Mathematical Responses to the Hole Argument: Then and Now

TL;DR

This paper unifies various mathematical responses to the hole argument, critiques their validity, and discusses the core issues in the philosophical debate about spacetime and mathematical formalism.

Contribution

It offers a unified perspective on mathematical responses to the hole argument and addresses key critiques, clarifying the philosophical significance.

Findings

Mathematical responses can be viewed as a unified approach.

Two major critiques of the mathematical response are rebutted.

The core philosophical issues in the debate are clarified.

Abstract

We argue that several apparently distinct responses to the hole argument, all invoking formal or mathematical considerations, should be viewed as a unified "mathematical response". We then consider and rebut two prominent critiques of the mathematical response before reflecting on what is ultimately at issue in this literature.