The Conceptual Contingency of Mathematical Objects

Abstract

Platonists often allege that mathematical entities exist of conceptual necessity, so that any view that treats such entities as fictitious is simply incoherent. (Position (i).) Less often, nominalists counter that such entities are in fact impossible, so that it is platonism that is incoherent. (Position (ii).) I think that on the most obvious construals of conceptually (and conceptually possible etc.), both of these views are false: the right view (position (iii)) is that it is whether mathematical entities exist. This means that the debate between platonism and nominalism on the existence of mathematical entities cannot be straightforwardly settled on conceptual grounds: other grounds are required. In formulating positions (i)-(iii) I have used the notion of conceptual necessity (and conceptual possibility etc.), and some philosophers would reject this notion as incoherent. But I find the position that there is nothing in this notion hard to take seriously. I don't dispute that talk of conceptual possibility is unclear in important ways; consequently, it could (prima facie) turn out that which of the three positions is true depends on how this notion is to be clarified. But my claim will be that in fact there is no reasonable construal of conceptually on which the existence of mathematical entities is necessary. (I'm not sure I want to take quite so strong a line against the position that it is impossible for mathematical entities to exist: this may admit some interpretation in which it's defensible, as we'll see.) I will focus my discussion on recent work by Bob Hale and Crispin Wright, especially theirjoint paper Nominalism and the Contingency ofAbstract Objects (1992). Hale and Wright are among those who hold that it is necessary that mathematical entities exist. But their joint paper is mostly directed against the nominalist alternatives: and primarily, against contingent nominalism, the view (iiib) that it is whether mathematical entities exist, and that in fact they don't. (They focus on the version of nominalism that I advanced in Realism, Mathematics and Modality (1989).) I think it is clear that if their arguments against nominalism were correct, they would also undermine contingent platonism, the view (iiia) that it is whether mathematical entities exist, and that in fact they do. The main burden of this paper will be to defend view (iiib) against their critique; but in fact almost nothing I will say bears on the choice between (iiia) and (iiib). In my view the decision between (iiia) and (iiib) turns in large part on whether the well-known indispensability arguments for the existence of mathematical entities can be undercut; or more generally, on whether we can undercut arguments for the existence