Scientific realism and mathematical nominalism: A marriage made in hell

Abstract

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter, I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine- Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni (2004), Hartry Field (1980, 1989), Alan Musgrave (1977, 1986), David Papineau (1993)); (ii) embracing mathematical realism (W.V.O. Quine (1981), Michael Resnik (1997), J.J.C. Smart (unpub.)); and (iii) embracing some form of scientific instrumentalism (Otvio Bueno (1999, 2000), Bas van Fraassen (1985)). Elsewhere (Colyvan 2001), I have argued for option (ii) and I wont repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave (1986) as to why option (i) is a plausible and promising approach. From the discussion of Musgraves argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be a holistic matterits whole theories (or significant parts thereof) that are confirmed in any experimentthen theres an inclination to opt for (ii) in order to resolve the marital tension outlined above. If, on the other hand, you take it that its a single hypothesis thats confirmed in a given experiment, then youll be more inclined towards option (i). As we shall see, Musgraves argument illuminates, in an interesting and original way, the important role confirmation has to play in realism debates in the philosophy of mathematics. © 2006 Springer.