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Thread: Zeno's paradox of Achilles and the tortoise

  1. #1
    won't be missed Array /DG/'s Avatar
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    Mar 2009

    Default Zeno's paradox of Achilles and the tortoise

    Many of you guys seem to be well versed in philosophy. Can anyone help me understand this? If you don't know about it, here is the Wikipedia page. Or you can just read this quote:
    "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." – as recounted by Aristotle, Physics VI:9, 239b15In the paradox of Achilles and the Tortoise,

    Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. 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 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. 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.
    The other day my philosophy professor told us about this. I still don't get it.

    Here is my attempt to explain this the way my professor did :
    Let's pretend that Achilles can go twice the speed of the tortoise. The turtle is given a head start of 128m before Achilles can start. When Achilles gets to 128m, he is not yet in the tortoise's position because it is now 64m in front of Achilles. When Achilles goes another 64m, he still hasn't caught up with the tortoise, which is now 32m away from Achilles. Again, when Achilles travels another 32m, the tortoise will be 16m in front of him. When Achilles goes another 16m, the tortoise will be 8m in front of him. Etc., etc. So you see, no matter how far Achilles goes, he can never pass the turtle. Blah blah something something attempted to show that the world is illusory or something.

    This doesn't make any sense to me. Why in the world is this considered a paradox? It is an error in the understanding of simple mathematics.

    *edit* Okay apparently it isn't quite simple mathematics. You need calculus to prove this problem incorrect. In the paradox, we are creating an infinite series of numbers. Zeno's paradox relies on the sum of an infinite series to always be an infinite number. This is not so. Through calculus, we can find that the sum of an infinite series can be an actual number. It is at this number where Achille's will pass the tortoise. It still doesn't explain why it is still a paradox today.

  2. #2
    Senior Member Array burymecloser's Avatar
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    You're kind of missing the point. Don't look at this as a math problem. Or if you insist on using that lens, you need to play by Zeno's rules: each time Achilles reaches the tortoise's previous position, it has advanced and maintained its lead. It's clever and interesting, and it conclusively proves that the world is illusory or something.
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  3. #3
    redundant descriptor Array netzealot's Avatar
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    Many people, and surprisingly enough perhaps even for Aristotle at this point, mistake "ideas without a vector between thought and finite reality" for philosophy. In this case, there is nothing to suggest that the vector occurs later in the metaphysical analogy, so it simply falls short of the standard of legitimate philosophy.

    Even with a very small discrepancy in speed, say 7 mph and 6 mph, and a mile-long head start, it would take only an hour for the faster runner to overtake the slower. Of course, someone will claim I'm not "getting it". I'd like to see them make up a reason why, though.

  4. #4
    .~ *aĉa virino* ~. Array Totenkindly's Avatar
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    To rephrase, they're just claiming that any time you make a calculation, the chaser needs a certain amount of time (even if it is a very very minute amount) to reach the position where the prey was positioned, but by that time the prey has always stepped a little bit further (even if it is only the slightest bit) and is technically no longer where the chaser was running to. That much people agree on.

    That's the "paradox" -- IRL, the guy obviously catches the turtle; mathematically, the way it works, the turtle will always be a bit ahead. Why? Zeno apparently is confused about it.

    The discrepancy is because the math problem and the real-life example are using two different frames of reference. The math problem cuts off the chase before the runner catches the turtle; meanwhile, IRL, there are no arbitrary endpoints that bracket the chase OR the runner can be considered to be running to the point where the turtle WILL be in the same space of time, and NOT to where the turtle WAS.

    So in that latter case, IRL, the point of overlap falls within the scenario, but the math problem arbitrarily places the point of conjunction outside the bounds of the problem. The "paradox" occurs because of the constraints on the math problem, which are unrealistic -- it "looks like" a paradox, but is basically a mathematical illusion.
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  5. #5
    Member Array EEW's Avatar
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    I agree with Jennifer. But it is a paradox. Not a mathematical illusion.
    This logic can be explained in mathematical ways: with limitation. To cut short:the limit of the runner will be (e.g.) 111,11111.... An infinite point number. That is when the runner would catch its prey. But because it is infinite, it would be near impossible to catch the prey. But that is wrong of course, - in reality, that is.
    That is exactly why it is a paradox.
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  6. #6


    Comes down to discrete math versus continuous math. Continuously we understand that Achilles will catch the Tortoise at some point. Discretely we understand he could never catch up to the Tortoise because we have an infinite amount of catching up to do (even though it converges). The paradox is one of reasoning reality. If you were Achilles and took the discrete approach, you would never succeed because your work would never end, but if you took the continuous approach, you would succeed.

    Of course, the discrete solution here does seem to warp ideas of space and time by giving the iterations more importance over time and space itself, whereas the continuous solution does not. So maybe you could consider it a logical fallacy, rather than a paradox.

  7. #7


    This is a key problem of spatial understanding, and why Einstein failed in his final days' pursuits.

    The universe is not geometrical, it is quantifiable. As soon as you stop thinking of things as lines and start thinking of it as counting beans the problem is solved.

  8. #8


    ^ How is that not what led to the paradox?

  9. #9


    These stories are so old, nothing we can say about these.

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