Notes from the underground
barang dot sg
Updated 2 December 2023
2. The wall

As we saw, before Achilles can reach the wall, he must first walk half the distance to the wall, and then half the remaining distance, and then half the new remaining distance, and so on, supposedly without end.

But does this process really have no end?

This question comes naturally to many and we should address it first.

Let’s call Achilles’s starting point 1, the first halfway point 2, the next halfway point 3, and so on, where each point stands midway between its predecessor and the wall.

Clearly, Achilles must first get to point 2, and then point 3, and then point 4, and so on. But why is Zeno so confident that this sequence of points has no end? Couldn’t there be a last point in the sequence – point 50 say – such that, when Achilles gets there, he has finally reached the wall?

Well, no, unfortunately Zeno is quite right that the sequence of points has no end. Due to the clever way in which the points are laid down, the numbers just go on forever, without reaching the wall. This is not hard to see, because each point is stipulated to lie midway between its predecessor and the wall. For example, point 5 above lies midway between point 4 and the wall.

This ensures two things. First, that point 5 is not at the wall. Second, there is space to lay down another point. These facts hold of any point of course. (Point 5 was just an example.) So no point will ever lie at the wall and there is no such thing as the last point: we can always lay down another one.

We can also see this with some simple numbers. Suppose that Achilles begins one meter from the wall. Then point 2 will lie half a meter from the wall, and so on, with the remaining distance being halved each time:

PointMeters from wall

The table clearly has no end. For that to happen, the number on the right would have to reach zero, indicating that Achilles has reached the wall. But it can never do that, since it is half of the number above it, and half of a positive quantity is never zero.

So infinitely many points indeed stand between Achilles and the wall and Zeno is quite right to say that, to reach the wall, Achilles must first get to point 2, and then point 3, and then point 4, and so on, without end.

Naturally, this “without end” is annoying since it threatens to put the wall beyond reach. If Achilles must move from one point to another in a sequence that has no end, then how on earth can he reach the wall? And yet, as we saw, the endlessness of the sequence cannot be denied. So if there is a flaw in Zeno’s reasoning, it must lie somewhere else!

At this point, a second thought comes naturally to some people’s minds. For even though Zeno’s points number an infinity, it is obvious that they edge ever closer to the wall.

Thus, the table shows that point 10 is only 1512 of a meter from the wall, which is less than 2 mm, a very small distance. And by the time Achilles reaches point 17, he is a mere 165536 of a meter away, which is within the breadth of a hair.

Now if Achilles is just a hair’s breadth from the wall, it may seem pedantic to insist that he is not yet there! Surely he has effectively reached the wall? On this view, Achilles need not cross every single point in Zeno’s infinite sequence in order to reach the wall. Beyond a certain point, he is so close to the wall that he has practically reached it. The remaining points in the sequence may therefore be ignored.

What should we make of this?