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Interesting Paradoxes: Achilles and the Tortoise

Mohammad Yasir is a physics graduate from the University of Delhi and currently enrolled in the master's programme at IIT.

Science and Philosophy have a strange love-hate relationship. While devout followers of the former scorn the idea of the latter, the fact remains that these two subjects are, in various instances, inseparable. The pursuit of philosophy leads to some pretty strange and outlandish situations which are, in the simplest of terms, really fun to examine and solve.

This is the first in a series of articles wherein I would like to discuss some interesting paradoxes that have boggled the human brain across the ages. Let us start with the most famous one: that of Achilles and the Tortoise.

"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."

— Aristotle, Physics, VI:9, 239b15

Achilles and the Tortoise

Achilles and the Tortoise

The Idea Behind the Paradox

Let us suppose that a tortoise is challenged into a race against the mighty warrior Achilles. It assures him that it is going to win as long as Achilles gives it a head start. Full of skepticism, the man laughs and grants the tortoise's wish.

"Sure. I'll wait till you are a hundred metres ahead of me," he says.

As an amused and bored Achilles watches, the tortoise starts crawling and, after a long while, reaches the hundred-metre mark. Now, consider the following steps.

  1. Achilles starts running, while the tortoise slowly ambles a little bit further.
  2. By the time Achilles catches up to the hundred-metre mark, the tortoise has moved forward. Let us say that it has covered an extra 1 metre.
  3. Once again, Achilles runs to catch up, but by the time he reaches the tortoise, the latter has stepped a little further again. Now, it is 0.1 metre ahead of Achilles.

This game of Catch Me If You Can continues, with the tortoise always a few steps ahead and Achilles always just a little bit behind until, eventually, the tortoise wins. The idea is beautifully demonstrated in the picture above. There is always some lead, albeit a tiny one, that the tortoise holds. But we know that to be impossible. Achilles must win. And therein lies the paradox.

What Do the Experts Say?

Interestingly enough, it took almost two millennia to arrive at a resolution to this paradox. It was only in 1821 that Augustin-Louis Cauchy developed a concept which could provide a solution to this problem. Let us look at it in detail.

interesting-paradox-achilles-tortoise
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The above equation demonstrates one of the most revolutionary realizations in the calculus of infinite series. The discrepancy in our theory and practice arises mathematically from the supposition that the above series will diverge to an infinite value, when, in fact, it converges. Don't worry, I'll explain the whole concept in layman's terms after this section. You can skip ahead if you find mathematics boring (I don't blame you if you do).

Augustin-Louis Cauchy provided a satisfactory definition of what the term limit means in mathematics and, further, was able to show that for 0 < r < 1:

a + ar + ar2 + ar3 + ar4 + ... = a/(1-r)

So, imagine that it takes Achilles 10 second to overcome the first hundred meters, 1 second to overcome the next bit, then 0.1 seconds for the next part, and so on. In this case, a = 10 seconds, r = 0.1, and we can use the above limit to calculate that Achilles will overtake the tortoise in 10/(1-0.1) = 11.11 seconds.

That's Too Much Maths!

If you lost me up there with the numeric mumbo-jumbo, here is an explanation in simpler terms.

We are starting with the idea that to catch up with the tortoise, Achilles must cover the first hundred metres, then the next bit, and so on. At first it may look like that this chase won't cease and Achilles will be forced to play catch-up till his defeat.

And That Is Where We Are Wrong

Instead, our statement doesn't mean that he can never overtake his opponent. An infinite number of terms can add up to a finite value. When you sum up all the infinitely many terms, it leads to a small, defined value instead of infinity. Thus, Achilles will take the lead in 11.11 seconds, which is what the mathematics in the previous section was about. After all, you can always say that to go 1 kilometre away from your house, you must first go half the distance (500m), and to do that, you must go half of that, and so on. And yet, the distance you must cover is still finite. Otherwise, you would never be able to reach anywhere at all.

Notice how I have resolved this paradox by considering the time it takes for Achilles to cover the distance between him and the tortoise. That is on purpose. The original problem was written in terms of distance and yet, we are talking about time. And that is precisely what solves our little conundrum.

While the distance can be divided into an infinite number of subintervals, the time it takes Achilles to cover that distance remains finite and thus, his victory is still assured.

Conclusion

The Achilles and the Tortoise problem baffled people for two almost two millennia before a suitable response was conceived. Interestingly enough, it took Mathematics to resolve it.

If you would like further reading about interesting paradoxes, stay tuned since there are more coming. Until then, au revoir.

© 2022 Mohammad Yasir

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