Wednesday, April 20, 2011

Special Relativity and Perdurantism

There seems to be a problem for the conjunction of Special Relativity and perdurantism. Maybe this is a standard problem that has a standard solution?

Let's say that being bent is an intrinsic property. Perdurantists of the sort I am interested in think that Socrates is bent at a time in virtue of an instantaneous temporal part of him being bent (I think the argument can be made to work with thin but not instantaneous parts, but it's a little more complicated). Therefore:
  1. x is bent at t only if the temporal part of x at t is bent simpliciter.
The following also seems like something perdurantists should say:
  1. x is bent simpliciter only if every temporal part of x is bent simpliciter.
Now, we need to add some premises about the interaction of Special Relativity and time.
  1. There is a one-to-one correspondence between times and maximal spacelike hypersurfaces such that one exists at a time if and only if one at least partly occupies the corresponding hypersurface.
Given a time t, let H(t) be the corresponding maximal spacelike hypersurface. And if h is a maximal spacelike hypersurface, then let T(h) be the corresponding time. Write P(x,t) for the temporal part of x at t. Then:
  1. P(x,t) is wholly contained within H(t) and if z is a spacetime point in H(t) and within x, then z is within P(x,t)
and, plausibly:
  1. If a point within x is within a maximal spacelike hypersurface h, then P(x,T(h)) exists.
Now suppose we have Special Relativity, so we're in a Minkowski spacetime. Then:
  1. For any point z in spacetime, there are three maximal spacelike hypersurfaces h1, h2 and h3 whose intersection contains no points other than z.
Add this obvious premise:
  1. No object wholly contained within a single spacetime point is bent simpliciter.
Finally, for a reductio, suppose:
  1. x is an object that is bent at t.
Choose a point z within P(x,t) and choose three spacelike hypersurfaces h1, h2 and h3 whose intersection contains z and only z (by 6). Now define the following sequence of objects, which exist by 4 and 5:
  • x1=P(x,t)
  • x2=P(x1,T(h1))
  • x3=P(x2,T(h2))
  • x4=P(x3,T(h3))
Observe that x4 is wholly contained in the intersection of the three hypersurfaces h1, h2 and h3, and hence:
  1. x4 is wholly at z.
  2. It is not the case that x4 is bent simpliciter.
  1. x1 is bent simpliciter. (By 1 and 8)
  2. x2 is bent simpliciter. (By 2 and 11)
  3. x3 is bent simpliciter. (By 2 and 12)
  4. x4 is bent simpliciter. (By 2 and 13)
    Since 14 contradicts 10, we have a problem. It seems the perdurantist cannot have any objects that are bent at any time in a Minkowski spacetime. This is a problem for the perdurantist.

    If I were a perdurantist, I'd deny 2, and maintain that an object can be bent simpliciter despite having temporal parts that are bent and temporal parts that are not bent. But I would not be comfortable with maintaining this. I would take this to increase the cost of perdurantism.

    What is ironic here is that it is often thought that endurantism is what has trouble with Relativity.


    1. In special relativity, there are no such things as times simpliciter, although there can be times relative to a foliation, or relative to an inertial path. So I'd expect the perdursntist to eschew unqualified talk of temporal parts, perhaps replacing it with (say) temporal-part-relative-to-an-inertial-path. Then he'll grant 1 and 2 when relativized to an inertial path, but resist the inferences to the conclusion that x4 is a temporal part relative to any path on the grounds that the argument for it illicitly shifts inertial-path parameters when we're not looking.

    2. Who thinks endurantists have trouble with relativity? It would be good to see cites on this.

      Lots of people think presentists have trouble with relativity, but that's a completely different argument. (Or is it?) I thought it was well known that perdurantists have to say some quirky things about relativity thanks to Hud Hudson's work.

    3. I normally think of times as members of a foliation, though in the post I don't suppose an identity between times and the members of a foliation but only a correspondence.

      If we relativize bentness to a foliation or inertial path, then bentness ceases to be monadic, and the perdurantist's advantage over other solutions to the problem of temporary intrinsics evaporates. So, yes, the perdurantist can get out of the argument by such relativization, but only at the cost of losing one of the main advantages of the theory.

    4. On endurantism and relativity, I think the main thing is: S. D. Hales and T. A. Johnson, “Endurantism, Perdurantism and Special Relativity”, The Philosophical Quarterly, 53 (2003), 524-539. (There are various replies to this. I think their argument is invalid.)

      The SEP article on Temporal Parts has a section on arguments from relativity to perdurantism, and it concludes that while it seems that relativity favors perdurantism, the arguments for this aren't very good.

      See also Gilmore's paper against an argument by Balashov for perdurantism from relativity.

      Also, it seems intuitively a bit weird that you are wholly present in each of two different spacetime regions that intersect within you, as relativistic endurantism will say--it seems to imply you has something like a self-intersection.

      It's been a couple of years since I looked at the persistence and relativity literature, and even then I was mainly looking at endurance and relativity, not perdurance and relativity, so I wouldn't be surprised if Hud has an argument like mine.

    5. This comment has been removed by the author.

    6. Ahh, I see, I think I was misunderstanding the original argument. What happens is you take the bent thing confined to hyperplaneplane h1, and note that, relative to the h2 foliation, there is a temporal part that intersects it just at h2, and repeat, until you get down to a point. By the inheritance principle (2), each of these parts is bent; reductio. Got it.

      I'm thinking that the perdurantist should reject the inheritance principle, and should insist that it's not as plausible as it looks in the first place. Consider in a Newtonian spacetime a particle that pops into existence, goes north for five minutes, veers off northwest and goes that way for five more minutes, and then pops out of existence. That particle (thought of as a 4d worm) is four-dimensionally bent, and so it is bent simpliciter. But none of its temporal parts are bent.

      I'd also like to hear more about why you think denying 2 is a cost. It doesn't seem like a dictate of common sense or anything (after all, it's not as though "temporal part" is something we have pre-theoretic commitments about!)

    7. Jason:

      Well, I take the perdurantist to think that temporary intrinsic properties are inherited from temporal part to whole, and that's how the problem of temporary intrinsics gets solved.

      Suppose x is F at t1 but x is non-F at t2. The perdurantist explanation is that there are two temporal parts, x1 and x2, such that x1 is F simpliciter and x2 is non-F simpliciter, but x cannot be said to be F simpliciter and x cannot be said to be non-F simpliciter.

      Now suppose that, in turn, there are times t3 and t4 such that x1 is F at t3 and x1 is non-F at t4. Then it seems that by the same token we shouldn't be allowed to say that x1 is F simpliciter. And that's what 2 says.

    8. Couple of questions. I'm new to these sorts of discussions, so you'll hopefully forgive me if I've misconstrued parts of your argument or some technical aspects.

      I take your strategy to show that you can decompose an object x bent simpliciter at t first into a temporal part x1 indexed to one hypersurface, h1, then decompose x1 into *its* temporal part x2 indexed to h2, and so on. But this sequence of decompositions seems ill-motivated. For one thing, why suppose we can take the temporal part of a temporal part? What would that even mean? Just because we can intersect the temporal part x1 of x indexed to h1 with the hypersurface h2 doesn't make it meaningful to then speak of *the temporal part* of x1 indexed to h2.

      Indeed, suppose h1 and h2 are distinct hypersurfaces and suppose x1 is a temporal part of x indexed to h1 and x2 a temporal part of x1 indexed to h2. h1 and h2 presumably correspond to distinct times t1 and t2 respectively, and then by (3) x1 exists at t1 and x2 at t2. But x2 is by specification an instantaneous temporal part of x1, and it seems furthermore x2 exists at t2 simpliciter. It should follow by (1) then that x1 at t1 exists at t2, and this seems quite strange.

      So I suppose my question is, why doesn't your argument simply reveal the untenability of (3)? The problem seems to be that objects inherit properties of their temporal parts, which then in turn inherit properties of *their* temporal parts if we assume that this notion makes sense. In this case, since a given object can be made to have a sequence of temporal parts each existing within distinct hypersurfaces, it then seems by inheritance that our original object in fact exists at multiple times in virtue of being located within a single hypersurface because we can examine its temporal parts (and their respective temporal parts) with respect to the intersection of distinct hypersurfaces. It seems then as though the correspondence shouldn't be 1-1, which seemed to be bothering Jason in the first place (I think his point is worth pressing further).

      Also, why suppose that the perdurantist is committed to that a spacetime point should be a part of an object x, much less a *temporal* part of an object x? It seems pretty reasonable for the perdurantist to reject both notions, and that it's at least arguable that ordinary objects are not *composed* of spacetime points.

      And I'm not sure how to interpret (7). What does it mean to be contained *within* a spacetime point? The upshot of the entire argument put together seems to be that under the perdurantist view an object x cannot be bent simpliciter at t because a spacetime point in a region x occupies cannot be bent at t. But surely a carefully worked out perdurantist view needn't validate *that* inference?

    9. I think one way to block Alex's argument would be to impose a reasonable restriction on (1) and (2):

      (1) x is bent at t only if the temporal part of x at t is bent simpliciter
      (2) x is bent simpliciter only if every temporal part of x is bent simpliciter

      as follows:

      (1') x is bent at t only if the temporal part of x at t that has the same number of spatial dimensions as x is bent simpliciter

      (2') x is bent simpliciter only if every temporal part of x that has the same number of spatial dimensions as x is bent simpliciter,

      or something like that.

      Here is a reason for it: shapes are 3D properties, so no one should think they can survive elimination of one or more spatial dimensions. A spatial cross-section of a bent stick is not necessarily bent itself. Similar restrictions would apply to other bona fide temporary properties.

      Notice that this does not deprive the perdurantist of the resources needed to deal with the problem of temporary intrinsics (if it is a problem). Alex's x1 is still bent. But x2 etc. need not be.

      A minor point on Alex's (6):

      (6) For any point z in spacetime, there are three maximal spacelike hypersurfaces h1, h2 and h3 whose intersection contains no points other than z.

      Shouldn't there be 'four' instead of 'three'?

      There is by now a fairly extensive literature on persistence and relativity, including several distinct arguments for and against relativistic endurance, as well as for and against relativistic perdurance, and various objections to these moves (see recent and not so recent works by Rea, Sider, Hudson, Balashov, Gilmore, Eagle, Sattig, Gibson & Pooley, Miller et al). Useful surveys and updates on earlier arguments, as well as new developments can be found in my book (Persistence and Relativity, Oxford 2010) and an entry on persistence in Callender's Oxford Handbook of Philosophy of Time (fresh from the press). More references are on websites of the above-mentioned authors. The latest development is probably yesternight's Pacific APA/PTS session (i.e., Sattig v. Gilmore).

    10. Yuri:

      Thanks for the response and for the references! Is this argument somewhere in all that literature?

      Yes, (6) is false as it stands. It should say:

      (6a) For any point z in spacetime and any spacelike hypersurface h0 containing z, there are three maximal spacelike hypersurfaces h1, h2 and h3 whose intersection with h0 contains no points other than z.

      (Then let h0=H(t).)

      The dimensionality constraint is clever. One little thought: This response won't work in the case of an analogue of this argument against versions of perdurance that use non-instantaneous temporal parts.

      A problem for (1') and (2') is beings that change in their number of spatial dimensions. We can imagine an x that which is normally spatially three-dimensional, but that gets thinner and thinner until at t it is spatially two-dimensional, and then maybe it starts to grow again.

      We certainly want to say that x is two-dimensional at t. By (1'), this will only be true if the temporal part at t with the same spatial dimensionality as
      x is two-dimensional. But what does it mean that "the temporal part at t has the same spatial dimensionality as x"? If it means: "the temporal part at t has the same spatial dimensionality as x at t", then (1') seems to collapse into (1). If it means: "the temporal part at t has the same spatial dimensionality as x at all times", then there is no temporal part satisfying that constraint, and on a Russellian reading of "the temporal part..." the right hand side of (1') is false.

      But you don't need to modify (1) into (1') to refute the argument. You could leave (1) alone, and still modify (2) into (2'), and that would block my argument.

      I think, though, (2') suffers from the same ambiguity. What is, in general, the spatial dimensionality of an entity that changes in spatial dimensionality? Suppose K is a species of jellyfish that started off spatially two-dimensional and then grew to be three-dimensional, and suppose that membership in K is essential. If the spacetime worm (with two-dimensional spacelike cross-section near one end and three-dimensional cross-section near the other) Sam is a K, we'd like to say that Sam is a K simpliciter. So we want (2') to apply in such cases (with "is a K" in place of "is bent").

      But what do we make of the right-hand side of (2') in the case of Sam? Which temporal parts of Sam count as having the same spatial dimensionality as Sam. There are four options: (1) all of them do; (2) precisely the two-dimensional ones; (3) precisely the three-dimensional ones; and (4) none of them. Options (2) and (3) are arbitrary (we can imagine Sam spends an equal amount of time in the two phases). Option (4) makes the right hand side of (2') be trivially satisfied in the case of dimension-changing beings. I suppose that's OK, since (2') is an "only if". But we really want a principle that can be strengthened into an "if and only if"--we want a story about what it is to be bent simpliciter. That leaves option (1). But now "has the same spatial dimensionality as Sam" seems to mean: has the same spatial dimensionality as some temporal part of Sam, and (2') collapses into (2).

      For even more fun, we can imagine fractalish entities whose temporal parts vary continuously in their Hausdorff dimensionality. :-)

      I have no argument that there isn't some way of fixing up the dimensionality constraint.

    11. Anonymous:

      "So I suppose my question is, why doesn't your argument simply reveal the untenability of (3)?"

      Well, I can reformulate the argument by quantifying over reference frames and times in that frame. Then, (1) and (2) become:

      (1a) x is bent at t in frame F only if the temporal part of x at t in F is bent simpliciter
      (2a) x is bent simpliciter only if for every frame F and every time t in F, P(x,t,F) is bent.

      And then I can still run a variant of argument, because I can find three frames, and times respectively in them, such that by successively applying the P(x,t,F) operation I get to a zero-dimensional entity.

      You could, of course, get out of this by saying that bentness is always frame-relative (and so the right hand side of (1a) should be "bent simpliciter in F"), but then we don't have an account of temporary intrinsics but a denial of temporary intrinsics (which I am fine with, but it was supposed to be an advantage of perdurantism that it let you have change in respect of genuinely monadic properties).

    12. Alex:

      I don't remember coming across an argument of this sort in the literature on relativistic persistence.

      And the move to beings that change the number of their spatial dimensions is intriguing. One quick thought on them. Scenarios of this sort may, perhaps, be possible in a broad sense (I really don't now). But are they physically possible? On the face of them, they involve topological changes that may conflict with conservation laws, such as the generalized form of matter conservation linking local density of matter with the its local flow. How would you define continuous local density across changes in which local volume goes to or from zero measure?

      Why is this worry about transgressing the boundaries of the physically possible important here? Because the case is initially motivated by relativistic considerations, which presuppose the validity of physical conservation laws. This makes scenarios that violate them irrelevant, even if they are not impossible tout court.

    13. Yuri:

      A lot of four-dimensionalists are fond of unrestricted compositionality principles that give credence to the idea that for any spacetime region, there is an object precisely occupying it.

      Also, I don't know if the spatial dimensionality of a four-dimensional object is defined simpliciter. If the spacetime region exactly occupied by the object is compact, some maximal spacelike hypersurfaces are likely (maybe even certain--I can't imagine a case where they don't) near the timelike beginning and end of the object are likely to intersect the object in a region of dimension less than three. And so such an object has temporal parts of dimension less than three. (This won't work for an object that exactly occupies an open region.)

      I've been away for the weekend at Jon Jacobs' power conference, but I requested the library to deliver your book to me (we've got a nice faculty book delivery arrangement), so I'll look at it as soon as I am at my Dept.

    14. You are quite correct.

      Pedurantism is unrigorous in any case of a physical three dimensional object with internally moving parts, which is pretty much a fatal flaw.

    15. I wrote a little paper on the implications of simultaneity on this:

    16. What an interesting and complex discussion. I'm a perdurantist by the way, though I'm new to philosophy and haven't read any of the referenced articles (yet). What is missing above is a definition of the meaning of "bent" though Yuri for one identifies that as the problem. Without attempting to be rigorous, one might say that something is bent if three points on the object are not collinear. That can apply to a 3D object either at a given time or over a period, or to three events along a worldline of a point-like entity, but it cannot apply to a single event by definition, it requires extension in at least one dimension.

      Imagine Socrates standing still and upright. We can identify three atoms, one in his head, one in his stomach and one in his foot which are collinear, hence he is not bent. If he then bends over to tie his shoelace, the three atoms are no longer collinear, he has become bent.

      The worldline of the atom in his foot is not bent since it is fixed to the earth but that in his head has a bend at the event where he started to bend forward and a second when he reached the angle at which he could reach his shoes and stopped leaning any farther forward. The atom in his stomach probably has opposing bends so he kept his balance, though may be straight if he was sitting against a wall.