Reducing binary distance relations to unary properties

Some philosophers say that space is fundamentally constituted by points. Others that it is fundamentally constituted by regions, and points are logical constructions out of regions. Here is an interesting advantage of an approach base on regions. Relations are more mysterious that properties. A point-based account is likely to involve distance relations: x and y are α units apart.

But a region-based account need not suppose a distance relation, but a diameter property. Intuitively, the diameter of a region is the largest distance between two points in the region, and hence is defined in terms of a distance relation (to account for regions that are not compact, we need to say that the diameter is the supremum of the distances between points in the region). But we could also suppose that the diameter property is more fundamental than distance, and just as we might define points as constructions out of a region-based ontology, we might define distances as constructions out of diameters plus region mereology.

How this would work depends on the details of the point construction. One kind of point construction identifies points with (equivalence classes of) sequences of regions that get smaller and smaller. Some have done this with special concentric regions like balls, but one can also do it with more general regions making use of the diameter D(A) of a region A. Specifically, we can let a point be (an equivalence class of) a sequence of A₁, A₂, ... of regions, where we requires that later regions in the sequence always being subregions of the earlier ones, and that the limit of D(A_n) is zero.(The equivalence relation can be defined by stipulating that the sequences A₁, A₂, ... and B₁, B₂, ... are equivalent just in case D(A_n+B_n) converges to zero where A_n + B_n is the fusion of A_n and B_n.) We can then stipulate the distance between the points defined by the sequences A₁, A₂, ... and B₁, B₂, ... is equal to the limit of D(A_n+B_n). We’re going to need some axioms concerning diameters and regions for all this to be well-defined and for the distance to be a metric.

Or we can take a version of Lewis’s construction where points are just identified with balls of a specific diameter δ₀, with the intuition that we identify a point with the ball of diameter δ₀ "centered on it". And we can again define distances in terms of diameters: d(A,B) = D(A+B) − δ₀.

This does not rid us of all relations. After all, we are supposing the mereological parthood relation (in its "subregion" special case). However, one might think that parthood is more of a fundamental binary predicate than a relation. And at least it’s not a determinable relation, in the way that distance is.

I am not myself fond of mereology. So the above is not something I am going to push. But it would be fun to work out the needed axioms if nobody’s done it (quite likely someone has—maybe Lewis, as I haven’t actually read his stuff on this, but am going on hearsay). It would make a nice paper for a grad student who likes technical stuff.