Radical Structural Essentialism for the Spacetime Substantivalist

Spacetime substantivalists insist that spatiotemporal points are fundamental entities and thus are ontologically independent from the physical objects occupying these points.This chapter argues that the best metaphysical option for the substantivalist is to adopt a radical version of structural essentialism, according to which participation in a certain relational structure is both a necessary and sufficient condition of identity for the elements characterized by this structure.It comparesthe author’sproposed variant of essentialism with some alternative conceptions, such as Tim Maudlin’s metric essentialism and David Glick’s Minimal Structural Essentialism. It discusses how the author’sRadical Structural Essentialism deals with the challenge of the hole argument as well as some less well-known challenges, such as the problem of contingent counterfactuals regarding the material contents of spacetime, and the ‘mole’ argument. Later the chapter explains why the structure consisting of the metric tensor is a genuinely relational structure, and itdefend the claim that in order to preserve the identity of its elements the actual structure may be merely embeddable in (and not necessarily isomorphic with) an alternative structure.

The Hole Argument in Homotopy Type Theory

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.