UCLA

Identifying the source of our mathematical knowledge is an old and challenging problem. In order to avoid postulating cognitive faculties that seem entirely mysterious, such as mathematical intuition, some philosophers have attempted to explain our mathematical knowledge by claiming that it is grounded in definitions, stipulations, or postulations. In this dissertation, I argue against a range of views of this sort in the domain of arithmetic. I argue that none of these views can account for our mathematical knowledge unless we already understand arithmetical structure when we introduce the relevant definitions, stipulations, or postulations. An adequate account of mathematical knowledge must explain how we have this antecedent cognitive grip on the structure of arithmetic. The views under consideration, I argue, are inadequate because they fail to explain this antecedent cognitive grip.

The dissertation begins with a discussion of the problem of mathematical knowledge. In the process, I introduce several desiderata for theories of mathematical knowledge. Definition-based accounts are attractive because they fare well on many of these desiderata. In the rest of the dissertation, I argue against three such accounts. First, it could not be the case that all of our mathematical knowledge is grounded in axiomatic definitions; we have arithmetical knowledge that could not emerge from axioms. Second, it could not be the case that our ability to think about the natural numbers is grounded in a stipulation of Hume's Principle as an implicit definition; such a stipulation could not uniquely fix referents for number-phrases, and ancient mathematicians could not have stipulated it in a suitable way. And third, our knowledge of basic arithmetical truths is not entirely grounded in the standard simple proofs of those truths, which rely on explicit definitions of the form `4=3+1'. While these proofs surely contribute to the security of our arithmetical knowledge, we can only understand these proofs if we already understand the structure of arithmetic. So, our most basic arithmetical knowledge is cognitively prior to these explicit definitions.