Achilles and the tortoise

Posted: April 28, 2009 in logic, paradox
Tags: ,

“In a race, the quickest runner can never overtake the slowest unless they start from the same point, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead”.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

PS: Adopted contents

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Comments
  1. Sahasranand says:

    Even simple mathematics tells us that Achilles will be able to overtake the tortoise. The reason is that, the distance covered by Achilles is like, say, 100 + 10 + 1 + 0.1 + 0.01 + ….Although this is an infinite series, it has a finite sum,S, like we have the value for ‘e’ (=2.072, I think).

    If Achilles is able to cover this distance S, surely has he overtook the tortoise.

    Pre requisite : The tortoise must not quit before Achilles gets there:)

  2. in fact there is no finite sum…even if there is that sum will not be less than the total distance to be covered…and more interestingly neither the tortoise can touch the finish line..ever..he he…guess how?

  3. Sahasranand says:

    I am unable to follow that reasoning of yours, I’m afraid. Perhaps this link is in place here.

    http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

  4. Sahasranand says:

    I am unable to follow your reasoning, I’m afraid. Perhaps this link is in place.

    http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

  5. Sahasranand says:

    Yes… I can follow that reasoning of yours. However, this link seems to relevant.

    http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

  6. Yeah u pointed out the rite case..i was refering to Zeno’s paradox only..may be u are interested with the dichotomy paradox as well..

  7. And I am sorry for that late response…ur comment was marked as spam by wordpress..so i missed it..

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