Shieva Kleinschmidt University of Southern California At It Again ...
Shieva Kleinschmidt University of Southern California At It Again ...
Shieva Kleinschmidt University of Southern California At It Again ...
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<strong>At</strong> <strong>It</strong> <strong>Again</strong>: Time-Travel And Motion<br />
<strong>Shieva</strong> <strong>Kleinschmidt</strong><br />
<strong>University</strong> <strong>of</strong> <strong>Southern</strong> <strong>California</strong><br />
The <strong>At</strong>-<strong>At</strong> account <strong>of</strong> motion has become extremely popular. First championed by<br />
Bertrand Russell in The Principles <strong>of</strong> Mathematics, it can be roughly stated as the view<br />
that: necessarily, something moves if and only if it’s at one place at one time, and at a<br />
distinct place at a distinct time. This, many believe, is all that motion consists in. I think<br />
this is incorrect. In this paper, I’ll present a case in which, intuitively, motion does not<br />
occur, though the <strong>At</strong>-<strong>At</strong> account <strong>of</strong> motion entails that it does. I will then turn to the only<br />
tenable response that avoids revising the <strong>At</strong>-<strong>At</strong> account: denying the possibility <strong>of</strong> my<br />
case. I will argue that the response is both contentious (since the possibility <strong>of</strong> my case is<br />
plausible and useful) and fails to defend the spirit <strong>of</strong> the <strong>At</strong>-<strong>At</strong> account qua reduction <strong>of</strong><br />
motion (rather than merely a listing <strong>of</strong> necessary and sufficient conditions for motion’s<br />
occurring). Since other responses require amending the <strong>At</strong>-<strong>At</strong> account, I will conclude<br />
that, in light <strong>of</strong> the case I present, one cannot accept the <strong>At</strong>-<strong>At</strong> account without<br />
amendment, especially when it is taken as a claim about what it means to move.<br />
1. The Cases<br />
Consider the following two cases. In the first, time-travelling Tom sits quietly from noon<br />
through 1:00pm in the living room. Then Tom time-travels back, and sits quietly from<br />
noon through 1:00pm in the kitchen. The case can be represented as follows:<br />
The Dull Case
Intuition tells us that there is not motion in this case (at least, not during the interval<br />
between and including noon and 1:00pm 1 ). But now consider this second case:<br />
Time-travelling Tom has become adept at running his time machine. Sitting in<br />
the living room waiting to watch Extreme Engineering, Tom realises that he would like<br />
some cookies. So he time-travels to be in the kitchen at 1:00pm, grabs a bunch <strong>of</strong> freshly<br />
baked cookies, time-travels to be in the kitchen at noon, bakes the cookies, and then time-<br />
travels to be in the living-room at 1, to watch his programme. This case can be<br />
represented thus:<br />
The Exciting Case<br />
Intuitively, there is motion in this case in the interval between and including noon and<br />
1:00pm. Even excluding facts about cookie-grabbing and cookie-making and time-travel<br />
machine operating, we think it is clear that Tom moves.<br />
Our intuitions about the cases seem to be grasping onto whatever it is that the<br />
arrows in the above diagrams represent. In spite <strong>of</strong> the fact that, in the two cases, Tom<br />
occupies the same regions (the living room at noon and 1:00pm, and the kitchen at noon<br />
and 1:00pm), we think he moves in one case and not the other. This, however, poses a<br />
problem for accounts that entail the following:<br />
• Motion Supervenience: Necessarily, facts about motion are determined by the<br />
location facts and the identity facts.<br />
1 We can even present a version <strong>of</strong> this case which lacks motion outside <strong>of</strong> this interval as well. Time<br />
travel needn’t require motion (let alone discontinuous motion) if it’s possible for spacetime to be bent and<br />
looped in certain ways. For instance, imagine that Tom is living in a manifold which is twisted and looped<br />
like a möbius strip. If the loop is temporally shorter than Tom’s life, he can simply sit and wait, and appear<br />
at the same time twice in different places.
The most popular account entailing Motion Supervenience is the <strong>At</strong>-<strong>At</strong> account <strong>of</strong><br />
motion. Is there a way <strong>of</strong> reading this account that will enable us to respect our intuitions<br />
about the motion facts in the above cases?<br />
2. The <strong>At</strong>-<strong>At</strong> Account <strong>of</strong> Motion<br />
Bertrand Russell presents us with this account <strong>of</strong> motion: “Motion consists merely in the<br />
occupation <strong>of</strong> different places at different times . . .” 2 That is:<br />
• (<strong>At</strong>-<strong>At</strong>) Necessarily, for any x, x is in motion iff there exist spatial regions s1 and s2,<br />
and times t1 and t2, such that s1 is distinct from s2, t1 is distinct from t2, and x is<br />
at s1 at t1, and at s2 at t2. 3<br />
In other words, objects move if and only if they’re at different places at different times.<br />
<strong>It</strong>’s clear that all the account appeals to in giving the necessary and sufficient conditions<br />
for motion are facts about the identity and location <strong>of</strong> objects.<br />
However, there are a lot <strong>of</strong> kinds <strong>of</strong> location facts. The <strong>At</strong>-<strong>At</strong> account tells us<br />
simply that what matters is where objects are at. To understand the <strong>At</strong>-<strong>At</strong> account, we<br />
must understand what this use <strong>of</strong> ‘at’ amounts to.<br />
2 Russell 1903, Ch. LIV section 447. One may worry (here as well as throughout this paper) about my<br />
reidentifying regions: making identity claims about spatial regions at different times, and temporal regions<br />
at different places. For doing this, I hope to appeal to nothing more than what is appealed to frequently in<br />
common discourse: people <strong>of</strong>ten talk <strong>of</strong>, e.g., returning to the same place, or leaving a location. We can, if<br />
we’d like, interpret them as making relative reidentifications <strong>of</strong> regions (e.g., “I’m in the same region<br />
relative to the room, as I was yesterday”). The central claims in this paper should be unaffected. (Thanks to<br />
Frank Arntzenius for bringing this worry, and the relativisation response, to my attention.)<br />
3 Russell also thought that all motion has to be continuous; hence, only certain types <strong>of</strong> series’ <strong>of</strong> spaces<br />
and times being occupied by an object would qualify that object as a mover. So the <strong>At</strong>-<strong>At</strong> account as<br />
formulated above doesn’t actually reflect Russell’s complete view <strong>of</strong> motion, but rather is a fragment <strong>of</strong> his<br />
view. But I think the <strong>At</strong>-<strong>At</strong> fragment does a better job <strong>of</strong> tracking our intuitions about strange cases.<br />
Consider, for example, how you’d react if you came into the presence <strong>of</strong> a goblin, who (it appears) can<br />
teletransport from one region to the next. When you naively mention to the goblin you’re on your way to<br />
the kitchen for a cookie, the goblin pops out <strong>of</strong> presence where you are, and pops into presence in the<br />
kitchen to get the cookie first. When asked why you didn’t get dessert, a natural response would be: “I<br />
wasn’t fast enough – some goblin got to it before I did.” That is, intuitively, the goblin moved to the<br />
kitchen, though discontinuously. (Discontinuous motion is portrayed all over in popular culture. I Dream <strong>of</strong><br />
Jeannie provides a good example.) Further, apart from fanciful cases like those, science may be pushing us<br />
to posit actual instances <strong>of</strong> discontinuous motion, with electrons making quantum leaps between energylevels<br />
and such (though the implications <strong>of</strong> quantum leaps are difficult to make out due to worries about<br />
wave/particle duality). Still, regardless <strong>of</strong> whether it should be preferred, it is the <strong>At</strong>-<strong>At</strong> account, rather than<br />
Russell’s full account, that is so <strong>of</strong>ten assumed to be the correct account <strong>of</strong> motion. And finally, the<br />
problematic case I discuss can be altered to involve continuous persistence, thereby raising a problem for<br />
Russell’s full account as well (though it’s worth noting that Russell’s statements in The Principles <strong>of</strong><br />
Mathematics strongly suggest that he would simply reject the possibility <strong>of</strong> my case). I will now set aside<br />
Russell’s full account without guilt.
Since the account is about motion <strong>of</strong> objects across space, the most natural<br />
reading <strong>of</strong> the account is as concerning the location <strong>of</strong> entire objects at times. However,<br />
what this amounts to will depend on what one thinks about persistence <strong>of</strong> objects through<br />
time. For the three-dimensionalist, it will amount to concerning the location <strong>of</strong> entire<br />
objects (though some may think that these facts are only expressible with some kind <strong>of</strong><br />
indexing to times; this issue shouldn’t be important for my paper, so I will set it aside).<br />
For four-dimensionalists, it will amount to concerning the location <strong>of</strong> entire temporal-<br />
parts, namely those temporal parts that are entirely present at the time in question. 4<br />
my two cases?<br />
Armed with our reading <strong>of</strong> ‘at’, what are the relevant location and identity facts in<br />
3. The Cases Revisited<br />
For ease <strong>of</strong> discussion, let’s label the locations and occupants <strong>of</strong> those locations as<br />
follows:<br />
4 Two other options for the interpretation <strong>of</strong> ‘at’ are: (i) ‘at’ means has a part present at; and (ii) ‘at’ means<br />
is such that the entire object is located at. The first option fails due to it causing the <strong>At</strong>-<strong>At</strong> account to give<br />
the result that an intuitively stationary blackboard is moving in virtue <strong>of</strong> its having one part at the right side<br />
<strong>of</strong> the room, and another part at the left side <strong>of</strong> the room. (These options, and the worry about the first<br />
option, were presented to me by Hud Hudson.)<br />
For the three-dimensionalist, who takes objects to be entirely present at each time, the second<br />
option will amount to the one I mentioned as the natural reading <strong>of</strong> what the <strong>At</strong>-<strong>At</strong> Account is concerned<br />
with. However, for those who reject Three-Dimensionalism (among whom are the four-dimensionalists,<br />
though others might reject both views), this option will come apart from the reading I prefer.<br />
For the four-dimensionalist, the <strong>At</strong>-<strong>At</strong> Account is not plausible on the second reading, because,<br />
generally, fourdimensionalists think ordinary objects are each entirely located in exactly one region. For<br />
stage theorists, this is a spatial part <strong>of</strong> an instantaneous sliver <strong>of</strong> spacetime; ordinary objects never, strictly<br />
speaking, are present at more than one time, and thus cannot ever meet the conditions for motion given by<br />
the <strong>At</strong>-<strong>At</strong> account on the second reading. For the worm-theorist, objects are entirely located at four (or<br />
more) dimensional regions <strong>of</strong> spacetime. If they were to meet the <strong>At</strong>-<strong>At</strong> Account’s conditions for motion,<br />
then, it would be in virtue <strong>of</strong> a single object being entirely located at more than one <strong>of</strong> these spacetime<br />
chunks. This, however, would conflict with intuitions about what motion occurs in virtue <strong>of</strong>, and it would<br />
fail to capture many cases which we believe involve motion in spite <strong>of</strong> not meeting these conditions.
We want to say that there is some sense in which Tom is at each <strong>of</strong> the locations L1-L4.<br />
But what is it in virtue <strong>of</strong> that Tom persists across time and space? We may answer this<br />
question in a variety <strong>of</strong> ways, though in what follows I’ll focus just on what I take to be<br />
the most mainstream answers to this question. These answers each produce views which<br />
lead to one <strong>of</strong> the following two answers to the question “Where is Tom at each time?”:<br />
(i) Tom is entirely in the kitchen, and also entirely in the living room, and (ii) Tom is<br />
entirely in the fusion <strong>of</strong> the occupied regions in the living room and kitchen, but is not<br />
entirely in either <strong>of</strong> those two proper subregions. To illustrate, consider the various<br />
(mainstream) ways <strong>of</strong> answering the question <strong>of</strong> how Tom persists across time and space:<br />
First, we might think that objects are present at more than one time or place in<br />
virtue <strong>of</strong> having a proper (temporal) part that is entirely present at each time, and<br />
similarly, that an object is present at more than one spatial region in virtue <strong>of</strong> having a<br />
proper spatial part at each proper subregion <strong>of</strong> the region the object occupies at a time. 5<br />
With this account, we would say that A, B, C, D and Tom are all distinct from one<br />
another, but that Tom has a temporal part, Tom1, which is identical to the fusion <strong>of</strong> A and<br />
C, AC, and another temporal part, Tom2, which is identical to the fusion <strong>of</strong> B and D, BD.<br />
The answer to the question “Where is Tom at each time?” is interpreted as being a<br />
question about Tom’s temporal parts, which are located in the same region at each time:<br />
the fusion <strong>of</strong> the occupied region <strong>of</strong> the kitchen, and the occupied region <strong>of</strong> the living<br />
room.<br />
5 This view <strong>of</strong> persistance across space and time is called Pertension, and is presented in Hud Hudson<br />
2006, pp. 99-103. He presents it as: “‘x pertends’ =df x is a material object that is entirely located at a nonpoint-sized<br />
region, r, and for each proper subregion <strong>of</strong> r, r*, x has a proper part entirely located at r*.”<br />
I worry about this definition, however. <strong>It</strong> is presented as describing the kind <strong>of</strong> spatial extension it<br />
is most natural for a four-dimensionalist to posit, because it is analogous to the four-dimensionalist’s<br />
conception <strong>of</strong> temporal extension, and four-dimensionalists are generally very sympathetic to (and find<br />
support for their view in) analogies between space and time. However, positing this kind <strong>of</strong> extension<br />
requires accepting a more liberal view <strong>of</strong> which regions can be occupied, as well as a more liberal view <strong>of</strong><br />
decomposition, than even many four-dimensionalists may be comfortable with.<br />
Unfortunately, it is not clear to me how a view about persistance across space can be stated in a<br />
way that both avoids the requirement <strong>of</strong> these liberal principles, and also precludes coextension with the<br />
spanning view, on which objects are present at more than one region iff the region is a proper subregion <strong>of</strong><br />
the region the entity is entirely present at, though the object does not have a proper part at any subregion <strong>of</strong><br />
the region the entity is entirely present at. (Discussed in Hudson 2006, pp. 99-103.) Nonetheless, I will set<br />
this issue aside because for the purposes <strong>of</strong> my discussion I need simply that Tom is entirely present at the<br />
fusion <strong>of</strong> L1 and L2 at T1, and at the fusion <strong>of</strong> L3 and L4 at T2, but is not entirely present at any one <strong>of</strong> L1-<br />
L4. Both the spanning view and the pertension view (and the variety <strong>of</strong> views occupying the spectrum<br />
between these) will give me this result.
Similarly, consider someone who takes objects to persist through time in virtue <strong>of</strong><br />
being entirely present at each time, but who agrees with the previous theorist about how<br />
objects extend across space? This view would yield the result that AC=BD=Tom, A≠C<br />
and B≠D, and Tom is distinct from each <strong>of</strong> A, B, C and D. <strong>It</strong> is left undetermined whether<br />
A=B and C=D, but that question does not influence what answer this theorist will need to<br />
give to the question <strong>of</strong> where Tom is at each time. Though, unlike for the previous<br />
theorist, this question does not amount to a question about temporal parts <strong>of</strong> Tom’s, but<br />
instead simply amounts to a question <strong>of</strong> what regions Tom is entirely present at, at each<br />
time, the theorist will give the same response: at each time, Tom is at the fusion <strong>of</strong> the<br />
occupied regions in the kitchen and in the living room.<br />
On one way <strong>of</strong> cashing out Stage Theory, the view will also yield the same<br />
answer to the question about Tom’s location at times. On this view, A-D are each<br />
distinct, and each is a counterpart <strong>of</strong> Tom’s. If this version <strong>of</strong> Stage Theory is one on<br />
which the question <strong>of</strong> where Tom is at a time reduces to a question <strong>of</strong> where Tom’s<br />
current temporal part is at a time, and a temporal part is taken to be the fusion <strong>of</strong> any<br />
counterparts <strong>of</strong> Tom’s at a time, then the facts about Tom’s location at times will be the<br />
same as before. However, if the question is taken to concern the location <strong>of</strong> Tom’s<br />
counterparts, then the answer will be the same as that given by the views that follow.<br />
Consider now a view according to which, as mentioned above, objects persist in<br />
virtue <strong>of</strong> being entirely present at each time at which they are present at all. That is: in<br />
some sense, an object is all there every time it’s present. Further, sometimes objects are<br />
entirely present in more than one spatial region as well. None <strong>of</strong> their parts are missing;<br />
they’re all there, even though there’s somewhere else where they’re all there as well. Just<br />
as the object is entirely located in more than one place in time, it is entirely located in<br />
more than one place in space. So, we can say, the object is multi-located. Unfortunately,<br />
this kind <strong>of</strong> multi-location is notoriously difficult to define, and many philosophers say it<br />
is inconceivable for them. 6 Because I believe (following Hudson 2006) that the notion <strong>of</strong><br />
6 Ted Sider (in Sider, 2001, p. 64) has pointed out difficulties in defining the notion <strong>of</strong> entire-presence, or<br />
wholelocation that the notion <strong>of</strong> multi-location seems to depend on. In what sense is the object in question<br />
all there when it is entirely present somewhere (or at some time)? If we cash this out in terms <strong>of</strong> all <strong>of</strong> the<br />
object’s parts being there, Sider says, we face a dilemma which turns on how we interpret ‘all <strong>of</strong> x’s parts’.<br />
If we take that phrase to refer to all <strong>of</strong> x’s parts at the current time or region, then everyone would accept<br />
the view. If we take ‘all <strong>of</strong> x’s parts’ to refer to all the parts x has at any time or region whatsoever, then the<br />
view entails Mereological Neighbourliness: For any ordinary object x, and any time or region, r, if x is<br />
present at r, then for all z, if there exists a time or region at which z is a part <strong>of</strong> x, then z is present at r.
located at that multi-location depends on (when construed as the view that an object can<br />
be located at more than one region) is not definable in terms <strong>of</strong> other, more easily<br />
recognisable primitives, I will take it as primitive and simply proceed from there.<br />
However, it may be helpful to mention some <strong>of</strong> the features <strong>of</strong> this kind <strong>of</strong> location that<br />
Hudson points out: I’m located at the person-shaped region right here, but not at the room<br />
I’m in, and I’m located at the smallest region containing my hand only derivatively, and<br />
in virtue <strong>of</strong> having a proper part located at the region. Finally, according to the theory<br />
we’ve been considering, when I time travel in order to be present at the same time “more<br />
than once”, I’m located at more than one region. (This sense <strong>of</strong> located at is the same<br />
sense in which I’ve been using entirely located at throughout the paper.) According to<br />
this theorist, then, the identity and location facts are these: A=B=C=D=Tom, and at each<br />
time Tom is entirely in the kitchen, and also entirely in the living room.<br />
Another potential view is one on which objects persist through time in virtue <strong>of</strong><br />
having proper temporal parts at each time, but they can sometimes be said to be entirely<br />
present in more than one spatial location at a time, in virtue <strong>of</strong> having a temporal part that<br />
is multi-located. The identity and location facts would thus be these: Taking Tom1 and<br />
Tom2 to be distinct temporal parts <strong>of</strong> Tom, A=C=Tom1, and B=D=Tom2. So, because<br />
we take the question <strong>of</strong> where Tom is located at each time to be a question <strong>of</strong> where<br />
Tom’s temporal parts are at each time, we have the result that Tom is entirely at L1 and<br />
L3 at T1, and also that Tom is entirely at L2 and L4 at T2.<br />
Some theorists will think I’ve misconstrued the notion <strong>of</strong> temporal part in taking<br />
it to be even conceivable that temporal parts can be multi-located. Some <strong>of</strong> these theorists<br />
will have in mind an account <strong>of</strong> temporal parts like the one Sider presents 7 , which is<br />
somewhat like:<br />
• ‘x is a temporal part <strong>of</strong> y at t’ =df x is a part <strong>of</strong> y at t, x exists only at t, and x overlaps<br />
everything that is a part <strong>of</strong> y at t.<br />
I do not think, however, that it is necessary to preclude multi-location <strong>of</strong> temporal parts<br />
by definition. This (and the tension between four-dimensionalism and multi-location <strong>of</strong><br />
temporal parts) is discussed elsewhere. [reference or addition <strong>of</strong> appendix?] Further, I<br />
Since neither <strong>of</strong> these options is acceptable, we should reject this account <strong>of</strong> entire-presence, or whole<br />
location. (Sider takes the view entailed on the second horn <strong>of</strong> the dilemma to be stronger than what I’ve<br />
presented here, but I believe Mereological Neighbourliness is all that really follows.)<br />
7 2001, pp. 59-60.
elieve that a more liberal notion <strong>of</strong> temporal parts will be necessary if the temporal parts<br />
theorist wishes to make sense <strong>of</strong> multi-location <strong>of</strong> an object (any object!) at times. This is<br />
because the locations <strong>of</strong> temporal parts are supposed to be what it is in virtue <strong>of</strong> that the<br />
location facts about the whole object obtain in virtue <strong>of</strong>. If an object is multilocated at a<br />
time, and we cannot appeal to multilocation <strong>of</strong> temporal parts <strong>of</strong> the object, and we can’t<br />
appeal to the existence <strong>of</strong> more than one temporal part at a time (which would also be in<br />
contradiction with the application <strong>of</strong> the above definition), then it seems we have nothing<br />
left to explain the location fact in terms <strong>of</strong>. In light <strong>of</strong> these (and other) considerations, I<br />
propose the following definition <strong>of</strong> ‘temporal part’:<br />
• ‘x is a temporal part <strong>of</strong> y’ =df there exists a time, t, and a region, r, such that (i) x is<br />
wholly located at a subregion <strong>of</strong> the intersection <strong>of</strong> r and t, (ii) y is wholly located<br />
at r, (iii) x is part <strong>of</strong> y, and (iv) for all z, if ((a) z is wholly located at a subregion <strong>of</strong><br />
r, (b) z is present at t, and (c) z is a part <strong>of</strong> y), then x overlaps z.<br />
The crucial point to take from the preceeding discussion in this section is simply<br />
this: there are a variety <strong>of</strong> ways to answer the question <strong>of</strong> what the location and identity<br />
facts in my two cases are. However, each <strong>of</strong> these determines one <strong>of</strong> two answers to the<br />
question <strong>of</strong> where Tom is at each time.<br />
Further, it should be noted that the answers we gave to what the location and<br />
identity facts are in the cases will not depend on any <strong>of</strong> the facts which differ between the<br />
cases as I originally described them. Any difference in our answers about location or<br />
identity facts between the cases would therefore be arbitrary. Thus, we cannot give a<br />
different answer to the question <strong>of</strong> where Tom is at each time without invoking a charge<br />
<strong>of</strong> arbitrariness.<br />
Finally, because the <strong>At</strong>-<strong>At</strong> account is blind to anything but the facts about identity<br />
and the location <strong>of</strong> the object whose motion is in question, the <strong>At</strong>-<strong>At</strong> account will entail<br />
that the same motion facts hold in the Dull Case and in the Exciting Case. But this runs in<br />
direct contradiction to our intuitions about motion in the cases. Thus, because we<br />
intuitively took one case to involve motion and the other to not, the <strong>At</strong>-<strong>At</strong> account will be<br />
too liberal or too conservative in its attribution <strong>of</strong> motion in the cases. Which it is will<br />
depend on which <strong>of</strong> the two answers we give to the question <strong>of</strong> where Tom is at each<br />
time.
If we claim that Tom is not entirely at any <strong>of</strong> L1-L4, but is instead at the fusion <strong>of</strong><br />
L1 and L3 at T1, and at L2 and L4 at T2, then the <strong>At</strong>-<strong>At</strong> account will be too restrictive<br />
because it will entail that Tom is not moving in the Exciting Case. In that case, as in the<br />
Dull Case, Tom is at the same region at each time (the fusion <strong>of</strong> R1 and R2), and thus<br />
does not meet the conditions for motion. When construed this way, my Exciting Case<br />
amounts to nothing more than a spinning disk case. These have been thoroughly covered<br />
in the literature, and so I will not discuss them in the body <strong>of</strong> this paper. However,<br />
because I find it puzzling that so many proponents <strong>of</strong> the <strong>At</strong>-<strong>At</strong> account set these cases<br />
quickly aside, I <strong>of</strong>fer my thoughts on some <strong>of</strong> their quick responses to the cases in<br />
Appendix 1, Spinning Disks: Is There Still Motion?<br />
For the remainder <strong>of</strong> this paper, I will focus on what happens if we claim that<br />
Tom is entirely at each <strong>of</strong> L1-L4. The most immediate result seems to be that the <strong>At</strong>-<strong>At</strong><br />
account will be too liberal because it will entail that Tom moves in the Dull Case, in<br />
virtue <strong>of</strong> his being entirely at one space at a time, and at a distinct space at a distinct time.<br />
(And it is worth noting that any case <strong>of</strong> a persisting object being multi-located at some<br />
time is sufficient to raise this problem for the <strong>At</strong>-<strong>At</strong> account.) <strong>It</strong> doesn’t matter to the <strong>At</strong>-<br />
<strong>At</strong> account that Tom simply sat in both regions as time passed. However, our intuitions<br />
seem to care, if not about that, then about something else that the <strong>At</strong>-<strong>At</strong> account has<br />
obviously missed.<br />
4. Replies to the Liberality Objection<br />
There are several ways to respond to this case, the first being to simply deny the<br />
possibility <strong>of</strong> the Dull Case. I have several worries about this. My first worry is that, for<br />
this to help, one would have to deny the possibility <strong>of</strong> something persisting which is, at<br />
some point in its existence, multi-located at a time (and which, intuitively, doesn’t move<br />
between the times at which it is multi-located and some other time), since that’s what<br />
causes the difficulty for the <strong>At</strong>-<strong>At</strong> account.<br />
That said, there are several ways to attempt to deny the possibility <strong>of</strong> my Dull<br />
Case as construed as involving multi-location. One way is to deny the possibility <strong>of</strong><br />
survivable timetravel. That is to say, no object can survive showing up at a time “more<br />
than once”. So in my cases above, the object would go out <strong>of</strong> existence as soon as it went<br />
back in time -- meaning that in the Dull Case, A=B and (perhaps) C=D, but B≠C. Which<br />
<strong>of</strong> 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<br />
intuition in cases where objects appear “more than once” at the same time due to<br />
spacetime being curved back on itself in appropriate ways. If Tom lives in a four-<br />
dimensional version <strong>of</strong> a möbius strip, and its temporal extension is shorter than his life,<br />
then we can imagine him at 63 visiting himself at 36. Nothing about this case seems<br />
particularly problematic. However, to save the <strong>At</strong>-<strong>At</strong> account via appeal to this response,<br />
we would have to claim that there’s something about this case which excludes it from the<br />
realm <strong>of</strong> the possible.<br />
Another way to try to deny the possibility <strong>of</strong> my case is to say that we are<br />
mistaken in thinking that the identity and location facts are even possibly as they would<br />
need to be to give the result that the <strong>At</strong>-<strong>At</strong> account entails there is motion in the Dull<br />
Case. That is, deny the possibility <strong>of</strong> multi-location in time-travel cases. However, this<br />
response comes with a cost. There is some work that multi-location does in explaining<br />
our intuitions about time-travel cases. For instance: When 63 year old Tom visits 36 year<br />
old Tom, we can ask some questions about Tom at that time. Where is Tom? <strong>It</strong> seems<br />
he’s entirely in the living room (after all, none <strong>of</strong> him seems to be missing from there),<br />
and also entirely in the kitchen. What is Tom shaped like? Plausibly, he’s person-shaped<br />
in the kitchen, and person-shaped in the living room. <strong>It</strong> definitely doesn’t seem that in<br />
virtue <strong>of</strong> time-travelling, he’s suddenly acquired an odd, noticeably gappy shape (that <strong>of</strong><br />
a fusion <strong>of</strong> two people). Finally, what are we to say about Tom’s weight? This, just like<br />
his shape, is plausibly region-relative. Suppose that Tom got dense in his old age (since<br />
my previous example requires him to fit into a region previously occupied by 36 year old<br />
Tom, I can’t have him gaining volume!), and at 63 he weighs more than he did at 36.<br />
How much does Tom weigh? <strong>It</strong> seems appropriate to say: more than he’d like to in the<br />
kitchen, and exactly what he’d like to in the living room. But it’s definitely not the case<br />
(at least, intuitively) that Tom’s weight almost doubled as a result <strong>of</strong> time-travelling to<br />
visit himself at 36. 8 How much does Tom weigh? <strong>It</strong> seems appropriate to say: more than<br />
he’d like to in the kitchen, and exactly what he’d like to in the living room. But it’s<br />
definitely not the case (at least, intuitively) that Tom’s weight almost doubled as a result<br />
<strong>of</strong> time-travelling to visit himself at 36.8 However, this, and the other implausible claims<br />
about shape and location (and a variety <strong>of</strong> other properties), is exactly what the<br />
8 These worries are not new, and have been raised as problems for the four-dimensionalist who accepts a<br />
notion <strong>of</strong> temporal parts like the one Sider puts forward.
pertension theorist (who must say Tom isn’t multilocated even though he time travelled)<br />
will have to claim.<br />
Finally, let us suppose that this interpretation <strong>of</strong> the identity and location facts in<br />
my Dull Case isn’t possible. Is this enough to ensure the safety <strong>of</strong> the <strong>At</strong>-<strong>At</strong> account from<br />
the threat <strong>of</strong> this latest worry? Sure, if we take the <strong>At</strong>-<strong>At</strong> account to be a mere listing <strong>of</strong><br />
necessary and sufficient conditions for motion. However, this isn’t what makes the<br />
account so appealing. People like it because it’s a claim about what motion consists <strong>of</strong>.<br />
<strong>It</strong>’s an analysis, maybe even a definition. But once that’s what we’re trying to defend,<br />
cases like mine become relevant once again, regardless <strong>of</strong> whether they’re possible – they<br />
need only be conceivable. If my case can’t obtain, it’s an interesting Metaphysical fact<br />
that it can’t, and not one that has to do with the <strong>At</strong>-<strong>At</strong> account. So when evaluating the<br />
account, it is fruitful (and, I think, necessary) to look at some counterpossibles and ask<br />
what our predications <strong>of</strong> motion would be in those cases. Do our predications <strong>of</strong> motion<br />
square with the Philosopher’s account <strong>of</strong> what motion consists <strong>of</strong>? If not, and if we take<br />
ourselves to be giving an account <strong>of</strong> a natural phenomenon that people grasp and are able<br />
to correctly identify, we have a problem and need to make some difficult choices. 9<br />
What else might someone say in responding to the case? There are two other<br />
options I take to be salient: (i) index motion to regions, or (ii) add a requirement for<br />
motion that the entity in question at a region at a time stands in certain (e.g., perhaps<br />
immanent causal) relations to itself at another region at a distinct time. I find the first<br />
option a bit compelling, and would be happy if it worked. The idea seems independently<br />
plausible: imagine a case where Tom in the kitchen is sitting still, and Tom in the living<br />
room is jumping on the furniture. We want to ask: Is Tom moving? The intuitive answer<br />
seems to be: he’s moving in the living room, but he’s not moving in the kitchen.<br />
However, I worry about how to pick out the relevant regions to index Tom’s motion to,<br />
without appealing to anything spooky. The second option also requires invoking<br />
something mysterious, and though it may be required due to other puzzles in<br />
Metaphysics, it will disappoint the <strong>At</strong>-<strong>At</strong> theorist to admit that motion consists partly in<br />
the obtaining <strong>of</strong> such spookiness. (I’m including appeals to personal time in this second<br />
9 Of course, there are worries about intuitions failing in strange enough cases, or <strong>of</strong> our everyday account<br />
being simply not made for extremely exotic cases, so that people are no longer able to correctly apply their<br />
concepts; however, I think these worries aren’t relevant here, as my case is quite easy to grasp, and in fact<br />
is relevantly similar to time-travel scenarios brought to people’s attention in pop culture.
category, because I do not believe that the notion <strong>of</strong> personal time can be cashed out just<br />
via appeal to location facts and identity facts; we will need to appeal to something more,<br />
like genidentity relations, immanent causation, or in-virtue-<strong>of</strong> dependence <strong>of</strong> properties<br />
had, on the having <strong>of</strong> particular previous properties.) Regardless <strong>of</strong> which option is<br />
chosen, the <strong>At</strong>-<strong>At</strong> theorist is faced with costs, and will need to revise the original account.<br />
As stated in its original formulation, the account is simply inadequate.<br />
Appendix 1, Spinning Disks: Is There Still Motion?<br />
Spinning disk cases threaten to show that the <strong>At</strong>-<strong>At</strong> account is too restrictive about when<br />
motion occurs. This problem, I take it, arises in virtue <strong>of</strong> the very feature that makes the<br />
account so attractive: the elegance with which it reduces something seemingly complex<br />
to a simple phenomenon we can all understand. No need to invoke something mysterious<br />
or complicated (like genidentity or irreducible vector quantities). However, the world<br />
doesn’t seem to be as simple as we’d like.<br />
Consider a spinning disk. 10 <strong>It</strong>’s in region R at T1, and R at T2. According to the<br />
<strong>At</strong>-<strong>At</strong> account, this disk isn’t moving. But intuitively it is moving – it’s turning within R.<br />
So why isn’t this case seen as a counterexample to the <strong>At</strong>-<strong>At</strong> account?<br />
There’s a sophisticated discussion <strong>of</strong> this case in the literature, and I won’t<br />
attempt to cover it here. However, in discussing the case with others, I’ve noticed a<br />
widespread (though not universal) lack <strong>of</strong> concern about the spinning disk case (at least,<br />
qua counterexample to the <strong>At</strong>-<strong>At</strong> account). From what I can tell, this is due to an<br />
assumption that there’s a straightforward amendment that allows us to still capture the<br />
spirit <strong>of</strong> the <strong>At</strong>-<strong>At</strong> account. I’ll now turn to this attempted quick-fix, argue that neither it<br />
nor other attempts like it succeed in producing the right results in spinning disk cases, and<br />
thereby, hopefully, provide cause to take the spinning-disk case seriously as a threat to<br />
the <strong>At</strong>-<strong>At</strong> account <strong>of</strong> motion (as, incidentally, it is seen in the literature on this topic).<br />
In observing the spinning disk case, what makes people think there’s an easy<br />
solution for the <strong>At</strong>-<strong>At</strong> theorist? There’s an intuitive response: Look, it’s said, the object is<br />
moving in virtue <strong>of</strong> all <strong>of</strong> its parts moving. This suggests the following account:<br />
10 The <strong>of</strong>t discussed spinning disk case was first presented by Saul Kripke, and is discussed in Sider 2001,<br />
pp. 224-236.
• (First Revised <strong>At</strong>-<strong>At</strong>) Necessarily, for any x, x is in motion iff there exist some ys such<br />
that x is wholly decomposable 11 into the ys, and for any one <strong>of</strong> the ys, yn there<br />
exists spatial regions s1 and s2, and times t1 and t2, such that s1 is distinct from<br />
s2, t1 is distinct from t2, and yn is at s1 at t1, and at s2 at t2.<br />
That is just to say: An object moves just in case it’s wholly decomposable into parts each<br />
<strong>of</strong> which are moving (in the way specified by the original <strong>At</strong>-<strong>At</strong> account).<br />
Unfortunately, this is an inadequate revision. For one thing, we may worry<br />
whether things do move whenever all <strong>of</strong> their proper parts (on some decomposition) are<br />
moving. Consider an example presented by James Blackmon in “A Dilemma for a Picture<br />
<strong>of</strong> Motion”: Suppose that (i) every region is such that it’s possible for something to<br />
wholly occupy it, 12 (ii) every material object is such that for any occupyable subregion <strong>of</strong><br />
the region it occupies, it has a part wholly located in that subregion, and (iii) for any two<br />
objects, there exists a fusion <strong>of</strong> those objects. 13 Now consider an apparently stationary<br />
disk. This disk is wholly decomposable into proper parts that each keep their locations<br />
throughout the disk’s existence – a fine candidate for staying still. However, the disk is<br />
also entirely decomposable into parts that move – for consider the part that’s the fusion <strong>of</strong><br />
the temporal part <strong>of</strong> sitting-part-1 at T1, and <strong>of</strong> the temporal part <strong>of</strong> sitting-part-2<br />
(directly to the left <strong>of</strong> sitting-part-1) at T2, and . . . This fusion will move, in virtue <strong>of</strong><br />
having temporal parts at different regions at different times. Further, the disk can be<br />
completely decomposed into fusions like this one. So, on the above account, it moves<br />
after all!<br />
I don’t know what to say about the above case. <strong>It</strong> seems clear to me that objects<br />
can be wholly decomposable into moving parts, without themselves moving. 14 For<br />
instance, suppose an object occupies the same region throughout my existence, but each<br />
one <strong>of</strong> the subatomic particles that composes it spins on its axis. Even if we didn’t think<br />
the particles were moving, they’re all (let’s say) composed <strong>of</strong> proper parts that are<br />
moving, and so the original object is wholly decomposable into moving parts. But this<br />
isn’t a way for an object to move! Still, I think there are also cases where objects do<br />
11<br />
For the purposes <strong>of</strong> this paper: x is wholly decomposable into the ys iff x is a fusion <strong>of</strong> the ys, and no two<br />
ys overlap one-another.<br />
12<br />
I.e., suppose Hud Hudson’s Liberal View <strong>of</strong> Receptacles is true (Hudson 2001).<br />
13<br />
For Kit Fine scholars: I’m requiring mere-sum aggregates.<br />
14<br />
Hudson made this claim in a commentary on Blackmon’s paper.
move in virtue <strong>of</strong> all <strong>of</strong> their proper parts moving, and I don’t know which kind <strong>of</strong> case<br />
Blackmon’s spinning disk example should be characterised as.<br />
However, taking a stand on that issue isn’t essential for having cause to reject the<br />
above First Revised <strong>At</strong>-<strong>At</strong> account. Here’s another worry: assume now that spacetime is<br />
pointy, and that the spinning disk is continuous and wholly decomposable into point-<br />
sized parts. When it spins, there will be a point in its centre which stays put. This isn’t a<br />
worry for someone who accepts the Doctrine <strong>of</strong> Arbitrary Undetached Parts along with<br />
the Liberal View <strong>of</strong> Receptacles; it doesn’t matter that there’s a decomposition <strong>of</strong> the<br />
object such that one <strong>of</strong> the parts in that decomposition isn’t moving – all we need is one<br />
decomposition that meets the above requirement to ensure the disk is spinning, and that<br />
won’t be hard to find. But suppose that we don’t accept DAUP and the LVR. Suppose, in<br />
fact, that we think the disk has only two decompositions: its decomposition into its<br />
improper part, and its decomposition into its point-sized parts. Now we’ll have the result<br />
that the disk is not moving. However, this is unacceptable: whether the disk is moving or<br />
not shouldn’t depend on whether we posit, e.g., disk-halves and disk-quarters in addition<br />
to the point-sized parts <strong>of</strong> the disk. Our Mereological commitments shouldn’t have<br />
implications for whether the whole thing is moving.<br />
What account could be given that would produce the right results regardless <strong>of</strong><br />
our Mereological commitments? I really don’t know: we might say that a thing moves<br />
just in case it’s wholly decomposable into parts most <strong>of</strong> which are moving (a second<br />
attempt at revising the account), but now we have to wonder how many moving parts are<br />
enough, and any line we draw will be arbitrary.<br />
One might respond by claiming that we should re-evaluate our reading <strong>of</strong> the <strong>At</strong>-<br />
<strong>At</strong> account, and use an interpretation <strong>of</strong> ‘at’ similar to the first one considered in section<br />
1.1. On this new reading, we take the <strong>At</strong>-<strong>At</strong> account to be claiming this:<br />
• (Reinterpreted <strong>At</strong>-<strong>At</strong>) Necessarily, for any x, x is in motion iff there exist some ys such<br />
that x is wholly decomposable into the ys, and for at least one <strong>of</strong> the ys, yn there<br />
existsspatial regions s1 and s2, and times t1 and t2, such that s1 is distinct from<br />
s2, t1 is distinct from t2, and yn is wholly at s1 at t1, and wholly at s2 at t2.<br />
That is, an object moves iff it has a part that moves (<strong>At</strong>-<strong>At</strong> style). This avoids the<br />
arbitrariness worry, but my worry about the first attempt to amend the account (claiming<br />
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<br />
posits less restrictions on when motion occurs than does the first. (The worry applies to<br />
the second attempt as well.) Also, consider a variant on the spinning disk case that’s been<br />
presented in the literature. The disk in this case is an extended simple; it has no parts that<br />
are moving <strong>At</strong>-<strong>At</strong> style. However, if we were to put a mark on it, we would observe the<br />
marked area <strong>of</strong> the disk rotating (though the disk wouldn’t have a proper part<br />
corresponding to that area). This case tells against all three attempts to provide a quick-<br />
fix for the <strong>At</strong>-<strong>At</strong> account. Nevertheless, if we don’t take this last case seriously (for<br />
instance, if we refuse to grant that extended simples are conceivable, or we want to bite<br />
the bullet in this case), the third attempted quick-fix may still be a live option. However,<br />
the account under this reading is still vulnerable to the worries I raise in the next section.<br />
Hopefully this section has provided motivation for viewing the spinning disk case<br />
as a challenging puzzle for anyone who accepts the <strong>At</strong>-<strong>At</strong> account.<br />
Appendix 2, Where To Go From Here?<br />
The <strong>At</strong>-<strong>At</strong> account is on shaky ground. If we posit even the conceivability <strong>of</strong> persisting<br />
multilocation, it’s not clear what we should say about motion. To make it even harder to<br />
give a clear, principled listing <strong>of</strong> necessary and sufficient conditions, consider the<br />
following series <strong>of</strong> cases.<br />
The First Case:<br />
The world consists <strong>of</strong> an object which is entirely located at R1 at T1, and entirely located<br />
at R2 at T2 (or which has temporal parts entirely located at those regions at the proper<br />
times)
Intuitively, there’s discontinuous motion in this case. And the <strong>At</strong>-<strong>At</strong> account<br />
accords with our intuitions: the object moves in virtue <strong>of</strong> being entirely at R1 at T1, and<br />
at R2 at T2. 15<br />
The Second Case:<br />
Just as in the last world, this world consists simply <strong>of</strong> an object entirely (or with a<br />
temporal part entirely) located at R1, and also entirely (or with a temporal part entirely)<br />
located at R2. But here the object persists in each region for a longer duration than in the<br />
previous world: the object is located at R1 at T1, and also at T2, and at R2 at T2, and also<br />
at T3. To make it seem more intuitive, imagine: I’m preparing to time travel, and I leave<br />
R1 exactly at T2. However, since I’m not good with machinery, I don’t end up going to<br />
the past or future at all – I simply “reappear” at T2, though in a different region (my<br />
machine won’t let me occupy a region that’s already filled). So I get to live the same<br />
instant twice. Discouraged, I stay put (until T3). The world looks like this:<br />
Has the object moved? Intuitions say it has: this case is just like the first, except<br />
that the object stays put in each region a bit longer. For another way to look at it: Cover<br />
up what’s happening at T2. Comparing just T1 and T3, we should definitely say there’s<br />
motion (at least, if that’s what we say in the first case). How could what happens at T2<br />
make a difference as to whether the object moves? The <strong>At</strong>-<strong>At</strong> account agrees with us.<br />
15 We could construct a similar example with the object entirely located at each <strong>of</strong> the relevant regions<br />
between R1 at T1 and R2 at T2, to achieve continuity, if we wanted. Or, another way <strong>of</strong> achieving a similar<br />
end: make the times extended, and one <strong>of</strong> them (say, T1) open while the other’s closed, and let R1 and R2<br />
be in contact.
The Third Case:<br />
This world is just like the one in the second case, except that the only times are T1 and<br />
T2. So the object is entirely (or has a temporal part entirely) at R1 at T1, and is also<br />
entirely (or has a temporal part entirely) at R1 at T2, and is also entirely (or has a<br />
temporal part entirely) at R2 at T2. To give a more intuitive picture <strong>of</strong> what’s happening,<br />
it’s just as in the last case, except Space (unbeknownst to me) the instant I get to live<br />
twice is actually the last instant <strong>of</strong> my existence. The world looks like this:<br />
Is there motion? Intuitions begin to waver here and many people are initially<br />
compelled to firmly deny that there is motion in this case. Still, once we note that the<br />
only difference between this case and the previous one is how long the object remains at<br />
R2, it’s much more difficult to deny that the object in this world moves. After all, surely<br />
it doesn’t make a difference to the motion <strong>of</strong> the object, whether it simply persists in a<br />
region for an extended period <strong>of</strong> time or not. Also, if we ignore R1 at T2, we have a case<br />
just like the first. The <strong>At</strong>-<strong>At</strong> account, once again, agrees that there is motion. 16<br />
The Fourth Case:<br />
This final world is just like the last, except that it’s “flipped upside-down” in the temporal<br />
dimension. So an object is entirely (or has a temporal part entirely) at R1 at T1, and also<br />
is entirely (or has a temporal part entirely) at R2 at T1, and finally also is entirely (or has<br />
a temporal part entirely) at R1 at T2 (but nothing occupies R2 at T2). Once again, to alter<br />
16 There is, however, something important that sets this case apart from the others: in the first and second<br />
cases, even one who thought temporal parts couldn’t be multilocated at a time would have accepted motion.<br />
They would have thought, e.g., that in the second case the object was first completely in R1, then was<br />
wholly located at the fusion <strong>of</strong> R1 and R2, and finally was simply in R2. But in this case, at least for the<br />
perdurance-theorist (i.e., one who thinks that any object that’s extended in a dimension is extended in virtue<br />
<strong>of</strong> having proper parts at each <strong>of</strong> the subregions, rather than by being multilocated across the dimension),<br />
the object never wholly occupies a region that doesn’t overlap with one the region wholly occupied<br />
previously. Accepting perdurantism won’t change what the <strong>At</strong>-<strong>At</strong> account says about the case, but it does<br />
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<br />
again, I’ve made a mistake with the machinery, one that takes me out <strong>of</strong> existence just<br />
after I get to the past. The world looks like this:<br />
Is there motion? As far as I can tell, the considerations weighing in on this case<br />
are just like those in the previous one, but oddly enough, people’s intuitions sometimes<br />
differ across these cases. Regardless, the <strong>At</strong>-<strong>At</strong> account gives the same result: motion all<br />
the way. 17<br />
Works Cited<br />
Blackmon, James. Unpublished manuscript. “A Dilemma for a Picture <strong>of</strong> Motion”,<br />
<br />
Hudson, Hud. 2006. The Metaphysics <strong>of</strong> Hyperspace, (Oxford <strong>University</strong> Press: Oxford).<br />
Hudson, Hud. 2002. “The Liberal View <strong>of</strong> Receptacles”, Australasian Journal <strong>of</strong><br />
Philosophy, vol. 80: pp. 432-439.<br />
Hudson, Hud. 2001. A Material Metaphysics <strong>of</strong> the Human Person, (Cornell <strong>University</strong><br />
Press: <strong>It</strong>haca).<br />
Russell, Bertrand. 1903. The Principles <strong>of</strong> Mathematics, (Cambridge: Cambridge<br />
<strong>University</strong> Press).<br />
Sider, Theodore. 2001. Four-Dimensionalism: An Ontology <strong>of</strong> Persistence and Time,<br />
(Oxford: Oxford <strong>University</strong> Press).<br />
van Inwagen. 1981. “The Doctrine <strong>of</strong> Arbitrary Undetached Parts”, Pacific Philosophical<br />
Quarterly 62: pp. 123-137.<br />
17 Thanks to Frank Arntzenius, Kit Fine, Cody Gilmore, Hud Hudson and Daniel Nolan for helpful<br />
discussion about these topics.