this weekOn the Clock

April 24, 2005

Zeno’s Paradoxes

Proof that motion unexists

Do you ever have one of those days when you just can’t seem to get yourself moving? Or maybe, no matter how hard you try to get caught up, you always seem to lag behind? I have those kinds of days all the time—and so, apparently, did ancient Greek philosophers. One of them, Zeno of Elea, devised an ingenious set of philosophical statements that amount to “proof” that motion is impossible, despite all evidence to the contrary. These statements are known as Zeno’s Paradoxes (or sometimes, collectively, as Zeno’s Paradox), and they continue to vex philosophers to this day.

I first became aware of Zeno and his ideas while working on my undergraduate degree in philosophy. I was reading Douglas Hofstadter’s Pulitzer-winning Gödel, Escher, Bach: an Eternal Golden Braid, in which philosophical issues are frequently presented in hypothetical dialogs between Achilles, the Greek warrior legendary for his swiftness, and a Tortoise. Lewis Carroll had used the same pair of characters, but it was Zeno who first put them together—in the fifth century B.C. In Hofstadter’s retelling of the story, Zeno himself makes a guest appearance in order to explain to Achilles and the Tortoise that motion is not merely impossible, it “unexists.” The story is based on one of Zeno’s eight so-called paradoxes, of which only three or four are usually mentioned. Allow me to give you a very brief taste.

Achilles and the Tortoise: Imagine that Achilles meets a Tortoise, who challenges him to a foot race. Achilles is amused when the Tortoise asks merely for a modest head start. But then the Tortoise explains that by agreeing to this demand, Achilles has already lost! The logic, says the Tortoise, is that if he starts ahead of Achilles at point A, Achilles will have to run to point A before he can overtake the Tortoise (which is, of course, obvious enough). Meanwhile, the Tortoise will have moved ahead slightly to point B. Again, Achilles must advance to point B before he can push ahead, by which time the Tortoise will have traveled farther (if only by inches), to point C. And so on. Although with each successive point in the race the Tortoise moves smaller and smaller distances, Achilles never quite catches up, always remaining one segment behind. And thus, says Zeno, the faster can never overtake the slower.

The Dichotomy: Another variation on the same theme is called the “dichotomy paradox” (or sometimes the “bisection paradox” or “race course paradox”). Suppose you want to cross a room. In order to get to the other side, you must first get to the halfway point, which will take you some finite amount of time. And before you can get halfway, you have to cross half of that distance, at which point you’d be a quarter of the way across. And before that, you’d have to cross half of a quarter, and so on infinitely. Each of these steps must take a finite amount of time. And yet, you have to cross an infinite number of distances to walk across the room—or indeed any distance at all. And since one cannot travel an infinite number of distances in a finite period of time, motion itself is impossible.

The Arrow: Just when you think motion is completely done for, Zeno makes matters even worse. Think of an arrow in motion, he says. At any particular instant during its flight, the arrow occupies just one position in space, which is how we define an object that’s at rest. So the arrow must, at that point, be at rest. At the next instant, whatever position the arrow is in, it’s also in just one spot, and thus, still at rest. Therefore, by definition, anything in motion is actually at rest!

Now, I know what you’re thinking: this is all very silly. A logical “proof” does not mean that motion is impossible, and whatever Zeno may have conjectured about such things at the office, he still certainly walked home at the end of the day. That is true. And yet, at some level, you have to admit that he does have a point, of sorts. Trying to tease apart Zeno’s logic from common sense has occupied a great many philosophers and mathematicians over the centuries. And if you’re willing to wrap your head around a bit of calculus, you can find some rather definitive mathematical explanations for why we can move after all. But that, say some people, is missing the whole point.

In the first place, there are philosophers who deny that these little stories are truly paradoxes. For example, it’s true that one can, in principle, divide a finite distance into an infinite number of points, but so what? They still add up to a finite distance. Meanwhile, the same is true of time: you can subdivide hours, minutes, seconds, and so on as much as you want, but that doesn’t make time grind to a halt. Both motion and time are, in reality, continuous. So if you don’t believe in the fiction that motion must occur in discrete steps of both distance and time, there’s no paradox at all. That, say critics, takes care of at least the first two statements; as for the arrow…you can define motion as a state that exists over successive points in time, which would mean Zeno’s idea of “rest” is fundamentally mistaken.

However, it may be that all the effort to debunk the paradoxes is misguided; by themselves, they’re nothing more than intellectual exercises that Zeno himself may not even have believed. Zeno, who lived from roughly 490 B.C. to 430 B.C., was a student of Parmenides, founder of a group of thinkers known as the Eleatics. Parmenides believed that the universe is fundamentally unchanging. Since everything is ultimately one, any motion or change must be merely an illusion. Although Zeno’s statements can be taken as defending Parmenides at face value, their intention was to do so in a more subtle way, using a logical technique known as reductio ad absurdum—reduction to the absurd—also known as proof by contradiction. In other words, if a logical argument yields an absurd conclusion, one of its premises must be wrong. So it’s not that Zeno believed motion in the everyday sense was impossible; he was trying to demonstrate, by way of these absurd stories, that time and distance are in fact not divisible—and in that way, support the claim that the universe is an unchanging whole.

Ultimately, though, no one will ever know for sure exactly what Zeno was getting at, because none of his writings survived. We know about Zeno only because later philosophers, such as Plato and Aristotle, mentioned his writings in their own works—and not, I should point out, in a very complimentary way. Although some philosophers still aren’t convinced that the paradoxes are resolved, most people believe now, as in Aristotle’s time, that Zeno was too clever for his own good. And personally, I just don’t find his stories moving. —Joe Kissell

More Information about Zeno’s Paradoxes…

This article was featured in Philosophers’ Carnival XXXI.

The Wikipedia has extensive articles on Zeno’s paradoxes, Zeno of Elea, and Eleatics.

Other discussions of Zeno’s paradoxes include:

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Read the earliest recorded versions of Zeno’s paradoxes in Plato’s Parmenides and Aristotle’s Physics. Also check out Douglas Hofstadter’s wonderful Gödel, Escher, Bach: An Eternal Golden Braid.