More on bilocation and movement

It is often said that the four-dimensionalist doesn’t have a good theory of movement beyond the at-at theory which holds that

1. to move is to be at x₁ at one time and at x₂ at a different time, where x₂ ≠ x₁.

However, I am inclined to think the at-at theory is false due to an argument that my son came up with: if an object is bilocated at both x₁ and x₂ at one time and stays unmoved in both locations until a later time, then it is true that the object is at x₁ at one time and at x₂ at another time, and yet has not moved.

It is interesting that this argument also works against the most natural tensed theory of movement, namely that:

2. an object has moved provided that it was at x₁ and is at x₂, where x₂ ≠ x₁.

For imagine that an object was and still is bilocated between x₁ and x₂ and has remained entirely unmoving. Nonetheless, it was at x₁ and is now at x₂, and x₂ ≠ x₁, so according to (2) it has moved.

Thus, my son’s argument against the at-at theory does not seem to confer an advantage on the A-theory of time.

It is tempting to tweak (2) to something like this:

3. an object has moved provided that the set of locations at which it is now present is different from a set of locations at which it was present.

But that fails. For cessation of bilocation is not movement. If an object was bilocated between two locations x₁ and x₂, and then ceased to exist at x₂, while remaining at x₁, the object nonetheless did not move, even though (3) says it did.

Furthermore, space at least could be discrete. So imagine a point particle that was bilocated at two neighboring points x₁ and x₂ in space. The particle then simultaneously moved from x₁ to x₂ and from x₂ to x₁. Yet the set of points occupied by the particle was the same as it is now. So (3) says it did not move, but it did move, twice over.

I suppose one can deny the possibility of bilocation. But that is a big price to pay, I think.

I suspect that any theory of change that the A-theorist comes up with that solves this problem will also solve the problem for the four-dimensionalist.

Movement

The following seem quite plausible:

1. It is possible for an object both (a) to have both a first and a last moment of its existence and (b) to be moving at every time during its existence.
2. It is not possible for an object (a) to exist at only one time and yet (b) be moving.

By (2), movement is not an instantaneous property: it is not a property an object has solely in virtue of how it is at one moment. By (1), however, movement is not a property defined in terms of the past and present states of an object (say, "an object moves at a time provided that it is a different location from where it was in the past"), since it can move at the first moment of its existence; nor is it a property defined in terms of the present and future states of the object since it can move at the last moment of its existence.

So what is movement? We could say that an object is moving at time t provided that there are arbitrarily close moments t* at which the object is in a different location. This would make sense of both (1) and (2). But this account falsifies the following intuition:

3. If a ball is thrown vertically into the air, then at the high point of its flight it is not moving.

(If it were moving, would it be moving upward or downward?) For at moments arbitrarily close to that top-point time, the ball is at different locations.

We could try to define movement in terms of there being a well-defined non-zero derivative of the position with respect to time, with the derivative being one-sided at the beginning and end of the object's existence. But then, given continuous time (which we need anyway to have time-derivatives), an object could continuously change location without ever moving, since there are continuous nowhere differentiable functions.

So what should we say? I think it is that the concept of "moving at time t" is underspecified, and specifications of it simply aren't going to cut nature at the joints. Being at different places at different times (at least relative to a reference frame) makes good and fairly precise sense. But moving (or changing) at a time does not. Zeno was right about that much.