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

- 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.
- 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.
- 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.
- 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.
- 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.
- The paradoxes that Zeno gave regarding motion are more perplexing.
- Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium.
- Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible.
- One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements.
- However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements.
- for time is not composed of indivisible 'nows', no more than is any other magnitude.
- However, this is considered by some to be irrelevant to Zeno's argument.
- Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either.
- Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution.
- Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position.
- Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.
- So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.
- As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals.
- Some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals.
- Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes.
- The most famous of Zeno's arguments is undoubtedly the Achilles.
- As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position.
- Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments.
- One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno.
- Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms.
- After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....
- Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus.
- It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics.
- realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.
- According to his description, Zeno proposed a universe consisting of several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or empty space.
- However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School.

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

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Analysis, Ancient Greek, Origin Italy, Physics, Set Theory, Special Numbers And Numerals

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**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive