Structure and Categoricity: Determinacy of Reference and Truth-Value in the Philosophy of Mathematics

Abstract: This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent `internal' renditions of the famous categoricity arguments for arithmetic and set theory.