is it that the natural numbers seem innate to humans. In the same way we can ask why language is innate in humans, or the skills to walk.
Regarding 0, historically, 0 has been much more controversial as a number. To the best of my knowledge, we have no evidence of it existing as a number prior to around 400AD India. Also, for the record, 0 being a natural number is denominational. So much so that many (most?) mathematicians (myself included) do not consider 0 to be a natural number. Math has not imploded because of this, we just waste a little bit of effort here and there to clarify what we mean when it is an important distinction.
>And I think 0 falls into this category unlike, say, complex numbers.
Funny you should bring up complex numbers. If we look at physics, we find that the complex numbers seem far more "natural" than the natural numbers do. In fact, I cannot think of a single physics theory defined over the natural numbers. In contrast, the complex numbers show up all the time. Even quantum mechanics, which (being quantized) would seem to be an ideal candidate for a natural number theory, ends up being defined heavily in terms of complex numbers.
> You can even do so in a sophisticated way such that your defined system is internally coherent
Rest assured, any mathematical or logical system capable of expressing arithmetic is incoherent, if only trying to prove so using it's own rules. (You can shift the burden of proof to an external system you trust, but that external system would suffer from the same problem if a similar proof was attempted)
But that sort of gives away the game, doesn't it? There is no perfect harmony in the real world, that only exists in the stylized world of scientific modeling. If you take scientific modeling on faith, or worse, if you adopt a "good enough is" position, then you've already smuggled in mathematical realism and given yourself a foregone conclusion.
I would hope so! The short answer is the philosophy of Math will help you determine whether what you're researching is true! Surely it would be very bizarre to research something with complete apathy regarding its truth value. A few examples:
The famous Peano axioms [1] are widely used to prove such things as the commutative property of multiplication (ab=ba). But as the name "axiom" suggests, you just have to accept them as true or the whole thing crumbles. So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism? If so, what are its limits? When is mathematical research warranted, and when can we simply regard mathematical beliefs as properly basic?
Also, consider the realist/anti-realist debate [2, 3] which seeks to answer the question "are numbers, sets, functions, etc. actual features of the real world, or are they all just in our heads?" (or some refined variation thereof). If they are real entities, how is it that these non-causal things (like 5) lie at the very heart of the laws governing the physical, causal universe? But if they aren't real, then what possible explanation can one give for the perfect harmony of the physical world and these functions, that are ultimately all in my head? Moreover, why is belief in these unreal entities so widespread (I know of no "amathists")?
[1] https://en.wikipedia.org/wiki/Peano_axioms
[2] https://plato.stanford.edu/entries/platonism-mathematics/
[3] https://plato.stanford.edu/entries/scientific-realism/