Thursday, 6 January 2011

Achillles and the Tortoise, which came first... The Easy Way, or The Hard Way

The Math Forum introduces Zeno's Paradox this way, "The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved. "

One of the most famous is the paradox of Achilles and the Tortoise. Here is one explanation of the tale, from a web site called, "Platonic Realms".
Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Iliad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles.
The original goes something like this:

The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.
“How big a head start do you need?” he asked the Tortoise with a smile.
“Ten meters,” the latter replied.
Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”
“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”
“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”
“Very quickly,” Achilles affirmed.
“And in that time, how far should I have gone, do you think?”
“Perhaps a meter – no more,” said Achilles after a moment's thought.
“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”
“Very quickly indeed!”
“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

“Ye-es,” said Achilles slowly.
“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.
Achilles said nothing.
“And so you see, in each moment you must be catching up the distance between us, and yet I – at the same time – will be adding a new distance, however small, for you to catch up again.”
“Indeed, it must be so,” said Achilles wearily.
“And so you can never catch up,” the Tortoise concluded sympathetically.
“You are right, as always,” said Achilles sadly – and conceded the race.

I imagine even in 450 BC it was common to have a faster person overtaking a slower person, so Zeno's real question was not about does Achilles really catch the tortoise, but more about the relationship of time and motion.

One way to solve it employs only simple algebra. The tortoise starts at position 10, and runs 1 meter per unit of time, t, so he will be at 10 + t meters for any t greater than zero. The speedy Achilles will start at 0 and advance 10 meters per unit of time, so he will be at 10 t meters. We seek the moment of capture when the two distances are equal, and the easy algebra gives t= 10/9 or 1 1/9 units of time, so Achilles will have covered 11 1/9 meters. The tortoise will have covered 1 1/9 meters.

"Easy Peasy", as the British folk say. But there is another way to do it, a somewhat harder way, using a geometric series. As the tortoise said, by the time Archimedes moves the ten yard head start, the tortoise will move one yard.. and in fact, during each interval that Achilles moves to catch up with the tortoise, the tortoise will move 1/10 of the distance Achilles moves. In successive pursuits then, Achilles will move 10, 1, .1, .01..... meters. The sum of the geometric series is the same as above.

It appears that the first person to solve this using the infinite series was Gregory St. Vincent in 1647. Ok, I can accept and appreciate that. In fact his method was sort of a "limit" process. Here is an explanation of the importance of his method from a website at Fairfield University.

Gregory (St Vincent) was the first to apply geometric series to the "Achilles" problem of Zeno (in which the tortoise always wins the race with the swift Achilles since he has an unbeatable head start) and to look upon the paradox as a question in the summation of an infinite series. Moreover, Gregory was the first to state the exact time and place of overtaking the tortoise. He spoke of the limit as an obstacle against further advance, similar to a rigid wall. Apparently, he was not troubled by the fact that in his theory the variable does not reach its limit. His exposition of the "Achilles" paradox was favorably received by Leibniz and by other geometers over a century later.

OK, now read the bold (I added the emphases) again. Can that be true? Viete had come and gone with his novel use of letters as variables; was it still possible that no one had solved the problem algebraically? As improbable as that seems, I found the exact same quote (obviously the uncredited source of the above) in Cajori's, "A History of Mathematics" on page 182.

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