A fun little argument against four-dimensionalism

1. Spinning a rigid object cannot affect its shape.

2. If four-dimensionalism is true, spinning a rigid object can affect its shape.

3. So, four-dimensionalism is not true.

The easiest way to see that 2 is true is to imagine that space is two-dimensional. Then if objects are considered to be extended in time, as the four-dimensionalist says, an object intuitively thought of as a rectangle that stays still is really a rectangular prism, while if that rectangle is spun by 90 degrees, it looks like a twisty thing.

I don’t think it’s too costly to deny 2. And perhaps one can make sense of some notion of internal shape that doesn’t change no matter how a rigid object moves around.

Change and potentiality

Aristotle defines motion or, more generally, change as the actuality of potentiality.

Imagine a helicopter hovering in one location, x. Its being at the same location x at time t₂ as it was at time t₁ is an actualization of its potentiality at t₁: namely, its potentiality to keep itself hovering in the same place by counteracting the force of gravity. Thus, by Aristotle’s definition it seems that the helicopter’s motionless hovering is motion.

Perhaps, though, we need to distinguish between potentiality and power. The helicopter, unlike a rock, has a power to stay in one place in mid-air. But neither the helicopter nor the rock has a potentiality to stay in one place, because a potentiality is necessarily for a state that does not yet obtain.

This suggests a view of potentiality like the following:

1. An object a has a potentiality for a state F just in case the object a has a possibility of being in state F and a is not in state F.

Here, “possibility” is used in the modern sense as not excluding actuality.

The helicopter has a possibility of being at location x in the future, but since it is already at location x, that possibility is not a potentiality.

Now, let’s go back to Aristotle’s definition. When are the actuality and potentiality predicated? Given that, as we saw, a necessary condition for a potentiality is lack of the corresponding actuality, it seems they cannot be predicated at the same time. This suggests that the Aristotelian account is:

2. An object changes provided it has a potentiality at one time and some other time actualizes that potentiality.

But now consider the simple at-at theory of change.

3. An object changes provided that it has a state at one time and lacks it at another.

We might call (2) “Aristotelian change” and (3) “at-at change”.

The following is trivially true:

4. Aristotelian change entails at-at change.

But what is curious is that the converse also seems to be true:

5. At-at change entails Aristotelian change.

For suppose that an object a is in state F at one time and not in state F at another. Swapping F and non-F if needed, we may assume for simplicity it is earlier in state non-F. Let t₁ be the earlier time. Since the object will later in be in state F, at t₁ it has a possibility for being in state F. That possibility is a potentiality by (1). And at t₂ that possibility is realized and hence is actual. Thus, at one time a has a potentiality for F and at another that potentiality is actualized. Hence, we have Aristotelian change.

So:

6. Necessarily, at-at change occurs if and only if Aristotelian change occurs.

So what does the Aristotelian account add?

Perhaps, though, we might say that (1) is too simplistic an account of potentiality. Perhaps not every unrealized possibility is a potentiality, but only an unrealized internally-grounded possibility. For instance, I have an internally-grounded possibility of standing up. But I do not have an internally-grounded possibility of instantly doubling in mass: rather, this possibility is grounded in the power of God.

On this view, however, the Aristotelian account of change appears to be false. For suppose that I have a possibility for a non-actual state F, but that possibility is not internally-grounded. Then if that possibility comes to be realized, clearly I have changed. Thus, if God miraculously doubles my mass, I have grown more massive, that’s a change. But that change isn’t a realization of an internally-grounded possibility.

One can escape this objection by insisting that every possibility for an object has to be internally-grounded. If so, then the Aristotelian account of change applies precisely to the same cases as the at-at account does, once again, but it adds a richer claim that change is always related to an internally-grounded possibility.

An argument that motion doesn't supervene on positions at times

In yesterday’s post, I offered an argument by my son that multilocation is incompatible with the at-at theory of motion. Today, I want to offer an argument for a stronger conclusion: multilocation shows that motion does not even supervene on the positions of objects at times. In other words, there are two possible worlds with the same positions of objects at all times, in one of which there is motion and in the other there isn’t.

The argument has two versions. The first supposes that space and time are discrete, which certainly seems to be logically possible. Imagine a world w₁ where space is a two-dimensional grid, labeled with coordinates (x, y) where x and y are integers. Suppose there is only one object, a particle quadlocated at the points (0, 0), (1, 0), (0, 1) and (1, 1). These points define a square. Suppose that for all time, the particle, in all its four locations, continually moves around the square, one spatial step at a temporal step, in this pattern:

(0, 0)→(1, 0)→(1, 1)→(0, 1)→(0, 0).
Then at every moment of time the particle is located at the same four grid points. But it is also moving all the time.

But there is a very similar world, w₂, with the same grid and the same multilocated particle at the same four grid points, but where the particle doesn’t move. The positions of all the objects at all the times in w₁ and w₂ are the same, but w₁ has motion and w₂ does not.

Suppose you don’t think space and time can be discrete. Then I have another example, but it involves infinite multilocation. Imagine a world w₃ where the universe contains a circular clock face plus a particle X. None of the particles making up the clock face move. But the particle X uniformly moves clockwise around the edge of the clock face, taking 12 hours to do the full circle. Suppose, further, that X is infinitely multilocated, so that it is located at every point of the edge of the clock face. In all its locations X moves around the circle. Then at every moment of time the particle is located at the same point, and yet it is moving all the time.

Now imagine a very similar world w₄ with the same unmoving clock face and the same spacetime, but where the particle X is eternally still at every point on the edge of the clock face. Then w₃ and w₄ have the same object positions at all times, but there is motion in w₃ and not in w₄.

I think the at-at theorist’s best bet is just to deny that there is any difference between w₁ and w₂ or between w₃ and w₄. That’s a big bullet to bite, I think.

It would be nice if there were some way of adding causation to the at-at story to solve these problems. Maybe this observation would help: When the particle in w₁ moves from (0, 0) to (1, 0), maybe this has to be because something exercises a causal power to make a particle that was at (0, 0) be at (1, 0). But there is no such exercise of a causal power in w₂.

Bilocation and the at-at theory of time

I was telling my teenage children about the at-at theory of motion: an object moves if and only if it is in one location at one time and in another location at another time. And then my son asked me a really cool question: How does this fit with the possibility of being multiply located at one time?

The answer is it doesn’t. Imagine that Alice is bilocated between disjoint locations A and B, and does not move at either location between times t₁ and t₂. Nonetheless, by the at-at theory, Alice counts as moving: for at t₁ she is in location A while at t₂ she is in location B.

My response to my son was that this was the best argument I heard against the at-at theory. My son responded that the argument doesn’t work if multilocation is impossible. That’s true. But there is good reason to think bilocation is possible. First, the real presence of Christ in the Eucharist appears to require multilocation. Second, God is present everywhere, but never moves. Third, there is testimonial evidence to saints bilocating. Fourth, the argument only needs the logical possibility of bilocation. Fifth, time-travel would make it possible to stand beside oneself.

(The time-travel case is probably the least compelling, though, as an argument against the at-at theory. For the at-at theorist could say that the times in the definition of motion are internal times rather than external ones, and time travel only allows one to be in two places at one external time.)

I’ve been inclining to think the at-at theory is inadequate. Now I am pretty much convinced, but I am not sure what alternative to embrace.

One might just try to tweak the at-at theory. Perhaps we say that an object moves if and only if the set of its locations is different between times. But that isn’t right. Suppose Alice is bilocated between locations A and B at t₁, but at t₂ she ceases to bilocate, defaulting to being in location A. Then the set of locations at t₁ is {A, B} while at t₂ it is {A}. But Alice hasn’t moved: cessation of bilocation isn’t motion. Nor will it help to require that the sets of locations at the two times have the same cardinalities. For imagine that Alice is bilocated at locations A and B at t₁, and then she ceases to be located at B, defaulting to A, and walks over to location A′ at t₂. Then Alice has moved, but the sets of locations at t₁ and t₂ have different cardinalities. I don’t know that there is no tweak to the at-at theory that might do the job, but I haven’t found one.

An open future precludes present motion

1. Whether an arrow is moving now depends on where it will be in the future.
2. If the future is open, there is no fact about where the arrow will be in the future.
3. If if whether p depends on how A is, and there is no fact about how A is, then it is not a fact that p.
4. So, if the future is open, no arrow is moving now.

Premise 1 is the most controversial one. Suppose that an arrow has been flying for half a second. There are two metaphysically possible worlds. In the first, it continues moving as usual. In the second, its motion instantly reverses after this moment, so that this moment is the furthest point in its flight. In the second world there is no more reason to say that the present moment is the last moment of forward motion than to say that it is the first moment of backward motion. So we shouldn't say the arrow is moving forward in the second world.

Hence there are worlds that differ on whether the arrow moves now and yet that differ only in the future positions of the arrow. That gives us reason to accept 1.

Premise 2 is particularly clear given theism: God can miraculously relocate the arrow if he so chooses. But I think premise 2 is going to be plausible on other views, too.

Experiencing present events and simultaneous causation

When I look at a rock, I see the rock and not just its outside surface. Of course, it is the outside layer that is causally responsible for my perception: a typical rock would look the same (except when intense light was shining through it) if suddenly all but the outer one millimeter of it disappeared. Similarly, when I see an event like a ball flying through the air, I see an event that includes the ball's presently flying through the air, even though it is only the temporal parts of the event fractions of a second prior to my perception that are causally responsible for my perception, since it takes light a few of nanoseconds to get to me from the ball and then it takes my visual apparatus rather longer to process it, and I need to process data from two different times in order to get the perception of motion. In both cases the data is processed without my being aware of it, and a rock or an event that includes the present is presented to me. If all goes well, this is veridical, though it could happen that there is no rock but a mere shell or that the ball was annihilated just before I had the perception.

So, in these experiences, when things go well, I have an experience of something extended through space and/or time caused by a very small proper part of the object of the experience. But veridical experiences must be caused by their objects (and in the right way). This means that a whole can count as causing something that is only caused by a proper part. (There are, of course, plenty of non-perceptual examples.) Moreover, notice that the case of the ball flying through the air, then, is a case of something like simultaneous causation when all goes well: a temporally extended event of the ball flying--including its flying now--causes my present perception, and the two events temporally overlap.

But this instance of simultaneous causation seems grounded in a case of non-simultaneous causation: a past temporal part of the ball's flight causing a present experience. That may be so. However, for all we know this non-simultaneous causation could be grounded in a finite sequence of fundamental simultaneous causations between temporally-extended temporally-simple events.

Uniform motion and relationalism

An old objection to relationalism about space--an objection going back to the Leibniz-Clarke correspondence--is that it seems possible for all things to move together at the same speed in the same direction. But since the relations between things don't change when they all move together, on a relationalist view of space it seems impossible to make sense of global uniform motion.

Here's a solution to the objection: The motion of an object x can be characterized by saying that x at t₂ is at a non-zero spatial distance from x at t₁. This allows one to characterize absolute motion in a relationalist account of space, which has typically been held impossible.

The above story works most neatly if we have eternalism and temporal parts: then x moves provided that it has temporal parts at a spatial distance from each other. But we can also do this with eternalism without temporal parts, provided that we index distance relations to two times. Whether a presentist who is a relationalist about space can make use of the solution depends on how well the presentist can solve the problem of cross-time relations.

I don't personally like this story, because I would prefer a relationalism based on spacetime relations rather than spatial ones.

Teleporting Zeno's arrow

Here are some plausible theses:

1. Necessarily, an object that is in the same place at time t as it has been for some non-zero period of time prior to t is not moving at t.
2. Necessarily, if an object is at one location at t₁ and at another at t₂ is moving at some time t at one of the two times or between them.
3. It is possible to have continuous time.
4. If it is possible to have continuous time, it is possible to have continuous time and instantaneous teleportation of the following sort: an object is in one place for some time up to and including t₁, then it is instantaneously teleported to a second place where it remains at all times after t₁ up to and including t₂.

These theses are logically incompatible. For, given (3) and (4), suppose we have a world with continuous time and instantaneous teleportation like in (4). Then by (2), this object moves at some time at or between the two times. But at t₁ the object is in the same place as it has been for some time, so by (1) it's not moving. And it's also not moving at any time after t₁ (up to t₂), since at any time after t₁, it's been sitting in the second location for some time.

In some ways, this is an improved version of Zeno's arrow paradox. Zeno had an implausibly strong version of (1) that implied that an object that stayed in the same place for an instant wasn't moving at that instant. That's implausible. But (1) is much weaker. The cost of this weakening is that we need to replace run-of-the-mill movement with teleportation.

Of the premises, I think (4) is the most secure, despite being the most complex. Surely God could teleport things. Here is an argument for (1). Whether an object is in motion at t should not be a future contingent at t. But if the answer to the question whether an object is in motion at t depends on what happens after t, then it would be a future contingent. So it only depends on what happens at or before t. Now if the object has been at the same place for some time prior to t, and is there at t, it should be possible (barring special cases like where God promised that the object will move) for the object to remain there for some time after t. In that case, the object would obviously not be moving at t. But since what happens after t is irrelevant to whether it's moving at t, we conclude that as long as the object has been standing in the same place for some time up to and including t, it's not moving at t.

That leaves (2) and (3). I am inclined to reject both of them myself, though of course the argument only requires one to reject one (given the reasons to believe (1) and (4)). Rejecting (2) seems to go hand-in-hand with seeing motion as something that doesn't happen at times, but only between times (the presentist may well have trouble with this).

Zeno's arrow, Newtonian mechanics and velocity

Start with Zeno's paradox of the arrow. Zeno notes that over every instant of time t₀, an arrow occupies one and the same spatial location. But an object that occupies one and the same spatial location over a time is not moving at that time. (One might want to refine this to handle a spinning sphere, but that's an exercise to the reader.) So the arrow is not moving at t₀. But the same argument applies to every time, so the arrow is not moving, indeed cannot move.

Here's a way to, ahem, sharpen The Arrow. Suppose in our world we have an arrow moving at t₀. Imagine a world w* where the arrow comes into existence at time t₀, in exactly the same state as it actually has at t₀, and ceases to exist right after t₀. At w* the arrow only ever occupies one position—the one it has at t₀. Something that only ever occupies one position never moves (subject to refinements about spinning spheres and the like). So at w* the arrow never moves, and in particular doesn't move at t₀. But in the actual world, the arrow is in the same state at t₀ as it is at w* at that time. So in the actual world, the arrow doesn't move at t₀.

A pretty standard response to The Arrow is that movement is not a function of how an object is at any particular time, it is a function of how, and more precisely where, an object is at multiple times. The velocity of an object at t₀ is the limit of (x(t₀+h)−x(t))/h as h goes to zero, where x(t) is the position at t, and hence the velocity at t₀ depends on both x(t₀) and on x(t₀+h) for small h.

Now consider a problem involving Newtonian mechanics. Suppose, contrary to fact, that Newtonian physics is correct.

Then how an object will behave at times t>t₀ depends on both the object's position at t₀ and on the object's velocity at t₀. This is basically because of inertia. The forces give rise to a change in velocity, i.e., the acceleration, rather than directly to a change in position: F(t)=dv(t)/dt.

Now here is the puzzle. Start with this plausible thought about how the past affects the future: it does so by means of the present as an intermediary. The Cold War continues to affect geopolitics tomorrow. How? Not by reaching out from the past across a temporal gap, but simply by means of our present memories of the Cold War and the present effects of it. This is a version of the Markov property: how a process will behave in the future depends solely on how it is now. Thus, it seems:

1. What happens at times after t₀ depends on what happens at time t₀, and only depends on what happens at times prior to t₀ by the mediation of what happens at time t₀.

But on Newtonian mechanics, how an object will move after time t₀ depends on its velocity at t₀. This velocity is defined in terms of where the object is at t₀ and where it is at times close to t₀. An initial problem is that it also depends on where the object is at times later than t₀. This problem can be removed. We can define the velocity here solely in terms of times less than t₀, as lim_h→0−(x(t+h)−x(t))/h, i.e., where we take the limit only over negative values of h.[note 1] But it still remains the case that the velocity at t₀ is defined in terms of where the object is at times prior to t₀, and so how the obejct wil behave at times after t₀ depends on what happens at times prior t₀ and not just on what happens at t₀, contrary to (1).

Here's another way to put the puzzle. Imagine that God creates a Newtonian world that starts at t₀. Then in order that the mechanics of the world get off the ground, the objects in the world must have a velocity at t₀. But any velocity they have at t₀ could only depend on how the world is after t₀, and that just won't do.

Here is a potential move. Take both position and velocity to be fundamental quantities. Then how an object behaves after time t₀ depends on the object's fundamental properties at t₀, including its velocity then. The fact that v(t₀)=lim_h→0(x(t₀+h)−x(t₀))/h, at least at times t₀ not on the boundary of the time sequence, now becomes a law of nature rather than definitional.

But this reneges on our solution to The Arrow. The point of that solution was that velocity is not just a matter of how an object is at one time. Here's one way to make the problematic nature of the present suggestion vivid, along the lines of my Sharpened Arrow. Suppose that the arrow is moving at t₀ with non-zero velocity. Imagine a world w* just like ours at t₀ but does not have any times other than t₀.[note 2] Then the arrow has a non-zero velocity at t₀ at w*, even though it is always at exactly the same position. And that sure seems absurd.

The more physically informed reader may have been tempted to scoff a bit as I talked of velocity as fundamental. Of course, there is a standard move in the close vicinity of the one I made, and that is not to take velocity as fundamental, but to take momentum as fundamental. If we make that move, then we can take it to be a matter of physical law that mlim_h→0(x(t₀+h)−x(t₀))/h=p(t₀), where p(t) is the momentum at t.

We still need to embrace the conclusion that an object could fail to ever move and yet at have a momentum (the conclusion comes from arguments like the Sharpened Arrow). But perhaps this conclusion only seems absurd to us non-physicists because we were early on in our education told that momentum is mass times velocity as if that were a definition. But that is definitely not a definition in quantum mechanics. On the suggestion that in Newtonian mechanics we take momentum as fundamental, a suggestion that some formalisms accept, we really should take the fact that momentum is the product of mass and velocity (where velocity is defined in terms of position) to be a law of nature, or a consequence of a law of nature, rather than a definitional truth.

Still, the down-side of this way of proceeding is that we had to multiply fundamental quantities—instead of just position being fundamental, now position and momentum are—and add a new law of nature, namely that momentum is the product of mass and velocity (i.e., of mass and the rate of change of position).

I think something is to be said for a different solution, and that is to reject (1). Then momentum can be a defined quantity—the product of mass and velocity. Granted, the dynamics now has non-Markovian cross-time dependencies. But that's fine. (I have a feeling that this move is a little more friendly to eternalism than to presentism.) If we take this route, then we have another reason to embrace Norton's conclusion that Newtonian mechanics is not always deterministic. For if a Newtonian world had a beginning time t₀, as in the example involving God creating a Newtonian world, then how the world is at and prior to t₀ will not determine how the world will behave at later times. God would have to bring about the initial movements of the objects, and not just the initial state as such.

Of course, this may all kind of seem to be a silly exercise, since Newtonian physics is false. But it is interesting to think what it would be like if Newtonian physics were true. Moreover, if there are possible worlds where Newtonian physics is true, the above line of thought might be thought to give one some reason to think that (1) is not a necessary truth, and hence give one some reason to think that there could be causation across temporal gaps, which is an interesting and substantive conclusion. Furthermore, the above line of thought also shows how even without thinking about formalisms like Hamiltonian mechanics one might be motivated to take momentum to be a fundamental quantity.

And so Zeno's Arrow continues to be interesting.

One problem for a moving present

Suppose that we think that the present moves, ever pushing into the future. Now the present is within a Friday. Tomorrow the present will be within a Saturday. On this theory, it is the same thing, the present, that today is within a Friday and tomorrow it will be within a Saturday.

It follows that the present is something that has always existed and will always exist. After all, if a rock will tomorrow be found in one cave, and today is present in another cave, then the rock exists both today and tomorrow. The present on this view has the same temporal extent as whole time sequence.

But this is absurd. Clearly, the present does not extend back to the Battle of Waterloo. Hence an A-theory on which the present relentlessly moves forward must be rejected.

Moving spotlight theorists should, thus, not reify the spotlight. So what should they do? Well, maybe they can say that events that are presently occurring have a special property, let's say L, for being lit up. And which events have this special property changes with time. Right now the writing of this post has L. In an hour, the writing of this post will no longer have L. This, I think, leads to the McTaggart paradoxes. Here's how. Let's ask: Is having L intrinsic or extrinsic to the writing of this post? If extrinsic, then there will be something else that has an L-like property in a more basic way, and we have failed to account for the present in terms of events having L. Let W be the event of the writing of this post. Suppose then that W intrinsically has L. But in an hour, W will not intrinsically have L. I think this is what triggers the McTaggart paradoxes: the idea that events change in respect of what intrinsic properties they have. Anyway, in an hour, the writing of this post will have L₁, the property of having been lit up an hour ago. The writing of this post will gain L₁ only at a time when W no longer exists. Hence, while L is intrinsic to W, L₁ is not intrinsic, since only extrinsic properties can be gained when one does not exist. Therefore, we need to define L₁ in terms of L and a B-relation of some sort.

OK, that's all I want to do in the way of helping moving spotlight theories.