Indispensability Arguments and Instrumental Nominalism

Abstract

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that mathematical objects exist on the grounds that we make indispensable reference to such objects in our best scientific theories (Quine, 1981a;Putnam, 1979a) and in our everyday reasoning (Ketland, 2005). I wish to defend a particular objection to such arguments called instrumental nominalism. Existing formulations of this objection are either insufficiently precise or themselves make reference to mathematical objects or possible worlds. I show how to formulate the position precisely without making any such reference. To do so, it is necessary to supplement the standard modal operators with two new operators that allow us to shift the locus of evaluation for a subformula. I motivate this move and give a semantics for the new operators.

What we talk about when we talk about numbers

“…At this point, we appeal to instrumental nominalism (Melia, 1995(Melia, , 2000Rosen, 2001;Pettigrew, 2012). Instrumental nominalism is usually deployed as a response to the indispenability arguments in the philosophy of mathematics.…”

Modal Structuralism and Reflection

“…28 See Uzquiano (2014), Sid, and the articles in Cotnoir & Baxter (2014) for discussion. 29 See, for example, Pettigrew (2012). He uses a new pair of modal operators to express the claim that the physical part of a world w has the same structure as the physical part of some other world w .…”