Intrinsic local distances: a mixed solution to Weyl's tile argument

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 the mixed account, according to which 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 (Synthese 103:327-354, 1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account.

Automated summary

The mixed account introduces a new concept of distance in atomistic space, where local distances are considered primitive and other distances are derived from them. In response to Weyl's tile argument, the mixed account is argued to be a more natural solution compared to other accounts, such as Forrest's restricted version of the mixed account.