ABSTRACT

What is ‘naturalism’, and what should a naturalist say about the abstract mathematical objects that are posited by our best empirical theories? Ontological naturalism is suspicious of abstracta, taking it that the only objects we should accept are those that enter into natural – causal – processes. On the other hand, methodological naturalism suggests that if mathematical posits are essential to our best scientific theories, we ought to believe in the mathematical objects posited. This tension between forms of naturalism is explored, and it is suggested that the argument from methodological naturalism to the existence of abstract mathematical objects, via the indispensability of mathematical posits in our best scientific theories, can be resisted. Finally, though, we consider the possibility raised by Maddy (2011) that there may be no fact of the matter as to whether methodological naturalism requires us to accept the existence of mathematical objects.