, 10:175

Endurance Per Se in B-time


    • Department of PhilosophyLund University

DOI: 10.1007/s12133-009-0049-z

Cite this article as:
Hansson Wahlberg, T. Int Ontology Metaphysics (2009) 10: 175. doi:10.1007/s12133-009-0049-z


Three arguments for the conclusion that objects cannot endure in B-time even if they remain intrinsically unchanged are examined: Carter and Hestevold’s enduring-objects-as-universals argument (American Philosophical Quarterly 31(4):269–283, 1994) and Barker and Dowe’s paradox 1 and paradox 2 (Analysis 63(2):106–114, 2003, Analysis 65(1):69–74, 2005). All three are shown to fail.



1 Introduction

Discussions of endurance in B-time (the kind of time posited in the B-theory of time) have tended to focus on the problem of temporary intrinsics, i.e., the question whether objects can endure through intrinsic change.1 However, that question is arguably subordinate to the question whether an object can endure in B-time without intrinsically changing. If objects cannot even endure in B-time without intrinsic change, discussion of temporary intrinsics is surely idle. Carter and Hestevold (1994) and Barker and Dowe (2003, 2005) have addressed the more fundamental question. They argue that objects cannot endure in B-time, irrespective of intrinsic change. In this paper, I shall take a close look at these arguments and see if they stand up to scrutiny. I shall argue that they do not.

2 The Enduring-Objects-as-Universals Argument

The first argument goes like this.2 In the B-theory of time, all times—times, which from our current location in time, appear to be either past, present, or future—and their contents, are ontologically on a par and interrelated by the B-relations earlier than, later than, and simultaneous with. (So-called A-properties, such as being past, present, and future, are rejected.)3 Since all times and their contents are equally real, if an object O (e.g. a stone, tree, car, or person) persists through time by enduring, i.e., by being wholly present at distinct times as numerically the same entity, O will have to be multiply located in time. In other words, O will have to be wholly present earlier and later than itself. But an object—a concrete particular—can only be repeated in this way, the argument goes, on pain of becoming a universal! A concrete particular can persist through B-time only by either perduring (i.e., being a 4-dimensional entity with distinct temporal parts at distinct times) or exduring (i.e., being a 3-dimensional entity with distinct temporal counterparts at distinct times).4

This is a weak argument. Why would an object, multiply located in B-time, have to be a universal? Universals—in the Aristotelian/Armstrongian tradition (e.g., Armstrong (1989)), which, unlike the Platonic tradition, allows them to be spatiotemporally located—are characteristically entities which are repeatable in time and in space (at a single time). However, an object enduring in a normal manner in B-time is multiply located only in time, not in space (at a single time). Hence, by simply enduring in B-time, such an object is quite far from behaving like a classical immanent universal.

An advocate of the argument may retort that proponents of endurance in B-time who endorse the possibility of backward time travel (e.g. Miller (2006)) are committed to the notion that objects are, or at least can be, wholly present at distinct places at a single time, apart from being wholly present at distinct times: if an enduring object “doubles back” in B-time to an earlier time where it is already located, it will be multiply located in space at the relevant time; and such an object would be universal-like indeed!

However, even the B-theorist endurantists who regard time travel as a genuine possibility5 (as many emphatically do not, e.g., Mellor (1998, Ch. 12)) can almost certainly resist the imputation that objects, on their view of time and persistence, turn out to be universals.6 For even if they are committed to the idea that both objects and universals are repeatables in space-time, they can still point to major differences between the two categories of being, and hence maintain that being a repeatable is not sufficient for being a universal.

To begin with, they can point out that for an object to be multiply located in space at a single time, that object has to enter a time machine or a wormhole in space-time and travel backwards in time. A classical, immanent universal need do no such a thing. For a universal to be repeated in space at a certain time, it is enough that it be multiply instantiated at the time in question. This difference in the way the relevant items are multiply located in space, they may further point out, is a by-product of the more fundamental and substantial metaphysical difference between immanent universals and objects, at least as traditionally understood: universals are entities which exist in space-time in virtue of being instantiated by some further entity (i.e., an object), but objects do not have to be instantiated in order to exist in space time—they are independent entities, in this respect. This crucial and classical difference between universals and objects has not been shown to collapse within the metaphysics we are considering. I conclude, therefore, that B-theorist endurantists are not obliged to concede that putative objects in fact are “universals in drag” (Carter and Hestevold (1994, p. 279))7.

3 The Argument from Paradox 1

Barker and Dowe (2003) do not press the enduring-objects-as-universals argument into service. Instead, they argue that if objects (and, equally, immanent universals) were to endure in B-time, they would be both 3-dimensional and 4-dimensional. Since no entity can be like this, endurance in B-time is impossible. (They consequently hold that not even universals can endure in B-time. Their argument, therefore, has a wider scope than the enduring-objects-as-universals argument.)

Barker and Dowe reason as follows:
Take a multi-located entity O, be it enduring entity or universal. Say that O is multi-located throughout a 4D space-time region R. Thus, there is a division of R into sub-regions r, such that O is wholly located at each r. [...] At each r that is a sub-region of R, there is an entity—a universal or an enduring entity—of a certain kind. Call it Or. Take the fusion, or mereological sum, of all such Ors. Call the fusion F(Or):
  1. (i)

    Each such Or is a 3D entity, since it is located at a 3D sub-region r. Or is an entity with non-zero spatial extent and zero temporal extent. Each Or is identical to every other. So each Or is identical with F(Or). So, F(Or) is a 3D entity.

  2. (ii)

    F(Or) has parts at every sub-region of R. So it has non-zero spatial and temporal extent. F(Or) is a 4D entity.


Conclusion: F(Or) is both 3D and 4D, but that is a contradiction since being 3D means having no temporal extent, and being 4D means having temporal extent. (Barker and Dowe (2003, p. 107))

This argument is flawed. If O is an enduring entity, it is 3D in the sense that it is extended in three spatial dimensions8 but not extended in the temporal dimension.9 In other words, it is 3D in the sense that it has spatial proper parts (at times) but no temporal proper parts—it is wholly present at each of the distinct times (instants) it is located. Given that O endures, each Or is identical with O. Supposing that it makes sense to mereologically fuse an entity with itself, the resulting fusion is simply the original entity. Thus, the mereological sum of the Ors, F(Or), is the old O. Since O (and each and every Or, i.e., O) is 3D, F(Or) is 3D. We can, therefore, agree with the conclusion arrived at in (i). However, in (ii) it is concluded that F(Or) is 4D because it has parts at every r. This is mistaken: it simply does not follow that F(Or) is 4D just because it has parts at every r. This would only follow if F(Or), i.e., O, had distinct temporal proper parts at the respective rs; but since, ex hypothesi, O endures, it has not. The parts of O at the respective rs are either spatial parts or else the single temporal part that O has (if it has any), namely O itself, which is wholly located at the distinct rs and is not a temporal proper part. Consequently, O/F(Or), is 3D and 3D only—it is not 4D. Notice, moreover, that this last fact does not entail that O is instantaneous. It just means that O has no temporal extent in the sense of having temporal proper parts. O may still persist—endure—for a certain positive duration, as indeed was presumed in the very argument.

I have not seen this response in print (other than in what can be discerned in the short quote below), but it seems that Barker and Dowe may have encountered some version of it verbally. In a follow-up paper (Barker and Dowe (2005)), where they respond to criticisms of their original paper, they write, after presenting an argument that is essentially the same as the one given above (but uses the example of Eric, the enduring cat):

Some people produce the following dismissive response to the first paradox: F(Ericr) has parts at every sub-region r of the 4-dimensional space R. But F(Ericr) is not 4-dimensional because these parts are not proper parts—they are all Eric!

We reply that there is a single thing at every subregion r of R. Eric is eternally at r1, r2, r3, etc. The region R is filled up! In terms of occupation there is no difference between this situation and that where each occupier of a 4-D space is non-identical to every other occupier. The fact of identity does not affect the matter of occupation and the whole space being occupied. If the space is fully occupied, how can the fusion of the occupiers not be a 4-dimensional object? (Barker and Dowe (2005, p. 70))

Answer: the fusion will fail to be 4D if the occupiers of the filled 4D space happen to be, not several, but a single enduring 3D object that is multiply located within the relevant region.

Is this answer tenable? In their original paper, Barker and Dowe use a spatial analogy to show that it is absurd to deny that the fusion of the Ors (Ericrs) is 4D.

Take a spatial analogy. Say a person time travels 10 times, and 10 time travelling individuals appear in the same room. Say they squash together on a bench. Then the bench is a ruler. Its parts are in a one-to-one correspondence, in the relevant sense, to human parts. The fusion of the human parts is an object. That 3D object is one-bench length. But we are told that the ten humans are identical. Their fusion then is simply a human. But a human is not one bench in width. Ergo, there can be no spatially multi-located individual. It seems to us nonsensical to attempt to rebut this argument by claiming that the fusion of individuals is not a bench-long thing. We think the response to the 4D case is in exactly the same boat. There is no relevant difference between the spatial case and the 4D case. (Barker and Dowe (2003, pp. 109–110))

Let us grant the possibility of time travel for the sake of the argument. Does the thought experiment establish that the fusion of the “ten” time traveling humans located at the relevant time is one bench in width at the time in question? I think not. Given endurantism, the allegedly nonsensical answer must be correct. If the time traveling person keeps a constant width of, say, 0.4 m (because he does not loose/gain weight) during his travels, the width of the fusion of the “ten” humans located at the relevant time, who are really just one and the same enduring human, is 0.4 m at the time in question.10 However, as the fusion, i.e., human, is multiply located in space at the relevant time, owing to repeated time travel, the fusion/human fills up a spatial region that is larger than 0.4 m in width: a region one bench in width. The situation may be described as one in which a fusion/human with a width of 0.4 m—extending in space for 0.4 m—s-endures (“endures” spatially—as a result of time travel) over a bench-long stretch at the time in question. Thus, the spatial case does not establish the impossibility of endurance in B-time, even granting that time travel is possible and that the spatial case and the 4D case are analogous.11 The spatial case is susceptible to the same kind of analysis as the original 4D case.

In their follow up paper, Barker and Dowe (2005) do not mention the spatial analogy. (Perhaps, at this point, they no longer found the spatial scenario more compelling than the ordinary 4D case.) Instead, they try to strengthen paradox 1 (in its Eric guise) by putting it in terms of plural reference:

We note that there is a form of the paradox that completely defuses the dismissive response anyway [the kind of response I offered above]. “The Ericrs” is a plural referring term. One can assert “The Ericrs occupy (fill up) R’’, since the Ericrs are scattered throughout R. The predication here is not distributed—each Ericr does not occupy or fill up R—but collective. How do we analyze collective predication? One analysis is through mereology: the mereological fusion of the Ericrs occupies R. Thus, we must admit that F(Ericr) is a 4-dimensional object, and the paradox remains. The other analysis is to treat plurals as plurally referring and collective predication as an irreducible form of predication. But this just means that the paradox takes on the following plural form:

The Ericrs occupy R (they fill up R)

Eric does not occupy R (by Endurantism—he is multi-located throughout R).

The Ericrs are one and the same; they are Eric (by Endurantism).

Therefore, the Ericrs occupy and don’t occupy R. (Barker and Dowe (2005, p. 70)).

Here Barker and Dowe claim that if Eric endures, he does not occupy R—he is just multi-located throughout R. This seems false, however. If Eric is multi-located throughout R, he surely occupies RR is filled up by Eric. Barker and Dowe concede this when discussing the original form of the paradox on the very same page: “[According to endurantism], Eric is eternally at r1, r2, r3, etc. The region is filled up! In terms of occupation there is no difference between this situation and that where each occupier of a 4-D space is non-identical to every other occupier. The fact of identity does not affect the matter of occupation and the whole space being occupied” ((2005, p. 70), my emphasis). Moreover, in the new form of the argument it is claimed that “each Ericr does not occupy or fill up R.” But given that each Ericr is identical with Eric, as endurantists have it, this claim is false. Finally, the fact that Barker and Dowe choose to use the plural expression “the Ericrs” shows little. When we say things such as “a and b are identical,” the plural “are” and the use of the two names, “a” and “b,” do not reveal that we are in fact referring to two entities as opposed to a single entity denoted by two distinct names. Similarly, if the Ericrs are in fact identical, the plural expression refers to a single object that is multiply located in time.

I conclude, then, that paradox 1, in its various guises, fails to generate a contradiction. Why do Barker and Dowe think it does? They believe the following “part/whole location” principle “underpins” (2003, p. 109) their argument:

WLP: if an entity W and a space-time region R are such that for some division of R into sub-regions r, W has a part p located at each sub-region r, then W is located at R and is a 3 or 4D entity according to the dimension of R itself. (Ibid.)

Notice, however, that if the endurance theory of persistence and the B-theory of time are true (or even if the combination is merely possible), WLP is false. In such a case, W is 3D–supposing that at any rate the entity W is an object and not a process–although W has a part located at each subregion r and R is 4D. The principle, then, is question-begging. If it indeed underpins the argument, the argument presupposes what it seeks to establish: that objects cannot endure in B-time.

4 The Argument from Paradox 2

Paradox 1 is no paradox. What about Barker and Dowe’s paradox 2?

Here is how it is formulated:

Say that O is multi-located throughout a 4D space-time region R. Intimately connected with O and R, there is, we submit, a 4D entity which we call the life of O, or L(O). […] Lives are part of common sense ontology; we speak of entities—be they people, animate entities, or inanimate—having long, interesting, varied, good, etc. lives. Lives are 4D things; they have beginnings, middles, and ends. L(O) is just like an event occurring at a region R; it is located at R with proper parts located at each sub-region r in R.

Assuming doctrines of multi-location, lives are paradoxical entities. Where there is a persisting thing, material object or universal, there is a life, L(O). There is a necessary connection between a persisting thing and its life. We should accept the principle, call it Independence, that there are no logically or metaphysically necessary connections between distinct existences. Thus, if there is a necessary connection between O’s persisting and L(O)’s existing, O and L(O) can’t be distinct things. That means, we think, that O is part of L(O) [...] But if O is part of L(O) then there is the following contradictory reasoning. [...] Split L(O) into two proper parts. Call them L(O)1 and L(O)2, which are located at distinct regions R1 and R2. In which case:

Paradox 2:
  1. (i)

    L(O)1 and L(O)2 are entities bounded by distinct regions R1 and R2. Qua event-like entities they occur or are located at R1 and R2, respectively, and nowhere else.

  2. (ii)

    L(O)1 and L(O)2 have as parts the entity O. O is located at rs in R2, so, as O is part of L(O)1, L(O)1 is not confined to R1; it is partly located in R2. O is located at rs in R1, so L(O)2 is not confined to R2; it is partly located in R1.


Conclusion: L(O)1 and L(O)2 are and are not bounded by R1 and R2. (Barker and Dowe (2003, pp. 110–111))

Given the supposition that O is a part of its life, it is tempting to respond to this alleged paradox by saying that what happens in the scenario is simply that O is part of L(O)1 as long, and only as long, as L(O)1 exists or perdures through time, and is part of L(O)2 as long, and only as long, as L(O)2 exists or perdures through time. After all, that parts can predate and outlive wholes of which they are parts—temporary parts—is a commonplace: my daughter stacks some building blocks on top of each other, creating a tower, but eventually, she tears it down, the blocks being spread out on the floor, still existing but no longer parts of the destroyed tower.

Tempting as this response may be, it is not the one I will defend (but see Beebe and Rush (2005), for this kind of response). The temporary parthood relation is most naturally wedded to an ontology of 3-dimensional enduring entities, i.e., an ontology where both the whole and its parts are 3-dimensional (see Simons (1987)); but in the case at hand, the wholes in question are 4-dimensional, perduring entities, and it is generally thought that the mereological apparatus most suitable for 4-dimensional entities is the one developed by Leonard and Goodman (1940). That mereology is “timeless” in the sense that mereological relations are borne, not relative to times, but simpliciter. Thus, it is doubtful that temporary parthood can be sensibly invoked here if we allow that L(O), L(O)1, and L(O)2 are 4-dimensional entities (cf. Barker and Dowe’s (2005) reply to Beebe and Rush).

My complaint, rather, is that Barker and Dowe have not established that an enduring object must, indeed, be a part of its life. (The relation might be nonmereological.) In particular, I will argue that the thesis underlying the claim—i.e., that “there is a necessary connection between a persisting thing and its life”—is simply not credible once it is agreed that a persisting thing/object is a 3D entity and that a life is a 4D entity.

It would generally be accepted that persisting, living objects might have had shorter or longer lives than they actually have. Descartes lived to fifty-four but he could have died at three (we think). If persisting objects are 3-dimensional and persist by enduring, such modal beliefs appear fairly unproblematic and straightforward: we simply imagine a possible world12 where the object of our concern fails for some reason to be wholly present at times (times later than a certain time, t) at which it is located in the actual world (or vice versa). However, in such a situation, the life of the object would have been shorter. If lives are indeed 4-dimensional perduring entities, they are to be understood as 4-dimensional aggregates of temporal parts (think of the standard perdurantism found in Lewis (1983), and Armstrong (1997, p. 102)). Aggregates, however, have their parts essentially (see Hansson Wahlberg 2009); and from this, it follows that a 3D object which, counterfactually, led a shorter life would lead a 4D life that is numerically distinct from its actual life, since its counterfactual life would lack some of the temporal parts that its actual life has.13 Hence, since the object in question exists in a possible world in which (with “its life” being understood rigidly) its life does not, there is no necessary connection between the enduring object and its life. (Descartes could, thus, truly have entertained the following pseudocontradictory thought: “I could have led a different life, but my life could not have been different!”.) It may be that an enduring object has to have a life—i.e., one life or another—but it does not appear necessary for the enduring object to have the particular life that it has. If I am right about this, an endurantist who accepts Barker and Dowe’s independence principle can consistently maintain that an enduring object and its life are distinct entities, as there is no necessary connection between them.

Notice, by the way, that if there were a necessary connection between a persisting thing and its life, then, given the independence principle, we would have to conclude that the thing and its life are numerically identical (the natural position given perdurantism). To conclude, with Barker and Dowe, that they must be related by a part-whole relation would be a very weak way of expressing their relationship, as proper parts, although not parts, are necessarily distinct from the wholes of which they are parts (although, they certainly overlap).

5 Conclusion

I conclude that all three arguments against endurance per se in B-time fail: they do not establish what they aim to establish. Although I would concede that endurance per se in B-time is counterintuitive to an extent, I do not think that this widespread intuition is sufficiently strong or reliable to render discussions about endurance through intrinsic change in B-time futile.


See e.g. Lewis (1986, pp. 202–204; 1988, 2002), Lowe (1988); Merricks (1995); Armstrong (1997, pp. 100–102); Mellor (1998, Ch. 8); Sider (2001, Ch. 4.6); Haslanger (2003); Wasserman (2003); and Hansson (2007).


See Carter and Hestevold (1994, pp. 278–279); see also Lewis (2002, p. 3); and Ingthorsson (2002, p. 135). The argument was also pressed by an anonymous reviewer of Hansson (2007). In the article, I said that I aimed to deal with the objection in a later paper. This is that paper.


For an introduction to and a defense of the B-theory of time (also known as “the tenseless theory of time” and “the static conception of time”), see Mellor (1998).


For the “endurance/perdurance” terminology, see Lewis (1986, p. 202); for “exdurance,” see Haslanger (2003, p. 319).


Persuaded by reasoning like that in Smith (1997).


It is often assumed that time travel is sensible only on the B-theory of time: it is thought that in order for time travel to take place there must be other times than the present one to travel to (the so-called destination requirement). Keller and Nelson (2001) have challenged this line of thought. If they are right, then the argument from time travel can be invoked against presentist endurantists as well (presentists hold that only the present moment exists).


I should mention that Carter and Hestevold (1994, pp. 278–279) back up their allegation by comparing endurance in B-time with “endurance” over possible worlds, assuming modal parity (i.e., the thesis that all possible worlds are ontologically on a par). They hold that only universals can be multiply located in different worlds given modal parity. They subsequently ask: why should the situation be any different for times given temporal parity (i.e., the thesis that all times are ontologically on a par)? It is not the purpose of this paper to discuss the modal case. However, I have two things to say about this analogy. First, Carter and Hestevold have not shown that only universals can be multiply located in different worlds given modal parity—they merely intuit that it is so. Second, the temporal case differs from the modal one in that times belong to a single, unified world: the contents of distinct times—in contrast with the contents of distinct possible worlds—are interrelated by earlier/later than relations and causal/nomological relations. Hence, the cases are significantly different.


That is, if it is extended in all spatial dimensions (neglecting the curled dimensions of string theory); otherwise, it is 2D or less. I suppose an immanent universal would not have to be 3D.


Here, I disagree with Bebee and Rush (2003, p. 313), who claim: “No sensible endurantist is going to equate being a 3D object and having zero temporal extent.” I think sensible endurantists should do just this.


If the time traveling human gains/loses weight and swells/shrinks during his time travel, it cannot be said that the fusion/human simply has a width of 0.4 m at the time in question: we must say that the fusion/human has distinct weights/widths at distinct places at the time in question (see Miller (2006)). In this paper, however, we are concerned with objects that do not change intrinsically.


Notice, though, that s-endurance for concrete objects is due to ordinary endurance plus time travel. Ordinary endurance, i.e., endurance in time, does not require some kind of “meta-endurance” to take place in a further dimension.


I take it that possible worlds can be invoked here even if we do not conceive of them realistically.


In Hansson Wahlberg (2009), I argue in detail that the 4D aggregate in the nonactual world is not even a counterpart of the actual 4D aggregate, pace van Inwagen (1990).



The paper was presented at the Swedish national philosophy congress, “Filosofidagarna”, Lund 12–14 June 2009. I thank the participants for helpful comments. Thanks are also due to Anna-Sofia Maurin, Nils-Eric Sahlin and Lena Wahlberg.

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