Philosophy paper
Aristotle ascribes to Zeno the ‘Achilles argument’: ‘This says that the slowest runner will never be overtaken by the fastest; for the pursuer must first come to the place from which the pursued set out, so that the slower must always be a little ahead.’ Explain the argument, and discuss whether it succeeds.
Aristotle inferred that motion is through a space that is infinitely divisible, but neither infinite in extent, nor actually infinitely divided. Bertrand
Russell (in his early and brilliant book Our Knowledge of the External World) inferred that space and time form a continuum, which is an actual infinity: between any two points, there exists a third point.
An actual infinity can be played in two ways: on the one hand, it can seem to imply that any movement through space requires performing an infinite series of tasks of progressing from one point to another; on the other, to every further division in space there corresponds a further division in time, so that there is always enough time to move a finite distance. It seems fine to allow that Achilles can traverse all the points necessary for his reaching a position just ahead of the tortoise so long as this doesn’t involve his performing, one after another, each of an infinite series of finite tasks.
This essay asks for an explanation and critical evaluation of Zeno’s so-called Achilles argument (or paradox). As such, it asks for an assessment of whether one of Zeno’s paradoxes genuinely challenges assumptions about space and time or is merely cleverly fallacious. Solid responses to this question will first of all set out the argument and explain why it is taken to present a paradox, describing the underlying premises as clearly as possible.
Good responses to the question will demonstrate an understanding of the relation between the assumption that space and time are infinitely divisible and motion that takes an infinite number of steps. Very good responses will consider the distinction between something that can be divided infinitely and something that is infinitely long (has infinite extension), and explain why the former does not imply the latter. Excellent responses will also show an understanding of Aristotle’s suggested resolution of the paradox by distinguishing between potential and actual divisibility, as well as show some familiarity with the secondary literature.
