So, I’ve had to revisit some old calculus lessons for a software development project I’m involved with. In this effort, I came across Zeno’s famous paradox of Achilles (or a rabbit) and the tortoise (that’s Zeno of Elea, from the 5th century B.C.). Fascinating! A paradox is a statement that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that is contradictory. For instance, “I always tell a lie” is a paradox because if it is true, it must be false.
For those of you who are math oriented, not I, this paradox is imbedded in the study of infinite series, and even more precisely, convergent series. I will not attempt to explain it mathematically. Alas, I am merely a storyteller (if even that, this is more or less copy and pasted from my textbook.)
Achilles is racing a tortoise. Our generous hero gives the tortoise a 100 yard head start. Achilles runs at 20 mph; the tortoise “runs” at 2 mph. Zeno used the following argument to “prove” that Achilles will never catch or pass the tortoise. If you’re persuaded by the “proof,” you’ve really got to get out more. Figure 1, below, provides two snapshots once the race is underway. They will be used in this explanation.
You take the first snapshot the instant Achilles reaches the point where the tortoise started. By the time Achilles gets there, the tortoise has “raced” forward and is now 10 yards ahead of Achilles. The tortoise moves a tenth as fast as Achilles, so in the time it takes Achilles to travel 100 yards, the tortoise covers a tenth as much ground, or 10 yards. If you do the math, you find that it took Achilles about 10 seconds to run the 100 yards (for the sake of argument, let’s call it exactly 10 seconds).
You have a really fast Polaroid, so you look at your first photo and note precisely where the tortoise is as Achilles crosses the tortoise’s starting point. The tortoise’s position is point A in the first photo. Then you take your second photo when Achilles reaches point A, which takes him about one more second. In that second, the tortoise has moved ahead to point B. You take your third photo (not shown) when Achilles reaches point B and the tortoise has moved ahead to point C.
Every time Achilles reaches the point where the tortoise was, you take another photo. There is no end to this series of photographs. Assuming you and your camera can work infinitely fast, you will take an infinite number of photos. And every single time Achilles reaches the point where the tortoise was, the tortoise has covered more ground – even if only a millimeter or millionth of a millimeter. Thus, the argument goes: because you can never get to the end of your infinite series of photos, Achilles can never catch the tortoise. The tortoise will always cover a tenth of the distance of Achilles.
Well, as everyone knows, Achilles does in fact reach and pass the tortoise. Therein lies the paradox.