# Zeno's paradoxes

Zeno's paradoxes are a series of paradoxes posed by the philosopher Zeno of Elea (c.490–c.425 BC).
Little is known about Zeno's life. He was born in Elea (now Lucania) in
southern Italy and was a friend and student of Parmenides. None of his writings
survive but he is known to have written a book, which Proclus says contained
40 paradoxes. Four of these, which all concern motion, have had a profound
influence on the development of mathematics. They are described in Aristotle's
great work *Physics* and are called the Dichotomy, Achilles (and
the Tortoise), the Arrow, and the Stadium.

The Dichotomy argues that "there is no motion because that which is moved must arrive at the middle of its course before it arrives at the end." In order the traverse a line segment it's necessary to reach the halfway point, but this requires first reach the 1/4-way point, which first requires reaching the 1/8-way point, and so on without end. Hence motion can never begin. This problem isn't alleviated by the well known infinite sum 1/2 + 1/4 + 1/8 + ... = 1 because Zeno is effectively insisting that the sum be tackled in the reverse direction. What is the first term in such a series?

Zeno's paradox of Achilles is told by Aristotle in this way: "The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead." Thus, Achilles, however fast he runs, will never catch the plodding Tortoise who started first. And yet, of course, in the real world, faster things do overtake slower ones. So how is the paradox to be solved? The German set theorist Adolf Frankel (1891-1965) is one of many modern mathematicians (Bertrand Russell is another) who have pointed out that 2,000 years of attempted explanations have not cleared away the mysteries of Zeno's paradoxes:

Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.