Ontology, modality, and mathematics: remarks on Chihara's constructibility theory

Abstract

Many theorems of mathematics appear to quantify over mathematical objects such as numbers and functions. But such objects, standardly conceived as causally inert and existing "outside" of space and time, do not seem to be the sorts of entities of which we could have knowledge, given that knowledge of an object seems to presuppose some sort of causal connection between knower and thing known. A number of philosophers have thus attempted to reconstruct mathematics in such a way that commitment to mathematical objects is avoided. One such philosopher is Charles Chihara, whose "constructibility theory" involves replacing traditional assertions of the existence of mathematical objects with assertions about possibilities of constructing certain open sentence tokens. I set forth Chihara's theory, explain why he rejects several alternatives, and consider whether constructibility theory fares any better. In the end, I conclude that, while Chihara's project can be defended against several important objections, it is no less epistemologically problematic than theories which involve mathematical objects.