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?

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.
