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 $\mathit{\text{the}}\mathit{\text{mixed}}\mathit{\text{account}}$, according to which $\mathit{\text{local}}\mathit{\text{distances}}$ 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.