THE JOURNAL OF PHILOSOPHY
… naturalism: abandonment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method … The naturalistic philosopher begins his reasoning within the inherited world theory as a going concern. He tentatively believes all of it, but believes also that some unidentified portions are wrong. He tries to improve, clarify, and understand the system from within. He is the busy sailor adrift on Neurath’s boat.4
Epistemological studies, in particular, are to be carried out within science, with the help of relevant psychological theories.
From the perspective of this scientific naturalism, a philosopher can criticize scientific practice, but only on scientific grounds, as a scientist might do, for good scientific reasons. This is enough to ratify an appeal to scientific practice in philosophical contexts: be- cause scientific practice can only be questioned on scientific grounds, a conflict between scientific practice and philosophy must be resolved by revising the philosophy. So, for example, if scientific practice holds that p does or does not count as evidence for q, to disagree on philosophical grounds is an offense against naturalism.
As we shall see, however, it is not clear that Quine intends to extend this naturalistic faith in practice to the practice of mathemat- ics. Leaving Quine aside for the moment, we must ask ourselves what the role of the philosophy of mathematics should be. Mathe- matics, after all, is an immensely successful enterprise in its own right, older, in fact, than experimental natural science. As such, it surely deserves a philosophical effort to understand it as practiced, as a going concern. Indeed, as in any discipline, there remain con- ceptual confusions in mathematics that might be clarified by philo- sophical analysis, providing that analysis is sensitive to the realities of actual mathematics. If it is to serve these purposes, a philosophical account of mathematics must not disregard the evidential relations of practice or recommend reforms on nonmathematical grounds.
These are, in my view, proper goals for the philosophy of mathe- matics. We, as philosophers of mathematics, should provide an ac- count of mathematics as practiced, and we should make a contribu- tion to unraveling the conceptual confusions of contemporary math- ematics. So it is against this backdrop that I shall assess the indispensability arguments, or rather, the view of mathematics the indispensability arguments generate. They will be judged by their ability to account for actual mathematics as practiced.