OK, so not so much a puzzle as a question this time.
I am currently co-teaching a graduate seminar on philosophy of mathematics this semester (structuralism versus logicism, to be more specific). We did a pretty good job of advertising the seminar, and as a result have a number of mathematicians sitting in the class (both faculty and graduate students).
The issue is this: As we talk about the philosophical questions and their possible solutions (for example, last week we read Benecerraf's "What Sets Could Not Be" and "Mathematical Truth", since these set up the issues at issue between modal structuralism and Scottish logicism quite nicely), the mathematicians kept coming back to the fact that none of these issues seem to have any bearing on what mathematicians actually do.
At one level I agree with this - when actually doing mathematics, mathematicians need not, and probably ought not, be thinking about whether their quantifiers range over abstract objects or something else. Rather, they should be worrying about what follows from what (to put it in an overly simplistic way).
There might be an exception to the above paragraph in moments of mathematical crisis - for example, if one were a nineteenth-century mathematician working in real analysis. But in general the point seems, on a certain level, right.
On the other hand, however, it seems obvious to me that mathematicians will benefit from thinking about philosophical issues (and benefit qua mathematician). But it is somewhat difficult to articulate why they would benefit.
So, any thoughts? In short, what should we say to mathematicians regarding why they ought to care about what philosophers say?