At It Again: Time-Travel and the At–At Account of Motion

Abstract

According to Russell’s At–At Account of Motion, necessarily, something moves if and only if it’s at one place at one time, and at a distinct place at a distinct time. This, many believe, is all that motion consists in. However, if it is possible for an entity to be at more than one place at more than one time (that is, to persist while multiply located), the At–At Account will entail that the entity is in motion even if, intuitively, the entity is simply at rest in two places at once. I will argue that these cases, if possible, give us reason to reject the At–At Account, and that if we endorse the At–At Account as an analysis, even the analytic possibility of the cases will be problematic. Further, we have reason to reject the stronger claim of Motion Supervenience, on which the facts about the motion of an individual within an interval are wholly determined by facts about the object’s location within that interval together with identity facts about regions and times.

The At–At Account of motion has become extremely popular. First championed by Bertrand Russell in The Principles of Mathematics, it can be roughly stated as the view that: necessarily, something moves if and only if it is at one place at one time, and at a distinct place at a distinct time. This, many believe, is all that motion consists in. It is what it is to move. Though some have attempted to object to this account, most notably by invoking spinning disk cases, the account continues to enjoy widespread acceptance. In this paper, I will present a new worry for the At–At Account: it is incompatible with the possibility of persisting, multilocated entities, exactly the sort that one may be tempted to posit in some metaphysically possible time-travel cases.

If it is possible for an entity to be at more than one place (i.e., to be multilocated) at more than one time, the At–At account will entail that the entity is in motion, even if intuitively the entity is at rest in each place it occupies. Thus, if such cases are possible, we have reason to reject the At–At account. Further, if we take the At–At account to be an analysis of motion rather than merely a listing of necessary and sufficient conditions for it, then even the analytic possibility of these cases will be problematic. Finally, some of these motion-free cases will have counterpart cases that do not differ in location or identity facts, but do intuitively differ in motion facts: intuitively, in these counterpart cases, there is motion. This gives us reason to reject the broader view of Motion Supervenience, according to which, necessarily, the facts about the motion of an individual within an interval are wholly determined by facts about the object’s location within that interval together with identity facts about regions and times. In addressing these worries, I will show that if we respond by denying the possibility of persisting, multilocated entities, the cases I’ve presented reduce to spinning-disk cases. Thus, if an At–At theorist believes she has an adequate response to spinning-disk cases, we can reduce this new problem to that more familiar one, but at the cost of endorsing a restriction on possibilities involving location.

This paper will proceed as follows. In Sect. 1 I will present the At–At account and make some remarks on what we mean by ‘at’. In Sect. 2, I will present a time-travel case that, if possible, is a counterexample to the At–At account. I will then note that this case (or at least, a similarly motion-free, slight variant of it) has a counterpart that differs with respect to facts about motion without differing with respect to facts about identity or location. This gives us ground to reject Motion Supervenience. In Sect. 3 I will present and reject several initial responses one may attempt to give on behalf of the At–At Account. Finally, in Sect. 4, I will discuss what I take to be the most promising response: accepting the possibility of the cases I describe, but denying that they involve persisting, multilocated entities. I will show that, with this response, my cases reduce to spinning disk cases. Thus, a restriction on possibilities involving location may give us a way to reduce this new objection to one widely taken to be more manageable.

1 The At–At Account of Motion

Bertrand Russell provided us with an elegant, reductive account of motion. He claimed: “Motion consists merely in the occupation of different places at different times…”¹ That is:

• The At–At Account Necessarily, for any x, x is in motion iff there exist spatial regions s1 and s2, and times t1 and t2, such that s1 is distinct from s2, t1 is distinct from t2, and x is at s1 at t1, and at s2 at t2.²

This account accords well with what we observe. Even motion’s relativity to reference frames requires only minor adjustments to the view. We might say, for instance:

• The Relative At–At Account Necessarily, for any x, x is in motion relative to reference frame r iff there exist (relative to r) spatial regions s1 and s2, and (relative to r) times t1 and t2, such that s1 is distinct from s2, (relative to r) t1 is distinct from t2, and x is at s1 at t1, and at s2 at t2.

The At–At Account is simple: it allows for an elegant reduction of motion facts to other, non-mysterious facts. All the account appeals to in giving the necessary and sufficient conditions for motion of an object are locative facts about that object, and identity facts about places and times. Thus, it entails Motion Supervenience.

• Motion Supervenience necessarily, the facts about the motion of an individual within an interval are wholly determined by facts about the object’s location within that interval together with identity facts about regions and times.

The At–At Account has wide appeal, with many contemporary philosophers among its proponents. For instance, Wesley Salmon appeals to the account in responding to Zeno’s Paradox of the Arrow.³ Richard Taylor uses the account to provide a description of moving forth and back in time, which is used in defense of his claim that space and time are analogous.⁴ Ted Sider uses the At–At Account to argue against Presentism.⁵ And Ulrich Meyer⁶ and Kenny Easwaran⁷ have published recent defenses of the At–At Account. The main competing account of motion, presented by Michael Tooley,⁸ involves positing non-reducible, first-order velocity properties. The simplicity and reductive nature of the At–At Account have made it a popular choice.

How we interpret the At–At Account will depend on how we think objects persist. Most straightforwardly, we could think that when the At–At account requires that the object is at one place at one time, and at another place at another time, the account means that the entire object is first at one location, and later at another location. Thus, on this interpretation, in order for an object to move it must be entirely present at each of two different times. This interpretation fits nicely with Three-Dimensionalism, according to which entities persist by being wholly present at each time at which they are present at all. That is, the whole object is present at each time; none of it is missing.

The interpretation does not give plausible results when combined with Four-Dimensionalism, however. According to four-dimensionalists, objects that persist do so by having proper temporal parts present at each time. Thus, at any given instant of time, any object that persists is only partly present; some of it is missing. The location of the whole object cannot fit within a single time; it is extended in both time and space. It is not plausible, then, for the four-dimensionalist to claim that something moves iff the whole object is entirely present at one place at one time, and entirely present at another place at another time.

However, four-dimensionalists can still speak of where objects are at times. The statement “x is at space s at time t” is true iff x has a temporal part entirely present at t, and that temporal part is located at s.⁹ And three-dimensionalists can also speak of where objects are at times: for them, the location of an object at a time is the location of the entire object at that time.

So, because the At–At account is presented in terms of where objects are at times, and the three- and four-dimensionalists can give accounts of what this means, both three- and four-dimensionalists can provide prima-facie plausible readings of the At–At account.

Now that we have worked out the content of the account, let us move on to problems for it.

2 The Cases

Consider the following case. Time-travelling Tom¹⁰ sits quietly from noon to 2:00 pm in the living room. Then Tom time-travels back,¹¹ and sits quietly from noon through 2:00 pm in the kitchen. The case can be represented as follows, with the circles indicating Tom’s locations at noon (T1) and 2:00 pm (T2), and the arrows tracking something like immanent causal relations between those stages of Tom at those times and places:

2.1 The Dull Case

Intuition tells us that there is not motion in this case (at least, not during the interval between and including noon and 2:00 pm¹²). Tom is just like any other entity that fails to move by staying perfectly still, except that Tom happens to stay still in two places at once.

However, consider what the At–At Account says about Tom in the case. At time 1, Tom is in the living room. And at time 2, Tom is in the kitchen. And times 1 and 2 are distinct, and the living room and kitchen are distinct. These facts, according to the At–At Account, are sufficient for there being motion within this interval of time, regardless of what the other facts are. In particular, they are sufficient regardless of the fact that Tom is in both the kitchen and the living room at both times. Because the At–At Account says that Tom is in motion between noon and 1 pm in this case, and because we believe that Tom does not move within that interval, this case is a counterexample to the At–At Account of motion. If it is possible, the At–At account is false. And if the At–At Account is to be analytically necessary, then even the conceivability of the case will be incompatible with it.

In fact, we are in a position to argue for an even stronger conclusion. Consider what I will call “The Exciting Case”. Suppose time-travelling Tom has become adept at running his time machine. Tom begins by sitting in the living room at noon, and stays in the living-room until just before 1:00 pm; that is, he is in the living-room for every instant between noon and 1:00 pm, but time-travels to have vacated the living-room by the first instant of 1:00 pm. Tom then uses his time-travel machine, to arrive in the kitchen at the instant of 1:00 pm (though he was in the living-room until just before then). Tom persists in the kitchen until 2:00 pm, and then time-travels back to arrive in the kitchen at noon. Tom persists in the kitchen until the instant of 1:00 pm, and uses his time-travel machine to have vacated the kitchen at that instant and to arrive in the living-room at the instant of 1:00 pm, and persists there for the rest of the afternoon.¹³ This case can be represented thus (again with T1 being noon, T2 being 2:00 pm, and the arrows representing causal connections between those stages):

2.2 The Exciting Case

Intuitively, there is motion in this case in the interval between and including noon and 2:00 pm. Tom is going all over the place! Just as we may think an electron can move discontinuousy between energy levels (with its presence at one energy level standing in the right causal relations to its presence at the previous energy level), we may think that Tom moves between R1 at T1 and R2 at T2.¹⁴ And the At–At Account gives the correct result, for Tom meets the requirement of being at one place at one time, and at a distinct place at a distinct time.

What’s important about the pair of cases is this: though the cases differ with respect to whether Tom moves between noon and 2:00 pm, the identity and location facts are held fixed between the cases. Both cases involve the same person, Tom, occupying the same regions at the same times. If this is correct, then we have more than a mere counterexample to the At–At Account of motion. This pair of cases gives us a counterexample to Motion Supervenience.

Reflecting on the cases, this result is not surprising. Because the identity and location facts, while they tell us where Tom is at any given time, do not tell us about many of the relations his stages at various locations stand into one another. In particular, they do not tell us about the relations that are represented by the arrows in the above diagrams. And when we think of whether Tom moves in the cases I’ve described, our intuitions about the cases seem to be grasping onto whatever it is that the arrows in the diagrams represent. Thus, we need to appeal to something beyond identity and location facts to account for motion.

3 Initial Responses

There are multiple ways that the multilocation theorist might respond on behalf of the At–At Account. The first is simply to deny the possibility of the Dull Case. Recall, in the Dull Case, Tom seems to persist motionlessly in two places at once. The At–At Account, however, gives the verdict that if Tom is indeed in two places for an extended period of time, then Tom counts as moving within that interval. The At–At Account, then, is too liberal. But, the defender of the At–At Account may claim, this problem will not arise if the Dull Case is not possible.

There are several worries we should have about this response. First, for this response to help, we must deny not just the possibility of time-travel cases of the sort described above, we must deny the possibility of any cases of persisting, multilocated entities that seem to be staying put at each region they occupy. So, for instance, suppose that we posit immanent universals. I have a black coat and a black shawl. The coat and shawl, and thus the immanent property of being black instantiated by them, may stay still for an extended period of time. This will be sufficient for the property to be in motion, according to the At–At Account.

That said, let us restrict our attention to cases like the Dull Case. If we think that multilocation, if it were to occur, would only occur in a time-travel scenario, one way to avoid the problem for the At–At Account would be to deny the possibility of survivable time travel. That is to say, no object can survive showing up at a time “more than once”. So in my cases above, the object would go out of existence as soon as it went back in time; no multilocation would occur, so there would be no multilocation-generated problem for the At–At Account. However, denying the possibility of survivable time-travel is more complicated than it appears. While it isn’t implausible to claim that an object cannot survive discontinuous travel to the past, not all travel to the past must be discontinuous. If spacetime could be curved so that it loops back upon itself, an entity could show up at a single time “more than once” simply in virtue of persisting through the loop. For instance, imagine that Tom lives in a four-dimensional version of a loop, and its temporal extension is shorter than his life; we can imagine that at 63, he visits himself at 36. To deny the possibility of this kind of time-travel, we would either need to deny that occupyable portions of spacetime can loop in this way, or claim that entities mysteriously all go out of existence sometime mid-loop.

Further, suppose we could succeed in showing the Dull Case is not metaphysically possible. This will only be enough to help the At–At Account if we take the account to be a mere listing of necessary and sufficient conditions for motion. However, this is not what makes the account so appealing. Philosophers seem to like it because it is a claim about what motion consists of. It is an analysis, maybe even a definition. But if we’re attempting to defend the account as an analysis of motion, cases like mine will be relevant even if they are not metaphysically possible: they need only be conceivable, or analytically possible. If my case cannot obtain, it is an interesting metaphysical fact, not one that has to do with the analytic At–At Account. So, if we take the account to be an analysis, then when evaluating it we should examine impossible as well as possible cases in making sure our account makes the right predictions.¹⁵

What else might a theorist say in responding to the case? There are two other options one might pursue: (i) relativising claims about motion to regions, or (ii) adding a requirement for motion that any moving entity stands in certain (perhaps immanent causal) relations to itself across the distinct regions and times at which it is located.

The option of relativising motion to regions seems independently plausible. Imagine a case where Tom in the kitchen is sitting still, and Tom in the living room is jumping on the furniture. We want to ask: is Tom moving? The intuitive answer seems to be that he is moving in the living room, but he is not moving in the kitchen. That is, we want to say that the statement “Tom is moving” is incomplete; we must say where it is that Tom is moving. However, it is not clear how to pick out the relevant regions to relativise Tom’s motion to without appealing to something spooky, such as immanent causal relations, gen-identity relations, or in-virtue-of property dependence relations (or perhaps whichever relations we use in picking out personal time as distinct from objective time). This brings us to our second alternative, which skips the region-relativising and simply incorporates something mysterious into the At–At Account itself. This response, though, will be of particular cost to the At–At theorist, whose account of motion is attractive largely because it does not contain anything mysterious. Perhaps this amendment to the account is welcome. Perhaps the correct response to my cases is to take them to draw our attention to something quite intuitive: there is more to motion than just where and when objects are. Motion involves some sort of whoosh.¹⁶ However, it should be stressed that this kind of revision to the At–At Account is not minor; the original account will be forfeit.

Thus, the possibility of my time-travel cases seems to raise problems for our elegant, simple account of motion. But perhaps in these discussions I have been too hasty. For I have been assuming that, in these cases, Tom is in two places at each time. But one may wish to respond to my cases by denying exactly this. Let us now turn to how one might give this response.

4 Denying the Possibility of Persisting, Mulitlocated Entities

We have already seen that how we interpret the At–At account of motion will depend on which view of persistence we endorse. In this section, I will show how Three-Dimensionalists and Four-Dimensionalists may go about accepting or rejecting multilocation in the time-travel cases I have presented.

The Exciting Case and the Dull Case can each be depicted with the following diagram. For ease of discussion, let us label the locations and occupants of those locations as follows:

Let us take this as our starting point: regions L1, L2, L3, and L4 are each distinct from one another,¹⁷ and each has something located at it. The questions we need to answer, then, are these: (i) What are the identity and distinctness relations between A, B, C, and D? (ii) What are the relationships A, B, C, and D stand into Tom? (iii) At each time, what is the answer to the question “Where is Tom at this time?” The most natural answers to these questions will cluster together.

Three-Dimensionalism ± Multilocation The most natural way to posit multilocation in my time-travel cases is to endorse Three-Dimensionalism. Recall, three-dimensionalists believe entities persist by being wholly present at each time at which they are present at all. If we think that entities can persist across time in such a way, it is natural to think that, at least in some possible cases, they can be multiply present across space in the same ways. Just as Tom may be all there at his 36th birthday and all there at his 63rd birthday, time-travelling Tom can be all there in the kitchen at noon, and all there in the living-room at noon. Tom is multilocated.¹⁸There is an alternative, however. The intuitions that drive us to say that none of Tom is missing at his 36th birthday (after all, he is not missing any feet or noses or the like) will drive us to say none of Tom is missing in the kitchen in my time-travel case. According to this theorist, then, the identity and location facts are these: Tom is wholly present at each of L1, L2, L3, and L4, and is identical to each of A, B, C, and D (though there also may be other entities at those locations that Tom is constituted by). And the answer to the question of where Tom is at any given time, is simply the region(s) Tom is located at, at that time. The semantics, then, is very straightforward.

However, one needn’t be a three-dimensionalist to claim that there would be multilocation in my time-travel cases.

Four-Dimensionalism ± Multilocation: Recall, four-dimensionalists believe entities persist across time in virtue of having proper temporal parts at each time. However, such a theorist can also claim that temporal parts of objects can sometimes be entirely present in more than one spatial location at a time. That is: four-dimensionalists can claim that, in my time-travel cases, some of Tom’s temporal parts are multilocated. The identity and location facts would thus be these: Taking Tom1 and Tom2 to be distinct temporal parts of Tom, A = C=Tom1, and B = D=Tom2. Four-dimensionalists typically claim persisting entities inherit their temporary properties from the properties their temporal parts have. Thus, because Tom’s temporal part is in two places at once at T1, we can say that Tom is located at more than one place at T1. In answering the question “Where is Tom at each time?” we can say: in the kitchen, and also in the living room.¹⁹ Again, the semantics is fairly straightforward and we have the answers we hoped for.²⁰

Suppose, however, that we wish to deny that time-travel cases involve multilocation. The most natural way to do this is the following.

Four-Dimensionalism ± Bloat If you believe that things persist across distinct times by having proper temporal parts at each time, it is natural to say that things (objects, but also temporal parts of those objects) persist across distinct spatial regions by having proper spatial parts at each place. This kind of four-dimensionalist will say that Tom is wholly present at the fusion of L1, L2, L3, and L4 (together with any other regions that Tom fills). Each of A, B, C, and D is distinct from the others, and each is a proper part of Tom. Further, Tom has a temporal part, Tom1, which is identical to the fusion of A and C, AC, and another temporal part, Tom2, which is identical to the fusion of B and D, BD. The answer to the question “Where is Tom at each time?” will be counterintuitive, because it is interpreted as a question about Tom’s temporal parts, which are located in the same region at each time: the fusion of the occupied region of the kitchen and the occupied region of the living room. Thus, at any given time, Tom bloats: for any time at which he is present, he is, at that time, located at the fusion of every region he fills at that time, and he is not located at a proper subregion of that region at that time.²¹

Three-Dimensionalism ± Bloat: Once again, we needn’t allow our view of persistence determine whether we take there to be multilocation in time-travel cases. A three-dimensionalist, while believing that entities persist by being entirely present at each time at which they are present at all, can agree with our four-dimensionalist bloat-theorist about how entities are spread across space. They can claim entities bloat: for any time at which the entity is present, the entity is, at that time, located at the fusion of every region the entity fills at that time, and it is not located at any proper subregions of that region at that time. This combination of views will yield the result that AC = BD = Tom, A ≠ C and B ≠ D, and Tom is distinct from each of A, B, C and D. It is left undetermined whether A = B and C = D, but that question does not influence what answer this theorist will give to the question of where Tom is at each time. Unlike for the four-dimensionalist, this question does not amount to a question about temporal parts of Tom’s, but instead simply amounts to a question of what regions Tom occupies at each time. Nonetheless, this theorist will give the same response: at each time, Tom is at the fusion of the Tom-filled regions in the kitchen and in the living room.

Notice, regardless of which of these four combinations of views we accept, the location and identity facts will not differ between the Dull Case and the Exciting Case. Thus, if facts about whether Tom moves supervene on identity and location facts, we cannot give a verdict on motion that differs between the Dull Case and the Exciting Case, regardless of which of our four combinations of views we endorse. Insofar as we think that Tom moves in exactly one of those two cases (in the Exciting Case but not in the Dull Case), distinguishing between the above views has not helped us to avoid our counterexample to the At–At Account of motion or to Motion Supervenience.

The distinctions we’ve drawn do, however, make a difference in how the counterexamples work. Regradless of which combination of views we endorse, in both the Dull Case and the Exciting Case the At–At Account will give the same verdict about motion. However, which verdict this is depends on whether we take the cases to involve multilocation or bloat.

If we say Tom (or his temporal part) is multilocated in both cases, this verdict will be that he moves in both cases, and the At–At account will be too liberal. This is the implication that we drew out above when noting that, when first examining the cases, it seems correct to say that Tom is at a region in the living room at noon, and at a region in the kitchen at 1 pm. Regardless of Tom’s also being at the same region in the living room at 1 pm, and at the same region in the kitchen at noon, the At–At Account’s conditions have been met and Tom is deemed to have been in motion within that interval. Any case of a persisting, multilocated entity will have the potential to cause these problems for the At–At Account of motion. It does not matter whether we are three-dimensionalists or four-dimensionalists; the temporally-extended multilocation is what generates this problem.

Suppose instead that we opt for bloat theory: we say that Tom (or his temporal part) bloats in both cases (being located at the fusion of all of the regions Tom fills at that time). Here the At–At account will say that, in both the Exciting Case and the Dull Case, Tom fails to move in the interval from noon to 1 pm. The At–At account will give this verdict because in each case, at both noon and 1 pm, the regions Tom is located at are the same: the fusion of a portion of the living room and a portion of the kitchen. Thus, if we endorse bloat theory, the At–At Account will be too conservative.

When construed this way, my Exciting Case amounts to a spinning disk case. Spinning disk cases have been presented, in part, as a means of showing that the At–At Account is too restrictive about when motion occurs. Suppose a disk is spinning in place for a full minute. It is in region R at T1, and still in region R at T2.²² According to the At–At account, this disk is not moving. But intuitively it is moving—it’s turning within R. Similarly, in the Exciting Case it seems that Tom is moving even though, according to Bloat Theory, he is located at the same region at each time. Tom is just like a disk that is spinning in place.

Perhaps we believe we have a way to defend the At–At Account from spinning disk cases. For instance, perhaps we believe that we can account for the motion in such cases via appeal to motion of the disk’s parts and minor amendments to the At–At Account. For the purposes of this project, I will remain neutral on whether these responses have promise. However, if one believes that spinning disk cases are manageable, they may hope to respond to my cases by denying that they involve multilocation, and then using bloat theory to reduce the cases to a more familiar, more manageable problem.

It is worth noting two difficulties with this response. First, there is a cost to adopting bloat theory over multilocation. Multilocation does some work in explaining our intuitions about time-travel cases. For instance: when 63 year old Tom visits 36 year old Tom, we can ask some questions about Tom at that time. We have already discussed the question of Tom’s location: intuitively, it seems he’s entirely in the living room (after all, none of him seems to be missing from there), and also entirely in the kitchen. We can also ask: What is Tom shaped like? Plausibly, he’s person-shaped in the kitchen, and person-shaped in the living room. It definitely does not seem that in virtue of time-travelling, he’s suddenly acquired an odd, noticeably gappy shape (that of a fusion of two people).²³ And we can ask (though it may be rude): How much does Tom weigh? Plausibly, he weighs roughly what one person would. Time-travelling to visit oneself is not a way to gain large amounts of weight. However, this, and the other implausible claims about shape, weight, and location (and a variety of other properties), is exactly what the bloat theorist (who must say Tom isn’t multilocated even though he time travels) will have to claim.

Here is a second worry about the anti-multilocation response. Earlier, I mentioned that the At–At Account is often taken to be an analysis of motion. Thus, it must be compatible not only with all metaphysical possibilities, but also with anything that is analytically possible. Thus, in order to protect the account from problems that arise from the possibility of persisting, multilocated individuals, we must deny not only their metaphysical possibility but also their analytic possibility. We must demonstrate that the claim that persisting multilocated entities exist somehow leads us to a contradiction in terms. And this is a much more challenging undertaking.

Thus, we have seen that cases involving persisting, multilocated entities are incompatible with the At–At Account of motion. We may attempt to respond to this problem by endorsing a restriction on possibilities involving location, but such a restriction would need to be not only metaphysically necessary, but also analytically necessary. Alternatively, if we wish to hold a more permissive view of location, it seems an amendment of the At–At Account is in order.