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9.12. Zeno of Elea (495?-435? B.C.)

Zeno of Elea was the first great doubter in mathematics. His paradoxes stumped mathematicians for millennia and provided enough aggravation to lead to numerous discoveries in the attempt to solve them.

Zeno was born in the Greek colony of Elea in southern Italy around 495 B.C. Very little is known about him. He was a student of the philosopher Parmenides and accompanied his teacher on a trip to Athens in 449 B.C. There he met a young Socrates and made enough of an impression to be included as a character in one of Plato's books Parmenides. On his return to Elea he became active in politics and eventually was arrested for taking part in a plot against the city's tyrant Nearchus. For his role in the conspiracy, he was tortured to death. Many stories have arisen about his interrogation. One anecdote claims that when his captors tried to force him to reveal the other conspirators, he named the tyrant's friends. Other stories state that he bit off his tongue and spit it at the tyrant or that he bit off the Nearchus' ear or nose.

Zeno was a philosopher and logician, not a mathematician. He is credited by Aristotle with the invention of the dialectic, a form of debate in which one arguer supports a premise while another one attempts to reduce the idea to nonsense. This style relied heavily on the process of reductio ad absurdum, which is the reduction of an idea to absurdity by finding an inherent contradiction. Zeno wrote only one known work, Epicheiremata, in which he attacks the opponents of his teacher Parmenides.

Zeno's greatest fame is from his paradoxes. With only 200 words surviving from his only book, the only records we have of his brainchilds are from secondary sources, mainly Aristotle. Originally, there were about forty but only eight have survived. The purpose of the arguments was a defense of his teacher's ideas. Parmenides believed that reality was one, immutable and unchanging. Motion, change, time and plurality were all mere illusions. This, of course, attracted many critics. Zeno's paradoxes attempted to show that holding the opposite position, that reality was many, was contradictory and absurd. Therefore, "the one" must be the correct philosophy. Curiously, using Zeno's methods, his own position can also be shown to be contradictory as well.

The four most famous paradoxes are the Dichotomy, the Achilles, the Arrow, and the Stadium.

  1. The Dichotomy: Motion cannot exist because before that which is in motion can reach its destination, it must reach the midpoint of its course, but before it can reach the middle, it must reach the quarterpoint, but before it reaches the quarterpoint, it first must reach the eigthpoint, etc. Hence, motion can never start.
  2. The Achilles: The running Achilles can never catch a crawling tortoise ahead of him because he must first reach where the tortoise started. However, when he reaches there, the tortoise has moved ahead, and Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. Hence the tortoise will always be ahead.
  3. The Arrow: Time is made up of instants, which are the smallest measure and indivisible. An arrow is either in motion or at rest. An arrow cannot move, because for motion to occur, the arrow would have to be in one position at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because by definition, instants are indivisible. Hence, the arrow is always at rest.
  4. The Stadium: Half the time is equal to twice the time. Take the three rows below. They start at the first position. Row A stays stationary while rows B & C move at equal speeds in opposite directions. When they have reached the second position, each B has passed twice as many C's as A's. Thus it takes row B twice as long to pass row A as it does to pass row C. However, the time for rows B & C to reach the position of row A is the same. So half the time is equal to twice the time.
Though all four arguments seem illogical, not to mention confusing, they are not that simple to explain away and lead to some very serious problems for mathematics. To the Greek mathematicians, who had no real concept of convergence or infinity, these reasonings were incomprehensible. Aristotle discarded them as "fallacies" without really showing why and Zeno's paradoxes were hidden away in the mathematical closet for the next 2500 years. For that time, they were reduced mainly as novelties of philosophy. However, they were revived mathematically in the twentieth century by the efforts of people like Bertrand Russell and Lewis Carroll. Today, armed with the tools of converging series and Cantor's theories on infinite sets, these paradoxes can be explained to some satisfaction. However, even today the debate continues on the validity of both the paradoxes and the rationalizations.

For related information of Zeno, see: Georg Cantor , Zeno's paradox.

Sources

Historical information compiled by Paul Golba
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