Intrinsic Local Distances: A Mixed Solution to Weyl's Tile Argument

TL;DR

This paper proposes a new 'mixed account' of local distances in atomistic space, allowing it to approximate Euclidean geometry effectively and offering a natural alternative to traditional distance notions, addressing Weyl's tile argument.

Contribution

It introduces the mixed account of local distances, providing a novel solution to Weyl's tile argument and demonstrating its advantages over Forrest's previous account.

Findings

Atomistic space can approximate Euclidean space using the mixed account.

The mixed account is as natural as the standard continuous space account.

It has advantages over Forrest's account in addressing Weyl's tile argument.

Abstract

Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called \textit{the mixed account}, according to which \textit{local distances} are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to Weyl's tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest's (1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account.