John asked:

Precisely what is wrong with Zeno’s Achilles and the Tortoise argument?

Answer by Craig Skinner

Only some 200 words of Zeno survive. We rely on later commentators such as Aristotle and Simplicius. The latter first called Achilles’ opponent the Tortoise.

Zeno was a pupil of Parmenides, and the 4 paradoxes of motion (Achilles, Dichotomy, Arrow, Stadium) attempt to show that motion is impossible in line with the nothing-changes view taught by the great man.

Since motion clearly is possible, indeed actual, there must be something wrong with the arguments as you suggest.

I will briefly outline the Achilles and the Dichotomy, which are logically equivalent, and then suggest how they might be refuted.

The Achilles:

Achilles (A) is a good runner. He sportingly gives his slower rival, the Tortoise (T), a start. The race begins. By the time A reaches T’s start point, T has moved on to a new point. By the time A reaches that new point, T has moves again to a further point. By the time A reaches that further point, T has again moved ahead, and so on endlessly. A can never catch T.

The Dichotomy:

Version 1: To travel any distance, I must first reach the halfway point. Then I must reach the halfway point of the remainder, then the halfway point of the new remainder, and so on endlessly. I can never complete the journey.

Version 2: To travel any distance, I must first cover half the distance. To do this I first have to travel half of that half (first 1/4 ). Before that, half of that quarter (first 1/8), before that, 1/16, and so on endlessly. I can never start the journey.

The paradox is not that we must travel an infinite distance, or for infinite time. Clearly, knowing the speeds of A and T, and the length of T’s start we can easily calculate where/ when A catches T, or when the Dichotomy runner completes the run. The paradox is that an infinite number of actions (tasks) seems necessary — A has to pass every one of the unending sequence of points where T once was.

What’s wrong?

To refute the argument we must deny at least one of its 3 presuppositions, which are:

  1. In travelling a distance we must cross each and all of the intervening points.
  2. A line consists of an infinity of points.
  3. It is impossible to complete an infinite series of actions (tasks).

Aristotle denied 1., saying a line can’t consist of points, they have no size, whereas a line has. A point is potential, becoming actual only if we divide the line there. The paradox invites us to repeatedly divide the line at an infinity of potential points, but A does not have to touch an infinity of actual points to catch T.

Others deny 2., saying a line doesn’t consist of an infinity of points, rather space is not continuous but consists of tiny discrete units (quantized). In covering a distance we traverse a finite number of  space quanta. Motion is jerky but this is undetectable due to the fantastically tiny size of the quanta.

Yet others deny 3., saying that an infinity of tasks is possible. This is a subtle business, bringing in Aristotle and Dedekind cuts, and dealing with it would make this answer too long (ask me if you’re interested).

After 2500 years there is still debate, and no closure, especially about 2. and 3.

Some modern mathematicians offer as a solution that that the time/ distance till A catches T is the (finite) limit of an infinite convergent series (1/2 +1/4 +1/8 + etc). But this simply tells us what Zeno already said, that the distance is finite, just infinitely divisible, and doesn’t explain how we complete the task.

I hope this helps.