Page 35 - GALIET PHYSICS BLOSSOMS I+

III. INFINITESIMALS AND AXIOMS.
(i) Explain Zeno’s parable of Achilles and the Tortoise, and why Zeno thought that this parable demonstrated a contradiction. Then, explain Archimedes’ idea of an infinitesimal, and how use of this idea allows us to resolve the paradox/contradiction raised by Zeno.
INFINITESIMALS
Zeno’s Paradox
Zeno’s paradox is expressed in a race between Achilles (in Homer, the Greatest and most wrathful of the Grecian Heroes that fought at Troy, but in Zeno, the fastest of all men) and a Tortoise. Briefly, a fast, but handicapped Achilles will never be able to reach the slowest of all animals, a Tortoise, if the tortoise is given an initial advantage or head start. Achilles’ effort is futile given that infinite divisibility exists between point A and B. In its simplest form, let’s suppose Achilles runs ten times faster than the Tortoise. If at the beginning of the race the Tortoise head starts by a meter, when Achilles has traversed this meter, the Tortoise would have traversed a decimeter. When Achilles has traversed this decimeter, the Tortoise would have journeyed a centimeter. When Achilles has traversed this centimeter, the Tortoise would have journeyed a millimeter, and so on, and so on, successively, ad infinitum, in such a way that Achilles will never be able to catch up to the Tortoise, even though, evidently, he is infinitely closer to it. That is, the Tortoise will always have an advantage over Achilles as infinitesimally close the Tortoise may be from Achilles. In other words, the infinite points traversed between a finite A-B interval, will approximate or get as close to zero, but never reach zero. Zeno’s aim was to defend Parmenides’ doctrine, which negated real motion, by positing that motion was illusory. The Achilles and the Tortoise is one of Zeno’s four arguments against motion: (1) the Dichotomy, (2) the Arrow (3) and the Stadium.
Why a contradiction?
The basic paradox or aporia Zeno’s Achilles and the Tortoise raises is how can the finite be actually infinite, or how can the finite or finite intervals be divided ad infinitum or be infinitely divisible. That is, how can we add an infinite number of things, or finite objects, and paradoxically still end up with something that is still finite?29 It seems self evident that to traverse an infinite number of points we need infinite, and not finite time.
29 Dr. Stamp. Lecture Blackboard Notes.
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