I wish some day a philosopher would come who would make my naive ways and habits of thinking more efficient but the more I look at their works, the more I am aware of the abyss that separates "natural sciences" and, say, "epistemology". I want to know and they want to know what it means to know. I ponder whether some particular way to approach the problem is right or wrong and they ponder over what "right" and "wrong" might mean. Sometimes it seems to me that, despite we (occasionally) use the same words, our languages describe two non-overlapping parts of "reality".
This observation makes me somewhat disagree with David: the philosophy of mathematics has about as much to do with the mathematics itself as history has to do with ruling a kingdom and as car reviews have to do with the engine manufacturing or winning a car race. A good historian may be a dismal warrior or organizer. A top journalist for "Car and driver" may have a very faint idea of how the internal combustion engine works and may skid into the ditch on the first patch of ice when driving any of the cars he recommends for Winter driving. So a graduate degree in mathematics is neither necessary, nor sufficient for a philosopher of mathematics, IMHO.
What is crucial for the outside look the philosophy takes (or, at least, pretends to take) is not the knowledge of the nitty-gritty of the thing under consideration but the knowledge of the place of that thing in some larger entity. A mechanic doesn't care about the atomic structure of the metal the pistons are made of but he makes sure they are in the right position when assembling the engine. The driver may have only a remote idea of how to assemble the engine but he rings an alarm if the response to controls is unusual. And so on. The philosophers are (or, at least, seem to be) near the top of that "chain of blissful ignorance". I have no more ability to see what they are looking at than a metallurgist has to see how the aluminum sheet he produced is connected with other sheets on the plane wing or the engineer has to see how the air traffic controllers direct the plane to a safe landing at JFK. Like any craftsman, all I can do is to say "I've made this thingy. It needs this to operate and will give you that if you use it in such a way".
This brings me to an uneasy question of what exactly is that larger entity that is seen by philosophers and where our craft fits in it. I guess the best thing to do would be to let the philosophers answer that themselves. Once that answer is given, we can, probably see much better which sides of mathematics have importance to them and which don't. Without that, we just risk to have an attitude of a miner who insists on a jewelry expert's being at ease with a sledgehammer.
Note that I am not talking about (ab)using mathematical methods in philosophy at all here. It is a separate story having nothing to do with the subject I tried to address in my unsophisticated English. All I'm really saying is that the word "mathematics" has a different meaning for them altogether and I'd better figure out what exactly it is before giving any opinion on the issue.
