Zeno's famous paradoxes of motion and space purport to show that motion is impossible (the arrow never moves) and that reality is not plural (space cannot be divided). In all discussions I have read on these puzzles a similar supposition is made: that points in space and instants in time exist.
Take the case of Achilles catching up with the tortoise. It is supposed that the gap between Achilles and the tortoise can never be closed because once Achilles has reached the point the from where the tortoise started it has moved on. But there is no starting point, halfway point, or point where the tortoise has moved to. Spatial points are approximations, convenient fictions imposed on the continuous fabric of reality by human minds.
In the case of the arrow which is at rest at each instant—there is no single moment of time. Again, the concept of 'instants' in time is a convenient fiction.
(see in this regard the work of Peter Lynds http://www.peterlynds.net.nz/)
