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

Read more

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

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.