[mikehain]

Philosophy often revolves around questions of purpose in life, and any non-mathematician might wonder why someone would devote their life to the study of advanced math. The question of purpose becomes even more interesting when a mathematician spends decades solving theoretical math problems that have no immediate practical use in engineering or programming. Any mathematician who's engaged in such problems wouldn't have any trouble understanding his own motives; for him, the philosophy is so obvious that it goes without saying. But an outsider might see his actions as inexplicable - as inexplicable as an ascetic monk spending a week meditating on a frigid mountaintop.

Philosophy functions as an aid in that understanding. The mathematician might read it and recognize truth, if it is written well, and a non-mathematician might be able to learn the rationale for devoting one's life to math. It might have a strong enough effect to make someone gravitate towards that type of life.

Different works in the philosophy of math could deal with different aspects of the mathematical life, such as the inherent beauty of an elegant proof, and it could also go into the larger, more long-term purposes of math, whose effects might not be recognized until long after the mathematician is dead.

I would think that the best philosophy of math would need to be written by a mathematician. An issue with philosophy is that it's quite difficult to write it well. There is a lot of philosophy out there that's poorly-written. It might suffer from a lack of clarity or simply be inaccurate. To have a brilliant philosophy of math, you would need a person who is both a skilled mathematician and a skilled writer. That is a rare thing.