Zeno's Paradox

Over two thousand years ago, a philosopher named Zeno made a startling discovery, one that had profound ramifications for life, the universe, and everything. In order to prove his idea, he needed two accomplices: Achilles, the mighty warrior, and a tortoise.

Zeno called Achilles and the tortoise together one day, and proposed a race between them. Achilles agreed immediately. Being the strongest warrior and fastest runner in all of Greece, he knew he could easily win a race against an animal that was essentially a paperweight with legs. He even suggested that he have both arms tied behind his back, that he be blindfolded, and that he hop through the race on one foot, so great was his confidence. Zeno declined Achilles’ suggestions, but gave the tortoise one concession: a head start of one stadium (a unit of length whose approximate size is obvious). Achilles laughed at the futility of the gesture, but Zeno smugly stated that he had given the tortoise the only concession he would need.

The runners lined up on their marks, Zeno gave the countdown, and with a quick signal, they were off. Achilles quickly made up half the distance between himself and the tortoise, who had barely moved. In even less time, he made up half the distance that remained, then half of that distance. That was where his problems started. In order for Achilles to run the distance that remained between himdelf and the tortoise, he fist had to run half that distance, taking a finite amount of time to do so. Then he had to run half that distance, then half that distance, and so on. Each of these half distances was finite in length, and therefore took a finite amount of time to travel. The distance between Achilles and the tortoise could be divided in half an infinite number of times, so Achilles found himself faced with the task of traveling an infinite number of finite distances, each one taking a finite amount of time. He soon realized that it would take him an infinite amount of time to catch up with the tortoise. Humiliated, he gave up, and the tortoise won the race.

When Zeno saw the result of the race, he was very troubled. It turned out that the tortoise had fallen asleep at the start of the race, and hadn’t moved at all. Yet Achilles still wasn’t able to catch him. Zeno followed his reasoning to its logical conclusion and realized that no matter how small the distance between Achilles and the tortoise was (as long as it was a finite distance), Achilles could never cross it because he would always have to travel an infinite number of half distances, each one being finite (half of any finite number is a finite number), thus taking a finite amount of time to travel. Zeno had proven that all motion is impossible.

S’il muovo! (But it moves!)

Where is the flaw in Zeno’s logic? One idea is fairly obvious: an infinite number of finite distances must be an infinite distance. Or is it? If you have a fine enough knife, you can divide something in half over and over again, an infinite number of times, without the object having to be infinite in length. Half of an object of finite size is still an object of finite size.

What about the time? It turns out we can divide time the same way we can divide distance. Half of a finite amount of time is still a finite amount of time, but an infinite number of finite amounts of time does not have to add up to an infinite amount of time any more than an infinite number of finite distances has to add up to an infinite distance, even though, logically, they both have to do just that!

Thus we not only have poor Achilles taking an infinite amount of time to catch the motionless tortoise, but we also have the tortoise being an infinite distance away from him! Because any finite distance can be divided in half an infinite number of times, and because an infinite number of finite distances has to be an infinite distance, any distance, no matter how small, is infinite!

Of course, Zeno never physically performed this experiment. It happened within the confines of his mind. But it is logically consistent. You could even write a formal proof of its validity, as I believe Zeno did. But that’s absurd, you say. It certainly is. But try to disprove it logically.

Now let’s look at a different situation. According to Einstein’s special theory of relativity, every observer must observe the same velocity for light, regardless of his state of motion. If you were traveling in a starship at the speed of light (relative to an observer on your home planet), and you shone a flashlight out the back of your ship, the light would travel at c, the speed of light, relative to you. So naturally, the observer you left behind would not see the light, because it would not be moving relative to him. But wait; according to special relativity, it is not that simple. The planetary observer would see the light traveling at its natural speed, 300,000 km/s, just as you would. So how would he see you? Following Einstein’s reasoning to its logical conclusion, as you approached the speed of light, the observer on the planet would see you as gradually slowing down, until the moment you reached the speed of light, when you would appear motionless. But that’s absurd, you say.

It certainly is. But try to disprove it logically.

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