Zeno's Paradoxes - Part 2

By Francis Moorcroft

This article was originally published in Issue 6 of The Philosophers' Magazine.

Last time we looked at the first two Paradoxes of Zeno, that attempted to show that if space is continuous and infinitely divisible then motion is impossible. This time we will consider the other two Paradoxes that attempt to show that motion is impossible if space is discrete.

What would it mean to say that space is discrete? It certainly doesn’t look as if things move in jerky little jumps. An analogy could be made with motion picture film: each separate frame is a still photograph, but when it is shown at a rate of 24 frames per second it looks as if the motion is continuous. So perhaps space could be like this, with only a finite number of states possible. Zeno’s Paradoxes are intended to show that this possibility also leads to contradictions, so that if we accept the conclusion of the first two Paradoxes - that motion is impossible if space is continuous - then Zeno will have established that motion is impossible whatever space is like.

The third Paradox is the Arrow. Consider the path of an arrow in flight: at each instant of its path the arrow occupies some position in space - this is what it means to say that space is discrete. But to occupy some position in space is to be at rest. So throughout the entire path of the arrow through space it is in fact at rest!

The fourth Paradox, the Moving Blocks, is the hardest to follow. Here we have three rows of blocks of uniform size and an equal distance apart. One row is at rest, the second row moves past the first row in one direction, the third row moves past the second row in the opposite direction. Both rows move at the same speed. We need only consider three blocks in each row and what must happen in the change from Position 1 to Position 2 in the following diagram:

For this change to happen, B1 and B2 and C3 and C2 each have to pass one member of the stationary row. But in the same time B1 and B2 will have passed two members of the C row and C3 and C2 will have passed two members of the B row. As the motion is uniform, it takes the Bs and Cs an equal amount of time to pass a given object. But the problem is that they pass two objects in the time it takes to pass only one.

This argument seems the least impressive of the four as it treats the Bs and Cs as if they are in motion when they pass each other but at rest when they are themselves being passed. However, the problem is deeper than this. Consider what would happen if the members of each row were as near to each other as they could possibly be, i.e., they occupy adjacent points in space, and that the change from position 1 to position 2 takes place in the smallest amount of time - an instant. Neither of these assumptions seem problematic. Now in the first instant C3 is opposite B1 and the next instant it is opposite B3. When did it get to pass B2? There is no time when this could have happened!

So time, it seems cannot come in instants and space cannot be discrete. We are left with the option that time and space must be continuous - but that was the assumption that was criticized by the first two Paradoxes. In fact the Arrow is deeper than it first appears as the flight of the arrow would be equally impossible even if space and time were continuous as the arrow would, in effect, have to be at rest an infinite amount of times during its flight. Thus, whether space is continuous or whether it is discrete, motion is impossible.