Shieva Kleinschmidt
University of Southern California
At It Again: Time-Travel And Motion1
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 §1 I will present the At-At account and make
some remarks on what we mean by ‘at’. In §2, I will present a time-travel case that, if
1
Thanks to Frank Arntzenius, Kit Fine, Cody Gilmore, Hud Hudson, Daniel Nolan, Mark Schroeder, Ted Sider, and James
Van Cleve for helpful discussion about these topics. Thank you especially to Erkenntnis Reviewers for extensive and
helpful feedback on this paper, which resulted in significant improvements.
2
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 §3 I will present and reject several initial
responses one may attempt to give on behalf of the At-At Account. Finally, in §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 . . .”2 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 t2.3
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:
2
Russell 1903, Ch. LIV section 447 (pp. 479). One may worry (here as well as throughout this paper) about my
reidentifying regions: making identity claims about spatial regions at different times, and temporal regions at different
places. For doing this, I hope to appeal to nothing more than what is appealed to frequently in common discourse: people
often talk of, e.g., returning to the same place, or leaving a location. We can, if we’d like, interpret them as making relative
reidentifications of regions (e.g., “I’m in the same region relative to the room, as I was yesterday”). The central claims in this
paper should be unaffected. (Thanks to Frank Arntzenius for bringing this worry, and the relativisation response, to my
attention.) If we wish to combine this with relativity, we may take being the same spatial region as to (a) not entail strict
identity (just as being the same species as does not), and (b) be relative to reference frames (for whether a region counts as
spatial will depend on which reference frame you have in mind).
3 Russell also thought that all motion has to be continuous; hence, only certain types of series’ of spaces and times being
occupied by an object would qualify that object as a mover. So the At-At account as formulated above doesn’t actually
reflect Russell’s complete view of motion, but rather is a fragment of his view. But I think the At-At fragment does a better
job of tracking our intuitions about strange cases. Consider, for example, how you’d react if you came into the presence of a
goblin, who (it appears) can teletransport from one region to the next. When you naively mention to the goblin you’re on
your way to the kitchen for a cookie, the goblin pops out of presence where you are, and pops into presence in the kitchen to
get the cookie first. When asked why you didn’t get dessert, a natural response would be: “I wasn’t fast enough – some
goblin got to it before I did.” That is, intuitively, the goblin moved to the kitchen, though discontinuously. (Discontinuous
motion is portrayed all over in popular culture. I Dream of Jeannie provides a good example.) Further, apart from fanciful
cases like those, science may be pushing us to posit actual instances of discontinuous motion, with electrons making
quantum leaps between energy-levels and such (though the implications of quantum leaps are difficult to make out due to
worries about wave/particle duality). Still, regardless of whether it should be preferred, it is the At-At account, rather than
Russell’s full account, that is so often assumed to be the correct account of motion. And finally, the Dull Case can be
presented as involving continuous persistence, and so will be a counterexample to Russell’s full account of motion (though
it’s worth noting that Russell’s statements in The Principles of Mathematics strongly suggest that he would simply reject the
possibility of the case).
3

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.4 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.5 Ted Sider uses the At-At Account to argue against Presentism.6 And Ulrich
Meyer7 and Kenny Easwaran8 have published recent defenses of the At-At Account. The
main competing account of motion, presented by Michael Tooley9, involves positing nonreducible, 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 FourDimensionalism, 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
4
Salmon 1991, p. 137.
Taylor 1992, p. 71.
6 Sider 2001, p. 34. In short, Sider claims that according to the At-At Account, motion involves cross-temporal relations,
Presentism cannot capture such relations, and so Presentism is false.
7 Meyer 2003.
8
Easwaran 2014.
9 Tooley 1988.
5
4
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.10 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 Tom11 sits quietly from noon to 2:00pm in the
living room. Then Tom time-travels back12, and sits quietly from noon through 2:00pm in
the kitchen. The case can be represented as follows, with the circles indicating Tom’s
locations at noon (T1) and 2:00pm (T2), and the arrows tracking something like immanent
causal relations between those stages of Tom at those times and places:
The Dull Case
10
For more on this, see chapter 3 of Sider 2001.
Tom bears some a striking resemblance to the traveller, Tim, discussed by Lewis (1976), but goes on quite different
adventures. He is proud to share a first name with an inspiring former professors of the author of this paper.
12 There are many pictures of how time-travel might occur. For instance, Tom might simply persist through a loop of
spacetime (e.g., by travelling through a wormhole) which takes him back to earlier times. Or Tom might move
discontinuously, his appearance in the past depending causally on his departure from the future. Perhaps Tom’s time-travel
involves utilisation of hypertime (a la van Inwagen, 2010). I wish to remain neutral on how Tom time-travels back.
11
5
Intuition tells us that there is not motion in this case (at least, not during the interval between
and including noon and 2:00pm13). 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 1pm 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 1:00pm; that is, he is in the living-room for every instant between noon and
1:00pm, but time-travels to have vacated the living-room by the first instant of 1:00pm. Tom
then uses his time-travel machine, to arrive in the kitchen at the instant of 1:00pm (though he
was in the living-room until just before then). Tom persists in the kitchen until 2:00pm, and
then time-travels back to arrive in the kitchen at noon. Tom persists in the kitchen until the
instant of 1:00pm, 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 1:00pm, and persists there for the rest of the
13
We can even present a version of this case which lacks motion outside of this interval as well, if we opt for a kind of timetravel that does not require motion. For instance, imagine that Tom is present in spacetime (or at least, a part of spacetime)
that is looped. If the loop is temporally shorter than Tom’s life, he can simply sit and wait, and appear at the same time
twice in different places.
6
afternoon.14 This case can be represented thus (again with T1 being noon, T2 being 2:00pm,
and the arrows representing causal connections between those stages):
The Exciting Case
Intuitively, there is motion in this case in the interval between and including noon and
2:00pm. 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 T2.15 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 2:00pm, 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 in to one another. In particular, they do not tell
14
In a simpler version of the Exciting Case, Tom is in the kitchen at noon, time-travels right then to be in the living-room at
2:00pm, stays less than a minute and then time-travels back to be in the living-room at noon, stays for less than a minute and
then time-travels forward to arrive in the kitchen at 2:00pm. In this simpler case, Tom is temporally gappy: there are times
between T1 and T2 when Tom is not present anywhere. If we wish to use this simpler version for our argument, we must
use a version of the Dull Case that matches it in locative and identity facts. Thus, we would have to give a version of the
Dull Case in which Tom persists discontinuously, and is completely absent at some times between T1 and T2, just as he is in
the simplified Exciting Case.
15 As I mention shortly, I believe our intuitions about motion in this case are grasping onto whatever is represented by the
arrows in the diagram for the Exciting Case. These may be causal relations between the occupants of the space/time
locations, or they may be gen-identity relations, or the arrows may be tracking a flow of personal time (and whatever
personal time itself tracks). I will discuss these options further in §3. Also, it should be noted that one may object to the
very notion of discontinuous motion. Here I can claim only that the intuition that discontinuous motion is, at least in
principle, possible seems to be widespread. See fn. 3 for more on this.
7
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 AtAt 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
8
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.16
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
16
Of course, as previously noted, there are worries about intuitions failing in strange enough cases, or of our everyday
account being simply not made for extremely exotic cases, so that people are no longer able to correctly apply their
concepts; however, I think these worries aren’t relevant here, as my case is quite easy to grasp, and in fact is relevantly
similar to time-travel scenarios brought to people’s attention in pop culture.
9
whoosh.17 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,18 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 in to 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.
17
Thanks to Dan Korman for this vivid description of the intuition.
That is: construed as spacetime regions, they are each distinct from one another. I do not want to claim that L1 is not the
same spatial region as L2.
18
10
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.19 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
19
This kind of multi-location is notoriously difficult to define, and many philosophers say it is inconceivable for them. For
instance, Ted Sider (in Sider, 2001, p. 64) has pointed out difficulties in defining the notion of entire-presence, or whole
location that the notion of multi-location seems to depend on. In what sense is the object in question all there when it is
entirely present somewhere (or at some time)? If we cash this out in terms of all of the object’s parts being there, Sider says,
we face a dilemma which turns on how we interpret ‘all of x’s parts’. If we take that phrase to refer to all of x’s parts at the
current time or region, then everyone would accept the view. If we take ‘all of x’s parts’ to refer to all the parts x has at any
time or region whatsoever, then the view entails Mereological Neighbourliness: For any ordinary object x, and any time or
region, r, if x is present at r, then for all z, if there exists a time or region at which z is a part of x, then z is present at r. Since
neither of these options is acceptable, we should reject this account of entire-presence, or whole location.
There is an alternative, however. We can follow Hudson (2006) in taking the notion of located at that
multilocation depends on (when construed as the view that an object can be located at more than one region) to not be
definable in terms of other, more easily recognisable primitives. Still, in spite of it being primitive, we can (again, following
Hudson) say some things to point to which relation we have in mind: I’m located at the person-shaped region right here, but
not at the room I’m in, and I’m located at the smallest region containing my hand only derivatively, and in virtue of having a
proper part located at the region.
11
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.20 Again, the semantics is fairly straightforward and
we have the answers we hoped for.21
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.22
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,
20
This view has a natural Stage-Theoretic variant, replacing talk of temporal parts with talk of counterparts.
Note that, if we hope to endorse the combination of four-dimensionalism and this kind of multilocation, we will have to
ensure that our account of temporal parts allows for multilocation This is not straightforward. For instance, Sider’s account
(2001, pp. 59-60) seems to straightforwardly preclude it: x is a temporal part of y at t’ =df (i) x is a part of y at t, (ii) x exists
only at t, and (iii) x overlaps (i.e., shares a part with) everything that is a part of y at t.
22 The Stage-Theoretic version of this view will only differ slightly. According to Stage Theory, ordinary objects are
instantaneous, and can be said to persist in virtue of having relevantly similar counterparts at other times. (See Sider 2001.)
Here, the stage theorist will not say that Tom is wholly present at the fusion of L1, L2, L3, and L4. And the temporal parts
mentioned above, AC and BD, will each be counterparts of Tom (one of which is, strictly speaking, identical to Tom). The
question “Where is Tom at each time?” will be interpreted as being a question about where Tom’s counterparts are at each
time, and will have the same answers as it would if it were about temporal parts.
21
12
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 1pm.
Regardless of Tom’s also being at the same region in the living room at 1pm, 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 temporallyextended 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 1pm. The At-At account will give this verdict because
in each case, at both noon and 1pm, the regions Tom is located at are the same: the fusion of
13
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. 23 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).24 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.
23
The oft discussed spinning disk case was first presented by Saul Kripke, and is discussed in Sider 2001, pp. 224-236.
Spinning disk cases are taken to involve reductive accounts of motion such as the At-At Account, and are often used to cause
problems for the Four-Dimensionalist, who has trouble distinguishing between spinning homogeneous disks and nonspinning homogeneous disks.
24
These worries are not new, and have been raised as problems for the four-dimensionalist who accepts a notion of temporal
parts like the one Sider puts forward.
14
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.
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