In order to get the desired result that the 3D and 4D views are equivalent, the authors need “sums-at-times” to satisfy two requisites: (a) sums-at-times are acceptable for endurantists, i.e. they are not additions to the endurantist ontology, they are nothing over and above the enduring particles that the endurantist already accepts (b) Sums-at-times are “timebound”, i.e. they exist at only one time. For any two different times t and t’ in which an object O exists, (O, t) is numerically distinct from (O, t'). (Because of problems with the blogger, I use brackets instead of > and < to represent sums-at-times...In my notation, (O, t)represents the sum of particles that constitute O at t).

The second requisite is necessary for the translation scheme they propose to work. If sums-at-times are not timebound, then something is true of them that is not true of temporal parts (namely, that they exist or may exist at more than one time). This is why, I think, the authors hasten to emphasize that

(O, t) [the sum of particles that constitute O at t] may be understood as a 3D object which exists only at time t and no other time. […] The upshot of this is that the intertranslatability of 3D and 4D descriptions rests ultimately upon entities which can be described indifferently as “instantaneous 4D temporal parts”, or “3D objects which exist at one time only”. (p. 574)

But in ensuring that sums-at-times satisfy (b), the authors compromise (a). Understood as entities that exist at only one time, sums-at-times are genuine additions to the endurantist ontology. And this is so independently of how ontologically promiscuous the endurantist decides to be about other issues (i.e. whether she accepts coincidence, arbitrary composition, etc) while still being endurantist.

Take an example. Suppose that there are two times t and t’ such that Tibbles does not change in its constituent particles from t to t’. Then the set of particles that constitute Tibbles at t is the same set that constitutes it at t’. However, given (b), (Tibbles, t) is not identical to (Tibbles, t’). They are two different entities, one existing only at t and the other only at t’. But why should the endurantist accept the existence of these two numerally distinct things, (Tibbles, t) and (Tibbles, t’)? She accepts the existence of Tibbles, the existence of times, and the existence of enduring particles that constitute Tibbles at different times. Let us assume that she will also accept the existence of sums of these particles. So she will accept the existence of a sum of particles that constitute Tibbles at t, and a sum that constitutes Tibbles at t’. But why should she say that these are two numerically distinct things? After all, they are composed of exactly the same enduring particles. Nothing in the endurantist’s position commits her with the existence of two things here. In fact, the endurantist position can be understood precisely as the negation of the existence of two distinct things in a case like this. So understood, the endurantist view is that there are sums-at-times, but not as many as the perdurantist think there are. Notice that the endurantist can have this view even if she accepts unrestricted mereological composition. The existence of two different sums-at-times in the example above does not follow from accepting arbitrary composition. It would follow from accepting arbitrary decomposition. But this is precisely the doctrine that the endurantist refuses to accept, and what makes her position non-equivalent to perdurantism. .

## 13 comments:

Hi Pablo!

In my own view, being appropriately promiscuous requires acknowledging that everything is a thing—and thus in particular that (O, t) is something, and hence something distinct from (O, t').

But be this at it may,

theyseem to share a concern like yours. That seems to be why they write:“Sums-of-particles-at-a-time need not be considered as new semantic entities, i.e. as new members of the domain, but can be identified with the semantic referents of predicates. As was seen, the basic building blocks of 3D semantics are 3D particles which exist through time. These are the sole elements of the domain.”(p. 573)And they exemplify how this works with Tibbles. What do you think?

Hi Dan,

thanks for replying! You say that:

“On my own view, being appropriately promiscuous requires acknowledging that everything is a thing—and thus in particular that (O, t) is something, and hence something distinct from (O, t')”.

If being promiscuous means accepting every putative entity in our ontology, then of course the endurantist cannot be completely promiscuous while still being an endurantist. Endurantism is a restrictive thesis in ontology, a bit like nominalism or presentism. Endurantists claim that there are fewer objects than perdurantists think there are. They reject at least some of the many short-lived objects countenanced by the perdurantist. That is the ontological disagreement between the two views. What I was trying to say is that this endurantist’s restrictive thesis is compatible with other “promiscuous” views, like unrestricted mereological composition or the acceptance of (some) coincident objects. So the endurantist can say, with you, that everything is a thing. She can also say, with you, that (O, t) is a thing. But she will not go with you (and the perdurantist) in saying that (in case like the one I have presented, where O does not change in its constituent parts from t to t’) (O, t) is something *different* form (O, t’). This extra claim is not forced upon her by her previous commitment to the existence of enduring particles, even assuming unrestricted mereological composition.

A source of confusion here may be this: the authors use ordered pairs to *represent* sums-at-times. The ordered pairs (O, t) and (O, t’) are of course different. But the sums at times are not really ordered pairs. They are concrete things that are merely represented by ordered pairs. Now, in order to get their result the authors need to assume that the sums-at-times represented by the pairs (O, t) and (O, t’) are different. This way they satisfy the requite (b) of my original posting. But in doing so, they go beyond the endurantist’s commitments. Endurantists can and should say that the pairs (O, t) and (O, t’) represent one and the same concrete thing.

As for the other part of your comment: I think what they say about sums of times being the semantic values of predicates is intended to satisfy the requisite (a) of my original posting, i.e. it is intended to show that sums-at-times are not genuine additions to the endurantist ontology. But it seems as if they are not consistent with what they say here, and that this way of understanding sums at times (as extensions of predicates) is abandoned later in the paper in order to satisfy requisite (b) and get the “translation” of 3D to 4D.

They say that the sum (O, t) is the semantic value of the predicate F:“is one of the constituent particles of O at t”. But if O does not change in its constituent parts from t to t’, then there is just one set of particles that constitutes O both at t and at t’. So the semantic value of predicate F is the same as the semantic value of G:“is one of the constituent particles of O at t’’”. There is only one and the same extension for both predicates: the sum (O, t) = the sum (O, t’). This would be ok for the endurantist, of course. But the authors do not stick to this way of understanding sums-at-times (as the extensions of predicates like F and G). In order to get their desired “equivalence of 3D and 4D”, they need sums-at-times to satisfy requisite (b), and be time-bound. So they commit themselves to the idea that (O, t) is not the same as (O, t’), even if only one thing is required is for the semantics of predicates F and G.

Hi again,

Just a couple of very brief comments.

Re “restrictivism”, this looks to me to be precisely the kind of view one may hold

in addition toone’s being a defender of 3D or 4D. (Of course, one is free to call ‘endurantist’ to someone who holds that objects persist by enduringand alsothinks that “there are fewer objects than perdurantists think there are.” But it’s not clear the point of such an alternative stipulation, is it?)Re the issue at hand, you say:

A source of confusion here may be this: the authors use ordered pairs to *represent* sums-at-times. The ordered pairs (O, t) and (O, t’) are of course different. But the sums at times are not really ordered pairs. They are concrete things that are merely represented by ordered pairs.Why do you say this? I thought of the quote I provided about their saying that sums-of-particles-at-a-time are not new members of the domain as supporting the interpretation that for them they are sort of abstractions, be that ordered pairs or alternative entities, as opposed to concrete things. But maybe I’m misreading them. Which part of the paper do you think supports your interpretation?

See also:

“The upshot of this is that the intertranslatability of 3D and 4D descriptions rests ultimately upon entities which can be described indifferently as “instantaneous 4D temporal parts”, or “3D objects which exist at one time only”. For the 4D ontologist these entities are primitive and basic; for the 3D ontologist they are defined as ordered pairs of sets of enduring particles and times.”(p. 574)Hi again,

As I understand the issue, what you call “restrictivism” (the idea that there are less “short-lived” objects than a perdurantist will countenance) is not a view that one can have “additionally” to being a defender of 3D (here I use ‘3D’ and ‘endurantism’ interchangeably). It is precisely what being a 3Dist means. This is a complicated issue because, as discussed by Sider (2001) ch. 3, it is not completely clear what 3D really amounts to. But if something is clear is that the disagreement between 3D and 4D concerns the existence of temporal parts and that the friend of 3D *at least* denies the 4D doctrine that “necessarily, every spatiotemporal object has a temporal part at each time in which it exist”. Some 3D will want to go further than merely denying 4D; some will want to say, for instance, that spatiotemporal objects do not have proper temporal parts at all. I do not know if this stronger view is tenable (Sider 2001 p. 64 argues that it is incoherent with other doctrines that many 3Dists will also want to hold). But in any case, I was thinking all along of an endurantist that just denies 4D and therefore may admit that there are some proper temporal parts, but just not as many as the 4D countenances –not one at each time in which an object exists. Another way of putting this weak form of 3D would be this: whereas objects may have some proper temporal parts, they are not “arbitrarily divisible” along the time axis into temporal parts. The point is that even this weak form of 3D is “restrictive” when compared with 4D. But it must be in order to count as 3D! (You say that a 3Dist is someone who believes that “objects persist by enduring”. But “persist by enduring” is generally explained in terms of lack of proper temporal parts, so the view turns into ontology again).

Also, the temporal parts whose existence is disputed by the 3D and 4D are concrete things. The 3D can perfectly accept the existence of “ersatz” temporal parts, like ordered pairs of objects and times. It is not over the existence of those abstract things that the 3D and the 4D disagree. So if sums-at-times are ordered pairs (rather than concrete objects represented by ordered pairs), the 3D can accept as many of them as you want, and in particular he can accept that (O, t) is not the same as (O, t’) in our example above. But so conceived, as mere abstractions, sums-at-times are not equivalent of to the perdurantist’s temporal parts.

With respect to the textual issue: in the passage you reproduce, the authors seem to identify sums at times with ordered pairs, but in a previous passage they say that ordered pairs *represent* sums-at-times. The point is clearer in another paper by Lowe (“Vagueness and Endurance”, Analysis 2005). There he says:

By a ‘momentary sum of molecules’ I mean a sum-at-a-moment of molecules, i.e. something that could be represented by an ordered pair of a sum of molecules and a moment of time, (S, t), and whose existence- and identity-conditions are correspondingly these: if M=(S, t), then M exists if and only if S exists at t, and if M exists then M exists only at t; and if M=(S, t) and M’=(S’, t’), then M=M’ if and only if S=S’ and t=t’. (Note: we shouldn’t take the identity sign in ‘M=(S, t)’ literally, of course –it’s just a shorthand for ‘is uniquely represented by’. (…) (p. 109)Here it is clear the sums at times are not ordered pairs but something represented by ordered pairs. He does not say here that they are concrete things, but it seems to me that they must be so, if they are going to play the job of temporal parts –which are concrete things! But then, understood as concrete instantaneous entities, sums-at-times are genuine additions to the endurantist ontology of enduring particles. The endurantist may be committed to accepting sums of enduring of particles constituting S, but not necessarily to one for each time in which S exists. In particular, if S does not change in its constituent particles from t to t’, the endurantist can and should refrain from having two different sums-at-times.

Not sure we're not talking past each other here, for I was going to repeat the essentials of my previous comments: that endurantism/3D better be understood in terms of the contention that objects persist by enduring (so that, if that means there turns out to be less disagreement with perdurantism/4D than some people like Sider might have thought, let this be!); and that, according to them, sums-of-particles-at-a-time have "existence- and identity-conditions" of the abstract ordered pairs, as opposed to concrete things.

But just to elaborate: (i) if "restrictivism" were not an additional claim, there would be an argument from 3D (understood as "persistence is endurance") in its favor. Any suggestion? (ii) You said that it is fine for 3D to acknowledge pairs of enduring particles and times. So would it be your objection against McCall&Lowe* who just

identifiedsums-of-particles-at-a-time with the pairs?(McCall&Lowe* make this explicit: “The upshot of this is that the intertranslatability of 3D and 4D descriptions rests ultimately upon entities which can be described indifferently as “instantaneous 4D temporal parts”, or “3D objects which exist at one time only”. For the 4D ontologist these entities are primitive and basic

and concrete; for the 3D ontologist they are defined as ordered pairs of sets of enduring particles and timesand thus abstract” (p. 574*))I meant of course "

whatwould it be your objection....".Hi Dan,

I took me a while to realize that your quote* was an imaginary one and that McCall&Lowe* were not McCall&Lowe. At first I thought that I had everything wrong!!! : ). Anyway, I think the problem of *identifying* sums-at-times with ordered pairs is that the identification would rather obviously misconstrue the dialectics between 3D and 4D. Both 3Ds and 4Ds can make the distinction between temporal parts (concrete things) and ordered pairs (abstract things). And both can agree to present their disagreement as concerning the cardinality of the first and not the second. They could say: we agree that there are n abstract things, but disagree about how many concrete things there are. I think that once the disagreement is stated in this way, it is not plausible to understand talk about temporal parts as equivalent to talk about ordered pairs.

I insist that the disagreement between 3D and 4D (or endurantism and perdurantism) is at the bottom about how many concrete things there are, but cannot say much more. This is the way Sider presents the debate, and I think this is also the standard way of understanding it. To say that objects persist by enduring (as you do) is generally taken, I think, as a shorthand of saying that objects do not have proper temporal parts, and hence that there are fewer concrete objects that perdurantists think there are. Do you have some other idea in mind when you say “persistence is endurance”?

Ups, sorry for the unclarity with the *s!!

Anyway, maybe this is a summary of the situation: McCall&Lowe, or at least McCall&Lowe*, contemplate a view where (i) objects persist by enduring (in the sense of Lewis; if you prefer: not “in virtue of” having different temporal parts); but (ii) in a way that seems of be, in a certain sense, equivalent to 4D (although, I agree, the sense of metaphysical equivalence in place is left underspecified). As a result, they seem to hold there is something to be dismissed in the apparent disputes among defenders of 3D vs 4D.

I don’t think you disagree with this (except perhaps about whether McCall&Lowe are essentially McCall&Lowe*). Rather, you seem to have a kind of transcendental argument against any such

indifferentismmeta-metaphysical attitude, as I would like to call it: there should be something additional to (i) in 3D for, otherwise, due to something like (ii), it would “rather obviously misconstrue the dialectics between 3D and 4D.” After all, people involved in this dialectics tend not to find their disputes as something to be dismissed.(Re-reading the whole discussion, what if sums-of-particles-at-a-time are indeed concrete, just represent

ableby the abstract pairs, as you alternatively understood? In your original post you said:Understood as entities that exist at only one time, sums-at-times are genuine additions to the endurantist ontology.But the picture also satisfies that objects persist by enduring. So even with the concrete-thing version, we seem to be in the position (i) and (ii) above.)

(Re-rereading, you said:

This extra claim [that (O, t) is something *different* form (O, t’)] is notforcedupon [the endurantist] by her previous commitment to the existence of enduring particles, even assuming unrestricted mereological composition.Maybe so (depending on one's views on the doctrine of ontological free-lunch). But the real issue seems to be whether that would

contradictthe contention that objects persist by enduring. And on the face of it, it would not.)Hi people! Sorry not to have participated sooner, I hope to do it from now on.

Just a little question: I agree with Pablo that there's a problem here for the authors if they try to establish a one-to-one correspondence between 3D (O,t) objects and 4D instantaneous objects for the reasons Pablo mentioned (if particles may endure and these 3D objects are nothing but sets of particles, it makes all the sense to say that sometimes the set remains unchanged between t and t', wich makes (O,t) and (O, t') the very same thing).

However, is this one-to-one correspondence really necessary for the translation scheme to go through? What would be the problem with having a many-to-one correspondence? (ie, one and the same 3D object corresponding to several instantaneous temporal parts).

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