Representation and Spacetime: The Hole Argument Revisited

ABSTRACT Ladyman and Presnell have recently argued that the Hole argument is naturally resolved when spacetime is represented within homotopy type theory rather than set theory. The core idea behind their proposal is that the argument does not confront us with any indeterminism, since the set-theoretically different representations of spacetime involved in the argument are homotopy type-theoretically identical. In this article, we will offer a new resolution based on ZFC set theory to the argument. It neither relies on a constructive-intuitionistic form of mathematics, as used by Ladyman and Presnell, nor is foundationally problematic, such as the existing set-theoretic suggestions.