What are the most important questions or areas of study in the philosophy of mathematics? This question is intended to complement What mathematical questions or areas have philosophical implications outside of mathematics?
 A: Here is a rather brief summary of the major positions on a couple of important philosophy of mathematics questions. First a list of questions:


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*Are we creating or discovering when we do mathematics?

*What are mathematical entities?

*How do you explain the unreasonable effectiveness of mathematics in natural science?
An early answer to these question was Plato's suggestion that mathematical entities (triangles, squares, numbers...) exist in the world of forms. So numbers really do exist, just in some kind of rarified universe that we only have partial access to. The idea of a "Platonic heaven" of numbers is not a particularly popular position any more.
Immanuel Kant thought that geometry and arithmetic were somehow innate (indeed, they were preconditions to experience of space and the passage of time). That is, without having an a priori understanding of space and number, we wouldn't be able to perceive the world around us as we do. Mathematical entities, for Kant, were real.
Logicism is a position that grew out of Russell and Whitehead's work on mathematical logic. The idea was that logic is somehow universal, and that all of mathematics is built up from these basic axioms, which are uncontroversially, universally, absolutely True. (With a capital "T").
Formalism started at roughly the same time with Hilbert. Here the idea is that mathematics is a game we play, with certain rules (axioms) laid down somewhat arbitrarily. This shares a lot with logicisim, but is importantly different in its attitude to the status of the axioms.
Structuralism is a much more recent position (although there are echoes of the ideas in Hilbert, Poincare and others). The idea here is that mathematics is about structures. We learn about structures through abstracting away incidental details to recognise patterns. What exactly structures are depends on what kind of structuralist you ask. Some (like Stewart Shapiro) think that structures really are out there in the world, somehow. That is, structures exist over and above the things that satisfy the structural relations. Others (like Michael Resnik) think they are not quite this real. Resnik's brand of structuralism looks a lot like nominalism: Nominalism about mathematical entities grows out of a more general position towards abstract entities.
As a commenter said above, philosophy isn't about results. I think the most important contribution of philosophy is to ask the big questions, like the ones above, and to map out the logical space of possible responses to them.
I have restricted myself to philosophy of mathematics and deliberately avoided philosophy of logic, which is its own cottage industry. Also, this summary reflects my own interests and biases and should not be taken as a representative summary of all the debates and positions taken in philosophy of maths. A more balanced summary can be found at the excellent Stanford Encyclopedia of Philosophy article by Leon Horsten.
A: Paul Benacerraf's identification problem is an argument against Platonism: Many different models satisfy Peano's arithmetic axioms. Which of these are really the Natural numbers? I.e., the uniquely intended set? We are unable to pick one account to the exclusion of all the others. Adding axioms won't help. Any such axiomatic system will be satisfied by multiple models. For example, is the number Three given by $3 = \{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}$ or by $3 = \{ \{ \{ \emptyset \} \} \}$? Benacerraf concludes that numbers are not sets, as most mathematicians believe.
A recent trend for dealing with Benacerraf's identification problem is fictionalism, a form of structuralism: "Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters." See http://plato.stanford.edu/entries/philosophy-mathematics/#Fic
Fictionalists include Hartry Fields, John Burgess, and Charles Chihara.
A: *

*Formal Undecidability and related topics (Tarski truth definability, Godel incompleteness, The hydra problem, Matiyasevich's negative solution of the 10th problem)

*Proof Theory (Gentzen's consistency proof, Girard's linear logic, Barendregt's Lambda Cube)

*The Continuum (Cantor's diagonalization, Cohen and Godel's independence proof, Robinson's analysis)

A: 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.
