Notes from the underground
barang dot sg
Updated 2 December 2023
3. Close but no cigar

Unfortunately, the solution just proposed is a little too easy! It is related to other similar proposals and we may treat them together.

Thus, some people say that Achilles doesn’t even need to be a hair’s breadth from the wall. An arm’s length will do, since he can then reach out and touch the wall.

Others say that once Achilles gets to a point that is close to the wall, he is bound to bump against the wall. After all, a point is a dimensionless entity, but Achilles himself must occupy substantial space around any point.

These views have the following in common. They deliberately skirt the matter of whether Achilles can move through the whole infinity of Zeno’s points, by claiming that he need not do so in order to reach the wall. In other words, it doesn’t matter whether Achilles can move through the whole lot of Zeno’s points. Even if he can’t, he can still (in some sense) reach the wall.

The examples above illustrate the idea and there are many variations on the theme.

Unfortunately, however, these solutions fail to engage the real issue and prove to be rather facile. In particular, the matter skirted above is not so easily skirted.

Suppose there is a doorway in the wall, and the question is whether Achilles can walk through the doorway and emerge on the other side.

Consider Achilles’s center of gravity and the corresponding point in the doorway (the two blue dots).

According to Zeno, Achilles can bring his center of gravity ever closer to the blue dot of the doorway, but he cannot quite reduce the distance between them to zero. But this implies that Achilles cannot get through the doorway.

Naturally, this is absurd, but the solutions above have no response, because they focus solely on whether Achilles can “reach” the doorway, and fail to ask if he can walk through the thing. In terms of the tortoise, they pronounce that Achilles can get close enough to grab the tortoise, but remain silent on whether he can overtake the thing.

Obviously, this is not good enough.

So the question of whether Achilles can move through the whole infinity of Zeno’s points cannot really be evaded. To get through the doorway, he must be able to do so. But how can he finish moving through the entire sequence of points, one after another, if the sequence has no end?

This is the crux of the paradox and it must be faced squarely.