Abstract

There is a powerful tradition in philosophy to regard mathematics as a premier example of a priori knowledge. As such, mathematics is typically distinguished from empirical science: because of its a priori character, the statements of mathematics are true or false independent of how matters stand in the world; in contrast, the propositions of empirical science must be justified, however indirectly, by evidence gathered through the senses, and are subject to revision in the face of recalcitrant experience. This tradition has been challenged by Quine, for example, who champions the view that there is no interesting sense in which mathematics is a priori and empirical science is not. ;I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons: His argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is related to an "expansion" of the possibilities of describing the empirical world; and This holistic conception does not clearly demarcate pure mathematics from applied mathematics. To justify , I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori proposition. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori. In the concluding chapter, I relate this formal account of applied mathematics to the practical employment of mathematics in empirical science