Zeno of Elea – Commentary

Arthur Fairbanks, ed. and trans.
The First Philosophers of Greece
(London: K. Paul, Trench, Trubner, 1898), Page 112-119.

Hanover Historical Texts Project
Scanned and proofread by Aaron Gulyas, May 1998.
Proofread and pages added by Jonathan Perry, March 2001.

Fairbanks’s Introduction
Simplicius’s account of Zeno’s arguments, including the translation of the Fragments
Zeno’s arguments as described by Aristotle
Passages relating to Zeno in the Doxographists

Fairbanks’s Introduction

[Page 112] Zeno of Elea, son of Teleutagoras, was born early in the-fifth century B.C. He was the pupil of Parmenides, and his relations with him were so intimate that Plato calls him Parmenides’s son (Soph. 241 D). Strabo (vi. 1, 1) applies to him as well as to his master the name Pythagorean, and gives him the credit of advancing the cause of law and order in Elea. Several writers say that he taught in Athens for a while. There are numerous accounts of his capture as party to a conspiracy; these accounts differ widely from each other, and the only point of agreement between them has reference to his determination in shielding his fellow conspirators. We find reference to one book which he wrote in prose (Plato, Parm. 127 c), each section of which showed the absurdity of some element in the popular belief.
Literature: Lohse, Halis 1794; Gerling, de Zenosin Paralogismis, Marburg 1825; Wellmann, Zenos Beweise, G.-Pr. Frkf. a. O. 1870; Raab, D. Zenonische Beweise, Schweinf. 1880; Schneider, Philol. xxxv. 1876; Tannery, Rev. Philos. Oct. 1885; Dunan, Les arguments de Zenon, Paris 1884; Brochard, Les arguments de Zenon, Paris 1888; Frontera, Etude sur les arguments de Zenon, Paris 1891

Simplicius’s account of Zeno’s arguments, including the translation of the Fragments

30 r 138, 30. For Eudemos says in his Physics, ‘Then does not this exist, and is there any one ? This was the problem. He reports Zeno as saying that if any one explains to him the one, what it is, he can tell him what things are. But he is puzzled, it seems, because each of the senses declares that there are many things, both absolutely, and as the result of division, but no one establishes the mathematical point. He thinks that what is not increased by receiving additions, or decreased as parts are taken away, is not one of the things that are.’ It was natural that Zeno, who, as if for the sake of exercise, argued both sides of a case (so that he is called double-tongued), should utter such statements raising difficulties about the one; but in his book which has many arguments in regard to each point, he shows that a man who affirms multiplicity naturally falls into contradictions. Among these arguments is one by which he shows that if there are many things, these are both small and great – great enough to be infinite in size, and small enough to be nothing in size. By this he shows that what has neither greatness nor thickness nor bulk could not even be. (Fr. 1)9 ‘For if, he says, anything were added to another being, it could not make it any greater; for since greatness does not exist, it is impossible to increase the greatness of a thing by adding to it. So that which is added would be nothing. If when something is taken away that which is left is no less, and if it becomes no greater by receiving additions, evidently that which has been added or taken away is nothing.’ These things Zeno says, not denying the one, but holding that each thing has the greatness of [Page 115] many and infinite things, since there is always something before that which is apprehended, by reason of its infinite divisibility; and this he proves by first showing that nothing has any greatness because each thing of the many is identical with itself and is one.
Ibid. 30 v 140, 27. And why is it necessary to say that there is a multiplicity of things when it is set, forth in Zeno’s own book? For again in showing that, if there is a multiplicity of things, the same things are both finite and infinite, Zeno writes as follows, to use his own words: (Fr. 2) ‘If there is a multiplicity of things; it is necessary that these should be just as many as exist, and not more nor fewer. If there are just as many as there are, then the number would be finite. If there is a multiplicity at all, the number is infinite, for there are always others between any two, and yet others between each pair of these. So the number of things is infinite.’ So by the process of division he shows that their number is infinite. And as to magnitude, he begins, with this same argument. For first showing that (Fr. 3) ‘if being did not have magnitude, it would not exist at all,’ he goes on, ‘if anything exists, it is necessary that each thing should have some magnitude and thickness, and that one part of it should be separated from another. The same argument applies to the thing that precedes this. That also will have magnitude and will have something before it. The same may be said of each thing once for all, for there will be no such thing as last, nor will one thing differ from another. So if there is a multiplicity of things, it is necessary that these should be great and small–small enough not to have any magnitude, and great enough to be infinite.’

Ibid. 130 v 562,.3. Zeno’s argument seems to deny that place exists, putting the question as follows: (Fr. 4) [Page 116] ‘If there is such a thing as place, it will be in something, for all being is in something, and that which is in something is in some place. Then this place will be in a place, and so on indefinitely. Accordingly there is no such thing as place.’

Ibid. 131 r 563, 17. Eudemos’ account of Zeno’s opinion runs as follows: ‘Zeno’s problem seems to come to the same thing. For it is natural that all being should be somewhere, and if there is a place for things, where would this place be? In some other place, and that in another, and so on indefinitely.’

Ibid. 236 v. Zeno’s argument that when anything is in a space equal to itself, it is either in motion or at rest, and that nothing is moved in the present moment, and that the moving body is always in a space equal to itself at each present moment, may, I think, be put in a syllogism as follows: The arrow which is moving forward is at every present moment in a space equal to itself, accordingly it is in a space equal to itself in all time; but that which is in a space equal to itself in the present moment is not in motion. Accordingly it is in a state of rest, since it is not moved in the present moment, and that which is not moving is at rest, since everything is either in motion or at rest. So the arrow which is moving forward is at rest while it is moving forward, in every moment of its motion.

237 r. The Achilles argument is so named because Achilles is named in it as the example, and the argument shows that if he pursued a tortoise it would be impossible for him to overtake it. 255 r, Aristotle accordingly solves the problem of Zeno the Eleatic, which he propounded to Protagoras the Sophist.11 Tell me, Protagoras, said he, does one grain of millet make a noise when it falls, or does the [Page 117] ten-thousandth part of a grain? On receiving the answer that it does not, he went on: Does a measure of millet grains make a noise when it falls, or not? He answered, it does make a noise. Well, said Zeno, does not the statement about the measure of millet apply to the one grain and the ten-thousandth part of a grain? He assented, and Zeno continued, Are not the statements as to the noise the same in regard to each? For as are the things that make a noise, so are the noises. Since this is the case, if the measure of millet makes a noise, the one grain and the ten-thousandth part of a grain make a noise.

Zeno’s arguments as described by Aristotle

Phys. iv. 1; 209 a 23. Zeno’s problem demands some consideration; if all being is in some place, evidently there must be a place of this place, and so on indefinitely. 3; 210 b 22. It is not difficult to solve Zeno’s problem, that if place is anything, it will be in some place; there is no reason why the first place should not be in something else, not however as in that place, but just as health exists in warm beings as a state while warmth exists in matter as a property of it. So it is not necessary to assume an indefinite series of places.
vi. 2; 233 a 21. (Time and space are continuous . . . the divisions of time and space are the same.) Accordingly Zeno’s argument is erroneous, that it is not possible to traverse infinite spaces, or to come in contact with infinite spaces successively in a finite time. Both space and time can be called infinite in two ways, either absolutely as a continuous whole, or by division into the smallest parts. With infinites in point of quantity, it is not possible for anything to come in contact in a finite time, but it is possible in the case of the infinites [Page 118] reached by division, for time itself is infinite from this standpoint. So the result is that it traverses the infinite in an infinite, not a finite time, and that infinites, not finites, come in contact with infinites.

vi. 9 ; 239 b 5. And Zeno’s reasoning is fallacious. For if, he says, everything is at rest [or in motion] when it is in a space equal to itself, and the moving body is always in the present moment then the moving arrow is still. This is false for time is not composed of present moments that are indivisible, nor indeed is any other quantity. Zeno presents four arguments concerning motion which involve puzzles to be solved, and the first of these shows that motion does not exist because the moving body must go half the distance before it goes the whole distance; of this we have spoken before (Phys. viii. 8; 263 a 5). And the second is called the Achilles argument; it is this: The slow runner will never be overtaken by the swiftest, for it is necessary that the pursuer should first reach the point from which the pursued started, so that necessarily the slower is always somewhat in advance. This argument is the same as the preceding, the only difference being that the distance is not divided each time into halves. . . . His opinion is false that the one in advance is not overtaken; he is not indeed overtaken while he is in advance; but nevertheless he is overtaken, if you will grant that he passes through the limited space. These are the first two arguments, and the third is the one that has been alluded to, that the arrow in its flight is stationary. This depends on the assumption that time is composed of present moments ; there will be no syllogism if this is not granted. And the fourth argument is with reference to equal bodies moving in opposite directions past equal bodies in the stadium with equal speed, some from the end of the stadium, others from [Page 119] the middle; in which case he thinks half the time equal to twice the time. The fallacy lies in the fact that while he postulates that bodies of equal size move forward with equal speed for an equal time, he compares the one with something in motion, the other with something at rest.

Passages relating to Zeno in the Doxographists

Plut. Strom. 6 ; Dox. 581. Zeno the Eleatic brought out nothing peculiar to himself, but he started farther difficulties about these things. Epiph. adv. Baer. iii. 11; Dox. 590. Zeno the Eleatic, a dialectician equal to the other Zeno, says that the earth does not move, and that no space is void of content. He speaks as follows:-That which is moved is moved in the place in which it is, or in the place in which it is not; it is neither moved in the place in which it is, nor in the place in which it is not ; accordingly it is not moved at all.
Galen, Hist. Phil. 3; Dox. 601. Zeno the Eleatic is said to have introduced the dialectic philosophy. 7 ; Dox. 604. He was a skeptic.

Aet. i. 7; Dox. 303. Melissos and Zeno say that the one is universal, and that it exists alone, eternal, and unlimited. And this one is necessity [Heeren inserts here the name Empedokles], and the material of it is the four elements, and the forms are strife and love. He says that the elements are gods, and the mixture of them is the world. The uniform will be resolved into them he thinks that souls are divine, and that pure men who share these things in a pure way are divine. 28; 320. Zeno et al. denied generation and destruc- tion, because they thought that the all is unmoved.

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Zeno of Elea was a Greek thinker, born around 490 B.C., about whom very little is known. Plato described him as a fair and tall man that was beloved byParmenides of Elea. Zeno became famous for writing a series of paradoxes that have intrigued, puzzled and irritated thinkers from his time until the present day. Most of what is known about his life and purposes comes from either Plato orAristotle, the first accusing him of being a mere defender of his mentorParmenides of Elea while the second has described Zeno as the inventor of dialectics in the lost work Sophist. Zeno’s propositions are quite literally para-doxes, from the Greek against, or contrary to, beliefs or opinions. His paradoxes challenged very basic common conceptions of plurality, space and motion. From its mathematical terms, Zeno’s paradoxes are believed to have been influenced by the efforts of Pythagoras, to apply mathematical concepts to the natural world. The ultimate reasons for Zeno’s paradoxes are most likely forever lost. The number of paradoxes written is also subject to dispute, mostly around 40, and none of them survive to our times in their original form. Most of the information we have on Zeno’s paradoxes comes from Aristotle‘s Physics, where Aristotle‘s develops on, and refutes some of, the paradoxes. Zeno is also known as one of the first examples of anti-logic or reductio ad absurdum (proof by contradiction). The paradoxes that most propelled thinkers from the past two millennium are the ones of the tortoise and Achilles, of the flying arrow, and of the dichotomy argument. Philosophers and mathematicians have argued extensively about the nature of the paradoxes, if metaphysical or mathematical.

Plato‘s account of Zeno and his paradoxes can be found in the dialogueParmenides of Elea, written on the occasion of the Athenian visit of Parmenides of Elea and his Pupil Zeno, both from Elea. From Plato‘s descriptions ofParmenides of Elea, Zeno, and Socrates, it is assumed that Zeno was born around 490 B.C. In the Parmenides of Elea dialogue by Plato, Socrates appears as a young listener to Zeno’s first readings of his paradoxes in Athens. There Zeno is said to have read the following paradox: “If the things that are are many, that then they must be both like and unlike, but this is impossible. For neither can unlike things be like, nor like things unlike.” Zeno further seems to argue that his overall aim with his paradoxes was to demonstrate the inconsistency of the common belief that there are many things and, although the passage is out of a fictional work by Plato, it is nevertheless taken as an evidence of Zeno’s overall project. Plato also claimed that all Zeno does is to repeat Parmenides of Elea but twisting the form so as to fool the others into thinking he is saying something altogether different. He claims that while Parmenides of Elea states that all is one, Zeno claims that there are not many things, which are virtually the same statement.

Aristotle, for his part, claimed that Zeno’s paradoxes were constantly reworked and rewritten by those who edited them so that it is hard to say what is the original and what are reworked versions by other authors. The paradoxes of motion, present in Aristotles Physics, do not have a direct connection with the supposed thesis that Zeno’s overall project was to question the common belief that there are many things. It may be said, based on the paradoxes as described by Aristotle that if Zeno were to have an overall project, it would be to question both plurality and motion.

One of the best know paradoxes attributed to Zeno is Achilles and the Tortoise, preserved in the following form in Aristotle‘s Physics: “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.” So if Achilles is running 1000m behind the tortoise, when he completes the distance, the tortoise will have already advanced to another point in space, say 100m ahead. When Achilles covers the following 100m, the tortoise will be already another 10m ahead and so on. However small the distance between Achilles and the tortoise, following Zeno’s logic, there will always remain a gap, the gap which corresponds to the distance the tortoise managed to advance while Achilles was busy reaching the previous position of the tortoise.

In another paradox classified as the paradox of motion, and named The Dichotomy Paradox, Zeno challenges the possibility of travelling from one point to another. As from Aristotle‘s Physics: “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” So if a car is to cover a distance of 100m, it will first have to cover 50m. Before covering 50m, it will have to cover 25m. This logic is then pushed to infinity making it impossible to start moving as well as impossible to define a first distance, as it could always be divided in half. Zeno concludes that all motion must therefore be a mere illusion. The paradox is called The Dichotomy Paradox because of its repeatedly split by two of the distance. Aristotle saw in this paradox a variation of the Achilles and the Tortoiseparadox. It is know as the Race Course Paradox as well.

The last of the motion paradoxes is The Arrow Paradox. Once more, fromAristotle‘s Physics: “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.” If in an instant of time, an object is occupying a specific place in space, it means that this object is still. But, even in the case of a flying arrow, it can always be said that, in a single, duration-less moment, it occupies a specific place in space and is therefore motionless. In the next moment in time, the arrow may well be in a different place, but at that moment, as in every moment, it is motionless. The conclusion is that if time is composed of instants and if in every instant everything occupies a place in space (motionless), then motion is impossible. Different then the Achilles and the Tortoise paradox and the Dichotomy Paradox, where time was always divided in slots, in The Arrow Paradox, time is divided into points, instants.

The paradox that is believed to have reached the present day closest to its original form is a paradox named The Antinomy of Limited and Unlimited, provided by Simplicius in his commentary on Aristotle‘s Physics. It says: “If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus the things that are are unlimited”. The argument seems to develop from the postulate that if two things are separate, then there must be a third thing in between them, and so on ad infinitum. It suggest a contradiction between things being definable and therefore countable, finite, and things being separate, having a third thing in between, and therefore logically infinite.

Thomas Aquinas, in the 13th Century, commented on Aristotle‘s comments of Zeno’s paradoxes, arguing that time is not made of instants. Bertrand Arthur William Russel for his part, agreed with Zeno’s proposal that in a duration-less instant, an object can only be still in space, but argued that what happens in between two such moments, given that the object has travelled in space, is motion. Peter Lynds simply dismisses Zeno’s paradoxes by claiming that they are founded on the assumption that such things as an instant exist, which Lynds objects. The early discoveries of quantum mechanics at the end of the 1970s, where it was noted that a quantum system can be inhibited or hindered through observation, were named “quantum Zeno effect” for its similarity with Zeno’s The Arrow Paradox. His paradoxes have also populated many novels, most notably works by Leo Tolstoy, Lewis Carroll, and Jorge Luis BorgesSlavoj Žižek also discusses some of them in his Looking Awry: an introduction to Jacques Lacan through popular culture


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