Thursday, January 12, 2017

Being different at different times without changing

Here’s a curious thing: an unchanging object can have one shape at one time and a different shape at a different time.

Example 1: In the context of special relativity, times are spacelike hyperplanes. Suppose a special relativistic universe, and suppose that an object is an unchanging cube. Well, being a cube is not invariant between reference frames. So there will be one reference frame F1 at which the object is an unchanging cube and another reference frame F2 where it has some other unchanging shape. Each reference frame defines a family of times, i.e., spacelike hyperplanes. At the times of F1, the object is cubical and at the times of F2, the object is not cubical. Hence, at one time the object has one shape and at another it has another.

One might think that this example can be handled as follows: the object unchangingly is a cube-relative-to-F1 and a non-cube-relative-to-F2, and it is a cube-relative-to-F1 even at the times of F2 and a non-cube-relative-to-F2 even at the times of F1. But that’s probably mistaken. It seems to make no sense to talk of the shape-relative-to-F1 at times in F2. So we still have a difference in relative shape: the shape-relative-to-F1 is well-defined at F1 times but not well-defined at F2 times.

Example 2: Different universes will have different spacetimes, and hence different times. Suppose an object that is wholly present simultaneously in multiple universes—after all, that seems no harder than multilocation within the universe, and we have some evidence of miracles where a saint is in more than one place at the same time (for an account of such possibilities, see this). In each universe the object is unchanging, but it has a different shape in different universes. Since the different universes come with different times, the object has one shape at one time and a different shape at a different time.

This seems to be a refutation of the at-at theory of change, on which change just is difference in properties across times. But while the cases, if possible, do indeed refute that theory, there is a slightly richer at-at theory that is unaffected by them:

  • an object changes from having P to having Q provided that it has P and not Q at an earlier time and has Q and not P at a later time

  • an object changes with respect to having a property P provided that it changes from having P to not having P or from having not-P to having P.

So it’s easy to fix the at-at theory. Still, I think something has been learned here: there is an essential directionality to change.


Michael Gonzalez said...
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Michael Gonzalez said...

Your conclusion is a step in the right.... Ok, that was a bad pun; never mind. But, I don't think it solves the fundamental problem.

For one thing, your second premise for the richer at-at theory uses the concept of "change" in the definition of change with respect to P. It doesn't seem that anything is clarified. It's circular.

And your first premise, while adding directionality, doesn't seem to do what you want it to. It still seems clear that a 4d object has all of its parts (spatially and temporally extended). It doesn't change just because it is different at different times.

In fact, I think your second premise is very deadly to the B-theory. It simply isn't true of ANYTHING on B-theory that it goes from "not having P to having P". Think of a 4d worm version of myself. If I grow a pimple tonight, one might be tempted to say that I went from not having it to having it, but really the 4d worm that I am has ALWAYS had that pimple right there in that particular 4-dimensional location (the first 3 dimensions being the exact coordinates of the tip of my nose, and the 4th being time). Nothing has ceased to be the case, nor did anything become true. It is exactly as true now that I have a pimple at that 4d location as it was the moment I was born.

Michael Gonzalez said...

Take the example of the road that gets narrower. REALLY, what's true is that the road is UNCHANGINGLY narrower in some locations and wider in others. It never changes. It only appears to us to do so because we are TRAVELLING on it, and so we are experiencing different parts of it at different times. So, unless we're talking about some sort of B-theory where the present is privileged and we are traveling along this dimension (in which case temporal becoming is real and we are not worms after all), I don't see how B-theory can survive this issue of change (unless we just discard the idea of change...).

Alexander R Pruss said...

The second statement depends on the first. The first defines what it is to change between properties, and the second what it is to change with respect to a property.

Brian Cutter said...

Interesting points. In case you're interested, I briefly discuss a point essentially equivalent to your "Example 1" in my recent paper, "Spatial Experience and Special Relativity." ( (The point is raised in footnote 14 as an objection to my identification of times with space-like hyperplanes. There I basically give the first response you offer on behalf of the "at-at" theory---namely, one of the times needs to be later than (or wholly in the future of) the other.

Alexander R Pruss said...

Scooped again!

Michael Gonzalez said...

Still, it seems painfully clear that no change occurs, since the 4d object remains exactly as it always is. Nothing ceases nor does anything become. It never actually goes from having non-P to having P; it ALWAYS had not-P at this spatiotemporal location and P at this other spatiotemporal location.

Alexander R Pruss said...


On reflection, the solution won't work in Special Relativity if we have a bounded space, since then we could have two bounded hyperplanes that are timelike separated. I am also not sure it works in General Relativity.

Probably the better move is to say that 3D shape isn't intrinsic.

Brian Cutter said...

Hm.. Nice point. (And yeah, I was also unsure whether the suggestion works in general relativity, partly because I'm not really sure what to identify with "times" in a GR setting.) One suggestion I float in the paper I mentioned earlier (though not in connection with this particular issue) is that we could say that an entity (e.g. a hyperplane) only counts as a time relative to a "temporal system," where a temporal system is roughly a collection of entities which jointly play the "time role," e.g. can be linearly ordered w.r.t. before/after, have so-and-so relations to the laws of nature, and so forth. In the context of SR, each foliation is a different temporal system, so we'd say that a hyperplane is only a time relative to the foliation to which it belongs. In the context of a world consisting of two Galilean spacetimes, we'd have two temporal systems, and each simultaneity hyperplane would count as a time only relative to the (unique, privileged) foliation of its own spacetime.

From here, the at-at theorist might say: just as "being a time" is relative to a temporal system, so too *change* is relative to a temporal system. The right place to start, then, is with the question: "What does it mean for something to change-relative-to-S" (for a given temporal system S). Answer: x changes relative to S just in case x has different properties at two different times-relative-to-S. From here, we could (if we like) analyze "x changes simpliciter" as "x changes relative to some temporal system."

I think this will handle your two cases, as well as the bounded Minkowski case. In all those cases, we have something which has P at t1 and lacks P at t2, but in each case t1 and t2 belong to different temporal systems, so that there will be no single temporal system S such that the object has different properties at two times-relative-to-S. (Then I think we'd get to preserve the intrinsicality of 3D shape, which I'd very much like to do.)

Alexander R Pruss said...

The system stuff does the job, I think. Of course, now we have the very interesting question of what constitutes a temporal system, and whether this can be characterized without talking about change.

In GR, maybe a temporal system would be a maximal foliation by maximal spacelike hypersurfaces?

Brian Cutter said...

Yeah, that seems promising for the GR case (at least given my very limited understanding of GR). As for the issue of what constitutes a temporal system, and whether this can be characterized without using the notion of change, I agree that's an interesting and difficult question. In a relativistic setting it might be possible to appeal to relations like "belongs to the forward light cone of," but if we want a notion of a temporal system that generalizes beyond the relativistic case, we might characterize it in terms of causation or causal priority. For example, we might say that something is eligible to belong to a temporal system only if it's a collection of events such that no member of the collection is causally prior to any other member. Say that an event-collection C1 is "fully before" another event-collection C2 just in case some event in C1 is causally prior to some event in C2, and no event in C2 is causally prior to any event in C1. Then a temporal system will be (at a minimum--and assuming for convenience that backward causation is impossible) a set of event-collections linearly ordered by the "fully before" relation. This won't quite be a sufficient condition for being a temporal system (otherwise you'd have temporal systems whose members are small proper parts of individual space-like hyperplanes, and probably other weird cases as well), but maybe with a few more tweaks we could get a sufficient condition. Maybe other tweaks would allow a modest amount of backward causation if we'd like to retain that possibility.