The Dialectic of Zeno

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This webpage reproduces a chapter of
A History of Philosophy
by
Frederick Copleston, S. J.
as reprinted by
Image Books — Doubleday
New York • London • Toronto • Sydney • Auckland
1993

The text, and illustrations except as noted,
are in the public domain.

Chapter 8

Part I: Pre‑Socratic Philosophy

p54 Chapter VII
The Dialectic of Zeno

Zeno is well known as the author of several ingenious arguments to prove the impossibility of motion, such as the riddle of Achilles and the tortoise; arguments which may tend to further the opinion that Zeno was no more than a clever riddler who delighted in using his wits in order to puzzle those who were less clever than himself. But in reality Zeno was not concerned simply to display his cleverness — though clever he undoubtedly was — but had a serious purpose in view. For the understanding of Zeno and the appreciation of his conundrums, it is therefore essential to grasp the character of this purpose, otherwise there is danger of altogether misapprehending his position and aim.

Zeno of Elea, born probably about 489 B.C., was a disciple of Parmenides, and it is from this point of view that he is to be understood. His arguments are not simply witty toys, but are calculated to prove the position of the Master. Parmenides had combated pluralism, and had declared change and motion to be illusion. Since plurality and motion seem to be such evident data of our sense-experience, this bold position was naturally such as to induce a certain amount of ridicule. Zeno, a firm adherent of the theory of Parmenides, endeavours to prove it, or at least to demonstrate that it is by no means ridiculous, by the expedient of showing that the pluralism of the Pythagoreans is involved in insoluble difficulties, and that change and motion are impossible even on their pluralistic hypothesis. The arguments of Zeno then are meant to refute the Pythagorean opponents of Parmenides by a series of collect reductiones ad absurdum. Plato makes this quite clear in the Parmenides, when he indicates the purpose of Zeno's (lost) book. "The truth is that these writings were meant to be some protection to the arguments of Parmenides against those who attack him and show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the p55 One."1 And Proclus informs us that "Zeno composed forty proofs to demonstrate that being is one, thinking it a good thing to come to the help of his master."2

I. Proofs against Pythagorean Pluralism

1. Let us suppose with the Pythagoreans that Reality is made up of units. These units are either with magnitude or without magnitude. If the former, then a line for example, as made up of units possessed of magnitude, will be infinitely divisible, since, however far you divide, the units will still have magnitude and so be divisible. But in this case the line will be made up an infinite number of units, each of which is possessed of magnitude. The line, then, must be infinitely great, as composed of an infinite number of bodies. Everything in the world, then, must be infinitely great, and a fortiori the world itself must be infinitely great. Suppose, on the other hand, that the units are without magnitude. In this case the whole universe will also be without magnitude, since, however many units you add together, if none of them has any magnitude, than the whole collection of them will also be without magnitude. But if the universe is without any magnitude, it must be infinitely small. Indeed, everything in the universe must be infinitely small.

The Pythagoreans are thus faced with this dilemma. Either everything in the universe is infinitely great, or everything in the universe is infinitely small. The conclusion which Zeno wishes us to draw from this argument is, of course, that the supposition from which the dilemma flows is an absurd supposition, namely, that the universe and everything in it are composed of units. If the Pythagoreans think that the hypothesis of the One is absurd and leads to ridiculous conclusions, it has now been shown that the contrary hypothesis, that of the many, is productive of equally ridiculous conclusions.3

2. If there is a many, then we ought to be able to say how many there are. At least, they should be numerable; if they are not numerable, how can they exist? On the other hand, they cannot possibly be numerable, but must be infinite. Why? Because between any two assigned units there will always be other units, just as a line is infinitely divisible. But it is absurd to say that the many are finite in number and infinite in number at the same time.4

p56 3. Does a bushel of cornº make a noise when it falls to the ground? Clearly. But what of a grain of corn, or the thousandth part of a grain of corn? It makes no noise. But the bushel of corn is composed only of the grains of corn or of the parts of the grains of corn. If, then, the parts make no sound when they fall, how can the whole make a sound, when the whole is composed of only of the parts?5

II. Arguments against the Pythagorean Doctrine of Space

Parmenides denied the existence of the void or empty space, and Zeno tries to support his denial by reducing the opposite view to absurdity. Suppose for a moment that there is a space in which things are. If it is nothing, then things cannot be in it. If, however, it is something, it will itself be in space, and that space will itself be in space, and so on indefinitely. But this is an absurdity. Things, therefore, are not in space or in an empty void, and Parmenides was quite right to deny the existence of a void.6

III. Arguments Concerning Motion

The most celebrated arguments of Zeno are those concerning motion. It should be remembered that what Zeno is attempting to show is this: that motion, which Parmenides denied, is equally impossible on the pluralistic theory of the Pythagoreans.

1. Let us suppose that you want to cross a stadium or race-course. In order to do so, you would have to traverse an infinite number of points — on the Pythagorean hypothesis, that is to say. Moreover, you would have to travel the distance in finite time, if you wanted to get to the other side at all. But how can you traverse an infinite number of points, and so an infinite distance, in a finite time? We must conclude that no object can traverse any distance whatsoever (for the same difficulty always recurs), and that all motion is consequently impossible.7

2. Let us suppose that Achilles and a tortoise are going to have a race. Since Achilles is a sportsman, he gives the tortoise a start. Now, by the time that Achilles has reached the place from which the tortoise started, the latter has again advanced to another p57 point; and when Achilles reaches that point, then the tortoise will have advanced still another distance, even if very short. Thus Achilles is always coming nearer to the tortoise, but never actually overtakes it — and never can do so, on the supposition that a line is made up of an infinite number of points, for then Achilles would have to traverse an infinite distance. On the Pythagorean hypothesis, then, Achilles will never catch up with the tortoise; and so, although they assert the reality of motion, they make it impossible on their own doctrine. For it follows that the slower moves as fast as the faster.8

3. Suppose a moving arrow. According to the Pythagorean theory the arrow should occupy a given position in space. But to occupy a given position in space is to be at rest. Therefore the flying arrow is at rest, which is a contradiction.9

4. The fourth argument of Zeno, which we know from Aristotle10 is, as Sir David Ross says, "very difficult to follow, partly owing to use of ambiguous language by Aristotle, partly owing to doubts as to the readings."11 We have to represent to ourselves three sets of bodies on a stadium or race-course. One set is stationary, the other two are moving in opposite directions to one another with equal velocity. FIGURE

The A's are stationary; the B's and C's are moving in opposite directions with the same velocity. They will come to occupy the following position: FIGURE

p58 In attaining this second position the front of B1 has passed four of the A's, while the front of C1 has passed all the B's. If a unit of length is passed in a unit of time, then the front of B1 has taken half the time taken by the front of C1 in order to reach the position of Fig. 2. On the other hand the front of B1 has passed all the C's, just as the front of C1 has passed all the B's. The time of their passage must then be equal. We are left with the absurd conclusion that the half of a certain time is equal to the whole of that time.

How are we to interpret these arguments of Zeno? It is important not to let oneself think: "These are mere sophistries on the part of Zeno. They are ingenious tricks, but they err in supposing that a line is composed of points and time of discrete moments." It may be that the solution of the riddle is to be found in showing that the line and time are continuous and not discrete; but, then, Zeno was not concerned to hold that they are discrete. On the contrary, he is concerned to show the absurd consequences which flow from supposing that they are discrete. Zeno, as a disciple of Parmenides, believed that motion is an illusion and is impossible, but in the foregoing arguments his aim is to prove that even on the pluralistic hypothesis motion is equally impossible, and that the assumption of its possibility leads to contradictory and absurd conclusions. Zeno's position was as follows: "The Real is a plenum, a complete continuum and motion is impossible. Our adversaries assert motion and try to explain it by an appeal to a pluralistic hypothesis. I propose to show that this hypothesis does nothing to explain motion, but only lands one in absurdities." Zeno thus reduced the hypothesis of his adversaries to absurdity and the real result of his dialectic was not so much to establish Parmenidean monism (which is exposed to insuperable objections), as to show the necessity of admitting the concept of continuous quantity.

The Eleatics, then, deny the reality of multiplicity and motion. There is one principle, Being, which is conceived of as material and motionless. They do not deny, of course, that we sense motion and multiplicity, but they declare that what we sense is illusion: it is mere appearance. True being is to be found, not by sense but by thought, and thought shows that there can be no plurality, no movement, no change.

p59 The Eleatics thus attempt, as the earlier Greek philosophers attempted before them, to discover the one principle of the world. The world, however, as it presents itself to us, is clearly a pluralistic world. The question is, therefore, how to reconcile the one principle with the plurality and change which we find in the world, i.e. the problem of the One and the Many, which Heraclitus had tried to solve in a philosophy that professed to do justice to both elements through a doctrine of Unity in Diversity, Identity in Difference. The Pythagoreans asserted plurality to the practical exclusion of the One — there are many ones; the Eleatics asserted the One to the exclusion of the many. But if you cling to the plurality which is suggested by sense-experience, then you must also admit change; and if you admit change of one thing into another, you cannot avoid the recurring problem as to the character of the common element in the things which change. If, on the other hand, you start with the doctrine of the One, you must — unless you are going to adopt a one‑sided position like that of the Eleatics, which cannot last — deduce plurality from the One, or at least show how the plurality which we observe in the world is consistent with the One. In other words, justice must be done to both factors — the One and the Many, Stability and Change. The one‑sided doctrine of Parmenides was unacceptable, as also was the one‑sided doctrine of the Pythagoreans. Yet the philosophy of Heraclitus was also unsatisfactory. Apart from the fact that it hardly accounted sufficiently for the stable element in things, it was bound up with materialistic monism. Ultimately it was bound to be suggested that the highest and truest being is immaterial. Meanwhile it is not surprising to find what Zeller calls "compromise-systems," trying to weld together the thought of their predecessors.

Note on "Pantheism" in pre‑Socratic Greek Philosophy

(i) If a Pantheist is a man who has a subjective religious attitude towards the universe, which latter he identifies with God, then the Pre‑Socratics are scarcely to be called pantheists. That Heraclitus speaks of the One as Zeus is true, but it does not appear that he adopted any religious attitude towards the One — Fire.

(ii) If a pantheist is a man who, while denying a Transcendent Principle of the universe, makes the universe to be ultimately Thought (unlike the materialist, who makes it Matter alone), then the Pre‑Socratics again scarcely merit the name of pantheists, for p60 they conceive or speak of the One in material terms (though it is true that the spirit-matter distinction had not yet been so clearly conceived that they could deny it in the way that the modern materialistic monist denies it).

(iii) In any case the One, the universe, could not be identified with the Greek gods. It has been remarked (by Schelling) that there is no supernatural in Homer, for the Homeric god is part of nature. This remark has its application in the present question. The Greek god was finite and anthropomorphically conceived; he could not possibly be identified with the One, nor would it occur to anyone to do so literally. The name of a god might be sometimes transferred to the One, e.g. Zeus, but the one is not to be thought of as identified with the "actual" Zeus of legend and mythology. The suggestion may be that the One is the only "god" there is, and that the Olympian deities are anthropomorphic fables; but even then it seems very uncertain if the philosopher ever worshipped the One. Stoics might with justice be called pantheists; but, as far as the early Pre‑Socratics are concerned, it seems decidedly preferable to call them monists, rather than pantheists.

The Author's Notes:

1 Parmen., 128B.

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2 Procl., in Parmen., 694, 23 (D. 29 A 15).

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3 Frags. 1, 2.

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4 Frag. 3.

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5 Arist., Phys., Η, 5, 250a19. Simplic. 1108, 18 (D. 29 A 29).

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6 Arist., Phys., Δ, 3, 210b22; 1, 209a23. Eudem., Phys., Frag. 42 (D. 29 A 24),

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7 Arist., Phys., Ζ, 9, 239b9; 2, 233a21. Top., Θ 8, 160b7.

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8 Arist., Phys., Ζ, 9, 239b14.

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9 Arist., Phys., Ζ, 9, 239b30.

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10 Arist., Phys., Ζ, 9, 239b33.

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11 Ross, Physics, p660.

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