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

TL;DR

This paper reviews recent debates on the role of categoricity in securing reference and truth in the philosophy of mathematics, analyzing the effectiveness of internal arguments for arithmetic and set theory.

Contribution

It critically assesses the sufficiency of categoricity arguments for establishing mathematical reference and evaluates recent internal approaches to these arguments.

Findings

Categoricity arguments are not always sufficient for securing reference.

Internal renditions of categoricity arguments offer new perspectives.

The effectiveness of categoricity in establishing truth-value remains debated.

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.