This seems to be a largely accurate statement, though of course the existence of dozens and dozens (ok, maybe even hundreds) of mathematicians actively working in set theory, model theory, recursion theory, and proof theory shows that it’s not totally true. It’s interesting that logic seems to have a slightly higher (though still quite low) status within philosophy than in mathematics. And things like modal logic and intuitionist logic seem to be far more widely known in philosophy than mathematics, even among logicians.
Anyway, an interesting post in a mathematics blog I just discovered on the list of academic blogs at Crooked Timber tries to introduce some more logic to mathematicians. Even more interestingly for me, the commenter sigfpe writes:
Mathematicians don’t care about about foundations because they don’t matter! It’s like using a well encapsulated software library. You don’t care what the inner workings are as long as the library performs as specified. If someone came along and completely rewrote the library, only the users who had been exploting the pathological cases might actually notice anything had changed. Same goes for foundations. Even if someone came along and found a contradiction in ZF, say, it would probably be in some really abstruse area, someone would come along with a ‘patch’, we’d have a new set of axioms and 99.99% of published mathematics would remain unaltered.
Something like this is a position I proposed a few months ago when I wanted to come up with a controversial position to take for a panel discussion with some of my fellow graduate students in logic at Berkeley on new axioms in mathematics. After working it out a little bit more, I actually think I believe something like this, though perhaps at a slightly lower level than this person. That is, the reason that we have any standard axiomatizations of mathematics at all is so that mathematicians don’t have to resolve all their disagreements about the philosophy of mathematics. If the platonist, nominalist, and structuralist can all agree that ZFC is a good set of axioms, then they can all return to being productive mathematicians – but if we didn’t have ZFC (or something like it), then they’d have to convince each other that their methods were valid and didn’t presuppose something about the nature of mathematical entities.
As for mathematicians not caring about the mathematical foundations (set theory, proof theory, recursion theory, model theory), that’s a different question. My account predicts that they should care about this stuff at least slightly more than the ontological and epistemological questions. But since specific results in set theory and proof theory are almost never applicable elsewhere, and only restricted amounts of proof theory and model theory seem to have applications, it seems that mathematicians outside of foundations may be able to apply my strategy at an even higher level and ignore what goes on beneath their axioms. Of course, in this case there isn’t disagreement beneath their shared axioms, unlike at the philosophical level.
Fortunately, model theorists seem to be making contact with algebraic geometers and number theorists these days, so maybe some day mathematicians will start caring about the rest of logic again.
(There’s also a connection to Penelope Maddy’s naturalism here – she suggests that mathematicians really shouldn’t care about the philosophical issues behind the axioms, for mathematical reasons. I suggest that the reasons are slightly more sociological.)