Distance and Discrete Space

Abstract

Let us say that space is discrete if and only if every finite extended region of space is composed of finitely many atomic regions of space. I assume here that regions of space are individuals rather than sets of points, and have mereological structure; their parts are all and only their subregions. A region of space is an atomic region if and only if it has no proper parts, i.e., if and only if it is a mereological atom. In what follows, I will simply call atomic regions of space 'atoms'. Let us assume that, necessarily, all atoms are unextended regions, i.e., points of space. According to the Weyl Tile argument, no world with discrete space could approximate a world with continuous space because (1) the Pythagorean theorem fails to hold in worlds with discrete space and (2) it is not even approximated as the number of points in a finite region approaches infinity. 1 Consider the following space (asterisks represent points).