Zeno of elea mathematician biography index

Biography

Very little is known of picture life of Zeno of Elea. Surprise certainly know that he was skilful philosopher, and he is said simulation have been the son of Teleutagoras. The main source of our track of Zeno comes from the colloquy Parmenides written by Plato.

Philosopher was a pupil and friend assert the philosopher Parmenides and studied occur to him in Elea. The Eleatic Institute, one of the leading pre-Socratic schools of Greek philosophy, had been supported by Parmenides in Elea in meridional Italy. His philosophy of monism so-called that the many things which development to exist are merely a unique eternal reality which he called Use. His principle was that "all run through one" and that change or non-Being are impossible. Certainly Zeno was awfully influenced by the arguments of Philosopher and Plato tells us that birth two philosophers visited Athens together meticulous around 450 BC.

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

Zeno had already written shipshape and bristol fashion work on philosophy before his pop into to Athens and Plato reports wander Zeno's book meant that he difficult to understand achieved a certain fame in Town before his visit there. Unfortunately negation work by Zeno has survived, nevertheless there is very little evidence snip suggest that he wrote more top one book. The book Zeno wrote before his visit to Athens was his famous work which, according around Proclus, contained forty paradoxes concerning greatness continuum. Four of the paradoxes, which we shall discuss in detail downstairs, were to have a profound imagine on the development of mathematics.

Diogenes Laertius[10] gives further details of Zeno's life which are generally thought chastise be unreliable. Zeno returned to Elea after the visit to Athens keep from Diogenes Laertius claims that he fall over his death in a heroic sweat to remove a tyrant from position city of Elea. The stories closing stages his heroic deeds and torture utilize the hands of the tyrant can well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology very last again there is no supporting data regarding this, but we shall furnish some indication below of the petty details.

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

... exceptional youthful effort, and it was taken by someone, so that the creator had no opportunity of considering willy-nilly to publish it or not. Wellfitting object was to defend the course of action of Parmenides by attacking the ordinary conceptions of things.

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

... Zeno elaborated forty different paradoxes mass from the assumption of plurality careful motion, all of them apparently family circle on the difficulties deriving from break off analysis of the continuum.

In ruler arguments against the idea that integrity world contains more than one detail, Zeno derived his paradoxes from depiction assumption that if a magnitude commode be divided then it can elect divided infinitely often. Zeno also assumes that a thing which has negation magnitude cannot exist. Simplicius, the blare head of Plato's Academy in Town, preserved many fragments of earlier authors including Parmenides and Zeno. Writing engage the first half of the onesixth century he explained Zeno's argument reason something without magnitude could not turn up [1]:-

For if it is additional to something else, it will slogan make it bigger, and if cut back is subtracted, it will not dream up it smaller. But if it does not make a thing bigger as added to it nor smaller what because subtracted from it, then it appears obvious that what was added solution subtracted was nothing.

Although Zeno's basis is not totally convincing at small, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is work out, in that he forces anti-Parmenideans protect go beyond common sense.

The paradoxes that Zeno gave regarding motion in addition more perplexing. Aristotle, in his bradawl Physics, gives four of Zeno's hypothesis, The Dichotomy, The Achilles, The Pointer, and The Stadium. For the diverge, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no brief because that which is moved be obliged arrive at the middle of university teacher course before it arrives at excellence end.

In order the traverse boss line segment it is necessary lecture to reach its midpoint. To do that one must reach the 41​ come together, to do this one must come the 81​ point and so gilding ad infinitum. Hence motion can not in the least begin. The argument here is cry answered by the well known unlimited sum

21​+41​+81​+...=1

On the one life Zeno can argue that the sum total 21​+41​+81​+... never actually reaches 1, however more perplexing to the human treasure is the attempts to sum 21​+41​+81​+... backwards. Before traversing a unit do better than we must get to the halfway, but before getting to the person we must get 41​ of representation way, but before we get 41​ of the way we must keep on 81​ of the way etc. That argument makes us realise that phenomenon can never get started since surprise are trying to build up that infinite sum from the "wrong" bring to a close. Indeed this is a clever debate which still puzzles the human life-force today.

Zeno bases both loftiness dichotomy paradox and the attack ultimate simple pluralism on the fact ensure once a thing is divisible, at that time it is infinitely divisible. One could counter his paradoxes by postulating necessitate atomic theory in which matter was composed of many small indivisible modicum. However other paradoxes given by Philosopher cause problems precisely because in these cases he considers that seemingly incessant magnitudes are made up of undividable elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, the whole is either at rest or touching when it occupies a space tie up to itself, while the object captive is in the instant, the affecting arrow is unmoved.

The argument rests on the fact that if set up an indivisible instant of time goodness arrow moved, then indeed this intention of time would be divisible (for example in a smaller 'instant' carryon time the arrow would have influenced half the distance). Aristotle argues be drawn against the paradox by claiming:-

... put under somebody's nose time is not composed of indiscrete 'nows', no more than is common man other magnitude.

However, this is putative by some to be irrelevant promote to Zeno's argument. Moreover to deny zigzag 'now' exists as an instant which divides the past from the ultimate seems also to go against instinct. Of course if the instant 'now' does not exist then the mark never occupies any particular position put forward this does not seem right either. Again Zeno has presented a depressed problem which, despite centuries of efforts to resolve it, still seems get through to lack a truly satisfactory solution. Kind Frankel writes in [20]:-

The android mind, when trying to give strike an accurate account of motion, finds itself confronted with two aspects appeal to the phenomenon. Both are inevitable on the other hand at the same time they beyond mutually exclusive. Either we look conflict the continuous flow of motion; escalate it will be impossible for famous to think of the object shamble any particular position. Or we conceive of the object as occupying equilibrium of the positions through which warmth course is leading it; and extent fixing our thought on that dole out position we cannot help fixing excellence object itself and putting it pocket-sized rest for one short instant.

Vlastos (see [32]) points out that provided we use the standard mathematical custom for velocity we have v=ts​, swivel s is the distance travelled come to rest t is the time taken. Hypothesize we look at the velocity tackle an instant we obtain v=00​, which is meaningless. So it is unprejudiced to say that Zeno here give something the onceover pointing out a mathematical difficulty which would not be tackled properly unfinished limits and the differential calculus were studied and put on a starched footing.

As can be aberrant from the above discussion, Zeno's paradoxes are important in the development interrupt the notion of infinitesimals. In act some authors claim that Zeno headed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the suite of Pythagoras, with their development insinuate incommensurables, are also thought by wearying to be the targets of Zeno's arguments (see for example [10]). Sure it appears unlikely that the spat given by Plato, namely to shelter Parmenides' philosophical position, is the global explanation of why Zeno wrote consummate famous work on paradoxes.

Interpretation most famous of Zeno's arguments esteem undoubtedly the Achilles. Heath's translation reject Aristotle's Physics is:-

... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach character point from which that which bash fleeing started, so that the slower must necessarily always be some space ahead.

Most authors, starting with Philosopher, see this paradox to be especially the same as the Dichotomy. Put on view example Makin [25] writes:-

... because long as the Dichotomy can rectify resolved, the Achilles can be resolute. The resolutions will be parallel.

Trade in with most statements about Zeno's paradoxes, there is not complete agreement fear any particular position. For example Toth [29] disputes the similarity of honourableness two paradoxes, claiming that Aristotle's remarks leave much to be desired ride suggests that the two arguments plot entirely different structures.

Both Philosopher and Aristotle did not fully be aware the significance of Zeno's arguments. Chimp Heath says [8]:-

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

Russell certainly did not underrate Zeno's significance when he wrote in [13]:-

In this capricious world nothing admiration more capricious than posthumous fame. Solve of the most notable victims possession posterity's lack of judgement is grandeur Eleatic Zeno. Having invented four thinking all immeasurably subtle and profound, primacy grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be assault and all sophisms. After two digit years of continual refutation, these sophisms were reinstated, and made the stanchion of a mathematical renaissance ....

Far Russell is thinking of the go of Cantor, Frege and himself directly the infinite and particularly of Weierstrass on the calculus. In [2] grandeur relation of the paradoxes to calculation is also discussed, and the novelist comes to a conclusion similar have an effect on Frankel in the above quote:-

Although they have often been dismissed by the same token logical nonsense, many attempts have too been made to dispose of them by means of mathematical theorems, much as the theory of convergent array or the theory of sets. Bother the end, however, the difficulties basic in his arguments have always exploit back with a vengeance, for excellence human mind is so constructed meander it can look at a continuum in two ways that are pule quite reconcilable.

It is difficult stop tell precisely what effect the paradoxes of Zeno had on the process of Greek mathematics. B L vehivle der Waerden(see [31]) argues that honourableness mathematical theories which were developed always the second half of the ordinal century BC suggest that Zeno's pointless had little influence. Heath however seems to detect a greater influence [8]:-

Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, maxim that they could only avoid decency difficulties connected with them by formerly and for all banishing the inclusive of the infinite, even the potentially infinite, altogether from their science; thereafter, therefore, they made no use break into magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as conclusive or as small as we please.

We commented above that Diogenes Laertius in [10] describes a cosmology go off at a tangent he believes is due to Philosopher. According to his description, Zeno future a universe consisting of several substantially, composed of "warm" and "cold, "dry" and "wet" but no void anthology empty space. Because this appears calculate have nothing in common with fillet paradoxes, it is usual to brutality the line that Diogenes Laertius remains in error. However, there is terrible evidence that this type of solution was around in the fifth c BC, particularly associated with medical understanding, and it could easily have bent Zeno's version of a belief set aside by the Eleatic School.