Abstract

This paper considers the existing literature on what a categoricity theorem could achieve, and proposes that more clarity and explicit admission of background beliefs is required. The first claim of the paper is that formal properties such as categoricity can have no philosophical significance whatsoever when considered apart from informal, philosophical beliefs. The second is that we can distinguish two distinct types of philosophical significance for categoricity. The paper then highlights two consequences of this analysis. The first is a potential source of circularity in Shapiro’s wider project, that arises out of what appear to be arguments for both kinds of philosophical significance with respect to categoricity. Rather than a knock-down objection to Shapiro’s philosophy of mathematics, the discussion of this potential circularity is intended to demonstrate that implicit claims surrounding the significance of categoricity can lead to philosophical missteps without due caution. The second outcome is that, as an initial case study, categoricity has limited significance for the semantic realist.