Illustrations of a Realist Methodology for the Philosophy of Mathematics

Abstract

This dissertation outlines and illustrates a methodological position most naturally described as realist. The kind of realism advanced and employed, however, has its most familiar analogue in legal and political philosophy.

The American legal realists are realist in the relevant sense. These philosophers argue that, when one looks at how courts really decide cases, intuitive judgments about a fair decision given the case’s facts play a more decisive role than reasoning based on distinctively legal rules. They note that judges may say otherwise, but that certainly does not imply this is the case. The political philosopher taking Raymond Geuss’s advice, “Don’t look just at what they say, think, believe, but at what they actually do, and what actually happens as a result,” also practices the relevant form of realism. These realists are alike in their attempts to replace a dependence on the unreliable self-reporting of political actors, whose own motivations may be opaque or self-deceived, with something more objective by appealing to the detailed examination of concrete cases and facts.

I argue that contemporary philosophy of mathematics needs to adopt a methodology realist in an analogous sense. Although the field has moved towards paying closer attention to mathematical practice, too much weight is still placed on the psychology of mathematicians and the analysis of the philosophical offerings found outside proofs and in prefaces.

In Chapter 1, I use Wittgenstein’s writings to aid in outlining the main tenets of the realist methodology illustrated in the following chapters. In these later chapters, I make use of this methodology to address the following: (1) The setting-dependence of mathematical questions, and problems with standard accounts of “mysterious” mathematical behaviors; (2) the role of geometric concepts and techniques in non-geometric contexts; (3) the puzzling concept of mathematical coincidence; and (4) the explanatory potential of proofs by mathematical induction. In each of these cases, a realistic approach to subject brings clarity to the questions being asked and also provides the means for satisfying answers or dissolutions.