Over at Matters of Substance, Alexander Pruss has an argument against perdurantism in the context of STR. It has, on my reconstruction, 5 assumptions:

- A perduring object is bent
*simpliciter*iff every temporal part is bent. - There is a one-one correspondence between maximal spacelike hyperplanes (flat hypersurfaces) in Minkowski spacetime and moments of time.
- A temporal part is the intersection of a time with the set of points occupied by an object (this captures Pruss’ (4) and (5)).
- For any point in Minkowski spacetime, there exist 4 hyperplanes that overlap at and only at that point.
- No point-sized object can be bent.

Given these assumptions, the argument runs like this. Consider an object *x* that is bent *simpliciter*, at some time *t*, such that some point *z* is in the intersection of *x* and *t*. By (D), there are 3 hyperplanes distinct from *t* that intersect on *z*: call them *h*_{1}, *h*_{2}, and *h*_{3}. By (B), each of these hyperplanes is a time. By (C), there is a thing, *x*‘s temporal part at *t*, call it *x _{t}*. By (A),

*x*is bent. By (C), there is a thing,

_{t}*x*‘s temporal part at

_{t}*h*

_{1}, call it

*x*

_{t,1}. By (A),

*x*

_{t,1}is bent. By (C), there is a thing,

*x*

_{t,1}‘s temporal part at

*h*

_{2}, call it

*x*

_{t,1,2}. By (A),

*x*

_{t,1,2}is bent. By (C), there is a thing,

*x*

_{t,1,2}‘s temporal part at

*h*

_{3}, call it

*x*

_{t,1,2,3}. By (A),

*x*

_{t,1,2,3}is bent. Since (B) involves taking successive

*intersections*, it follows that the only points in the location of

*x*

_{t,1,2,3}are in the intersection of all of these hyperplanes—i.e.,

*z*is the only point in this location. So

*x*

_{t,1,2,3}is a a bent point object; by (E), this is impossible. So there can be no objects that are bent

*simpliciter*. (The argument against perdurance then I suppose is that they too cannot have objects that are bent

*simpliciter*, and need to relativise just like an endurantist does faced with the same problem of temporary intrinsics, thus removing a supposed advantage of perdurance over endurance.)

Note that the argument isn’t really an argument against perdurance at all. For there is a simple variant that is an argument against relativistic *endurance*. We only need these principles, just as plausible as (A) and (C):

~~ ~~

~~An enduring object is bent~~*constantly*iff it is bent at each of its locations (the regions in which it is wholly present).~~An object which exists at a time is wholly present at the intersection of that time with the set of points occupied by the object at that time.~~

~~So now consider an enduring object ~~*x* that is bent *constantly* and wholly present at *t*, and which has a spatial part at *z*. By (D) our familiar hyperplanes exist and overlap only on *z*; by (B), they are times; by successive applications of (G), *x* is wholly present at the region which is the intersection of *x*‘s location and each of the hyperplanes; so *x* is wholly present at a point. By (F), *x* is bent at that point; by (E), that is impossible. So there are no enduring objects that are bent *constantly*. But it is surely possible that there are constantly bent objects.

- An enduring object is bent
*constantly*iff it is bent at each time at which it exists. - If an enduring object is bent at some temporally extended region which it occupies, then it’s bent at every time in that region (i.e., bent at every subregion obtained by intersecting the region with a time).

So now consider an enduring object *x* that is bent *constantly* and wholly present at *t*. By (D) our familiar hyperplanes exist and overlap only on *z*; by (B), each of those hyperplanes is a time, so by (F), *x* is bent at each hyperplane. In particular, it is bent on *t*. Since many parallel spacelike hypersurfaces intersect with *t*, by (B), many times intersect with *t*, so *t* is temporally extended. By (G), *x* is bent at every time in *t*, and so in particular is bent at the intersection of *t* and *h*_{1}. Since *t* ∩ *h*_{1} is itself temporally extended (since many non-overlapping entities that, by (B), are times intersect with it), by (G), *x* is bent at the intersection of *t* ∩ *h*_{1} with some time in it, perhaps *h*_{2}. By the same argument, *t* ∩ *h*_{1} ∩ *h*_{2} is temporally extended, and so *x* is bent at every time in that region, so will be bent at *t* ∩ *h*_{1} ∩ *h*_{2} ∩ *h*_{3}—i.e., it will be bent at *z*. But that conflicts with (E). So there are no enduring objects that are bent *constantly*. But it is surely possible that there are constantly bent objects.

What’s gone wrong? The problem is that allowing overlapping times, in conjunction with ordinary definitions of temporal part and wholly present, entails that there are all sorts of weird regions which end up being eligible to contain objects or temporal parts.

I think the most natural response is to deny (B). It is just not true that there is a correspondence between moments of time and spacelike hyperplanes. (In Pruss’ original formulation, this will mean that the function from hypersurfaces to times he introduces, *T*(·), is only a partial function, so that some instances of his claim (5) will fail because the definite description ‘the temporal part of *x* at *T*(*h*)’ misfires.) For ‘moment of time’ is a frame-relative notion; a maximal spacelike hyperplane *h* has that maximal spacelike status frame-invariantly, but it is only a moment of time relative to those inertial frames in which it is a hyperplane of simultaneity. Of course moments of time in distinct frames can overlap; but (following Balashov’s *Persistence and Spacetime*, §5.1), the correct definitions of the notions ‘temporal part’, ‘eligible location of an object’, ‘achronal part’, etc. are all *frame-relative*. This is so for endurantists and perdurantists alike—all need to give relativistic analogues of these familiar metaphysical notions, and giving a frame-relative translation of these things is the only way to get anything that deserves to play the ‘temporal part’-role, or the ‘temporary location-role, etc. For example—to be a good deserver of the description, ‘moment of time’, an entity better not overlap with any other entity correctly described as a moment of time. On (B), this is false—moments of time overlap all over the place. If ‘moment of time’ is however translated to the relativistic ‘moment of time in frame *F*‘, then moments of time do not overlap.

The alternative to frame-relativising is to abandon the notion of temporal part as a distinctive category in relativistic theories of persistence, and just make do with spatiotemporal part. This is actually a strategy I prefer; if I remember rightly, it’s the strategy pursued by Ian Gibson and Oliver Pooley in their paper ‘Relativistic Persistence‘. But on this strategy, (A) and (F) are going to be the culprits—if temporal part, or temporary location, don’t play an interesting role in relativistic metaphysics, then we are hardly going to be tempted to think that bentness, constant or *simpliciter*, of an object is to be best understood making use of these notions. (No more than we think combustion is best understood by making use of phlogiston, no matter how tempting the association between combustion and phlogiston might have been on a prior but now superseded theory.)

One couldn’t preserve the argument by simply replacing (C) with this claim:

- A temporal part is the intersection of a spacelike hyperplane with the set of points occupied by an object.

For there is no plausibility in associating temporal parts with arbitrary spacelike hyperplanes, if such hyperplanes aren’t themselves times! While, given DAUP, there will be objects that exactly occupy the intersections of perduring objects and arbitrary hyperplanes, there is no reason to think that such objects are of interest from the point of view of change or persistence, no more than other fusions of proper spatial parts of me at different times are of interest for change or persistence in a classical spacetime. In particular, the objects whose existence is guaranteed by (H), namely, *x*_{t,1} and the like, aren’t going to be temporal parts, because they don’t occupy just one time, and so don’t need to be bent in order for the things they are parts of to be bent.

Rejecting (B) in this way still involves relativisation. As Balashov puts it, ‘in the Minkowskian framework the possession of temporary properties is relativised, in effect, to times-in-frames’ (p. 94). This relativisation might seem like a cost—isn’t that kind of the whole point of the problem of temporary intrinsics? But as the example of length contraction shows, it’s unavoidable in a relativistic framework. Length at a time is a frame-relative notion; there is just no way of treating it as an intrinsic property. For the points included in the relevant times vary from frame to frame, which is why an object can occupy regions with different volume (manifesting as different length) from frame to frame, from the perspective of a single given point. Now perhaps we can in some ways avoid this; perhaps there is a frame, the rest frame for an object, which gives the proper length; maybe length can be a temporary intrinsic when evaluated with respect to a rest frame. But the problem for the endurantist is that they need to relativise to times and frames; even if we allow this move, that an object’s rest frame is privileged *for that object*, and length in that frame is the intrinsic proper length, then variations in length can be had without further relativisation by temporal parts of the object, but we still need additional relativisation for variations in length of enduring objects.

Part of this is due to our use of ‘bent’ and ‘length’, etc. If we talk about the variation from region to region in frame-independent relativistic properties—those properties of objects that are more *natural*, closer to appearing in fundamental physics—then we never need to relativise these properties to times or frames. They are had, unrelativisedly, by whole 4D objects in virtue of the arrangement of their parts simpliciter, without regard to whether those parts are spatial or temporal or both. Alteration from region to region can be perfectly well understood to ground all other change without further need of relativisation. As I said elsewhere,

[For the perdurantist] no fundamental natural property is relativised. The situation thus remains quite different to what must happen in the case of endurantist intrinsic change, which does involve relativising such properties. Since perdurantists can treat all facts in relativistic spacetimes—including all facts about the location and identity of persisting objects—as supervening on the distribution of perfectly natural intrinsic properties over spacetime, no fact in the supervenience base is relativised. (‘Duration in Relativistic Spacetime‘,

OSM5)

*[Post edited October 24, 2011, in response to a comment by Alexander Pruss; old text is struck-through, new text in green.]*

I don’t think the endurantist version of the argument works. The problem is that at although x wholly exists at h1 and x wholly exists at h2 and x wholly exists at h3, you don’t get to conclude from that that x wholly exists at the intersection of h1, h2 and h3. If that were a good inference, then you’d be able to conclude that endurantists are committed to objects wholly existing at every spacetime point within them.

By the way, I’m neither an endurantist nor a perdurantist, so I’m not going to be devastated if the endurantist version works. I am fond of the view (which is saying less than that I endorse it) that we’re four-dimensional extended simples.

Right. What I need instead is something like:

Then we get that a constantly bent object

xis bent at h1; by (B), h1 is temporally extended (it intersects with many times parallel to h2), so by (G*)xis bent at the intersection of h1 and h2. By (B), this intersection too is temporally extended, since it intersects with many times parallel to h3; so by (G*)xis bent at the intersection of h1 and h2 and h3. But that intersection is a point, soxis bent at a point.The object

xisn’t located at a point, but the idea of a constantly bent object should have been of one that is bent whenever it exists, and on that alternative conception, (G*) is plausible.And here it is (B) that does the work: without frame-relativisation (which if introduced makes (G*) very plausible), we get weird consequences because times can be said in a to overlap, and moments of time turn out to be temporally extended themselves, etc. Better to do away with times altogether, or to frame-relativise them.