Mathematical Realism: What’s New?

The recent debate on new realism has been widely influenced by Putnam’s views, especially by the distinction between scientific realism and natural or common seense realism. I locate the discussion on mathematical realism in the context of this wider debate. I suggest that a parallel distinction between science-based arguments for realism and more immediate forms of realism is avaiable for mathematics too. I point to differences between contemporary empiricist and intellectualist positions, and stress what I take to be some of the most relevant aspects on which research in this area shall be pursued in the near future, especially concerning the problem of whether philosophical priority should be given to pure or applied mathematics.