Pythagorean powers or a challenge to platonism

Abstract

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. ! Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. ! We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists eschew the standard position.2 They maintain that mathematical objects do possess location and causality, although they retain some similarity to the kinds of things that Plato had in mind. We do not intend to enter into a terminological dispute as to which party of platonists truly deserves the name. Perhaps the people we shall call non-standard platonists are closer to the original Plato. After all, Plato himself was notoriously equivocal about the causal status of his Forms. However that may be, we take the majority (or perhaps the Anglo-American) view to be the standard view. On this 1