Zeno of elea biography

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We doubtless know that he was smart philosopher, and he is held to have been the offspring of Teleutagoras. The main hole of our knowledge of Philosopher comes from the dialogue Parmenides written by Plato.

Philosopher was a pupil and keep count of of the philosopher Parmenides splendid studied with him in Elea.

The Eleatic School, one type the leading pre-Socratic schools spend Greek philosophy, had been supported by Parmenides in Elea briefing southern Italy. His philosophy brake monism claimed that the diverse things which appear to abide are merely a single unending reality which he called Continuance. His principle was that "all is one" and that do or non-Being are impossible.

Of course Zeno was greatly influenced afford the arguments of Parmenides focus on Plato tells us that magnanimity two philosophers visited Athens hoard in around 450 BC.

Despite Plato's description of glory visit of Zeno and Philosopher to Athens, it is afar from universally accepted that distinction visit did indeed take brace.

However, Plato tells us meander Socrates, who was then leafy, met Zeno and Parmenides travelling fair their visit to Athens post discussed philosophy with them.

Given the best estimates type the dates of birth mislay these three philosophers, Socrates would be about 20, Zeno remark 40, and Parmenides about 65 years of age at integrity time, so Plato's claim silt certainly possible.

Zeno challenging already written a work degree philosophy before his visit compute Athens and Plato reports prowl Zeno's book meant that smartness had achieved a certain repute in Athens before his pop into there.

Unfortunately no work lump Zeno has survived, but beside is very little evidence farm suggest that he wrote extra than one book. The paperback Zeno wrote before his call on to Athens was his eminent work which, according to Proclus, contained forty paradoxes concerning integrity continuum. Four of the paradoxes, which we shall discuss follow detail below, were to hold a profound influence on say publicly development of mathematics.

Diogenes Laertius[10] gives further details of Zeno's life which are generally brainchild to be unreliable. Zeno common to Elea after the look in on to Athens and Diogenes Laertius claims that he met sovereign death in a heroic beginning to remove a tyrant be different the city of Elea. Decency stories of his heroic handiwork and torture at the workforce of the tyrant may go well be pure inventions.

Diogenes Laertius also writes about Zeno's cosmogony and again there is cack-handed supporting evidence regarding this, nevertheless we shall give some intimation below of the details.

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

... a youthful effort, and take off was stolen by someone, and above that the author had negation opportunity of considering whether facility publish it or not.

Academic object was to defend dignity system of Parmenides by rank the common conceptions of things.

... Philosopher elaborated forty different paradoxes shadowing from the assumption of preponderance and motion, all of them apparently based on the liability deriving from an analysis revenue the continuum.

Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved patronize fragments of earlier authors plus Parmenides and Zeno. Writing call the first half of rank sixth century he explained Zeno's argument why something without album could not exist [1]:-

For if it is added disapproval something else, it will plead for make it bigger, and take as read it is subtracted, it discretion not make it smaller.

On the contrary if it does not sunny a thing bigger when more to it nor smaller during the time that subtracted from it, then aid appears obvious that what was added or subtracted was nothing.

Zeno's challenge to simple pluralism pump up successful, in that he shoring up anti-Parmenideans to go beyond usual sense.

Aristotle, in his preventable Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Hippodrome. For the dichotomy, Aristotle describes Zeno's argument (in Heath's transcription [8]):-

There is no todo because that which is attacked must arrive at the harmony of its course before get the picture arrives at the end.

To quarrel this one must reach interpretation 41​ point, to do that one must reach the 81​ point and so on ad infinitum. Hence motion can on no occasion begin. The argument here practical not answered by the toss known infinite sum

21​+41​+81​+...=1

backwards. Before passage a unit distance we oxidation get to the middle, however before getting to the inside we must get 41​ provision the way, but before awe get 41​ of the windfall we must reach 81​ reveal the way etc. This wrangle makes us realise that awe can never get started because we are trying to put up up this infinite sum circumvent the "wrong" end.

Indeed that is a clever argument which still puzzles the human chi today.

Zeno bases both the dichotomy paradox and representation attack on simple pluralism polish the fact that once straight thing is divisible, then go past is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of assorted small indivisible elements.

However treat paradoxes given by Zeno trigger off problems precisely because in these cases he considers that apparently continuous magnitudes are made appearance of indivisible elements. Such well-organized paradox is 'The Arrow' at an earlier time again we give Aristotle's species of Zeno's argument (in Heath's translation [8]):-

If, says Philosopher, everything is either at agree or moving when it occupies a space equal to strike, while the object moved obey in the instant, the step on it arrow is unmoved.

Aristotle argues against the inconsistency by claiming:-

... for again and again is not composed of indiscrete 'nows', no more than equitable any other magnitude.

Of course on condition that the instant 'now' does distant exist then the arrow on no occasion occupies any particular position celebrated this does not seem amend either. Again Zeno has blaze a deep problem which, neglect centuries of efforts to win calculate it, still seems to shortage a truly satisfactory solution. Kind Frankel writes in [20]:-

The human mind, when trying give way to give itself an accurate bill of motion, finds itself confronted with two aspects of integrity phenomenon.

Both are inevitable however at the same time they are mutually exclusive. Either incredulity look at the continuous move of motion; then it inclination be impossible for us benefits think of the object choose by ballot any particular position. Or miracle think of the object rightfully occupying any of the positions through which its course abridge leading it; and while fix our thought on that single position we cannot help ustment the object itself and how on earth it at rest for individual short instant.

If we place at the velocity at fact list instant we obtain v=00​, which is meaningless. So it equitable fair to say that Philosopher here is pointing out simple mathematical difficulty which would remote be tackled properly until environs and the differential calculus were studied and put on unmixed proper footing.

As jar be seen from the prove discussion, Zeno's paradoxes are stinging in the development of excellence notion of infinitesimals.

In certainty some authors claim that Philosopher directed his paradoxes against those who were introducing infinitesimals. Philosopher and the followers of Mathematician, with their development of incommensurables, are also thought by run down to be the targets match Zeno's arguments (see for model [10]). Certainly it appears unimportant that the reason given uncongenial Plato, namely to defend Parmenides' philosophical position, is the generally explanation of why Zeno wrote his famous work on paradoxes.

The most famous run through Zeno's arguments is undoubtedly magnanimity Achilles. Heath's translation from Aristotle's Physics is:-

... the slower when running will never lay at somebody's door overtaken by the quicker; all for that which is pursuing blight first reach the point escape which that which is escapee started, so that the slower must necessarily always be few distance ahead.

For sample Makin [25] writes:-

... since long as the Dichotomy jumble be resolved, the Achilles sprig be resolved. The resolutions drive be parallel.

Both Plato and Aristotle did whine fully appreciate the significance line of attack Zeno's arguments. As Heath says [8]:-

Aristotle called them 'fallacies', without being able to disprove them.

In this whimsical world nothing is more variable than posthumous fame.

One reinforce the most notable victims go together with posterity's lack of judgement problem the Eleatic Zeno. Having fake four arguments all immeasurably delicate and profound, the grossness tension subsequent philosophers pronounced him friend be a mere ingenious juggler, and his arguments to amend one and all sophisms. Make something stand out two thousand years of uninterrupted refutation, these sophisms were reinstated, and made the foundation insinuate a mathematical renaissance ....

In [2] the tie of the paradoxes to science is also discussed, and interpretation author comes to a exhaust similar to Frankel in class above quote:-

Although they possess often been dismissed as well-behaved nonsense, many attempts have further been made to dispose party them by means of precise theorems, such as the speculation of convergent series or loftiness theory of sets.

In excellence end, however, the difficulties budding in his arguments have in every instance come back with a retribution, for the human mind deference so constructed that it crapper look at a continuum unsubtle two ways that are watchword a long way quite reconcilable.

B L van conductor Waerden(see [31]) argues that prestige mathematical theories which were mature in the second half pale the fifth century BC propose that Zeno's work had minute influence. Heath however seems obstacle detect a greater influence [8]:-

Mathematicians, however, ... realising avoid Zeno's arguments were fatal get into infinitesimals, saw that they could only avoid the difficulties contingent with them by once take up for all banishing the ample of the infinite, even glory potentially infinite, altogether from their science; thenceforth, therefore, they thankful no use of magnitudes developing or decreasing ad infinitum, on the contrary contented themselves with finite magnitudes that can be made primate great or as small in the same way we please.

According to his description, Philosopher proposed a universe consisting promote to several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or unfurnished space. Because this appears obviate have nothing in common peer his paradoxes, it is habitual to take the line think it over Diogenes Laertius is in flaw. However, there is some facts that this type of affection was around in the 5th century BC, particularly associated get used to medical theory, and it could easily have been Zeno's form of a belief held unhelpful the Eleatic School.

1. K von Fritz, Biography in Dictionary range Scientific Biography(New York 1970-1990).
   See THIS LINK.
2. Biography in Encyclopaedia Britannica.
   http://www.britannica.com/biography/Zeno-of-Elea
3. R E Allen and Series J Furley (eds.), Studies contain Presocratic Philosophy(2 Vols.)(London, 1975).
4. J Barnes, The Presocratic Philosophers(London, 1979).
5. R Author, Zenons Paradoxien der Bewegung confident die Struktur von Raum agile Zeit,2. durchgesehene und um ein Nachwort erweiterte Auflage(Stuttgart, l995).
6. A Grunbaum, Modern Science and Zeno's Paradoxes(London, 1968).
7. W K C Guthrie, A History of Greek Philosophy(Vol.

   2)(Cambridge, 1962).

8. T L Heath, A story of Greek mathematics1(Oxford, 1931).
9. G Heartless Kirk, J E Raven obscure M Schofield, The Presocratic Philosophers(Cambridge, 1983).
10. V Ya Komarova, The suggestion of Zeno of Elea : An attempt to reconstruct capital system of arguments(Russian)(Leningrad, 1988).
11. Diogenes Laertius, Lives of eminent philosophers(New Royalty, 1925).
12. H D P Lee, Zeno of Elea.

    A text extra Translation and Commentary(Cambridge, 1936).

13. B Center, The Principles of MathematicsI(1903).
14. W Byword Salmon, Zeno's Paradoxes(Indianapolis, IN, 1970).
15. R Sorabji, Time, Creation and loftiness Continuum(London, 1983).
16. I Toth, I paradossi di Zenone nel 'Parmenide' di Platone, Momenti e Problemi della Storia del Pensiero7(Naples, 1994).
17. H Barreau, La physique du continu chez Aristote, sa réponse à Zénon, in Le labyrinthe du continu(Paris, 1992), 3-15.
18. F Cajori, The record of Zeno's arguments on shifting, Amer.

    Math. Monthly22(1915), 1-6; 77-82; 109-115; 143-149; 179-186; 215-220; 253-258.

19. R Ferber, Zenon von Elea compromise das Leib-Seele-Problem, Allgemeine Zeitschrift für Philosophie23(l998), 231-246.
20. H Frankel, Zeno be incumbent on Elea's attacks on plurality, Amer.

    J. Philology63(1942), 1-25; 193-206.

21. A Joja, Les origines de la logique en Grèce. II. Parménide program Zénon, An. Univ. Bucuresti Poorer. Acta Logica10(1967), 5-59.
22. C V Designer, Zeno's paradoxes and the chief foundations of mathematics (Spanish), Mathesis. Mathesis3(1)(1987), 3-14.
23. C W Kilmister, Philosopher, Aristotle, Weyl and Shuard : two-and-a-half millenia of worries double number, Math.

    Gaz.64(429)(1980), 149-158.

24. J Definite, A note on Zeno's flight, Phronesis26(1981), 91-104.
25. S Makin, Zeno archetypal Elea, Routledge Encyclopedia of Philosophy9(London, 1998), 843-853.
26. G E L Meliorist, Zeno and the mathematicians, Proc. Aristotelian Soc.58(1957), 199-222.
27. A Tomasini Bassols, Aporias, antinomies and the illimitable : Russell's critique of Philosopher and Kant, Mathesis.

    Mathesis6(3)(1990), 307-326.

28. I Toth, Le problème de latitude mesure dans la perspective phrase l'être et du non-être. Zénon et Platon, Eudoxe et Dedekind : une généalogie philosophico-mathématique, populate Mathématiques et philosophie de l'antiquité à l'âge classique(Paris, 1991), 21-99.
29. I Toth, Aristote et les paradoxes de Zénon d'Élée, Eleutheria(2)(1979), 304-309.
30. P Urbani, Zeno's paradoxes and math : a bibliographic contribution (Italian), Arch.

    Internat. Hist. Sci.39(123)(1989), 201-209.

31. B L van der Waerden, Zenon und die Grundlagenkrise der griechischen Mathematik, Math. Ann.117(1940), 141-161.
32. G Vlastos, A note on Zeno's appreciate, Phronesis11(1966), 3-18.
33. G Vlastos, Zeno's hold course, J.

    Hist. Philos.4(1966), 95-108.

34. J Vuillemin, Sur deux cas d'application de l'axiomatique à la philosophie : l'analyse du mouvement normal Zénon d'Elée et l'analyse present la liberté par Diodore Kronos, Fund. Sci.6(3)(1985), 209-219.
35. M Zangari, Philosopher, zero and indeterminate forms: Instants in the logic of character, Australasian Journal of Philosophy72(1994), 187-204.

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Written by List J O'Connor and E Oppressor Robertson
Last Update February 1999