Achilles and the Tortoise Paradox

Very well; I shall accept and make the first argument, thus intending to pass on the last round as you have requested. I would like to thank you for this interesting debate which is sure to reveal some interesting thoughts and ideas, as well as to wish you the best of luck. My understanding of this debate is that I am to defend the paradox of Achilles and the Tortoise. According to dictionary.com, there are four definitions of "paradox" (1): 1. a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth. 2. a self-contradictory and false proposition. 3. any person, thing, or situation exhibiting an apparently contradictory nature. 4. an opinion or statement contrary to commonly accepted opinion. Definitions three and four merely state that something is apparently contradictory or contrary to something, and make no statements about their actual truth. Thus, I do not think this is what my honorable opponent wishes to debate. This leaves definitions one and two which, rather ironically, are apparent opposites. However, based on what I have inferred by Con's argument, I assume definition one is the one intended for me to defend. In order to eschew confusion, I shall henceforth refer to something meeting the conditions of definition one as a Paradox, and the example of Achilles and the Tortoise as a paradox, as we have not yet ascertained that it is a Paradox. As such, I have the burden of proving: [1] that the paradox of Achilles and the Tortoise seems self-contradictory or absurd, and [2] that the paradox of Achilles and the Tortoise expresses a possible truth. I think that there is little dispute over [1], but I shall nonetheless make my case. Achilles can run faster than the tortoise and, as I'm sure nearly anyone will agree, a faster person (or creature) will always manage to overtake a slower one, provided that we assume the rates of each are constant, neither one tiring or requiring a break of any sort. I think that these assumptions are fairly valid, as they seem to have been implicitly assumed in the original stating of the paradox. This paradox states that Achilles will not catch the tortoise, and since Achilles can run faster, this in contradictory. Thus, this paradox meets the first condition of being a Paradox. I shall now consider [2]. If we consider in turn each iteration of Achilles running to the tortoise's previous position, Achilles gradually will get closer to the tortoise. My contention is that there are two ways of looking at this. First, we could consider that the amount of time each successful iteration takes also approaches zero, and so Achilles will reach the tortoise after a finite amount of time. We could go on to prove this, with one method or another, but I do not feel that is necessary. However, there appears to be little conflict over whether or not this is paradoxical and, indeed, this interpretation is not my main point. The other interpretation, however, is. Regardless of the number of iterations considered, Achilles will not catch the tortoise. One could argue that he would get close enough for most practical purposes, but this is not the debate. The most relevant definition of overtake that I find is "to catch up with and pass, as in a race; move by" (2); by this definition, even if Achilles reaches the tortoise after an infinite number of iterations, he will still not be ahead and thus not have overtaken the tortoise. Either way, this interpretation clearly shows some "possible truth" that, after an infinite number of iterations, the tortoise will still not have been overtaken. Thus, my contention is that Zeno's paradox of Achilles and the Tortoise is, indeed, a Paradox. (1)http://dictionary.reference.com... (2)http://dictionary.reference.com...