Bet on Achilles to Beat the Tortoise

Many are the times I’ve thought about Achilles and the tortoise:

Zeno concluded from this and other paradoxes that motion is an illusion, so it’s important to show why Achilles will beat the tortoise. Otherwise, we’ll all have to sit very still.

Zeno’s argument goes something like this:

1) When Achilles starts running at point A, the tortoise is already at B.
2) By the time Achilles reaches point B, the tortoise is at point C.
3) By the time Achilles reaches point C, the tortoise is at point D.
4) This series can be extended forever.
5) In order to catch up, Achilles will have to perform an infinite series of tasks in a finite amount of time.
6) It’s impossible to perform an infinite series of tasks in a finite amount of time.
7) Therefore, Achilles can never catch up to the tortoise.

But we know that mighty Achilles is going to catch up to the tortoise and win the race. Fast runners beat slow walkers. Hence, the paradox.

In his book A Brief History of the Philosophy of Time, Adrian Bardon suggests that Aristotle had a good response to Zeno. Aristotle apparently argued that Zeno’s paradox rests on confusing “an abstract value (i.e. time), which is mathematically divisible into instants, with actual change, which is not literally composed of infinitesimal units of change”. In other words, Aristotle distinguished between “the rules for time (as a mere abstraction) and those for change (as a real phenomenon)”. Maybe Aristotle was right, but I’m not sure Zeno would have been convinced.

Bardon also discusses the mathematical concept of a “limit”, which “allows for an infinite number of finite quantities to add up to a finite sum”. That’s the idea mentioned at the end of the video. Many have concluded that Zeno can be answered using this concept, although Bardon asks: “Can a limit be a real endpoint to a real process, or is it just a new mathematical convention that disregards the metaphysical question about time and change with which Aristotle and Zeno are struggling? Does it really help matters to say that [motion or change] represents convergence on a limit? That wouldn’t have sounded like real motion to either Zeno or Aristotle”.

Maybe Zeno’s argument does require a subtle and sophisticated response. But what struck me as weak about his argument was one of the premises listed above:

6) It’s impossible to perform an infinite series of tasks in a finite amount of time.

Really? Who says? Isn’t that what we do every time we move from point A to point B? Having arrived at point B, haven’t we also traveled 1/2 of the distance, 1/3 of the distance, 1/4 of the distance, 1/5 of the distance and so on? Isn’t this a clear example of performing an infinite series of tasks in a finite amount of time? Granted that the tasks overlap, but it seems fair (albeit boring) to describe what we’ve done this way, without having to explain what the mathematical concept of a “limit” is or draw a distinction between the rules for speaking about time and the rules for speaking about change.

Perhaps this is mere sophistry and Zeno would have considered it such, since he and the Sophists were contemporaries. I think it’s a simple truth that moving around involves doing many little things by doing one big thing. Take that, Zeno!

On a similar note, the English philosopher G. E. Moore wrote a famous article called “Proof of an External World”. In that article, he said he could prove the existence of the world outside our minds by drawing our attention to his two hands: “by doing this, ipso facto, I have proved the existence of external things”. Whether that’s a great argument or not is an open question. Moore defended his argument, however, by pointing out that skeptical philosophical arguments (such as proving motion to be an illusion) often rely on philosophical intuitions or generalities (such as premise 6 above) that we have much less reason to accept than the common sense beliefs they supposedly refute.

All right, it’s safe to start moving again.

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