I am a huge fan of Gian-Carlo Rota, who has been said to be the founding father of modern algebraic combinatorics. (He is also my mathematical grandfather-to-be.)
Rota was a philosopher as well as a mathematician, and wrote an entire book primarily concerning the philosophy of mathematics. His book is called Indiscrete Thoughts.
I’ve been reading this recently, and I highly recommend it. It reads like a novel; he motivates everything with enticing examples regarding mathematicians that he has known or familiar mathematical theorems and proofs. He brings up a lot of interesting points and questions, including:
Is mathematics ``created’’ or ``discovered’’? This is a common point of debate among mathematicians, and Rota addresses it beautifully. He gives clear and precise examples of mathematical work that is obviously one or the other, and then goes on to show how the two notions can, and do, naturally coexist.
How can we make rigorous some of the notions that mathematicians use all the time, but can never formally write about? There are plenty of processes that go on in our mind, leaps of faith and intuition, that we cannot easily talk about and use in a formal mathematical setting, because they are not part of established formal logic.
What is mathematical beauty, and why does it seem to depend on context and historical era?
Even if you don’t agree with Rota’s conclusions, his examples are so vivid and revealing that it’s impossible not to get something out of this book. I personally am coming away with a clearer perspective on mathematics and what it actually is.