PHILOSOPHY

Philosophy: Zeno's Achilles argument



Philosophy: Zeno's Achilles argument

Question: Explain Zeno's Achilles argument. Is Aristotle's criticism of the argument accurate? Does Aristotle beg the question? Is there another answer to Zeno?

In Zeno's Achilles argument, he argues, “The second is the one called the Achilles. This is to the effect that the slowest as it runs will never be caught by the quickest. The pursuer must first reach the point from which the pursued departed, so that the slower must always be some distance in front. This is the same argument as the Dichotomy, but it differs in not dividing the given magnitude in half.” (A Pre-Socratic's Reader, p.68)

In this argument, Achilles and tortoise have a footrace. Achilles allows tortoise to start 500 meters head. Assume Achilles and tortoise keep running at a constant speed. Tortoise runs very low, but, Achilles runs very fast. After some finite time, Achilles has run and reached the 500 meters point where the tortoise started. At this finite time, the Tortoise has run a relatively shorter distance, about 50 meters. So it will take Achilles more time to run those 50 meters, in the mean time, the Tortoise will run further distance. So it will always take Achilles sometime to reach the point where the Tortoise has passed. Since there are infinitely points that Achilles needs to reach where the Tortoise has passed, and the time is also finite. Therefore, Achilles will never catch up the Tortoise in finite time.

The paradox of this argument is that it is impossible to finish innumerable things at a finite time. While, Aristotle criticizes that “Zeno's argument is falsely assumes that it is impossible to traverse or come into contact with an infinite number of things individually in a finite time” (Aristotle, Physics 6.2 233a21-31). Aristotle believes that things are continuous called infinite in two ways: infinite in the division and infinite in extremities. This means things that are divisible into infinitely small things and can be added up to infinitely large things. So it is impossible to contact with infinitely things at a finite time, but it is possible to contact with infinitely divisible things in finite time, because time itself can be divided into infinity. Then Aristotle raises two concepts based on his infinite definition: actual infinity and potentially infinity. While actual infinity suggests that things can be divided into infinitely small things, also things can be made up to infinitely large things. For instance, there exists a set of all natural numbers since all natural numbers can be calculated. Potentially infinity believes that things cannot be divided into infinitely small things, if this is the case then infinitely small things can also be divided into smaller things. In the same way, potentially infinity believes that there are no infinitely large things. Therefore, Aristotle criticizes that if Zeno's Achilles argument based on the actual infinite system, then Achilles will never overtake the tortoise because it is impossible for a thing to catch up things by passing over ...