Two implications of Aristotle's theory of time and locality

Aristotle thinks that time is infinitely subdivisible but only finitely subdivided. Thus, there will be moments t₁ < t₂ such that there is actually no moment between t₁ and t₂, but there could have been. What would make there have been a moment between t₁ and t₂? Presumably, this would be if something happened strictly between those times. Time is the measure of change, so if, say, some object started or finished changing at a time between t₁ and t₂, then there would have been a time between them, say t_1.5.

But here is a curious consequence. Suppose that in the actual world, w₀, I am living from t₁ to t₂, which are so close together in time that there are no time between them. But in another world, t₁, where everything in our galaxy was the same, in some other galaxy indeterministically an event happened between t₁ and t₂, namely at t_1.5. Then:

• In w₀, it is not true that I exist at t_1.5 (because there is no t_1.5).

• In w₁, it is true that I exist at t_1.5.

And what is responsible for that difference is that indeterministic event in another galaxy. So it seems that something in another galaxy is responsible, in a faster-than-light way, for whether I exist at t_1.5. In other words, the Aristotelian theory seems to imply highly non-local influences.

There is perhaps a way out. Perhaps fundamentally time sequences are internal to substances. Thus, I have a time sequence internal to me, you have one internal to you, and things in that other galaxy have time sequences internal to them. There are, additionally, connections (probably causal ones) between objects that allow one to form a global time sequence. That global time sequence will include moments that don’t correspond to any moments internal to me. For instance, it will include moments earlier than my conception, but more interestingly, it could be that for me t₂ immediately succeeds t₁, but something else has a time that fits between t₁ and t₂, and so global time could have times corresponding to t₁ and t₂, but also some intermediate time between them.

The difference between w₀ and w₁, then, would not be a difference in my internal time sequence. What happened in that other galaxy wouldn’t affect my internal time except perhaps once the light from that galaxy could reach me.

On this account, while it is true that I exist at t_1.5, my existing at t_1.5 is not an intrinsic feature of me. The difference between my existing at t_1.5 in w₁ and my not existing at t_1.5 in w₀ is a merely Cambridge difference.

I think it is hard to make this story fit with presentism. When t_1.5 is present, then it had better be intrinsic to me that I exist presently, i.e., at t_1.5. A similar point applies to growing block.

Maybe, though, there is a way of making this story fit with a moving spotlight A-theory. We could suppose that at global time t_1.5, what is “lit up” by the spotlight is the time t_1.5 for the thing in the other galaxy that has something happening to it then, but for me what is lit up is the entire interval between t₁ and t₂.

If I am right, then

1. Locality, and

2. Aristotle’s theory of time

seem to imply:

3. Internal time is primary

4. Eternalism is true.

Substantivalism and locality

I find myself going back and forth between substantivalism and relationalism about spacetime. On a substantivalist theory, the points of spacetime are real.

But here is a problem. It seems essential to the concept of a point that geometric relations between points are essential to them. If two points are a certain distance apart, say, then they couldn’t be a different distance apart. But on General Relativity, where geometric properties are determined by the distribution of mass-energy in the universe, if geometric relations between points are essential to them, locality is violated. For imagine two events that are distantly spacelike separated. Then the geometric relation between the points at which the events are found depends on the distribution of mass-energy between the events. If the geometric properties are essential to the points, then influencing the mass-energy between the events will affect which points these events happen at. And that will be a non-local influence.

Perhaps we can say that only local geometric relations are essential to points. Perhaps the way to say this is that if a point x exists in worlds w₁ and w₂, then there is a set N of points such that every member of N exists in both worlds, and N is a neighborhood of x in both worlds, and the geometry on N is the same in both worlds.

A deterministic collapsing local quantum mechanics without hidden variables beyond the wavefunction

I will give a really, really wacky version of quantum mechanics as a proof of concept that if one wants, one can have all of the following:

1. Compatibility with experiment

2. Determinism

3. Collapse

4. No “hidden variables” beyond the wavefunction: the wavefunction encompasses all the information about the world

5. Locality

6. Schroedinger evolution between collapes.

Here’s the idea. We suppose that the Hilbert space for quantum mechanics is separable (i.e., has a countable basis). A separable Hilbert space has continuum-many vectors, so each quantum state vector can be encoded as a single real number. We suppose, further, that collapse occurs countably many times over the history of the universe. We can now encode all the times and outcomes of the collapses over the history of the universe as a single real number: the outcome of a collapse is a quantum state vector, encodable as a real number, the time of collapse is of course a real number, and a countable sequence of pairs of real numbers can be encoded as a single real number.

We now consider the wavefunction ψ of the universe. For simplicity, consider this as a function on R³ⁿ × R where n is the number of particles (if the number of particles changes over time, we will need to tweak this). Say that x ∈ R³ⁿ is rational provided that every coordinate of it is a rational number. We now add a new law of nature: ψ(x,t) has the same value for every rational x and every time t, which value encodes the history of all the collapses that ever happen in the history of the universe.

Since standard quantum mechanics does not care about what happens to the wavefunction on sets of measure zero, and the set of rational points of R³ⁿ has measure zero, this does not affect Schroedinger evolution between collapses, and so we have 6. We also clearly have 2, 3 and 4. If we suppose a prior probability distribution on the collapses that fits with the Born rule, we get 1. We also have 5, since any open region of space that contains an experiment will also contain the real number encoding the collapse history.

Of course, this is rather nutty. It just shows that because the wavefunction has more room for information than just the quantum state vector—the quantum state vector can be thought of as an equivalence class of wavefunctions differing on sets of measure zero—we can stuff the hidden variables into the wavefunction. Those of us who think that the state vector is the real thing, not the wavefunction, will be quite unimpressed.

Does general relativity lead to non-locality all on its own?

A five kilogram object has the determinable mass with the determinate mass of 5 kg. The determinate mass of 5 kg is a property that is one among many determinate properties that together have a mathematical structure isomorphic to a subset of real numbers from 0 to infinity (both inclusive, I expect). Something similar is true for electric charge, except now we can have negative values. Human-visible color, on the other hand, lies in a three-dimensional space.

I think one can have a Platonic version of this theory, on which all the possible determinate properties exist, and an Aristotelian one on which there are no unexemplified properties. There will be important differences, but that is not what I am interested in in this post.

I find it an attractive idea that spatial location works the same way. In a Newtonian setting the idea would be that for a point particle (for simplicity) to occupy a location is just to have a determinate position property, and the determinate position properties have the mathematical structure of a subset of three-dimensional Euclidean space.

But there is an interesting challenge when one tries to extend this to the setting of general relativity. The obvious extension of the story is that determinate instantaneous particle position properties have the mathematical structure of a subset of a four-dimensional pseudo-Riemannian manifold. But which manifold? Here is the problem: The nature of the manifold—i.e., its metric—is affected by the movements of the particles. If I step forward rather than back, the difference in gravitational fields affects which mathematical manifold our spacetime is isomorphic to. If determinate position properties are tied to a particular manifold, it means that the position of any massive object affects which manifold all objects are in and have always been in. In other words, the account seems to yield a story that is massively non-local.

(Indeed, the story may even involve backwards causation. Since the manifold is four-dimensional, by stepping forward rather than backwards I affect which four-dimensional manifold is exemplified, and hence which manifold particles were in. )

This is interesting: it suggests that, on a certain picture of the metaphysics of location, general relativity by itself yields non-locality.