The Hole Argument in Homotopy Type Theory

James Ladyman,Stuart Presnell

doi:10.1007/s10701-019-00293-9

Abstract

The Hole Argument is primarily about the meaning of general covariance in general relativity. As such it raises many deep issues about identity in mathematics and physics, the ontology of space–time, and how scientific representation works. This paper is about the application of a new foundational programme in mathematics, namely homotopy type theory (HoTT), to the Hole Argument. It is argued that the framework of HoTT provides a natural resolution of the Hole Argument. The role of the Univalence Axiom in the treatment of the Hole Argument in HoTT is clarified.

Highlights

The Hole Argument is primarily about the meaning of general covariance in general relativity (GR)
This paper is about the application of a new foundational programme in mathematics, namely homotopy type theory (HoTT), to the Hole Argument
It is argued that the framework of HoTT provides a natural resolution of the Hole Argument

Summary

Introduction

The Hole Argument is primarily about the meaning of general covariance in general relativity (GR). Einstein said that“[t]he general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates [...] (generally co-variant)” [4] This does not mean that the laws of physics take the same form in all coordinate systems ( that remains a common and not unreasonable rough formulation (see, for example, Dainton [2], [372]), because, for example, classical electromagnetism can be formulated in a generally covariant way What makes GR special is that the invariance group of the fields that satisfy the Einstein equation is the diffeomorphism group, so the solutions of GR are closed under arbitrary coordinate transformations. This is why general covariance seems essential to GR, but not to Newtonian gravity, special relativitistic mechanics, or classical electrodynamics. The section explains the Hole Argument as it is usually formulated

The Hole Argument

Homotopy Type Theory to the Rescue?

Overview

Internal and External Identity HoTT has two different identity relations:

Multiple Identifications

Products and Dependent Pairs

Univalence

HoTT and the Hole Argument

Conclusion