Naturalizing Mathematical Methodology
'Naturalism' is a term much bandied-about these days, especially in discussions of scientific methodology. Many have supported the incursion of naturalistic thinking into the philosophy of mathematics as well; I myself have attempted to adopt a naturalistic approach to the study of settheoretic methodology. However, in recent efforts to spell out more precisely what 'naturalism' comes to in these contexts, I have come to suspect that much of what I once regarded as 'naturalistic' philosophy of mathematics does not deserve that title, that the act of 'naturalizing' has deeper consequences than have heretofore been acknowledged.
My goal here is to trace these consequences. I begin with a discussion of naturalism itself in Section I, describe a case-study and criticize one purportedly naturalistic approach to it in Section II, and conclude, in the final Section III, with an analysis of what's gone wrong and a hint at how to go right.
Like most popular philosophical notions, 'naturalism' means different things to different people. Consider, for example, the recent survey issue of the Philosophical Review: Tyler Burge, in his overview of philosophy of language and mind since mid-century, uses 'naturalism' as a synonym for 'physicalism', the view that there are no mental states over and above ordinary physical ones and that mental phenomena should be explained within materialistic physical science;1 only pages later, Philip Kitcher devotes his entire essay to a history and analysis of 'naturalism', this time understood, crudely, as the claim that scientific results in psychology and elsewhere are central to the pursuit of epistemology.2 My own usage differs from both of these, so I begin with a brief characterization.____________________