Two versions of Zeno’s paradox

  1. The story of the hare and the tortoise is well-known. The hare lost because he took a long nap on the way. But, here is an argument that shows that even if the hare were running fast, he could not overtake the tortoise, provided that the tortoise was ahead to begin with. Suppose the hare was at a point A and the tortoise was at point B ahead of A on the race track. By the time, the hare reached B, the tortoise would have moved to C ahead of B. Again when the hare reaches C; the tortoise would have moved to D ahead of C. And, so on, ad fnfinitum. So, the tortoise will always be ahead of the hare! What is the fallacy in this argument? This is one of the paradoxes of the Greek philosopher Zeno relating to infinity.
  2. Here is another of Zeno’s paradoxes. A runner wishes to cover a distance of 1 km. Zeno uses the following argument to show that he/she will never succeed if, at each stage, he/she aims at covering half the remaining distance. For to do so, first he/she will have to cover half the remaining distance. That leaves \frac{1}{4} km, of which he/she next covers half the distance. if he/she adopts the strategy of covering half the remaining distance each time. He/she has an infinity of operations lying ahead of him/her, with the destination always lying just that little bit ahead. What is wrong with this reasoning?

Ref: Fun and Fundamentals of Mathematics, Jayant V Narlikar and Mangala Narlikar, Universities Press.

More fun later,

Nalin Pithwa

2 Comments

  1. Anubhav C. Singh
    Posted August 10, 2016 at 3:00 pm | Permalink | Reply

    Ah……. This question. I can’t say how many times I have foxed my mates by asking them this riddle (ask them a standard math question- B starts 5 minutes later than A as speed two times that of A, find when B overtakes A- and then bring Zeno’s paradox in to counter whatever answer they give, and in conclusion end up with no friends). Technically, I agree with this paradox, one is not supposed to (as in example 1), reach their destination; but in real life the distances soon become so short then one ends up ignoring it. This is not a mathematical answer, just a statement that 0.00000001 cm will be taken as 0 cm in regular use.
    So in the question 1: there appears a point where point (k+1) is barely any length away from point k so the rabbit (jumps?) the distance and ignores it. Similarly for problem two. This is the reasoning I give myself, but am still not satisfied with it.

    I still don’t know a mathematical answer/proof though.

  2. Posted August 11, 2016 at 12:13 am | Permalink | Reply

    It’s the definition of a limit of a sequence/function. Or, a converging GP !

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