Warning: the views expressed here are caricatured representations of papers that I’ve read through somewhat, but not carefully enough yet. They’re probably not actually the views of the philosophers mentioned.
Penelope Maddy makes a distinction in the philosophy of mathematics between questions that are “methodological” and ones that are “philosophical”. On her preferred view (which she calls “naturalism”), it is primarily the former that are actually interesting questions. The idea is that questions that tell us which sets of axioms to pursue are useful to actual mathematical practice, whereas questions about how we can know that 2+2=4 don’t seem to be. (That is, if they tell us that we don’t know this fact, then we can’t really do any math, and if they tell us that we do, then we’re right back where we started.) However, these methodological questions are principally the domain of mathematicians and “naturalistic philosophers of math”, who have basically become real mathematicians without the occasional philosophical worries. In a sense, this would suggest that philosophical questions properly stated can’t be relevant to mathematical practice.
However, during FEW I came up with what seems to be a reductio of this position:
1. If mathematicians care about question Q, then the answer should be relevant to their practice. (This seems clear if we’re talking about mathematicians caring about the question qua mathematician, and if they’re doing their job right.)
2. Philosophical questions can’t be relevant to mathematical practice. (This is my caricature of Maddy’s naturalism.)
3. No question mathematicians care about is philosophical.
However, this seems clearly false. I recently stumbled upon an article in the 1994 Mathematical Intelligencer by Bonnie Gold that basically just lists dozens of philosophical questions about mathematics that are interesting to mathematicians (and in many cases, much more interesting to mathematicians to philosophers). For example, “What is common to those subjects (e.g., algebra, analysis, topology, geometry, combinatorics, category theory) that are classified as mathematics which causes us to classify them, but not other subjects, as mathematics?” Or “Is mathematical knowledge more sure than other forms of knowledge? It seems so …, but is that simply an illusion?” Many of the other questions in the article seem more directly sociological than philosophical, but these ones seem philosophical to me. And I don’t think she’s being derelict in her duties as a mathematician to wonder about these questions.
I think premise 1 seems pretty clear. If we were to come up with a good classification of some subjects as mathematics and some as not, it seems that this would provide some insight into useful methods for mathematical work, and ideas for new fields of mathematics that haven’t been studied yet. And if mathematical knowledge can be shown to be no more sure than other forms of knowledge, this would increase the importance of alternate proofs and double-checking in mathematics. So the fault must lie with premise 2, which I’m guessing is therefore a misreading of Maddy’s position. Perhaps it’s because she’s using “philosophical” as almost a technical term, separating off all the “methodological” questions from it. If philosophical questions are just defined to be the ones that have no relevance to the working mathematician, then there’s no problem. But the questions mentioned above do strike most people as at least somewhat philosophical.
I think the real answer is that I’ll have to re-read some of Maddy’s more recent papers, and actually read her book “Naturalism in Mathematics”. I think she’s right to reject at least some seeming questions in the philosophy of mathematics, but it does generally seem to me that she goes a bit farther than I’d like.