Does the Differential Structure of Space-Time Follow from Physical Principles?

TL;DR

This paper explores whether the structure of space-time arises from physical principles, focusing on the role of mathematics and causality.

Contribution

It introduces a novel argument linking Einstein causality to the emergence of a Tychonoff topology and potential differentiable structure.

Findings

Einstein causality can define a Tychonoff topology on a countably infinite point set.

Shirota’s theorem allows embedding this set as a closed subspace of RJ.

Completion processes from QJ to RJ remain empirically untestable.

Abstract

This article examines Wigner’s view on the unreasonable effectiveness of mathematics in the natural sciences, which was based on Cantor’s claim that ‘mathematics is a free creation of the human mind’. It is contended that Cantor’s claim is not relevant to physics because it was based on his power set construction, which does not preserve neighborhoods of geometrical points. It is pointed out that the physical notion of Einstein causality can be defined on a countably infinite point set M with no predefined mathematical structure on it, and this definition endows M with a Tychonoff topology. Under Shirota’s theorem, M can therefore be embedded as a closed subspace of RJ for some J. While this suggests that the differentiable structure of RJ may follow from the principle of causality, the argument is constrained by the fact that the completion processes (analyzed here in some detail) required for the passage from QJ to RJ remain empirically untestable.