I am wanting to begin a self-study concerning mathematical philosophy, and I was wondering if anyone would suggest reading material of a modern introduction to mathematical philosophy. I stress on modern, because reading something like Russell's Introduction to Mathematical Philosophy was difficult to understand, because of the way mathematical objects or properties of those objects were expressed or defined according to how they were defined during his time, as opposed to how they are define in the present.
No reference request on philosophy and mathematics would be complete without the work of Douglas Hofstadter. Godel, Escher, Bach: an eternal golden braid is a remarkably accessible effort to explain the philosophical implications of set theory and paradoxes. I am a Strange Loop is even more philosophical, though I do believe GEB is more famous. Depending on what you seek, his work may not be as in depth as you are looking for, but he's always a good starting point for building a list, especially when your list of mathematical philosophy documents may or may not include a SE question listing all documents that are not related to mathematical philosophy.
In the title of your question, you're mentioning both "mathematical philosophy" and "philosophy of mathematics", but these two subjects do not necessarily coincide. Mathematical philosophy is more of a methodology --a mathematical/formal approach to philosophical problems. Within mathematical philosophy one can be dealing with traditional problems from various fields, such as epistemology, metaphysics, philosophy of science, philosophy of language, ethics, political philosophy and (of course) philosophy of mathematics.
On the other hand, philosophy of mathematics is a specific area of philosophy (and not a methodological approach) which deals with problems pertaining to mathematics, such as the ontological status of mathematical objects (e.g., are mathematical objects real? do we discover or construct mathematical truths?), epistemology (how do we gain mathematical knowledge?), foundations (e.g., can mathematics be reduced to pure logic? Can we prove the consistency of our theories? Does category theory or set theory provide a foundation for all mathematics? Is mathematics justified by its succesful application to the natural sciences?) and so on, so forth.
Regarding mathematical philosophy, an excellent place to start is the following link with a great collection of resources (intended to offer a formal background/training in order for someone to be able to deal with philosophical issues by means of formal methods), provided by the Munich Center for Mathematical Philosophy:
Also, two great books, more or less to the same effect, are:
- E. Steinhart, More Precisely: The Math You Need to Do Philosophy, Broadview Press
- D. Papineau, Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets, Oxford UP
Regarding philosophy of mathematics, a great place to start is (as always) the corresponding entry on the Stanford Encyclopedia of Philosophy:
Great books that can be used as a general survey through the subject are:
- S. Shapiro, Thinking About Mathematics: The Philosophy of Mathematics, Oxford UP
D. Bostock, Philosophy of Mathematics: An Introduction, Wiley-Blackwell
S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford UP
and the classic collection:
- P. Benacerraf and H. Putnam, Philosophy of Mathematics 2ed: Selected Readings, Cambridge UP