Wednesday, March 16, 2016

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).

9 comments:

Baxter White said...

I think (1) is wrong. Consider an object that is stationary until t and then begins moving. (An arrow held at rest on a tense bowstring, then released.) I would say that its motion begins in the place it was resting.

Heath White said...

Sorry, that should be Heath White.

William said...

"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."

Assume a frame of reference at a located at a relatively close, nonzero spacelike interval from all positions of the object.

If the object is teleported and is then somewhere else at a time AFTER t1 but BEFORE t2 then it arrives (at the location it is later, at t2) at some other time, say tz. The sequence of times is then

t1 -> tz -> t2 where these are different times, from our frame of reference. So the "motion" occurs during the (maybe infinitesimal) interval tz - t1.

Alexander R Pruss said...

Heath:
On the other hand, think of a ball you toss vertically up. At the top point in the flight it's like the bow string at the moment of release. But at the top point, the ball is at rest, I think.

SMatthewStolte said...

A ball which begins to fall may be said to be at rest at time t=0, but it is still accelerating. Given that modern physics treats acceleration as the *real* change-of-state (whereas constant velocity is equivalent to rest), perhaps this is important. I don’t know.

Alexander R Pruss said...
This comment has been removed by the author.
Alexander R Pruss said...

Good point. One can, though, rerun the argument with change of velocity in place of change of position.

IanS said...

In a Newtonian world with finite forces,(4) would be false. In a non-Newtonian world, our intuitions about position and motion would be unreliable.

Alexander R Pruss said...

Ian:

Motion should just be a special case of change in general. Even in a Newtonian world, there can be instantaneous changes in properties that go beyond the Newtonian scheme. For instance, one could replace movement with change in the degree of pain that some agent experiences.