Towards a Thomistic theory of fundamental distributional properties

Recently, various metaphysicians (e.g., Parsons, and Arntzenius and Hawthorne) have tried to give an account of spatially nonuniform properties that would work for extended simples or gunky objects (i.e., ones that have no smallest parts). I think there is an interesting account that has in an important way a Thomistic root, and that's no surprise because Aquinas did not believe that substances had substantial parts, so he faced the problems that people thinking about extended simples face. I will develop a partial account for shape, location and color. The version I will give in moderate detail is Pythagorean, because mathematical objects are involved in physical reality itself. The Pythagorean account is easier to wrap one's mind around. I think it may be possible to use Category Theory to de-Pythagorize the account, but I will only sketch the beginning of that line of thought.

A basic insight Aquinas has is that material objects have a special accident called "dimensive quantity", which accident in turn provides a basis for further accidents, such as color. Moreover, objects normally are located in a place by having their dimensive quantity be located there.

On to the Pythagorean account. Suppose that each extended object O has fundamentally associated with it a manifold G of some appropriate smoothness type (we may in the end want to generalize this, perhaps to a metric space, perhaps a topological space, but let's stick to manifolds for now). This manifold I will call the object's (internal) geometry. The fundamental relation between the object and the manifold that associates the manifold to the object is being geometrized by: the object is geometrized by the manifold. This manifold is a purely mathematical object existing in the Platonic heaven. Nonetheless, which manifold an object is geometrized by significantly affects its nomic interaction with other objects. The shape properties of an object are grounded in the fact that the object is geometrized by such-and-such a manifold.

Next, we need location. There is a fundamental relation between an object O and a function L from the object's geometry G to another mathematical manifold called "spacetime", which relation I will call being located by. The function L describes how the object's geometry is located within spacetime. We can now say that two objects O₁ and O₂ overlap if and only if there are L₁ and L₂ such that O_i is located by L_i, for i=1,2, and the ranges of the functions L₁ and L₂ overlap.

Now, let's add some color into the picture. There is an abstract object which is a colorspace. Maybe it's some kind of a three-dimensional manifold. There is a fundamental relation between an object O and a function c from the object's geometry to the colorspace, which we may call being colored by. This function describes the distribution of color over the object's geometry.

This is the Pythagorean version of the view. Now we should de-Pythagorize it. Suppose a fundamental determinable of objects: being geometrized. A maximally specific determinate of being geometrized will be called a geometrization. And then—this is getting sketchier—one makes the geometrizations, and maybe other Platonic things like geometrizations, into a category isomorphic to an appropriate category of manifolds. I don't know what, if any, classical ontological category arrows correspond to. Maybe some kinds of token relations. If we're substantivalists about spacetime, we can suppose a special object, S, the spacetime. And then there is a fundamental relation of being located by between an object O and a morphism L of the category of geometrizations whose domain is O's geometrization and whose codomain is S's geometrization. It's harder to bring colors into the picture. This is far as I got. And even if I finish the de-Pythagorization, I will still want to de-Platonize it.

More generally, the de-Pythagorization proceeds by replacing mathematical objects associated with an object with maximally specific determinates of a determinable that, nonetheless, stand in the same structural relations as the mathematical objects did. Category Theory is a promising way to capture that structural sameness, but it might not be the only way.