Zeno

Last weekend my friend Mark gave me his college textbook on paradoxes. I suppose I have a reputation for appreciating these kinds of gifts. I appreciated this one, and I managed to skim through the first chapter, about Zeno's paradoxes, despite being rather busy with the wedding goings-on at the time.

What are Zeno's paradoxes? They are paradoxes that all have to do with a regression to infinity that seems to lead to the conclusion that all change, or motion, is impossible. My favorite variant is the one about the race between Achilles and the Tortoise. Here's how it goes.

Achilles and the Tortoise have a footrace. Achilles, being fleet of foot, is much faster than the Tortoise, so to make things a bit less unfair, the Tortoise is given a head start. Each racer takes his mark, and the starting gun fires. Achilles, fast as he is, soon reaches the Tortoise's starting position. However, during the time it takes Achilles to reach that position, the Tortoise has inched farther along, so the Tortoise remains ahead of Achilles. However, Achilles soon catches up to that second notable position of the Tortoise—i.e., where the Tortoise was when Achilles reached the Tortoise's starting position—but during the time it takes Achilles to move to that second spot, the Tortoise has moved yet farther ahead. And so it goes, with each time Achilles catching up to where the Tortoise was, only to have the Tortoise meanwhile move forward and maintain his lead. This sort of reasoning is then taken ad infinitum, leading some old Greek sages to the conclusion that Achilles can never overtake the Tortoise.

As I said, there are many variants on this paradox, as you can use it to show that motion in general is impossible. For example, you can show that for an airplane that's taking off from a runway, either there's no last moment that it's on the ground or there's no first moment it's in the air. And you can prove that it's impossible to push a pencil a full inch along a table. And that it's impossible to transition from one second to the next. And that it's impossible to eat a cookie. Hopefully you get the point.

Zeno's paradoxes had people baffled for at least a good two thousand years, until calculus was invented. The crux of the paradoxes' arguments goes like this:

1. You can take any change and subdivide it into an infinite number of progressively smaller changes.
2. It's impossible to realize an infinite number of changes, however small they may be.

According to calculus, the second point is wrong. For example, if you move an inch today, half an inch tomorrow, a quarter of an inch the day after that, and so on, with each day you moving half as far as the day before, you won't go an infinite distance. Instead, you'll merely get ever closer to having moved two inches. This is an amazing fact that is learned, ho-hum, by thousands of high-school students each year—that sometimes an infinite number of positive numbers add up to a finite number. But it's exactly what allows Achilles to indeed catch up to the Tortoise, overtake him, and win the race—so long as Achilles manages to stay awake for the whole race.

Nevertheless, calculus, like all of mathematics, is just a model of the real thing. Calculus, in its continuous form, assumes an infinite divisibility of all things, just as Zeno's paradoxes do in point 1, above. Whether reality truly is continuous or else is innately discrete is left up to the physicists to try to figure out.