Friday, December 19, 2008

Zeno's Paradoxes

Paremenides was a philosopher who rejected change and motion as illusory (I've mentioned him before). One of his students was Zeno, who set up a series of paradoxes (some 40) to show that the concept of motion was absurd. Most of them ultimately stem from the problem of the infinitely divisibility of finite spans of space and time. Nonetheless, they present perplexing problems, which certainly weren't soluble with Greek mathematics, and even strain modern mathematical theory (George Cantor notwithstanding).

The two that are the most characteristic are the arrow paradox and Achilles and the tortoise. The arrow paradox first defines a thing at rest as something that occupies its own space, then says that when you throw an arrow through the air, then it always occupies its own space, therefore it must always be at rest. Why we should say that a thing that occupies its own space must be at rest and can't also be in motion is unclear. It might have something to do with the idea that if it occupies its own space and nothing more, then how can it continue to propel itself forward after it leaves the hand. This was a problem that vexed Aristotle, but which we can now can take for granted, leaving Zeno's arrow paradox not so vexing to us.

The Achilles and the Tortoise paradox is more interesting though. Achilles was reputed to be a fast runner. If Achilles is racing with a tortoise and he gives him a head start, then when Achilles starts running he will catch up to where the tortoise was when he first started running, but since time will have elapsed the tortoise will no longer be there, so he'll run to where the tortoise is now, but he won't be there by the time he reaches that spot, and so on. Thus, Achilles would never reach the tortoise. We should remind ourselves of the ultimate purpose of this argument, which is to show that motion is absurd; therefore, nothing is in motion, there is no change.

Aristotle's response to this paradox, is that it is irrelevant, since we can clearly see that things move. But Aristotle's refutation wouldn't convince Zeno since he and Parmenides both realize that we perceive motion, but they think that the perceived motion is an illusion. George Cantor, will also come along with a resolution to this with new mathematics of infinite sets and actual infinities.

I think, though, even without this sophisticated mechanism we can already see that it is problematic. The problem with Achilles and the tortoise is that, if the paradox is valid, then we should actually see Achilles being unable to catch up with the tortoise. The way the paradox is set up, it should affect both real and apparent motion. And yet we have many times seen faster runners catch up to faster and pass by them. We shouldn't even be able to see this occur. And yet we do see it occur.

In other words, Zeno's paradoxes prove too much. They prove both the absurdity of real and apparent motion. Whereas we don't observe any problem with apparent motion. This shows us that there is a problem with Zeno's paradoxes, though it doesn't show us what that problem is. It is up to later and more sophisticated mathematics to point out those problems.

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