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.