From a determinable-determinate model of location to a privileged spacetime foliation

Here’s a three-level determinable-determinate model of spacetime that seems somewhat attractive to me, particularly in a multiverse context. The levels are:

1. Spatiotemporality

2. Being in a specific spacetime manifold

3. Specific location in a specific spacetime manifold.

Here, levels 2 and 3 are each a determinate of the level above it.

Thus, Alice has the property of being at spatiotemporal location x, which is a determinate of the determinable of being in manifold M, and being in manifold M is a determinate of the determinable of spatiotemporality.

This story yields a simple account of the universemate relation: objects x and y are universemates provided that they have the same Level 2 location. And spatiotemporal structure—say, lightcone and proper distance—is somehow grounded in the internal structure of the Level 2 location determinable. (The “somehow” flags that there be dragons here.)

The theory has some problematic, but very interesting, consequences. First, massive nonlocality, both in space and in time, both backwards and forwards. What spacetime manifold the past dinosaurs of Earth and the present denizens of the Andromeda Galaxy inhabit is partly up to us now. If I raise my right hand, that affects the curvature of spacetime in my vicinity, and hence affects which manifold we all have always been inhabiting.

Second, it is not possible to have a multiverse with two universes that have the same spacetime structure, say, two classical Newtonian ones, or two Minkowskian ones.

To me, the most counterintuitive of the above consequences is the backwards temporal nonlocality: that by raising my hand, I affect the level 2 locational properties, and hence the level 3 ones as well, of the dinosaurs. The dinosaurs would literally have been elsewhere had I not raised my hand!

What’s worse, we get a loop in the partial causal explanation relation. The movement of my hand affects which manifold we all live in. But which manifold we all live in affects the movement of the objects in the manifold—including that of my hand.

The only way I can think of avoiding such backwards causation on something like the above model is to shift to some model that privileges a foliation into spacelike hypersurfaces, and then has something like this structure:

1. Spatiotemporality

2. Being in a specific branching spacetime

3. Being in a specific spacelike hypersurface inside one branch

4. Specific location within the specific spacelike hypersurface.

We also need some way to handle persistence over time. Perhaps we can suppose that the fundamentally located objects are slices or slice-like accidents.

I wonder if one can separate the above line of thought from the admittedly wacky determinate-determinable model and make it into a general metaphysical argument for a privileged foliation.

Eliminating contingent relations

Here’s a Pythagorean way to eliminate contingent relations from a theory. Let’s say that we want to eliminate the relation of love from a theory of persons. We suppose instead that each person has two fundamental contingent numerical determinables: personal number and love number, both of which are positive integers, with each individual’s personal number being prime and an essential property of theirs. Then we say that x loves y iff x’s love number is divisible by y’s personal number. For instance suppose we have three people: Alice, Bob and Carl, and Alice loves herself and Carl, Bob loves himself and Alice, and Carl loves no one. This can be made true by the following setup:

	Alice	Bob	Carl
Personal number	2	3	5
Love number	10	6	1

While of course my illustration of this in terms of love is unserious, and only really works assuming love-finitism (each person can only love finitely many people), the point generalizes: we can replace non-mathematical relations with mathematizable determinables and mathematical relations. For instance, spatial relations can be analyzed by supposing that objects have a location determinable whose possible values are regions of a mathematical manifold.

This requires two kinds of non-contingent relations: mathematical relations between the values and the determinable–determinate relation. One may worry that these are just as puzzling as the contingent ones. I don't know. I've always found contingent relations really puzzling.

Spacetime and Aristotelianism

For a long time I’ve been inclining towards relationalism about space (or more generally spacetime), but lately my intuitions have been shifting. And here is an argument that seems to move me pretty far from it.

Given general relativity, the most plausible relationalism is about spacetime, not about space.

Given Aristotelianism, relations must be grounded in substances.

Here is one possibility for this grounding:

1. All spatiotemporal relations are symmetrically grounded: if x and y are spatiotemporally related, then there is an x-to-y token relation inherent in x and a y-to-x token relation inherent in y.

But this has the implausible consequence that there is routine backwards causation, because if I walk a step to the right, then that causes different tokens of Napoleon-to-me spatiotemporal relations to be found in Napoleon than would have been found in him had I walked a step to the left.

So, we need to suppose:

2. Properly timelike spatiotemporal relations are grounded only in the later substance.

But what about spacelike spatiotemporal relations? Presumably, they are symmetrically or asymmetrically grounded.

If they are symmetrically grounded, then we have routine faster-than-light causation, because if I walk a step to the right, then that causes different tokens of x-to-me spatiotemporal relations to be found in distant objects throughout the universe.

Moreover, on the symmetric grounding, we get the odd consequence that it is only the goodness of God that guarantees that you are the same distance from me as I am from you.

If they are asymmetrically grounded, then we have arbitrariness as to which side they are grounded on, and it is a regulative ideal to avoid arbitrariness. And we still have routine faster-than-light causation. For presumably it often happens that I make a voluntary movement and someone on the other side of the earth makes a voluntary movement spacelike related to my movement (because there are so many people!), and now wherever the spatiotemporal relations is grounded, it will have to be affected by the other’s movement.

I suppose routine faster-than-light causation isn’t too terrible if it can’t be used to send signals, but it still does seem implausible. It seems to me to be another regulative ideal to avoid nonlocality in our theories.

What are the alternatives to relationalism? Substantivalism is one. We can think of spacetime as a substance with an accident corresponding to every point. And then we have relationships to these accidents. There is a lot of technical detail to work out here as to how the causal relationships between objects and spacetime points and the geometry of spacetime work out, and whether it fits with an Aristotelian view. I am mildly optimistic.

Another approach I like is a view on which spacetime position is a nonrelational position determinable accident. Determinable accidents have determinates which one can represent as values. These values may be numerical (e.g., mass or charge), but they may be more complex than that. It’s easiest in a flat spacetime: spacetime position is then a determinable whose determinates can be represented as quadruples of real numbers. In a non-flat spacetime, it’s more complicated. One option for the values of determinate positions is that they are “pointed spacetime manifold portions”, i.e., intersections of a Lorentzian manifold with a backwards lightcone (with the intended interpretation that the position of the object is at the tip of the lightcone). (What we don’t want is for the positions to be points in a single fixed manifold, because then we have backwards causation problems, since as I walk around, the shifting of my mass affects which spacetime manifold Napoleon lived in.)

Spacetime: Beyond substantivalism and relationalism

According to substantivalism, spacetime or its points or regions is a substance, and location is a relation between material things and spacetime or its points or regions. According to relationalism, location is constituted by relations between material things. Often, the two views are treated as an exhaustive division of the territory.

But they're not. Lately, I've found myself attracted to a tertium quid which I know is not original (it's a story other people, too, have come to by thinking about the analogy between location and physical qualities like charge or mass). On a simplified version of this view, being located is a determinable unary property. Locations are simply determinates of being located. This picture is natural for other physical qualities like charge. Having charge of 7 coulombs is not a matter of being related to some other substances--whether other charged substances or some kind of substantial "chargespace" or its points or regions. It's just a determinate of the determinable having charge.

This determinate-property view is more like the absolutism of substantivalism, but differs from substantivalism by not positing any "spacetime substance", or by making the locations into substances. Locations are determinates of a property, and hence are properties rather than substances. If nominalism is tenable for things like charge or mass, the theory won't even require realism about locations.

A Platonic theory of determinables

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.