Space

Abstract

In standard geometry, space is said to be composed of indivisible parts called ‘points’. There are continuum many of these in any region with a nonzero size. All geometric objects — lines, planes, triangles, circles, and so on — are said to be built out of points. It is usually said that these geometric objects are sets of points; however, it makes more sense to regard them as fusions of points, so that is how I shall henceforth speak. The fusion of a and b is understood as an object that has a and b as parts and has no other parts that don’t overlap with a or b. (The generalization to cover fusions of any number of objects should be obvious.) Fusions differ importantly from sets: for example, the fusion of two physical objects is itself a physical object, whereas a set of physical objects would be an abstract, mathematical object. Similarly, the fusion of some spatial regions would plausibly be a spatial region, whereas the set containing some spatial regions would not itself be a region but would instead be some abstract object.