Mathematics and Modality

It has been suggested in a number of places that a platonist account of mathematical truth raises difficult problems for a theory of mathematical knowledge.1 A "platonist theory of mathematical truth" is, roughly, a theory of mathematical truth which contains explicit reference to peculiarly mathematical objects such as sets or numbers. In this paper I consider several "alternatives" to a platonist account of mathematical truth. They can all be called "modal views of mathematics" in that they purport to analyze mathematical statements in such a way that reference to abstract mathematical objects is eliminated in favor of talk about necessity and possibility. The details of this elimination vary significantly from case to case. They all appear to agree, however, on the claim that elementary number theory (and, indeed, all of mathematics) need not be understood as a theory about a special domain of abstract mathematical objects. One proponent of a modal approach to mathematics is Putnam. He is quite explicit about what he takes to be the point of viewing mathematics from a modal perspective. In "Mathematics Without Foundations" he suggests that we are too much in the grip of the "mathematics as set theory" picture ([7]: 11). What is notable about this picture is not just that it identifies numbers with sets. Rather "the important thing about the picture is that [according to it] mathematics describes 'objects'" ([7]: 9). Putnam feels that a "modal view of mathematics" provides us with an "equivalent description" of the same mathematical facts as are described by a "mathematics as set theory" view. However, by focussing on a modal picture, we see that the emphasis on mathematical objects (be they-numbers or sets) is not essential. We need not analyze the content of mathe-