Object realism versus mathematical structuralism

In this article, two possible standpoints concerning the nature of mathematical entities are discussed: object realism (according to which the identity of mathematical objects is determined by their intrinsic properties); and structuralism (according to which it is determined only by their relations to other mathematical objects). I discuss some motivations that underlie the structuralist view of mathematical entities, and indicate some philosophical drawbacks of this position. In particular, I argue, that these motivations and intuitions violate basic ontological intuitions concerning the notions of set and membership. I also claim that, from the point of view of mathematical structuralism, the main problem of object realism is the reference problem. On the other hand, from the point of view of object realism, mathematical structuralism has to accept very strong ontological assumptions (reifying classes of isomorphism). It is also not quite clear how the notion of identity is understood from the point of mathematical structuralism. Finally, I argue that the difference between mathematical structuralism and object realism is (at least partly) a matter of formulation.