Why the Indispensability Argument Does Not Justify Belief in Mathematical Objects

The Quine-Putnam indispensability argument concludes that we should believe that mathematical objects exist because of their ineliminable use in scientific theory. I argue that none of the objects in which the indispensability argument justifies belief are mathematical objects. I first present a traditional characterization of mathematical objects. I then propose a general formulation of the indispensability argument. I argue that the objects to which the indispensability argument refers do not have six important properties traditionally ascribed to mathematical objects. One can not justify a platonist mathematical ontology with an empiricist epistemology. Why the Indispensability Argument Does Not Justify Belief in Mathematical Objects, p 1 Aside from Quine and Putnam’s work, see Resnik 1993: “A Naturalized Epistemology 1 for a Platonist Mathematical Ontology.” Nothing I say in the paper depends on the reductive presumption; all results could be 2 generalized. Critics of the reductive presumption include structuralists motivated by Benacerraf 1965, which denies a unique reduction. In another direction, category theorists may hold that reduction should be instead to more fundamental categories. §1: Mathematical Objects The Quine-Putnam indispensability argument may be used to try to justify beliefs in the existence of mathematical objects on the basis of their ineliminable use in science. The indispensabilist claims that we can eschew traditional appeals to mathematical intuition, or other platonist epistemology, yet maintain a substantial, abstract mathematical ontology. The 1 argument is most often criticized for its claim that mathematical objects are ineliminable from scientific theory. In this paper, I argue that the so-called mathematical objects to which the indispensability argument refers, the objects to which scientific theory may be taken to be committed, are not really mathematical objects. Thus, the indispensability argument does not justify beliefs in mathematical objects. I will discuss the indispensability argument shortly, but it will help to start with a characterization of mathematical objects. For simplicity, the only mathematical objects I shall consider will be sets, due to common, though not universal, presumptions about the reducibility of all mathematical objects to sets. The characterization of sets that I will provide is traditional. 2 Readers may find the traditional conception contentious. My point is that the indispensability argument does not justify beliefs in mathematical objects, as traditionally conceived. Why the Indispensability Argument Does Not Justify Belief in Mathematical Objects, p 2 Attempts to locate mathematical objects with their concrete members lead to substantial 3 difficulties, as Frege argued against Mill (see Frege 1980, §7-§9), and as Mark Balaguer argues against Penelope Maddy (see Balaguer 1994). Or universes. Balaguer 1998 argues that any consistent axiomatization truly describes a 4 universe of sets, even if it conflicts with other consistent axiomatizations. I speak here of pure sets. Sets with ur-elements may exist contingently. 5 We do sometimes pursue mathematics in order to solve specific problems in empirical 6 science. My claim is that we have criteria for determining whether to accept a mathematical assertion which are independent from the application of that assertion to empirical science. There are empirical aspects to mathematical methods, of course: knowledge of who 7 proved which theorems, say, and observations of inscriptions. But, such empirical claims do not suffice for mathematical justification. They are consistent, for example, with what James Robert Brown calls the 8 “mathematical image” (Brown 1999: 1-7) and with Stuart Shapiro’s traditional picture (Shapiro 2000: 21-23). I take it that sets are abstract objects, lacking any spatio-temporal location. The universe 3 of sets is described by various standard axiomatizations; where different axiomatizations conflict we find disagreement about the nature and extent of the set-theoretic universe. Their existence 4 is not contingent on our existence, nor is it contingent on the existence of any physical objects. 5 Furthermore, I take it that mathematics is a discipline autonomous from empirical science; mathematical standards are independent of application. Mathematical methodology is a priori. 6 7 Each of the properties of sets that I have mentioned has been denied of mathematical objects, just as the existence of mathematical objects has been denied. Still, these characteristics constitute, at least in part, the standard starting point for discussions of the nature of mathematics and mathematical objects. In this paper, I remain agnostic on whether mathematical objects 8 exist. I also remain agnostic on whether sets must have all of the above characteristics. My claim is merely that any objects which lack all of the above characteristics should not be called Why the Indispensability Argument Does Not Justify Belief in Mathematical Objects, p 3 Field 1980 and its legacy remain the focus of the debate over whether such reference is 9 eliminable. Burgess and Rosen 1997 collects a variety of strategies for removing mathematical elements from scientific theory. For a selection of such allusions, see Quines 1939, 1948, 1951, 1955, 1958, 1960, 1