Hyperdeterminism? Spacetime 'Analyzed'
Lu Chen, Tobias Fritz
TL;DR
This paper investigates how choosing analytic versus smooth regularity for spacetime fields alters the philosophical interpretation of classical field theories and general relativity. By formulating analytic general relativity and applying the identity theorem, the author shows that the hole argument would fail and a form of hyperdeterminism would arise, where local data determine the entire spacetime. The work argues that neither regularity choice is obviously superior and emphasizes the risk of drawing metaphysical conclusions from mathematical formalisms alone. It also proposes using a sheaf-condition approach to reconcile free recombination with hyperdeterminism and highlights the broader need to scrutinize foundational assumptions in physics beyond mere technical convenience.
Abstract
When modelling spacetime and classical physical fields, one typically assumes smoothness (infinite differentiability). But this assumption and its philosophical implications have not been sufficiently scrutinized. For example, we can appeal to analytic functions instead, which are also often used by physicists. Doing so leads to very different philosophical interpretations of a theory. For instance, our world would be 'hyperdeterministic' with analytic functions, in the sense that every field configuration is uniquely determined by its restriction to an arbitrarily small region. Relatedly, the hole argument of general relativity does not get off the ground. We argue that such an appeal to analytic functions is technically feasible and, conceptually, not obviously objectionable. The moral is to warn against rushing to draw philosophical conclusions from physical theories, given their drastic sensitivity to mathematical formalisms.