### Motion and Temporal Density

Today I was introduced to an interesting problem. The following five theses are jointly inconsistent:

1. Some object changes from being at rest to being in motion.

2. Any object which changes from being at rest to being in motion has a first instant during which it is in motion.

3. Any object which changes from being at rest to being in motion has a last instant during which it is at rest.

4. No object is both at rest and in motion during the same instant.

5. For any object and any instant, that object is either in motion during that instant or at rest during that instant.

6. Time is dense (for any instants t1 and t3 where t3 is later than t1, there is an instant t2 that is earlier than t3 and later than t1)

Let's prove that these six statements are inconsistent. (1) says that there is an object which changes from being at rest to being in motion. Call that object 'O'. According to (2) and (3), there is a last instant during which O is at rest and a first instant during which O is in motion. Call the last instant during which O is at rest 'Tr' and call the first instant during which O is in motion 'Tm'. So, Tr is the last moment during which O is at rest and Tm is the first moment during which O is in motion. Moreover, according to (4), Tr is not identical to Tm. So, according to (6), there is an instant, Tx, later than Tr but earlier than Tm. By (5), The object is either at rest or in motion during Tx. If it is at rest and Tx is later than Tr, then Tr is not the last instant during which it is at rest. But, since Tr is the last moment during which O is at rest, it follows that O is not at rest during Tx. So, O is in motion during Tx. But, if O is in motion during Tx and tx is earlier than Tm, then Tm is not the first moment during which it is in motion. But, Tm is the first moment during which O is in motion. So, O is not in motion during Tx. We have arrived at a contradiction.

All of these claims seem plausible. I should admit that there is an a strong claim implied by (5). That claim is that for any object and any time that object exists during that time. Although this is very strong, we can weaken (5) so that it only implicitly implies that an object does not go out of existence between its last moment of rest and its first moment of motion. But that claim seems plausible enough. It would be very strange, for example, if an object briefly went out of existence every time it changed from being at rest to being in motion. so, I am going to keep (5) the way it is in order to avoid making the argument too complicated.

My guess is that the weakest claims are (2) and (3). The friend of dense time should give up on one of these. But, it seems that to give up on one rather than the other would be arbitrary. at first I thought that there are probably possible worlds where (2) is true and (3) false and there are other possible worlds where (3) is true and (2) is false. But, now I am not sure about this solution. Now, I am beginning to think this is a problem of indeterminacy.

It seems to me that some version of the At-At theory of motion is true. There are various problems with simple formulations of the At-At theory of motion, but we can ignore those problems for the purposes of this discussion. So, let's consider the following Simple At-At view:

(AT-AT) An object O is in motion during an extended interval T iff O is located at one region during one instant of T and O is located at a different region during a different instant of T.

Now instantaneous motion is a derivative notion that can be spelled out in one of two ways:

(IM1) An object O is in instantaneous motion during an instant t iff there is an extended interval T such that t is an instant in T and O is in motion during T and for any extended sub-interval of T, O is in motion during that sub-interval as well.

(IM2) An object O is in instantaneous motion during an instant t iff there is an extended open interval T such that t is an instant in T and O is in motion during T and for any extended sub-interval of T, O is in motion during that sub-interval as well.

One difference between these (IM1) and (IM2) is that according to (IM1) any object that changes from being at rest to being in motion has a first instant of being in motion whereas according to (IM2) any such object has a last instant of being at rest. Let me show each of these consequences in turn.

First, let's focus on (IM1) and consider an object that changes from being at rest to being in motion. Suppose for reductio that that object has no first moment of motion. It follows that it has a last instant during which it is at rest. Call that instant 't'. If the object has a last moment during which it is at rest, then there will be an open interval during which it is in motion. Moreover, that open interval during which the object is in motion will have t as a boundary point. Call that open interval 'T'. The union of T and t is an interval of time. Moreover, the object in question is in motion during the union of T and t. Moreover, during any extended sub-interval of the union of T and t, it is in motion. So, by (IM1) the object is in instantaneous motion during t. But, if it is in motion during t, then it is not at rest during t. but, we already said that it is at rest during t. So, we have arrived at a contradiction. It follows that if (IM1) is true, then any object which changes from being at rest to being in motion will have a first instant of motion.

Now, let's focus on (IM2). I said that (IM2) implies that any object that changes from being at rest to being in motion will have a last insant of being at rest. To show that this is true, let's suppose that (IM2) is true and suppose that there is an object, O, that changes from being at rest to being in motion. Now consider an arbitrary instant t during which O is in instantaneous motion. I will show that there is a time before t during which O is in motion as well. Since O is in instantaneous motion during t, it follows by (IM2) that both (A) there is an extended open interval T such that t is an instant in T and O is in motion during T and (B) for any extended sub-interval of T, O is in motion during that sub-interval as well. Since, by (A), there is an open interval that includes t during which O is in motion, it follows that there is an open sub-interval of that open interval which is before t. but, by (B), O is in motion during that open sub-interval that is before t. So, O is in motion before t. So, t is not the first instant during which O is in motion. But, since t was an arbitrarily chosen instant, it follows that there cannot be a first instant during which O is in motion. So, there must be a last moment during which O is at rest. Since O was arbitrarily chosen as well, we can conclude that any object that changes from being at rest to being in motion will have a last insant of being at rest. So, (IM2) implies that any object that changes from being at rest to being in motion will have a last insant of being at rest.

Now, my current belief is that use of 'is in instantaneous motion' is indeterminate between (IM1) and (IM2). So, currently, I think that either (2) or (3) from our original puzzle is false and necessarily so. But, it is indeterminate which is false.

Quick Side Note:

You might think that any legitimate precisification of our language has to obey the following constraint: any sentence of the form 'if O is in motion during T then for any time in T, O is in motion during that time as well'. Moreover, you might think that if this is right, then we have some reason to prefer (IM1) over (IM2).

Although this sounds plausible at first, It seems to me that in addition to the following traditional At-At view:

(AT-AT) An object O is in motion during an extended interval T iff O is located at one region during one instant of T and O is located at a different region during a different instant of T.

there is an alternative, open interval At-At view:

(Open At-At) An object O is in motion during an extended interval T iff T is an open interval and O is located at one region during one instant of T and O is located at a different region during a different instant of T.

It seems to me that our use of 'is in motion' is probably indeterminate between (At-At) and (open At-At). But, that also supports the claim that it is indeterminate which of (2) or (3) is false. So, the suggestion above just seems to push the solution back a level.

1. Some object changes from being at rest to being in motion.

2. Any object which changes from being at rest to being in motion has a first instant during which it is in motion.

3. Any object which changes from being at rest to being in motion has a last instant during which it is at rest.

4. No object is both at rest and in motion during the same instant.

5. For any object and any instant, that object is either in motion during that instant or at rest during that instant.

6. Time is dense (for any instants t1 and t3 where t3 is later than t1, there is an instant t2 that is earlier than t3 and later than t1)

Let's prove that these six statements are inconsistent. (1) says that there is an object which changes from being at rest to being in motion. Call that object 'O'. According to (2) and (3), there is a last instant during which O is at rest and a first instant during which O is in motion. Call the last instant during which O is at rest 'Tr' and call the first instant during which O is in motion 'Tm'. So, Tr is the last moment during which O is at rest and Tm is the first moment during which O is in motion. Moreover, according to (4), Tr is not identical to Tm. So, according to (6), there is an instant, Tx, later than Tr but earlier than Tm. By (5), The object is either at rest or in motion during Tx. If it is at rest and Tx is later than Tr, then Tr is not the last instant during which it is at rest. But, since Tr is the last moment during which O is at rest, it follows that O is not at rest during Tx. So, O is in motion during Tx. But, if O is in motion during Tx and tx is earlier than Tm, then Tm is not the first moment during which it is in motion. But, Tm is the first moment during which O is in motion. So, O is not in motion during Tx. We have arrived at a contradiction.

All of these claims seem plausible. I should admit that there is an a strong claim implied by (5). That claim is that for any object and any time that object exists during that time. Although this is very strong, we can weaken (5) so that it only implicitly implies that an object does not go out of existence between its last moment of rest and its first moment of motion. But that claim seems plausible enough. It would be very strange, for example, if an object briefly went out of existence every time it changed from being at rest to being in motion. so, I am going to keep (5) the way it is in order to avoid making the argument too complicated.

My guess is that the weakest claims are (2) and (3). The friend of dense time should give up on one of these. But, it seems that to give up on one rather than the other would be arbitrary. at first I thought that there are probably possible worlds where (2) is true and (3) false and there are other possible worlds where (3) is true and (2) is false. But, now I am not sure about this solution. Now, I am beginning to think this is a problem of indeterminacy.

It seems to me that some version of the At-At theory of motion is true. There are various problems with simple formulations of the At-At theory of motion, but we can ignore those problems for the purposes of this discussion. So, let's consider the following Simple At-At view:

(AT-AT) An object O is in motion during an extended interval T iff O is located at one region during one instant of T and O is located at a different region during a different instant of T.

Now instantaneous motion is a derivative notion that can be spelled out in one of two ways:

(IM1) An object O is in instantaneous motion during an instant t iff there is an extended interval T such that t is an instant in T and O is in motion during T and for any extended sub-interval of T, O is in motion during that sub-interval as well.

(IM2) An object O is in instantaneous motion during an instant t iff there is an extended open interval T such that t is an instant in T and O is in motion during T and for any extended sub-interval of T, O is in motion during that sub-interval as well.

One difference between these (IM1) and (IM2) is that according to (IM1) any object that changes from being at rest to being in motion has a first instant of being in motion whereas according to (IM2) any such object has a last instant of being at rest. Let me show each of these consequences in turn.

First, let's focus on (IM1) and consider an object that changes from being at rest to being in motion. Suppose for reductio that that object has no first moment of motion. It follows that it has a last instant during which it is at rest. Call that instant 't'. If the object has a last moment during which it is at rest, then there will be an open interval during which it is in motion. Moreover, that open interval during which the object is in motion will have t as a boundary point. Call that open interval 'T'. The union of T and t is an interval of time. Moreover, the object in question is in motion during the union of T and t. Moreover, during any extended sub-interval of the union of T and t, it is in motion. So, by (IM1) the object is in instantaneous motion during t. But, if it is in motion during t, then it is not at rest during t. but, we already said that it is at rest during t. So, we have arrived at a contradiction. It follows that if (IM1) is true, then any object which changes from being at rest to being in motion will have a first instant of motion.

Now, let's focus on (IM2). I said that (IM2) implies that any object that changes from being at rest to being in motion will have a last insant of being at rest. To show that this is true, let's suppose that (IM2) is true and suppose that there is an object, O, that changes from being at rest to being in motion. Now consider an arbitrary instant t during which O is in instantaneous motion. I will show that there is a time before t during which O is in motion as well. Since O is in instantaneous motion during t, it follows by (IM2) that both (A) there is an extended open interval T such that t is an instant in T and O is in motion during T and (B) for any extended sub-interval of T, O is in motion during that sub-interval as well. Since, by (A), there is an open interval that includes t during which O is in motion, it follows that there is an open sub-interval of that open interval which is before t. but, by (B), O is in motion during that open sub-interval that is before t. So, O is in motion before t. So, t is not the first instant during which O is in motion. But, since t was an arbitrarily chosen instant, it follows that there cannot be a first instant during which O is in motion. So, there must be a last moment during which O is at rest. Since O was arbitrarily chosen as well, we can conclude that any object that changes from being at rest to being in motion will have a last insant of being at rest. So, (IM2) implies that any object that changes from being at rest to being in motion will have a last insant of being at rest.

Now, my current belief is that use of 'is in instantaneous motion' is indeterminate between (IM1) and (IM2). So, currently, I think that either (2) or (3) from our original puzzle is false and necessarily so. But, it is indeterminate which is false.

Quick Side Note:

You might think that any legitimate precisification of our language has to obey the following constraint: any sentence of the form 'if O is in motion during T then for any time in T, O is in motion during that time as well'. Moreover, you might think that if this is right, then we have some reason to prefer (IM1) over (IM2).

Although this sounds plausible at first, It seems to me that in addition to the following traditional At-At view:

(AT-AT) An object O is in motion during an extended interval T iff O is located at one region during one instant of T and O is located at a different region during a different instant of T.

there is an alternative, open interval At-At view:

(Open At-At) An object O is in motion during an extended interval T iff T is an open interval and O is located at one region during one instant of T and O is located at a different region during a different instant of T.

It seems to me that our use of 'is in motion' is probably indeterminate between (At-At) and (open At-At). But, that also supports the claim that it is indeterminate which of (2) or (3) is false. So, the suggestion above just seems to push the solution back a level.

## 5 Comments:

Hi Joshua,

Nice post. Two quick observations:

First, those who say that the topological structure of time is

continuousalso say that between any two distinct instants is a third instant distinct from both. So the problem is not just for those who say that time is dense. Proponents of both views accept (6).The second concerns your argument that (IM1) implies that any object that goes from rest to motion has a first instant of motion. At one point you claim that if the union of

tandTis a temporal interval and if an object is in motion through the union oftandT, then (IM1) implies that it is in motion att. But (IM1) only implies that it is in motion at everyextendedsub-interval of the union oftandT. Buttis of course not extended: it is an instant. Thus (IM1) is silent about whether that object is in motion att. So the argument is invalid.As a promissory note, I'd like to argue that if you accept a Russellian theory of velocity (which would be completely natural thing to do if you accept the at-at theory of motion), then you have independent reason to reject

allof (2) through (4); and that doing so does not commit you to any indeterminacy regarding states of motion.But that's for later; right now it's dinnertime!

So here's the tact I'd prefer to take. In summary: those who hold the at-at theory of motion say that facts about rest and motion are fixed by facts about an object's trajectory. But once we're clear about the extent to which this is the case, we can show that either (3) is false or (4) is false.

Roughly, and for sake of reminder: to endorse the Russellian theory of velocity (RTV) is to say that the instantaneous velocity of an object at an instant,

t, is just the first time derivative of that object's position function att. (RTV) is compatible with, and almost always taken to supplement, the at-at theory: the latter stateswhenan object is in motion at a time, while the former states therateof its motion at that time.But if we accept the at-at theory and (RTV), then we should also accept that what the velocity of an object is at an instant,

t, depends upon whether we evaluate that derivative 'from the left' (i.e., whether we approximate the slope of lines tangent to the trajectory beforet) or instead 'from the right' (i.e., whether we approximate the slope of lines tangent to the trajectory aftert).Let me explain. Consider two objects,

xandy, whose trajectories are given by the following two position functions:x(t) = 2tiftis less than or equal tot0; and 0 iftis greater thant0.y(t) = 2tiftis greater than or equal tot0; and 0 iftis less thant0.Presumably the proponent of (RTV) will want to say that these two trajectories are possible: the first is simply a case in which an object goes from motion to rest, and the second is simply a case in which an object goes from rest to motion. Notice, though, that these are cases in which the the derivative from the left

does not equalthe derivative from the right.For instance, consider

x. In this case, there is a last instant at whichxis in motion (i.e., has a non-zero velocity). It'st0. But the derivative from the left att0 is 2 units, while the derivative from the left att0 is 0 units. And it's precisely the opposite fory.So what's this to do with the problem? Here's the thought: there

isno fact of the matter whether an object is at rest or in motion at an instantsimpliciter. An object is at rest or in motion at an instant only relative to its trajectory either immediately before or immediately after that instant.(Note that, according to view, it's not that 'the' instantaneous velocity of an object is indeterminate between these two segments of its trajectory. The view is that there

isno such velocity. All instantaneous velocity is velocity relative to the trajectory either before or after the relevant instant. We're simply confused into thinking that there is 'the' instantaneous velocity of an object because in many cases these 'forward-looking' and 'backward-looking' velocities have the same value.)If that's the case, then we can proceed as follows. First, we need to disambiguate (4) as either (4*) or (4**):

(4*) No object is both at rest 'from the left' and in motion 'from the right' during the same instant.

(4**) No object is both at rest 'from the left' and in motion 'from the left' during the same instant.

Now (4*) is straightforwardly false according to the modified Russellian view under consideration: the object

yhas zero derivative from the left of an instant but a non-zero derivative from the right.So consider (4**) instead, which the modified Russellian will accept. In that case, we still need to disambiguate (3). But as far as I can tell, the case of

ywill be a counterexample to any way of disambiguating (3). (I'll leave you all to verify or disconfirm this last claim; this post is painfully long as it is!)What do you think?

Hi Alex,

I think your suggestion is interesting. But, I have one quick question. I was wondering if we thought that time had a direction to it, then would that give us a reason to favor one direction of evaluating the derivative?

Actually, now that I think about it, I wondering if the mere seeming perception of a direction to time would give us a reason to favor one evaluation of the derivative over the other. My thought here is that our use of the term will be partly constrained by our seeming perceptions. If that is the case, then perhaps one direction of evaluating the derivative would be a better fit for our use of the phrase 'is in motion' than the other.

I know this question is a bit unclear, but I hope you see what my thought is. What do you think?

Hi Josh,

I think that's a cool idea that's worth investigating on its own. It hadn't occurred to me that whatever facts fix (our experience of) the directionality of time may also metaphysically privilege one of the trajectory-relative velocities, or semantically privilege one them as the more eligible candidate extension of 'velocity' in our thought and talk.

An alternative view is to hold the reverse: that divergences between the two trajectory-relative velocities are what explain (our experience of) the directionality of time. It would be really interesting if that type of view could be made to work.

All that said, I'm not sure how to answer your original question: whether, if time is asymmetric, we therefore have a reason to privilege one velocity over the other. Certainly it will hinge upon what type of fact time is asymmetric in virtue of, an issue that's still very much up in the air in the literature on this.

But let's suppose the answer is 'yes'. Do you think that would pose a difficulty for the solution I offered?

I don't think it poses a problem for the solution you are offering. Rather, I think it pushes us toward one of the horns or your dilemma rather than the other. If we can figure out which velocity is privileged, then we'll be able to figure out which of (3) or (4) is false. I think we might be pushed toward rejecting (3) rather than (4).

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