On the Metaphysics of Numbers

We reason that if there are nurnbers, they are abstract and therefore acausal. The question thus arises as to how we have knowledge of these objects. My thesis consists of three parts, each of which explores a response to this question. Charles Parsons has argued that we know about the numbers because we can apprehend (or "intuit") some abstract objects such as expression-types which form a model for arithmetic and which therefore stand as representations of the numbers. Although I think we can intuit some abstract objects in the way Parsons describes, I show that our knowledge that arithmetic has a model cannot be based on our intuition of these objects. I conclude that intuition does not offer a route to the numbers. Constructivists such as the traditional intultionists offer a different sort of answer. They argue that numbers are constructions which describe the mental processes one goes through in doing mathematics, albeit at a certain level of abstraction. Our knowledge of these constructions is thought to be straightforward since, after all, they are objects of our own experience. I show that this account of the numbers Is circular, and that the only way to avoid this circularity commits the intuitionist to a version of mathematical finitism. I conclude that numbers are not mental constructions. Mathematical antirealists such as Michael Dummett argue that the truth of a mathematical statement cannot intelligibly transcend evidence for the truth of that statement, and that an account nf mathematical truth which recognizes this fact commits us to an intuitionistic logic and mathematics. I show that Dummett's arguments for intuitionism involve the crucial assumption that one's understanding of a language is constituted by one's linguistic behavior. I argue that Dummett's reasons for making this assumption are not well-grounded and, therefore, that his arguments for intuitionism fail. Thesis Supervisor: George Boolos Title: Professor of Philosophy