Tuesday, October 19, 2021

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:

  1. 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.)

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