Why the Weyl Tile Argument is Wrong

TL;DR

This paper critiques Weyl's argument against discrete space by demonstrating that Euclidean geometry can be approximately recovered through fundamental spacetime structures, using examples from physics.

Contribution

It reveals a flawed assumption in Weyl's argument and provides rigorous examples from physics to challenge the claim that discrete space cannot support Euclidean geometry.

Findings

Random walks can approximate Euclidean geometry in discrete space.

Quantum mechanics models support the viability of discrete spacetime structures.

Weyl's assumption about geometry and dynamical laws is flawed.

Abstract

Weyl famously argued that if space were discrete, then Euclidean geometry could not hold even approximately. Since then, many philosophers have responded to this argument by advancing alternative accounts of discrete geometry that recover approximately Euclidean space. However, they have missed an importantly flawed assumption in Weyl's argument: physical geometry is determined by fundamental spacetime structures independently from dynamical laws. In this paper, I aim to show its falsity through two rigorous examples: random walks in statistical physics and quantum mechanics.