The Problem of Zeno: Part I

There is an argument called Zeno's Paradox which shows unequivocally that motion cannot exist. Hold on, you say, I can move about quite nicely, so how can it be that motion does not exist? Consider this post and see for yourself. Achilles the hero and the tortoise were very good friends in Zeno's day. Achilles always had the upper hand when it came to physical toughness, but the tortoise had a very crafty mind. One day, Achilles challenged the tortoise to a race. Achilles was so sure that he could beat the tortoise, he offered to give the tortoise a head start. The tortoise then politely let Achilles in on our paradox. He said that if Achilles were to give the tortoise a head start, he would never pass the tortoise and thus the tortoise would win the race. Achilles thought this was foolish and demanded an explanation from the tortoise. His explanation may have gone something like this.

"Now Achilles, is it not true that when the race begins, I will be some distance ahead of you?"

"Yes that is true, for I will give you a head start."

"Indeed, but before you can pass me, you must first pass the point where I start the race, correct?"

"Yes, this is true."

"But in that time, I will have moved forward."

"If you intend on participating in the race!"

"Quite. However, in order to pass me, you will then have to reach my new position."

"Agreed."

"But by that time, I again will have moved forward some distance."

(It is at this point that Achilles begins to realize that he will not win the race.)

"But-"

"And so you see Achilles, you will never pass me, because you will have to reach a point I was at previously before passing me."

Begrudgingly, Achilles responded, "Tortoise, you are correct, I cannot win if I give you a head start. I concede the race."

Examine this argument, try to understand what is going on here. See how the tortoise wins. Is there a flaw in his argument? If so, what is it? And if there is a flaw in his argument, why did Achilles miss it? I will publish my interpretation in my next post two weeks from now. In the interim, I am presenting a talk for Student Research Week at Texas A&M discussing this subject. It will be Wednesday the 29th around 9:00 AM in Rudder tower. I will present my interpretation in the hopes of showing, under a careful formulation, that this is not a paradox after all, but a genuine theorem of mathematics.