I've been interested in the philosophy of maths for quite a long time. In the last year of my PhD I started writing a pair of seminars on the subject to be given at the Maths department at the University of Warwick. Unfortunately, writing up my thesis got in the way and I never got round to writing the second part.
The first part is largely finished though, and briefly covers the classical views on mathematical philosophy from the early half of the twentieth century (foundationalism), as well as a slightly more recent and radical alternative called quasi-empiricism, due to Imre Lakatos (a student of Karl Popper). I was planning to go on to talk about Wittgenstein and Putnam.
The second part was going to focus more on mathematical practice and how things might develop in the future. I was going to compare the broadly syntactical approach characterised by computer proof and projects like Mizar, and the broadly semantical approach advocated by people like William Thurston (whose attitude to fully worked out rigorous proofs is notorious).
Finally, I was going to talk about what advances in computer technology and algorithms might do to mathematics in the future. Roughly, I was going to say that proof assistants and guided computer proof will likely become a serious part of mathematics within the next, say, 30 years, and that this will change the nature of mathematical practice considerably. It will become possible to focus less on the mechanics of proof (which will be left to the computers), and more on the meaning and underlying ideas. This sounds like a trivial point, but the implications for the teaching and practice of mathematics are enormous. Almost all current mathematical training at present focuses on the mechanics of proof, and much of the expended effort in mathematical research is geared towards the production and checking of proofs.