Space, Supervenience and Substantivalism

Consider a straight line on a flat surface running from point A to C and passing though B. Suppose the distance AB to be four inches, and the distance BC to be six inches. We can infer that the distance AC is ten inches. Of all geometrical inferences, this is surely one of the simplest. Of course, things are a little more complicated if the surface is not flat. If A, B and C are points on a sphere, then the shortest distance between A and C may be smaller (it may even be zero). We can make our inference immune from concerns about non-Euclidean spaces, however, by qualifying it as follows: if AB = n, and BC = m, then, in the direction A ⇒ B ⇒ C, the distance AC is n + m. This is apparently entirely trivial. But trivial truths can hide significant ontological ones. Let us translate our mathematical example to the physical world, and suppose A, B and C to be points, still in a straight line, but now at the centre of gravity of three physical objects: