Gunk and Points

Gunky space contains no spatial points. That is, Gunky space contains no regions that are unextended and have no further subregions. However, there are points. After all, when we say things like "given any two points, there is exactly one line that passes through them both" we speak truly and what we say is in some sense about space.

"Have no fear", says the gunky space theorist, "of course there are points, they just have different features than you expect". According to the gunky space theorist, a point is an infinite series of nested solid spheres. Tarski showed us that we can recover all of geometry within a gunky space by using these series of nested spheres in place of points.

But, now I have a question. What exactly are the points? Are they fusions of nested spheres? It seems that can't be right since the fusion of nested spheres that converges on the "point" at the end of my cats nose just is the fusion of the nested spheres that converges on the "point" at the end of the statue of liberty's torch. Moreover, this fusion just is all of space. Since we think that the "point" at the end of my cat's nose and the "point" at the end of the statue of liberty's torch are different "points", and (moreover) since we are not point monists (who believe there is one and only one point that is identical to all of space), we must reject the claim that points are fusions of series of convergent solid spheres.

Might we take the original view at face value and say that points are ordered sets of convergent solid spheres? I think not. First, we will run into a Benacerraf problem with respect to points. But, more intuitively, the "point" at the end of my cat's nose is located (in some sense) at the point of my cat's nose. No abstract entity like a set is so located. So, points are not ordered sets.

I think that the gunky space theorist has a reason to accept structure in composition of spatial regions. Perhaps, points are special kinds of fusions of convergent series of solid spheres. They are fusions that preserve a certain kind of structure, an ordered structure to the parts. One consequence of this view seems to be the following. There are uncountably many things that are co-located with the entirety of space, each one of which is a point and each one of which has a different structure than the others. This seems like quite a robust ontology of spatial regions.