In an earlier post, I explored, without endorsing, a Platonic theory of spacetime, on which spacetime is an abstract Platonic entity, and objects are located by virtue of standing in a relation to abstract points of that entity.
This could extend to other determinables. Consider, for instance, mass, and simplify by supposing presentism or lack of time variation. An object o could have mass x, where x is some real number (we need a natural unit system for that), precisely in virtue of o's being M-related to the real number x, a Platonic entity, where M is a natural "mass relation". This works even for much more complex determinables like wavefunctions. Thus, an object o could have a wavefunction in virtue of being W-related to some abstract function from R3 to C, again assuming presentism or lack of time variation. To get time variation into the picture, we could suppose that the mass relation relates objects to functions from a time sequence (an internal time sequence?) to reals.
This would help with regard to the epistemology of abstracta even if (contrary to fact, I am inclined to say) abstracta are causally inert. For even if the number x is causally inert, the event of o being M-related to x is not causally inert (it causes gravitational influences, for instance).
One intuitive difficulty for this theory is that it is now looking logically possible for an object to have two masses or two wavefunctions at any given time. I do not think this consequence absurd myself. If the second person of the Trinity became incarnate as two different humans at the same time, which Aquinas thinks is possible (a possibility that we may care about if it turns out that there are fallen non-human rational beings), he might have two different masses at a given time. Alternately, one can just say that there are brutely necessary restrictions here.
Notice an interesting consequence of this theory. If a naturalist were to adopt this theory, it might make it easier to get her to accept a non-reductionist theory of mind on which for us to believe a proposition just is to stand in an irreducible belief relation to a proposition. After all, it is no more philosophically puzzling how one can stand in an irreducible mass relation to a number or function than it is how one can stand in an irreducible belief relation to a proposition. And it is no more philosophically puzzling how one's standing in a belief relation to a proposition could causally affect one's behavior than how one's standing in a mass relation could.
What bothers me about this theory, as well as the earlier theory of spacetime, is that abstracta are divine ideas. But it seems wrong to say that mass and location facts are constituted by a relation to God. That sounds too panentheistic. But here's one interesting philosophical/theological question. Aquinas insists that things are the way they are by participation in God. Thus, Socrates is wise by (natural) participation in God (and Paul is wise by supernatural participation in God). Does this mean that (a) Socrates' accidental form of wisdom is identical with a participating in God or does it mean that (b) Socrates' accidental form of wisdom is something distinct from but dependent on Socrates' participating in God? If the former, then the Platonic theory I offered will be no more problematic than Aquinas' view (but of course I'll want to say something like what Aquinas says about one-sided relations, so that the mass relation is a relation to God but there is no corresponding relation of God to the object--maybe the suggestion in this post helps), and in fact Aquinas' view might just be a variant of the Platonic theory. If the latter, then the Platonic theory is more panentheistic than Aquinas', and insofar as Aquinas seems to me to be as close as one can orthodoxly come to panentheism, I would then reject the Platonic theory.
There is also going to be some trickiness coordinating the location determinable with the other determinables. We want to be able to say things like "x is beige on its left side". Working this out may require me to abandon the heuristic that there is nothing special about location--that it's just another relation.