TITLE: What are the most important questions or areas of study in the philosophy of mathematics?
QUESTION [11 upvotes]: This question is intended to complement What mathematical questions or areas have philosophical implications outside of mathematics?

REPLY [3 votes]: George Lakoff and Rafael Núñez started studying embodied mathematics pretty recently.  The idea is that our ideas of mathematics are inextricable from our humanity as opposed to being platonic truths, and can all be understood in terms of metaphors for real-world concepts and our learning/acquisition of these metaphors.  To take a simple example, whenever we add numbers, our brains will always essentially be adding things to a pile, because that's what addition is to us.  If I've understood it correctly.  
Their theories really haven't caught on among mathematicians, in part because they're not mathematicians themselves, which is honestly a pretty big weakness among philosophers of math.  In my opinion, a lot of what they say is really silly (my brain has nothing to do with $\mathbb{R}$ being the unique completion of $\mathbb{Q}$ as well as the largest Archimedean field), but there is a kernel of truth there: we don't study math by taking logical step after logical step, we study it by thinking about things we have ways of comprehending.  For instance, we live in three-dimensional space, and so it's hard to study higher-dimensional things that we therefore can't visualize.  I think it would be a useful philosophical project to understand how the mathematicians who study such objects conceptualize them, and how that relates to our human nature, but that would obviously require a larger mathematical background.
Another important question is the origin of mathematical taste and beauty.  This is probably more important to mathematicians than to outsiders who don't have as much of a sense of it.  (Yet I'm convinced that a cool proof can be appreciated by anyone, if it's explained properly!)  I don't know who, if anyone, has written about this, but it definitely exists and is worthy of study.