How much math should I know in order to delve really deeply into philosophy of mathematics? Must philosophers of mathematics be mathematicians, as well?
Philosophers of mathematics don't have to be mathematicians, but it would be helpful to know a fair amount of math. Here are some more specific suggestions: 1. You need to study enough math to appreciate the role of proofs in mathematics. Usually students don't see this until they get beyond calculus. Courses on analysis and abstract algebra would show you this side of mathematics. (Abstract algebra would also allow you to see the role of abstraction in math.) 2. There are ideas from logic, such as Godel's incompleteness theorems, that are important in philosophy of math, so you should study logic. 3. You should know about how some of the fundamental objects studied in mathematics are defined. For example, how are the real numbers defined? Usually mathematicians trace the ideas behind the definitions of these fundamental mathematical objects back to set theory, so it would be good to learn some set theory.