Abstract

According to the so-called "Indispensability Argument", the central role played by mathematics in science gives us sufficient reason to believe in the existence of abstract mathematical objects such as numbers, sets, and functions. The Indispensability Argument may be formulated as follows: We ought rationally to believe our best available scientific theories. Mathematics is indispensable for science. we ought to believe in the existence of mathematical objects. Platonism is the view that there exist enough abstract mathematical objects to make the bulk of the mathematical claims we accept true. The contrary view is nominalism, which denies the existence of abstract objects. This dissertation focuses on two key questions: whether the Indispensability Argument can refute nominalism, and whether it can establish full-blown platonism. ;Chapter 1 surveys the historical background of the Indispensability Argument in the writings of Quine and Putnam. The aim is to lay bare the major philosophical presuppositions of the Argument. Chapter 2 examines premise of the Argument, which summarizes the philosophical stance commonly referred to as "scientific naturalism", and defends it against the objection that our best available scientific theories may turn out to be flawed in ways which make it rational not to believe them. Chapter 3 focuses on premise; I argue that this indispensability claim is plausible because mathematics is required for the development of new scientific theories and for the discovery of new results. Two historical case studies are presented in support of this claim. Chapter 4 addresses the issue of whether Occam's razor is a genuine principle of scientific theory choice, and---if so---whether scientists' preference for theories that postulate fewer things might decisively tip the balance in favor of the nominalist. In Chapter 5 I argue that although the Indispensability Argument is sufficient to establish that mathematical objects exist, it cannot establish the existence of any specific kind of mathematical object because there are always other kinds of mathematical object that can fulfil the same role. Chapter 6 briefly explores two possible ways of developing the Indispensability Argument in response to this predicament.