Mathematics

How much do you need to know about mathematics to begin learning about the philosophy of mathematics or, for example, read something like The Principles of Mathematics or Principia Mathematica by Bertrand Russell?

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How much do you need to know about science to begin learning about the philosophy of science? Some but not a great deal if you are interested in very general metaphysical questions about e.g. the nature of explanation, laws and causation, and in very general methodological questions about how scientific theories are confirmed and refuted. You'll need to know quite a lot more if you are interested in understanding more specific foundational questions about the interpretation of quantum mechanics or are worrying about the nature of natural selection.

Similarly, how much do you need to know about mathematics to begin learning about the philosophy of mathematics? Some but not a great deal if you are interested in very general metaphysical and epistemological questions about e.g. the nature of numbers and the nature of our knowledge of such things (if they are "things"). Quite a lot more if you are bugged by more specific questions about how we are to settle axioms for set theory or to decide whether category theory offers a new kind of foundational framework for mathematics.

Most (but not all) beginners get into the philosophy of science via the pretty general metaphysical and methodological questions -- on which typical introductory books concentrate. Likewise most but not all beginners get into the philosophy of mathematics via the pretty general questions which require little mathematical background to grasp. And you can read a really excellent introduction like Stewart Shapiro's Thinking about Mathematics, which concentrates on the more general questions, without knowing much mathematics at all.

For more on this general theme, see also my answer to Qn 2938.

You also mention Russell. Well, you don't need a great deal of mathematical background to read The Principles of Mathematics -- but you do need a lot of historical background to understand what on earth is going on in some of Russell's pretty puzzling philosophical discussions: it is a confused and confusing great shambles of a book. That's not the place to start! As for Principia that is best admired from a great distance, and is pretty much now a historical curiosity -- a noble failure.

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