How good does one need to be in mathematics to do good work in philosophy of
You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!):
- "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that.
- "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too.
- "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about what can be reconstructed in various weak subsystems of full analysis, and about what infinitary assumptions these subsystems need.
- "Is there a unified justification for all the axioms of the standard set theory ZFC?" You better know a bit of set theory to tackle this -- but you perhaps needn't know much about e.g. large cardinal axioms that take us beyond ZFC!
- "Do we need axioms that take us beyond ZFC? -- and if so, what?" For this, by contrast, you will need to know more about the fancier reaches of set theory.
- "Does category theory in any sense provide 'foundations' for mathematics rivalling the set theoretic approach?" Well, plainly, you better know some category theory to tackle this one.
- "Why is the question whether P = NP so hard?" And you'll need to know qutie an amount about complexity theory for this one.
- "How satisfactory is Lakatos's model of the growth of mathematical knowledge?" Here, by contrast, you'd be better equipped if you know a little about a lot of mathematics, rather than a lot about a little, so your discussions aren't to suffer from an impoverished diet of examples.
And so it goes. What maths you need to know will vary a lot with which philosophical questions bug you. And sometimes it is difficult to tell in advance, when you start thinking about a topic, just how embroiled you will need to get in the mathematical nitty-gritty. But that's part of the fun.