Benacerraf's Identification Problem: Hartry Field and Bertrand Russell on the Ontology of the Natural Numbers

Since numbers are so useful, we assume that they have some measure of being. This immediately gives rise to a number of philosophical questions. Do numbers exist in any real sense? How can we come to know these objects? This thesis presents an examination of popular answers to these questions.

Reductive mathematical realists, like Bertrand Russell, hold that numbers exist as observer-independent and causally inert objects, and their contentions are motivated by the ultimate nature of truth claims as grammatical. Since what we can know is presumed to be real, and since Russell (like his mentor, G. E. Moore) contends that there is a real world, then human beings have an ontological commitment to the reality of mathematical objects, inasmuch as we know the truth of the propositions in which they feature. While I find Moore's proof of an external world appealing, I find Russell's application of it, in this context, wanting. Specifically: the boundary between abstracta and concrete objects, to which we refer, is not nearly so well-defined as Russell assumes. As such, Russell's system does not offer us a reasonable explanation of either the ontology of mathematical objects, or the existence of mathematical knowledge.

Mathematical fictionalists, like Hartry Field, appear to agree with Bertrand Russell about the nature of grammatical truth-claims as reflective of ontological reality, but they invert his specific claim about numbers. Since numbers have no spatiotemporal location, the fictionalist asserts that propositions featuring mathematical objects are either false, or incapable of carrying a truth value.

Unable to deny the usefulness of mathematics, the typical fictionalist argument hinges on the disambiguation of truth and utility. Field goes further, and attempts to reconstruct Newton's theory of universal gravitation in a non-mathematical language. I will argue here that this attempt, while initially compelling, is ultimately incomplete, and Field is guilty of smuggling abstracta into his methodology. Moreover, the attractiveness of mathematical language is a necessary condition of indispensability, and I will argue that his project fails to dispense with abstract formulations, due to the inelegance of its own application.