# Achilles and the Tortoise

Zeno of Elea (5^{th} century BC) came up with paradoxes that have been debated ever since. The one, perhaps the most famous, concerns the race between Achilles, the greatest warrior of Homer's Iliad, and a tortoise. Here's a Description in the words of Bertrand Russell

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*We can now understand why Zeno believed that Achilles cannot overtake the tortoise and why as a matter of fact he can overtake it. We shall see that all the people who disagreed with Zeno had no right to do so, because they all accepted premises from which his conclusion followed. The argument is this: Let Achilles and the tortoise start along a road at the same time, the tortoise (as is only fair) being allowed a handicap. Let Achilles go twice as fast as the tortoise, or ten times or a hundred times as fast. Then he will never reach the tortoise. For at every moment the tortoise is somewhere and Achilles is somewhere; and neither is ever twice in the same place while the race is going on. Thus the tortoise goes to just as many places as Achilles does, because each is in one place at one moment, and in another at any other moment. But if Achilles were to catch up with the tortoise, the places where the tortoise would have been would be only part of the places where Achilles would have been. Here, we must suppose, Zeno appealed to the maxim that the whole has more terms than the part. Thus if Achilles were to overtake the tortoise, he would have been in more places than the tortoise; but we saw that he must, in any period, be in exactly as many places as the tortoise. Hence we infer that he can never catch the tortoise. This argument is strictly correct, if we allow the axiom that the whole has more terms than the part. As the conclusion is absurd, the axiom must be rejected, and then all goes well. But there is no good word to be said for the philosophers of the past two thousand years and more, who have all allowed the axiom and denied the conclusion.*

The retention of this axiom leads to absolute contradictions, while its rejection leads only to oddities.

Book VI in Aristotle's *Physica* is practically devoted to resolving Zeno's paradoxes. In particular, he wrote

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*Zeno's reasoning, however, is fallacious, ...*

*The second is the so-called "Achilles", and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. ... so that the solution must be the same. And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true while it holds a lead: but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed.
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Aristotle argument is at least two-fold. If this is possible to divide the length into infinitely many pieces the same holds for time. And, if the runner proceeds at a constant speed the time intervals are proportional to the corresponding intervals of length. If the latter accumulate to a finite distance so do the former adding up to a finite interval of time. Choosing any finite time interval and the corresponding distance as the units of time and length it is possible to measure in finite terms any interval and any distance that Achilles may need to overtake the tortoise.

This is very close to the solution that accepted nowadays in mathematical circles. The sequences of ever smaller time intervals and distances form a geometric series, both convergent to finite values. Zeno plays on an idea of infinity as something too big to be reached, while the modern view is that, in this instance, infinity is rather tame.

Bertrand Russell continues his discourse on the paradox:

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*The retention of this axiom leads to absolute contradictions, while its rejection leads only to oddities. Some of these oddities, it must be confessed, are very odd. One of them, which I call the paradox of Tristram Shandy, is the converse of the Achilles, and shows that the tortoise, if you give him time, will go just as far as Achilles. Tristram Shandy, as we know, employed two years in chronicling the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that, as years went by, he would be farther and farther from the end of his history. Now I maintain that, if he had lived for ever, and had not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For consider: the hundredth day will be described in the hundredth year, the thousandth in the thousandth year, and so on. Whatever day we may choose as so far on that he cannot hope to reach it, that day will be described in the corresponding year. Thus any day that may be mentioned will be written up sooner or later, and therefore no part of the biography will remain permanently unwritten. This paradoxical but perfectly true proposition depends upon the fact that the number of days in all time is no greater than the number of years.*

Tristram Shandy's paradox takes a curious twist in a probabilistic variant.

### References

- R. McKeon,
*The Basic Works of Aristotle*, HarperOne, 1957 - B. Russell,
*Mathematics and the Metaphysics*, in J. R. Newman,*The World of Mathematics*, Dover Publications, 2003 - L. Sterne,
*The Life and Opinions of Tristram Shandy, Gentleman*, Penguin Classics, 2003

- What Is Infinity?
- What Is Finite?
- Infinity As a Limit
- Cardinal Numbers
- Ordinal Numbers
- Surreal Numbers
- Infinitesimals. Non-standard Analysis
- Various Geometric Infinities
- Paradoxes of Infinity

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