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.

Agency Question

Some composite things are agents and some are not. For example, I am an agent but a rock is not (unless it is rock*). One might wonder what features distinguish the agents from the non-agents. Although this is an interesting question, I will not be pursuing that question in this post. Instead I want to consider what might be called derivative agency. Consider the Supreme Court. That entity has some kind of agency; it makes decisions and is morally blameworthy for some of those decisions. It has what we might call derivative agency. However, consider the fusion of me and all the furniture in my office. That thing does not make decisions and is not morally blameworthy for anything. It does not have derivative agency. But, what exactly is the difference between the Supreme Court and the fusion of me and all the furniture in my room?

Here is one thought: Perhaps it is because the Supreme Court completely decomposes into agents that it has derivative agency. The fusion of me and all the furniture in my office, however, does not completely decompose into agents. So, that is why it does not have agency.

Although perhaps appealing, I don't think this initial thought is right. For we can consider the thing that happens to be composed of me and some alien millions of light-years away. It does not seem to me that that latter thing has derivative agency and yet it (just like the Supreme Court) completely decomposes into agents.

Here is a second idea. Perhaps the Supreme Court essentially decomposes into agents whereas the thing that happens to be composed of me and that distant alien does not. If agents, derivative or otherwise, are essentially agents and derivative agents completely decompose into agents, then we can say that the Supreme Court is a derivative agent whereas the thing that happens to be composed of me and the alien is not. We can say that this is the case because the Supreme Court could not have failed to be composed of agents whereas the thing composed of me and the alien could have failed to be composed of agents (it might have had me and the alien and an extra particle as a part).

SCQ and the Brutality of Parthood

The Special Composition Question is often formulated as follows:

The Special Composition Question (SCQ): What necessary and jointly sufficient conditions must any xs satisfy in order for it to be the case that there is an object composed of xs?

And answers to the Special Composition Question often take the following form:

The Special Composition Schema (SCS): Necessarily, for any non-overlapping xxs, there is a y such that y is composed of xs iff ____________________.

Often, however, those who are concerned with SCQ take SCQ to be a question about what it is in virtue of which some things compose something. The formulations of SCQ and SCS given above do not make this clear.

I will not attempt to formulate more satisfactory versions of SCQ and SCS. Rather, I want to investigate whether we should expect there to be an answer to SCQ on the assumption that it is a question about what it is in virtue of which some things compose something.

Consider the simples that compose me. Call these simples 'ggs'. ggs compose something. We may now ask: In virtue of what is it the case that there is an x such that ggs compose x?

It seems to me that there is an obvious answer to this question: There is an x such that ggs compose x in virtue of the fact that ggs compose me. (Compare with this case. In virtue of what is it the case that there is someone to whom I am married? Answer: There is someone to whom I am married in virtue of the fact that I am married to Mary.)

If, then, SCQ is a question about what it is in virtue of which some things compose something else, it appears that the answer to SCQ will look something like this:

ggs compose something in virtue of the fact that ggs compose Greg, ees compose something in virtue of the fact that ees compose the Eiffel Tower, etc.

There is a further question that can be asked, however. Suppose that ggs compose something in virtue of the fact that ggs compose me. In virtue of what do ggs compose me? This is an interesting question as well.

Now to say that ggs compose me is to say that ggs are all parts of me, no two of ggs have a part in common, and every part of me has a part in common with at least one of ggs. So, we can reformulate our question as follows: In virtue of what is it the case that ggs are all parts of me, no two of ggs have a part in common, and every part of me has a part in common with at least one of ggs?

But then consider the following thesis:

The Brutality of Parthood (BT): Necessarily, for any x and y, if x is a part of y, then there is nothing in virtue of which x is a part of y.

BT is relatively plausible. But it seems to me that if BT is true, then the correct answer to the question "In virtue of what is it the case that ggs are all parts of me, no two of ggs have a part in common, and every part of me has a part in common with at least one of ggs?" is "Nothing".