Abstract

Since we know that there are four prime numbers less than 8 we know that there are numbers. This ‘short argument’ is correct but it is not an ontological claim or part of philosophy of mathematics. Both realists (Quine) and nominalists (Field) reject the short argument and adopt the idea that the existence of numbers might be posited to explain known mathematical truths. Philosophers operate with a negative conception of what numbers are: they are not in space and time, not related causally to us, not perceivable, etc. This preliminary outlook does not actually characterize a kind of existing thing at all. It creates the atmosphere of weirdness characteristic of both fictionalism and Platonism. Positing things for the sake of explanation makes sense in empirical contexts, but the intelligibility of positing cannot not survive the move to philosophy of mathematics. Modal realism is a model for the unsatisfactory thinking that generates ontological commitment in mathematics