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© Copyright 1997, Jim Loy
Among the most famous of Zeno's "paradoxes" involves Achilles and the tortoise, who are going to run a race. Achilles, being confident of victory, gives the tortoise a head start. Zeno supposedly proves that Achilles can never overtake the tortoise. Here, I paraphrase Zeno's argument:
Before Achilles can overtake the tortoise, he must first run to point A, where the tortoise started. But then the tortoise has crawled to point B. Now Achilles must run to point B. But the tortoise has gone to point C, etc. Achilles is stuck in a situation in which he gets closer and closer to the tortoise, but never catches him.
What Zeno is doing here, and in one of his other paradoxes, is to divide Achilles' journey into an infinite number of pieces. This is certainly permissible, as any line segment can be divided into an infinite number of points or line segments. This, in effect, divides Achilles' run into an infinite number of tasks. He must pass point A, then B, then C, etc. And what Zeno is arguing is that you can't do an infinite number of tasks in a finite amount of time. Why not?
Zeno says that you can divide a line into an infinite number of pieces. And then he says that you cannot divide a time interval into an infinite number of pieces. This is inconsistent.
There is no paradox here. Zeno was just showing (pretending?) some ignorance of the nature of time. A time interval is just another line segment (when you graph it), that you can divide up in any way you want.
Addendum #1:
Here is an animation which counts to infinity in 8 seconds. Actually, I cheated. There are not an infinite number of frames in this animation. I skipped 1/100 second (approximately). "1" takes 4 seconds, "2" takes 2 seconds, "3" is 1 sec., then .5 sec., .25 sec., etc. If our computers were infinitely faster, we could get in infinitely many frames in that 8 seconds.
I think that Zeno, and Euclid, and Archimedes all had a firm grasp of infinity. 90% of our knowledge of infinity is from these three people. We did not have to wait for Newton and Cantor to explain it to us. They merely clarified some of the details. Zeno may have been puzzled, somewhat. But, I think he had infinity mostly figured out. Euclid (in defining pi) and Archimedes (in estimating pi) used geometric objects with an infinite number of sides, as a limit (without using the term "limit"), many centuries before Newton and Leibniz. Paradox (self-contradiction) is an important way in which Euclid and Archimedes disproved things. They showed no doubts about the legitimacy of dividing something into an infinite number of pieces.
A finite length can be divided up into an infinite number of pieces, all of zero length. You can imagine that, can't you. Just divide a length into halves, then fourths, then eighths, etc. But, in the Zeno story above, we find that none of the pieces is of zero length. They are all, infinitely many of them, longer than zero length. That may be counter-intuitive. But, it obviously is no paradox, as the mathematics is simple and clear.
Little-known story: Achilles didn't win the above race. So, he challenged the tortoise to a pole vault competition, double or nothing. The tortoise's pole bent impressively, before it catapulted him out of Greece, never to be seen again. I made that one up.
Addendum #2:
It seems that there are plenty of people who think that Zeno's paradoxes are real paradoxes which show a basic inconsistency of science and the universe. It is certainly possible that Zeno thought so, too. It turns out that there is some reason to believe that Zeno thought that his paradoxes were real paradoxes.
There were apparently three Zenos. One was an emperor of the Eastern Roman Empire, in the 5th Century AD. Zeno of Citium founded Stoicism in the 4th Century BC. And Zeno of Elea (5th Century BC) was the Zeno of the paradoxes.
To me, Zeno's arrow paradox seems much more interesting than his other paradoxes. It essentially says that if you examine an arrow in flight at one instant in time, it would appear to be no different from an arrow just hanging in the air. What is different about the two arrows that gives one motion and the other no motion? In other words, if you could stop the action (freeze the movie), what is the difference between a moving and a nonmoving object?
Addendum #3:
I have neglected to present all four of Zeno's paradoxes, as they are all really the same "paradox." And, of course, they are not really paradoxes, as they do not show the contradictions that they pretend to show. Anyway, here they are:
Thinking about this unclear stadium paradox, it would seem that Zeno thinks that both time and distance have some smallest irreducible size (atoms of time and distance, if you will). And if you divide motion at twice the speed (one observer as seen by the observer going in the opposite direction) into these smallest pieces of time and distance then, from the point of view of a stationary observer, the moving people go half the smallest possible piece of distance. And we have the opposite effect, from the point of view of a moving observer, the other moving person moves the same smallest possible distance in half the smallest possible time.
Above, I thought that Zeno accepted that you could divide up any distance into infinitely many pieces, and that he rejected dividing up time into infinitely many pieces. Here it would seem that he rejected both. That is his invalid assumption in all four "paradoxes."
Addendum #4:
There is a site on the WWW which says that the above paradoxes are real, and that they show that motion is not continuous but discrete. I would say that saying that motion is discrete is where Zeno went wrong, especially in the Stadium paradox. Actually Zeno assumed that distance is continuous, and time is discrete. He may have assumed that both were discrete in the Stadium paradox. Assuming that both distance and time are discrete will lead to the same problems. But if both distance and time are continuous, then there is no problem, and none of the above is a real paradox.
Also see Aristotle's Wheel Paradox, which is similar.