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Zeno's paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides' doctrine that "all is one" and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.

Zeno conceived …not less than forty arguments revealing contradictions… (Proclus, Commentary on Plato's Parmenides p. 29)

For what Parmenides had uttered in an intricate and concentrated style. Zeno unfolds and transmits to us in extended discourse. (Proclus, Commentary on Plato's Parmenides p. 65)

… these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one. (Plato, Parmenides, 128c,d [1])

Sources

No copies of Zeno's writings survive. Of the forty paradoxes referred to by Proclus we have second-hand knowledge of eight which are generally credited to Zeno.

Aristotle in his Physics [2], describes six. These include the four so-called "arguments against motion", the most famous and historically important of Zeno's paradoxes. Aristotle only gives paraphrases of Zeno's arguments. And it is not known how he came to know them.

Simplicius in his Commentary of Aristotle's Physics tells of two more. Simplicius may have been quoting from a copy (or fragments of a copy) of Zeno's writings.

There is one more surviving argument described by Aristotle, called the "Argument from Complete Divisibility" which may be Zeno's.

The Pardoxes of Motion

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. (Bertrand Russell, The Principles of Mathematics (1903)

The Dichotomy

yo dawgsPhysics VI:9, 239b10 [3])

The Achilles

"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." (Aristotle Physics VI:9, 239b15)

The Arrow

"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." (Aristotle Physics VI:9, 239b5)

The Pardoxes of Plurality

The Argument from Denseness (second paradox of plurality)

"If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited." (Simplicius(a) On Aristotle's Physics , 140.29)

The Argument from Finite Size

"… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing." (Simplicius(a) On Aristotle's Physics ,139.9)

"But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited." (Simplicius(a) On Aristotle's Physics , 141.2)

Paradox of Place

"… if everything that exists has a place, place too will have a place, and so on ad infinitum". (Aristotle Physics IV:1, 209a25)

Paradox of the Millet Seed

"… there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially." (Aristotle Physics VII:5, 250a20)

The Argument from Complete Divisibility

There is another arguement described by Aristotle which may be Zeno's (Simplicious thinks so):

… whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided …  though perhaps nobody in fact could so divide it.

What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through . But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude. (Aristotle On Generation and Corruption, 316a19)

(These words are Aristotle's not Zeno's, and indeed the argument is not even attributed to Zeno by Aristotle. However we have Simplicius' opinion ((a) On Aristotle's Physics, 139.24) that it originates with Zeno)

Responses to Zeno

Zeno's paradoxes can be seen as posing problems of a philosophical, mathematical, and physical nature.

Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found no satisfactory solution to them. Plato's and Aristotle's attempts to refute Zeno remain unconvincing.

Mathematicians thought they had disposed of Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again, when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century, by Weierstrass, Dedekind and Cantor.

Although Zeno's paradoxes are still hotly debated by some philosophers, most have accepted the solutions offered by modern mathematics.

Nevertheless, infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker an earlier 19th century mathematician. It would be incorrect to say that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrauss and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has resolved forever all problems involving infinities, including Zeno's.

And, It should be understood, that whatever mathematicians say about continuity, or infinity, applies only to mathematics and mathematical models of the real world - not necessarily to the real world itself. In other words, a line as a mathematical object is whatever mathematicians say it is. A line as a physical object is whatever the universe says it is.

As a practical matter, however, since the invention of the calculus, no engineer has been concerned about them. In ordinary life, very few people have ever been much concerned.

Simplicius Text

(see: http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Philosophical%20Texts/Zeno/Zeno.html )

Simplicius, Commentary on Aristotle's Physics, 138.29-139.23:

But Alexander seems to have taken his opinion about how Zeno established the one from the accounts of Eudemus.  For Eudemus says in the Physics, "And so is it that this does not exist, but is some one?  For he raised this puzzle, and they say that Zeno says that if someone should propose to him whatever the one is, he will have to speak of real things.  But he raised puzzles, as it seems, from the fact that each of the perceptibles is said to be both categorically many and many by division, but supposes the point to be nothing--namely he did not consider to be a real thing that which neither increases something by being added nor diminishes it by being taken away."  And it is likely that Zeno did exercises in taking each side (whence this is called "double-tongued") and produced some such puzzling arguments about the one.  In his treatise which contains many dialectical proofs on each side, however, he shows that it follows for anyone who says that there are many that he must say opposite things of which one dialectical proofs is where he shows that if there are manny, they will be both big and small, big so as to be infinite in size and small so as to have no size.  In this argument, he shows that where there is neither size nor bulk nor mass there is nothing, nor would this exist.

(Zeno, frag. DK B2) "For if it were added to another thing," he says, "it would not make it larger.  For given it has not size, when it is added it could not contribute anything to its magnitude.  And so what is added in this way would be nothing.  If when it is removed, the other would be not be any smaller, and when added the other is not increased, it is clear that what is added is isn't anything and what is removed isn't either."

And  Zeno doesn't say this to destroy the one, but to destroy the view that each of the many and infinite things has size, since for each thing taken there is always another because of infinite division.  He shows this after showing that nothing has size from the fact that each of the many is the same as itself and one.  And Themistius says that the argument of Zeno establishes that what-is is onefrom the fact that it is continuous and indivisible, "since if it were divided," he says, "there will not be anything which is precisely one due to the infinite divisibility of bodies."  But Zeno seems rather to say that there is not be many things.

Simplicius, Commentary on Aristotle's Physics, 140.27-141.8:

And what should we say of the many, considering what is comes up in the book of Zeno.  For once more he shows that if there are many the same things are finite and infinite.  Zeno writes these things, (verbatim),

(Zeno, frag. DK B3) "If there are many, it is necessary that they be as many as they are and neither more than they are nor less.  If they are many, the existents will be infinite.  For there will always be other things between each of the existents, and again other things between them.  And way the existents are in this way infinite."

And in this way he showed the infinite in multitude from dichotomy.  But as to the infinite in magnitude, he earlier proved with the same dialectical reasoning.

(Zeno, frag. DK B1) Having first proved that "if the existent does not have magnitude it would not exist," he infers that "if it is, each thing must have some size and bulk, and that one part of it can be apart from another.  And with regard to what there is at this stage*, the same argument follows.  For that will have size and some part of it will be there at this stage*.  To say this once is like saying it constantly.  For there will not be some sort of limit to this, nor will one thing not be related to another [as a part].  In this way if there are many, they must be both small and large, small so as not to have size and large so as to be infinite."

• The Greek word, 'proekhein' can mean 'protrude' or 'be outstanding'.  However, in the context, it is contrasted with 'apekhein' (be apart), and so, I think, the sense is  'be there before some part of it departs from it or later 'be there as a part before it departs'.

Simlpicius text - (another translation)

(see: http://history.hanover.edu/texts/presoc/zeno.htm)

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.

http://www.ub.rug.nl/eldoc/dis/fil/p.s.hasper/c1.pdf

References

• Aristotle, 1984, On Generation and Corruption, A. A. Joachim (trans), in The Complete Works of Aristotle, J. Barnes (ed.), Princeton: Princeton University Press
• Proclus, Comentary on Plato's Parmenides, translated by Glenn R. Morrow and John M. Dillon, Princeton University Press; Reprint edition (1992) ISBN 0691020892
• Russell, Bertrand, The Principles of Mathematics, W. W. Norton & Company; Reissue edition (1996) ISBN 0393314049
• Simplicius(a), 1995, On Aristotle's Physics, in Readings in Ancient Greek Philosophy From Thales to Aristotle, S. M. Chohen, P. Curd and C. D. C. Reeve (eds), Indianapolis/Cambridge: Hackett Publishing Co. Inc. 58-59

External links to online texts

1. Plato's Parmenides
2. Aristotle's Physics

Consider a thing. Now suppose the thing has parts. Then divide the thing into all its parts. Now consider one of these parts. Suppose that this part is divisible. Then divide the part. We now have two new parts. But this impossible since we had all the parts already. It must be then that all the parts are indivisible. But if a part is indivisible then it must have zero size. but if all the parts of a thing have zero size, the thing must have zero size. Thus the thing has no parts.

The best-known example of a current-day Zeno type paradox is the Thomson Lamp, named after James F. Thomson.