Shieva Kleinschmidt University of Southern California At It Again ...

At It Again: Time-Travel And Motion
Shieva Kleinschmidt
University of Southern California
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’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. I think
this is incorrect. In this paper, I’ll present a case in which, intuitively, motion does not
occur, though the At-At account of motion entails that it does. I will then turn to the only
tenable response that avoids revising the At-At account: denying the possibility of my
case. I will argue that the response is both contentious (since the possibility of my case is
plausible and useful) and fails to defend the spirit of the At-At account qua reduction of
motion (rather than merely a listing of necessary and sufficient conditions for motion’s
occurring). Since other responses require amending the At-At account, I will conclude
that, in light of the case I present, one cannot accept the At-At account without
amendment, especially when it is taken as a claim about what it means to move.
1. The Cases
Consider the following two cases. In the first, time-travelling Tom sits quietly from noon
through 1:00pm in the living room. Then Tom time-travels back, and sits quietly from
noon through 1:00pm in the kitchen. The case can be represented as follows:
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 1:00pm 1 ). But now consider this second case:
Time-travelling Tom has become adept at running his time machine. Sitting in
the living room waiting to watch Extreme Engineering, Tom realises that he would like
some cookies. So he time-travels to be in the kitchen at 1:00pm, grabs a bunch of freshly
baked cookies, time-travels to be in the kitchen at noon, bakes the cookies, and then time-
travels to be in the living-room at 1, to watch his programme. This case can be
represented thus:
The Exciting Case
Intuitively, there is motion in this case in the interval between and including noon and
1:00pm. Even excluding facts about cookie-grabbing and cookie-making and time-travel
machine operating, we think it is clear that Tom moves.
Our intuitions about the cases seem to be grasping onto whatever it is that the
arrows in the above diagrams represent. In spite of the fact that, in the two cases, Tom
occupies the same regions (the living room at noon and 1:00pm, and the kitchen at noon
and 1:00pm), we think he moves in one case and not the other. This, however, poses a
problem for accounts that entail the following:
• Motion Supervenience: Necessarily, facts about motion are determined by the
location facts and the identity facts.
1 We can even present a version of this case which lacks motion outside of this interval as well. Time
travel needn’t require motion (let alone discontinuous motion) if it’s possible for spacetime to be bent and
looped in certain ways. For instance, imagine that Tom is living in a manifold which is twisted and looped
like a möbius strip. 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.

The most popular account entailing Motion Supervenience is the At-At account of
motion. Is there a way of reading this account that will enable us to respect our intuitions
about the motion facts in the above cases?
2. The At-At Account of Motion
Bertrand Russell presents us with this account of motion: “Motion consists merely in the
occupation of different places at different times . . .” 2 That is:
• (At-At) 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
In other words, objects move if and only if they’re at different places at different times.
It’s clear that all the account appeals to in giving the necessary and sufficient conditions
for motion are facts about the identity and location of objects.
However, there are a lot of kinds of location facts. The At-At account tells us
simply that what matters is where objects are at. To understand the At-At account, we
must understand what this use of ‘at’ amounts to.
2 Russell 1903, Ch. LIV section 447. 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.)
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 energylevels
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
problematic case I discuss can be altered to involve continuous persistence, thereby raising a problem for
Russell’s full account as well (though it’s worth noting that Russell’s statements in The Principles of
Mathematics strongly suggest that he would simply reject the possibility of my case). I will now set aside
Russell’s full account without guilt.

Since the account is about motion of objects across space, the most natural
reading of the account is as concerning the location of entire objects at times. However,
what this amounts to will depend on what one thinks about persistence of objects through
time. For the three-dimensionalist, it will amount to concerning the location of entire
objects (though some may think that these facts are only expressible with some kind of
indexing to times; this issue shouldn’t be important for my paper, so I will set it aside).
For four-dimensionalists, it will amount to concerning the location of entire temporal-
parts, namely those temporal parts that are entirely present at the time in question. 4
my two cases?
Armed with our reading of ‘at’, what are the relevant location and identity facts in
3. The Cases Revisited
For ease of discussion, let’s label the locations and occupants of those locations as
follows:
4 Two other options for the interpretation of ‘at’ are: (i) ‘at’ means has a part present at; and (ii) ‘at’ means
is such that the entire object is located at. The first option fails due to it causing the At-At account to give
the result that an intuitively stationary blackboard is moving in virtue of its having one part at the right side
of the room, and another part at the left side of the room. (These options, and the worry about the first
option, were presented to me by Hud Hudson.)
For the three-dimensionalist, who takes objects to be entirely present at each time, the second
option will amount to the one I mentioned as the natural reading of what the At-At Account is concerned
with. However, for those who reject Three-Dimensionalism (among whom are the four-dimensionalists,
though others might reject both views), this option will come apart from the reading I prefer.
For the four-dimensionalist, the At-At Account is not plausible on the second reading, because,
generally, fourdimensionalists think ordinary objects are each entirely located in exactly one region. For
stage theorists, this is a spatial part of an instantaneous sliver of spacetime; ordinary objects never, strictly
speaking, are present at more than one time, and thus cannot ever meet the conditions for motion given by
the At-At account on the second reading. For the worm-theorist, objects are entirely located at four (or
more) dimensional regions of spacetime. If they were to meet the At-At Account’s conditions for motion,
then, it would be in virtue of a single object being entirely located at more than one of these spacetime
chunks. This, however, would conflict with intuitions about what motion occurs in virtue of, and it would
fail to capture many cases which we believe involve motion in spite of not meeting these conditions.

We want to say that there is some sense in which Tom is at each of the locations L1-L4.
But what is it in virtue of that Tom persists across time and space? We may answer this
question in a variety of ways, though in what follows I’ll focus just on what I take to be
the most mainstream answers to this question. These answers each produce views which
lead to one of the following two answers to the question “Where is Tom at each time?”:
(i) Tom is entirely in the kitchen, and also entirely in the living room, and (ii) Tom is
entirely in the fusion of the occupied regions in the living room and kitchen, but is not
entirely in either of those two proper subregions. To illustrate, consider the various
(mainstream) ways of answering the question of how Tom persists across time and space:
First, we might think that objects are present at more than one time or place in
virtue of having a proper (temporal) part that is entirely present at each time, and
similarly, that an object is present at more than one spatial region in virtue of having a
proper spatial part at each proper subregion of the region the object occupies at a time. 5
With this account, we would say that A, B, C, D and Tom are all distinct from one
another, but that 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?” is interpreted as being 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.
5 This view of persistance across space and time is called Pertension, and is presented in Hud Hudson
2006, pp. 99-103. He presents it as: “‘x pertends’ =df x is a material object that is entirely located at a nonpoint-sized
region, r, and for each proper subregion of r, r*, x has a proper part entirely located at r*.”
I worry about this definition, however. It is presented as describing the kind of spatial extension it
is most natural for a four-dimensionalist to posit, because it is analogous to the four-dimensionalist’s
conception of temporal extension, and four-dimensionalists are generally very sympathetic to (and find
support for their view in) analogies between space and time. However, positing this kind of extension
requires accepting a more liberal view of which regions can be occupied, as well as a more liberal view of
decomposition, than even many four-dimensionalists may be comfortable with.
Unfortunately, it is not clear to me how a view about persistance across space can be stated in a
way that both avoids the requirement of these liberal principles, and also precludes coextension with the
spanning view, on which objects are present at more than one region iff the region is a proper subregion of
the region the entity is entirely present at, though the object does not have a proper part at any subregion of
the region the entity is entirely present at. (Discussed in Hudson 2006, pp. 99-103.) Nonetheless, I will set
this issue aside because for the purposes of my discussion I need simply that Tom is entirely present at the
fusion of L1 and L2 at T1, and at the fusion of L3 and L4 at T2, but is not entirely present at any one of L1-
L4. Both the spanning view and the pertension view (and the variety of views occupying the spectrum
between these) will give me this result.

Similarly, consider someone who takes objects to persist through time in virtue of
being entirely present at each time, but who agrees with the previous theorist about how
objects extend across space? This view would 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 need to
give to the question of where Tom is at each time. Though, unlike for the previous
theorist, 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 is entirely present at, at each
time, the theorist will give the same response: at each time, Tom is at the fusion of the
occupied regions in the kitchen and in the living room.
On one way of cashing out Stage Theory, the view will also yield the same
answer to the question about Tom’s location at times. On this view, A-D are each
distinct, and each is a counterpart of Tom’s. If this version of Stage Theory is one on
which the question of where Tom is at a time reduces to a question of where Tom’s
current temporal part is at a time, and a temporal part is taken to be the fusion of any
counterparts of Tom’s at a time, then the facts about Tom’s location at times will be the
same as before. However, if the question is taken to concern the location of Tom’s
counterparts, then the answer will be the same as that given by the views that follow.
Consider now a view according to which, as mentioned above, objects persist in
virtue of being entirely present at each time at which they are present at all. That is: in
some sense, an object is all there every time it’s present. Further, sometimes objects are
entirely present in more than one spatial region as well. None of their parts are missing;
they’re all there, even though there’s somewhere else where they’re all there as well. Just
as the object is entirely located in more than one place in time, it is entirely located in
more than one place in space. So, we can say, the object is multi-located. Unfortunately,
this kind of multi-location is notoriously difficult to define, and many philosophers say it
is inconceivable for them. 6 Because I believe (following Hudson 2006) that the notion of
6 Ted Sider (in Sider, 2001, p. 64) has pointed out difficulties in defining the notion of entire-presence, or
wholelocation 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.

located at that multi-location depends on (when construed as the view that an object can
be located at more than one region) is not definable in terms of other, more easily
recognisable primitives, I will take it as primitive and simply proceed from there.
However, it may be helpful to mention some of the features of this kind of location that
Hudson points out: 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. Finally, according to the theory
we’ve been considering, when I time travel in order to be present at the same time “more
than once”, I’m located at more than one region. (This sense of located at is the same
sense in which I’ve been using entirely located at throughout the paper.) According to
this theorist, then, the identity and location facts are these: A=B=C=D=Tom, and at each
time Tom is entirely in the kitchen, and also entirely in the living room.
Another potential view is one on which objects persist through time in virtue of
having proper temporal parts at each time, but they can sometimes be said to be entirely
present in more than one spatial location at a time, in virtue of having a temporal part that
is multi-located. 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. So, because
we take the question of where Tom is located at each time to be a question of where
Tom’s temporal parts are at each time, we have the result that Tom is entirely at L1 and
L3 at T1, and also that Tom is entirely at L2 and L4 at T2.
Some theorists will think I’ve misconstrued the notion of temporal part in taking
it to be even conceivable that temporal parts can be multi-located. Some of these theorists
will have in mind an account of temporal parts like the one Sider presents 7 , which is
somewhat like:
• ‘x is a temporal part of y at t’ =df x is a part of y at t, x exists only at t, and x overlaps
everything that is a part of y at t.
I do not think, however, that it is necessary to preclude multi-location of temporal parts
by definition. This (and the tension between four-dimensionalism and multi-location of
temporal parts) is discussed elsewhere. [reference or addition of appendix?] Further, I
Since neither of these options is acceptable, we should reject this account of entire-presence, or whole
location. (Sider takes the view entailed on the second horn of the dilemma to be stronger than what I’ve
presented here, but I believe Mereological Neighbourliness is all that really follows.)
7 2001, pp. 59-60.

elieve that a more liberal notion of temporal parts will be necessary if the temporal parts
theorist wishes to make sense of multi-location of an object (any object!) at times. This is
because the locations of temporal parts are supposed to be what it is in virtue of that the
location facts about the whole object obtain in virtue of. If an object is multilocated at a
time, and we cannot appeal to multilocation of temporal parts of the object, and we can’t
appeal to the existence of more than one temporal part at a time (which would also be in
contradiction with the application of the above definition), then it seems we have nothing
left to explain the location fact in terms of. In light of these (and other) considerations, I
propose the following definition of ‘temporal part’:
• ‘x is a temporal part of y’ =df there exists a time, t, and a region, r, such that (i) x is
wholly located at a subregion of the intersection of r and t, (ii) y is wholly located
at r, (iii) x is part of y, and (iv) for all z, if ((a) z is wholly located at a subregion of
r, (b) z is present at t, and (c) z is a part of y), then x overlaps z.
The crucial point to take from the preceeding discussion in this section is simply
this: there are a variety of ways to answer the question of what the location and identity
facts in my two cases are. However, each of these determines one of two answers to the
question of where Tom is at each time.
Further, it should be noted that the answers we gave to what the location and
identity facts are in the cases will not depend on any of the facts which differ between the
cases as I originally described them. Any difference in our answers about location or
identity facts between the cases would therefore be arbitrary. Thus, we cannot give a
different answer to the question of where Tom is at each time without invoking a charge
of arbitrariness.
Finally, because the At-At account is blind to anything but the facts about identity
and the location of the object whose motion is in question, the At-At account will entail
that the same motion facts hold in the Dull Case and in the Exciting Case. But this runs in
direct contradiction to our intuitions about motion in the cases. Thus, because we
intuitively took one case to involve motion and the other to not, the At-At account will be
too liberal or too conservative in its attribution of motion in the cases. Which it is will
depend on which of the two answers we give to the question of where Tom is at each
time.

If we claim that Tom is not entirely at any of L1-L4, but is instead at the fusion of
L1 and L3 at T1, and at L2 and L4 at T2, then the At-At account will be too restrictive
because it will entail that Tom is not moving in the Exciting Case. In that case, as in the
Dull Case, Tom is at the same region at each time (the fusion of R1 and R2), and thus
does not meet the conditions for motion. When construed this way, my Exciting Case
amounts to nothing more than a spinning disk case. These have been thoroughly covered
in the literature, and so I will not discuss them in the body of this paper. However,
because I find it puzzling that so many proponents of the At-At account set these cases
quickly aside, I offer my thoughts on some of their quick responses to the cases in
Appendix 1, Spinning Disks: Is There Still Motion?
For the remainder of this paper, I will focus on what happens if we claim that
Tom is entirely at each of L1-L4. The most immediate result seems to be that the At-At
account will be too liberal because it will entail that Tom moves in the Dull Case, in
virtue of his being entirely at one space at a time, and at a distinct space at a distinct time.
(And it is worth noting that any case of a persisting object being multi-located at some
time is sufficient to raise this problem for the At-At account.) It doesn’t matter to the At-
At account that Tom simply sat in both regions as time passed. However, our intuitions
seem to care, if not about that, then about something else that the At-At account has
obviously missed.
4. Replies to the Liberality Objection
There are several ways to respond to this case, the first being to simply deny the
possibility of the Dull Case. I have several worries about this. My first worry is that, for
this to help, one would have to deny the possibility of something persisting which is, at
some point in its existence, multi-located at a time (and which, intuitively, doesn’t move
between the times at which it is multi-located and some other time), since that’s what
causes the difficulty for the At-At account.
That said, there are several ways to attempt to deny the possibility of my Dull
Case as construed as involving multi-location. One way is to deny the possibility of
survivable timetravel. 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 -- meaning that in the Dull Case, A=B and (perhaps) C=D, but B≠C. Which
of B or C we take to be Tom is up to us. I find this solution problematic, however,

ecause (i) it seems to lack any independent motivation, and (ii) it seems to conflict with
intuition in cases where objects appear “more than once” at the same time due to
spacetime being curved back on itself in appropriate ways. If Tom lives in a four-
dimensional version of a möbius strip, and its temporal extension is shorter than his life,
then we can imagine him at 63 visiting himself at 36. Nothing about this case seems
particularly problematic. However, to save the At-At account via appeal to this response,
we would have to claim that there’s something about this case which excludes it from the
realm of the possible.
Another way to try to deny the possibility of my case is to say that we are
mistaken in thinking that the identity and location facts are even possibly as they would
need to be to give the result that the At-At account entails there is motion in the Dull
Case. That is, deny the possibility of multi-location in time-travel cases. However, this
response comes with a cost. There is some work that multi-location does 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. Where is Tom? 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. What is Tom shaped like? Plausibly, he’s person-shaped
in the kitchen, and person-shaped in the living room. It definitely doesn’t seem that in
virtue of time-travelling, he’s suddenly acquired an odd, noticeably gappy shape (that of
a fusion of two people). Finally, what are we to say about Tom’s weight? This, just like
his shape, is plausibly region-relative. Suppose that Tom got dense in his old age (since
my previous example requires him to fit into a region previously occupied by 36 year old
Tom, I can’t have him gaining volume!), and at 63 he weighs more than he did at 36.
How much does Tom weigh? It seems appropriate to say: more than he’d like to in the
kitchen, and exactly what he’d like to in the living room. But it’s definitely not the case
(at least, intuitively) that Tom’s weight almost doubled as a result of time-travelling to
visit himself at 36. 8 How much does Tom weigh? It seems appropriate to say: more than
he’d like to in the kitchen, and exactly what he’d like to in the living room. But it’s
definitely not the case (at least, intuitively) that Tom’s weight almost doubled as a result
of time-travelling to visit himself at 36.8 However, this, and the other implausible claims
about shape and location (and a variety of other properties), is exactly what the
8 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.

pertension theorist (who must say Tom isn’t multilocated even though he time travelled)
will have to claim.
Finally, let us suppose that this interpretation of the identity and location facts in
my Dull Case isn’t possible. Is this enough to ensure the safety of the At-At account from
the threat of this latest worry? Sure, if we take the At-At account to be a mere listing of
necessary and sufficient conditions for motion. However, this isn’t what makes the
account so appealing. People like it because it’s a claim about what motion consists of.
It’s an analysis, maybe even a definition. But once that’s what we’re trying to defend,
cases like mine become relevant once again, regardless of whether they’re possible – they
need only be conceivable. If my case can’t obtain, it’s an interesting Metaphysical fact
that it can’t, and not one that has to do with the At-At account. So when evaluating the
account, it is fruitful (and, I think, necessary) to look at some counterpossibles and ask
what our predications of motion would be in those cases. Do our predications of motion
square with the Philosopher’s account of what motion consists of? If not, and if we take
ourselves to be giving an account of a natural phenomenon that people grasp and are able
to correctly identify, we have a problem and need to make some difficult choices. 9
What else might someone say in responding to the case? There are two other
options I take to be salient: (i) index motion to regions, or (ii) add a requirement for
motion that the entity in question at a region at a time stands in certain (e.g., perhaps
immanent causal) relations to itself at another region at a distinct time. I find the first
option a bit compelling, and would be happy if it worked. The idea 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: he’s moving in the living room, but he’s not moving in the kitchen.
However, I worry about how to pick out the relevant regions to index Tom’s motion to,
without appealing to anything spooky. The second option also requires invoking
something mysterious, and though it may be required due to other puzzles in
Metaphysics, it will disappoint the At-At theorist to admit that motion consists partly in
the obtaining of such spookiness. (I’m including appeals to personal time in this second
9 Of course, 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.

category, because I do not believe that the notion of personal time can be cashed out just
via appeal to location facts and identity facts; we will need to appeal to something more,
like genidentity relations, immanent causation, or in-virtue-of dependence of properties
had, on the having of particular previous properties.) Regardless of which option is
chosen, the At-At theorist is faced with costs, and will need to revise the original account.
As stated in its original formulation, the account is simply inadequate.
Appendix 1, Spinning Disks: Is There Still Motion?
Spinning disk cases threaten to show that the At-At account is too restrictive about when
motion occurs. This problem, I take it, arises in virtue of the very feature that makes the
account so attractive: the elegance with which it reduces something seemingly complex
to a simple phenomenon we can all understand. No need to invoke something mysterious
or complicated (like genidentity or irreducible vector quantities). However, the world
doesn’t seem to be as simple as we’d like.
Consider a spinning disk. 10 It’s in region R at T1, and R at T2. According to the
At-At account, this disk isn’t moving. But intuitively it is moving – it’s turning within R.
So why isn’t this case seen as a counterexample to the At-At account?
There’s a sophisticated discussion of this case in the literature, and I won’t
attempt to cover it here. However, in discussing the case with others, I’ve noticed a
widespread (though not universal) lack of concern about the spinning disk case (at least,
qua counterexample to the At-At account). From what I can tell, this is due to an
assumption that there’s a straightforward amendment that allows us to still capture the
spirit of the At-At account. I’ll now turn to this attempted quick-fix, argue that neither it
nor other attempts like it succeed in producing the right results in spinning disk cases, and
thereby, hopefully, provide cause to take the spinning-disk case seriously as a threat to
the At-At account of motion (as, incidentally, it is seen in the literature on this topic).
In observing the spinning disk case, what makes people think there’s an easy
solution for the At-At theorist? There’s an intuitive response: Look, it’s said, the object is
moving in virtue of all of its parts moving. This suggests the following account:
10 The oft discussed spinning disk case was first presented by Saul Kripke, and is discussed in Sider 2001,
pp. 224-236.

• (First Revised At-At) Necessarily, for any x, x is in motion iff there exist some ys such
that x is wholly decomposable 11 into the ys, and for any one of the ys, yn there
exists spatial regions s1 and s2, and times t1 and t2, such that s1 is distinct from
s2, t1 is distinct from t2, and yn is at s1 at t1, and at s2 at t2.
That is just to say: An object moves just in case it’s wholly decomposable into parts each
of which are moving (in the way specified by the original At-At account).
Unfortunately, this is an inadequate revision. For one thing, we may worry
whether things do move whenever all of their proper parts (on some decomposition) are
moving. Consider an example presented by James Blackmon in “A Dilemma for a Picture
of Motion”: Suppose that (i) every region is such that it’s possible for something to
wholly occupy it, 12 (ii) every material object is such that for any occupyable subregion of
the region it occupies, it has a part wholly located in that subregion, and (iii) for any two
objects, there exists a fusion of those objects. 13 Now consider an apparently stationary
disk. This disk is wholly decomposable into proper parts that each keep their locations
throughout the disk’s existence – a fine candidate for staying still. However, the disk is
also entirely decomposable into parts that move – for consider the part that’s the fusion of
the temporal part of sitting-part-1 at T1, and of the temporal part of sitting-part-2
(directly to the left of sitting-part-1) at T2, and . . . This fusion will move, in virtue of
having temporal parts at different regions at different times. Further, the disk can be
completely decomposed into fusions like this one. So, on the above account, it moves
after all!
I don’t know what to say about the above case. It seems clear to me that objects
can be wholly decomposable into moving parts, without themselves moving. 14 For
instance, suppose an object occupies the same region throughout my existence, but each
one of the subatomic particles that composes it spins on its axis. Even if we didn’t think
the particles were moving, they’re all (let’s say) composed of proper parts that are
moving, and so the original object is wholly decomposable into moving parts. But this
isn’t a way for an object to move! Still, I think there are also cases where objects do
11
For the purposes of this paper: x is wholly decomposable into the ys iff x is a fusion of the ys, and no two
ys overlap one-another.
12
I.e., suppose Hud Hudson’s Liberal View of Receptacles is true (Hudson 2001).
13
For Kit Fine scholars: I’m requiring mere-sum aggregates.
14
Hudson made this claim in a commentary on Blackmon’s paper.

move in virtue of all of their proper parts moving, and I don’t know which kind of case
Blackmon’s spinning disk example should be characterised as.
However, taking a stand on that issue isn’t essential for having cause to reject the
above First Revised At-At account. Here’s another worry: assume now that spacetime is
pointy, and that the spinning disk is continuous and wholly decomposable into point-
sized parts. When it spins, there will be a point in its centre which stays put. This isn’t a
worry for someone who accepts the Doctrine of Arbitrary Undetached Parts along with
the Liberal View of Receptacles; it doesn’t matter that there’s a decomposition of the
object such that one of the parts in that decomposition isn’t moving – all we need is one
decomposition that meets the above requirement to ensure the disk is spinning, and that
won’t be hard to find. But suppose that we don’t accept DAUP and the LVR. Suppose, in
fact, that we think the disk has only two decompositions: its decomposition into its
improper part, and its decomposition into its point-sized parts. Now we’ll have the result
that the disk is not moving. However, this is unacceptable: whether the disk is moving or
not shouldn’t depend on whether we posit, e.g., disk-halves and disk-quarters in addition
to the point-sized parts of the disk. Our Mereological commitments shouldn’t have
implications for whether the whole thing is moving.
What account could be given that would produce the right results regardless of
our Mereological commitments? I really don’t know: we might say that a thing moves
just in case it’s wholly decomposable into parts most of which are moving (a second
attempt at revising the account), but now we have to wonder how many moving parts are
enough, and any line we draw will be arbitrary.
One might respond by claiming that we should re-evaluate our reading of the At-
At account, and use an interpretation of ‘at’ similar to the first one considered in section
1.1. On this new reading, we take the At-At account to be claiming this:
• (Reinterpreted At-At) Necessarily, for any x, x is in motion iff there exist some ys such
that x is wholly decomposable into the ys, and for at least one of the ys, yn there
existsspatial regions s1 and s2, and times t1 and t2, such that s1 is distinct from
s2, t1 is distinct from t2, and yn is wholly at s1 at t1, and wholly at s2 at t2.
That is, an object moves iff it has a part that moves (At-At style). This avoids the
arbitrariness worry, but my worry about the first attempt to amend the account (claiming
that a thing moves iff it’s wholly decomposable into parts that move), that the account

ecomes too liberal, resurfaces and is intensified here. After all, this third amendment
posits less restrictions on when motion occurs than does the first. (The worry applies to
the second attempt as well.) Also, consider a variant on the spinning disk case that’s been
presented in the literature. The disk in this case is an extended simple; it has no parts that
are moving At-At style. However, if we were to put a mark on it, we would observe the
marked area of the disk rotating (though the disk wouldn’t have a proper part
corresponding to that area). This case tells against all three attempts to provide a quick-
fix for the At-At account. Nevertheless, if we don’t take this last case seriously (for
instance, if we refuse to grant that extended simples are conceivable, or we want to bite
the bullet in this case), the third attempted quick-fix may still be a live option. However,
the account under this reading is still vulnerable to the worries I raise in the next section.
Hopefully this section has provided motivation for viewing the spinning disk case
as a challenging puzzle for anyone who accepts the At-At account.
Appendix 2, Where To Go From Here?
The At-At account is on shaky ground. If we posit even the conceivability of persisting
multilocation, it’s not clear what we should say about motion. To make it even harder to
give a clear, principled listing of necessary and sufficient conditions, consider the
following series of cases.
The First Case:
The world consists of an object which is entirely located at R1 at T1, and entirely located
at R2 at T2 (or which has temporal parts entirely located at those regions at the proper
times)

Intuitively, there’s discontinuous motion in this case. And the At-At account
accords with our intuitions: the object moves in virtue of being entirely at R1 at T1, and
at R2 at T2. 15
The Second Case:
Just as in the last world, this world consists simply of an object entirely (or with a
temporal part entirely) located at R1, and also entirely (or with a temporal part entirely)
located at R2. But here the object persists in each region for a longer duration than in the
previous world: the object is located at R1 at T1, and also at T2, and at R2 at T2, and also
at T3. To make it seem more intuitive, imagine: I’m preparing to time travel, and I leave
R1 exactly at T2. However, since I’m not good with machinery, I don’t end up going to
the past or future at all – I simply “reappear” at T2, though in a different region (my
machine won’t let me occupy a region that’s already filled). So I get to live the same
instant twice. Discouraged, I stay put (until T3). The world looks like this:
Has the object moved? Intuitions say it has: this case is just like the first, except
that the object stays put in each region a bit longer. For another way to look at it: Cover
up what’s happening at T2. Comparing just T1 and T3, we should definitely say there’s
motion (at least, if that’s what we say in the first case). How could what happens at T2
make a difference as to whether the object moves? The At-At account agrees with us.
15 We could construct a similar example with the object entirely located at each of the relevant regions
between R1 at T1 and R2 at T2, to achieve continuity, if we wanted. Or, another way of achieving a similar
end: make the times extended, and one of them (say, T1) open while the other’s closed, and let R1 and R2
be in contact.

The Third Case:
This world is just like the one in the second case, except that the only times are T1 and
T2. So the object is entirely (or has a temporal part entirely) at R1 at T1, and is also
entirely (or has a temporal part entirely) at R1 at T2, and is also entirely (or has a
temporal part entirely) at R2 at T2. To give a more intuitive picture of what’s happening,
it’s just as in the last case, except Space (unbeknownst to me) the instant I get to live
twice is actually the last instant of my existence. The world looks like this:
Is there motion? Intuitions begin to waver here and many people are initially
compelled to firmly deny that there is motion in this case. Still, once we note that the
only difference between this case and the previous one is how long the object remains at
R2, it’s much more difficult to deny that the object in this world moves. After all, surely
it doesn’t make a difference to the motion of the object, whether it simply persists in a
region for an extended period of time or not. Also, if we ignore R1 at T2, we have a case
just like the first. The At-At account, once again, agrees that there is motion. 16
The Fourth Case:
This final world is just like the last, except that it’s “flipped upside-down” in the temporal
dimension. So an object is entirely (or has a temporal part entirely) at R1 at T1, and also
is entirely (or has a temporal part entirely) at R2 at T1, and finally also is entirely (or has
a temporal part entirely) at R1 at T2 (but nothing occupies R2 at T2). Once again, to alter
16 There is, however, something important that sets this case apart from the others: in the first and second
cases, even one who thought temporal parts couldn’t be multilocated at a time would have accepted motion.
They would have thought, e.g., that in the second case the object was first completely in R1, then was
wholly located at the fusion of R1 and R2, and finally was simply in R2. But in this case, at least for the
perdurance-theorist (i.e., one who thinks that any object that’s extended in a dimension is extended in virtue
of having proper parts at each of the subregions, rather than by being multilocated across the dimension),
the object never wholly occupies a region that doesn’t overlap with one the region wholly occupied
previously. Accepting perdurantism won’t change what the At-At account says about the case, but it does
make more clear some intuitions against motion in the case.

the time-travel case: this time, I get lucky. I’m able to go back in time. However, once
again, I’ve made a mistake with the machinery, one that takes me out of existence just
after I get to the past. The world looks like this:
Is there motion? As far as I can tell, the considerations weighing in on this case
are just like those in the previous one, but oddly enough, people’s intuitions sometimes
differ across these cases. Regardless, the At-At account gives the same result: motion all
the way. 17
Works Cited
Blackmon, James. Unpublished manuscript. “A Dilemma for a Picture of Motion”,
Hudson, Hud. 2006. The Metaphysics of Hyperspace, (Oxford University Press: Oxford).
Hudson, Hud. 2002. “The Liberal View of Receptacles”, Australasian Journal of
Philosophy, vol. 80: pp. 432-439.
Hudson, Hud. 2001. A Material Metaphysics of the Human Person, (Cornell University
Press: Ithaca).
Russell, Bertrand. 1903. The Principles of Mathematics, (Cambridge: Cambridge
University Press).
Sider, Theodore. 2001. Four-Dimensionalism: An Ontology of Persistence and Time,
(Oxford: Oxford University Press).
van Inwagen. 1981. “The Doctrine of Arbitrary Undetached Parts”, Pacific Philosophical
Quarterly 62: pp. 123-137.
17 Thanks to Frank Arntzenius, Kit Fine, Cody Gilmore, Hud Hudson and Daniel Nolan for helpful
discussion about these topics.