Mathematicians and their Contributions

Zeno of Elea

Zeno of Elea

Born: about 490 BC in Elea, Lucania (now southern Italy)
Died: about 425 BC in Elea, Lucania (now southern Italy)

Zeno of Elea was a Greek philosopher famous for posing so-called paradoxes which challenged mathematicians' view of the real world for many centuries.

Very little is known of the life of Zeno of Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato.

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.

Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.

Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details.

Zeno's book of forty paradoxes was, according to Plato [8]:-

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Proclus also described the work and confirms that [1]:-

... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.

In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-