Suppose a rock is flying through the air northward, and God miraculously and instantaneously teleports the rock, without changing any of its intrinsic properties other than perhaps position, one meter to the west. Will the rock continue flying northward due to inertia?
If velocity is defined as the rate of change of position, then no. For the rate of change of position is now westward and the magnitude is one meter divided by zero seconds, i.e., infinite. So we cannot expect inertia to propel the rock northward any more. In fact, at this point physics would break down, since the motion of an object with infinite velocity cannot be predicted.
But if velocity (or perhaps momentum) is an intrinsic feature that is logically independent of position, and it is merely a law of physics that the rate of change of position equals the velocity, then even after the miraculous teleportation, the rock will have a northward velocity, and hence by inertia will continue moving northward.
I find the second option to be the more intuitive one. Here is an argument for it. In the ordinary course of physics, the causal impact of physical events at times prior to t1 on physical events after t1 is fully mediated by the physical state of things at t1. Hence whether an object moves after time t1 must depend on its state at t1, and only indirectly on its state prior to t1. But if velocity is the rate of change of position, then whether an object moves via inertia after t1 would depend on the position of the object prior to t1 as well as at t1. So velocity is not the rate of change of position, but rather a quality that it makes sense to attribute to an object just in virtue of how it is at one time.
This would have the very interesting consequence that it is logically possible for an object to have non-zero velocity while not moving: God could just constantly prevent it from moving without changing its velocity.
