I. Zeno of Elea
Life.
155. According to Apollodoros,[[851]] Zeno flourished in Ol. LXXIX. (464-460 B.C.). This date is arrived at by making him forty years younger than his master Parmenides. We have seen already ([§ 84]) that the meeting of Parmenides and Zeno with the young Sokrates cannot well have occurred before 449 B.C., and Plato tells us that Zeno was at that time “nearly forty years old.”[[852]] He must, then, have been born about 489 B.C., some twenty-five years after Parmenides. He was the son of Teleutagoras, and the statement of Apollodoros that he had been adopted by Parmenides is only a misunderstanding of an expression of Plato’s Sophist.[[853]] He was, Plato further tells us,[[854]] tall and of a graceful appearance.
Like Parmenides and most other early philosophers, Zeno seems to have played a part in the politics of his native city. Strabo ascribes to him some share of the credit for the good government of Elea, and says that he was a Pythagorean.[[855]] This statement can easily be explained. Parmenides, we have seen, was originally a Pythagorean, and the school of Elea was no doubt popularly regarded as a mere branch of the larger society. We hear also that Zeno conspired against a tyrant, whose name is differently given, and the story of his courage under torture is often repeated, though with varying details.[[856]]
Writings.
156. Diogenes speaks of Zeno’s “books,” and Souidas gives some titles which probably come from the Alexandrian librarians through Hesychios of Miletos.[[857]] In the Parmenides, Plato makes Zeno say that the work by which he is best known was written in his youth and published against his will.[[858]] As he is supposed to be forty years old at the time of the dialogue, this must mean that the book was written before 460 B.C. ([§ 84]), and it is very possible that he wrote others after it. The most remarkable title which has come down to us is that of the Interpretation of Empedokles. It is not to be supposed, of course, that Zeno wrote a commentary on the Poem of Empedokles; but, as Diels has pointed out,[[859]] it is quite credible that he should have written an attack on it, which was afterwards called by that name. If he wrote a work against the “philosophers,” that must mean the Pythagoreans, who, as we have seen, made use of the term in a sense of their own.[[860]] The Disputations and the Treatise on Nature may, or may not, be the same as the book described in Plato’s Parmenides.
It is not likely that Zeno wrote dialogues, though certain references in Aristotle have been supposed to imply this. In the Physics[[861]] we hear of an argument of Zeno’s, that any part of a heap of millet makes a sound, and Simplicius illustrates this by quoting a passage from a dialogue between Zeno and Protagoras.[[862]] If our chronology is right, there is nothing impossible in the idea that the two men may have met; but it is most unlikely that Zeno should have made himself a personage in a dialogue of his own. That was a later fashion. In another place Aristotle refers to a passage where “the answerer and Zeno the questioner” occurred,[[863]] a reference which is most easily to be understood in the same way. Alkidamas seems to have written a dialogue in which Gorgias figured,[[864]] and the exposition of Zeno’s arguments in dialogue form must always have been a tempting exercise. It appears also that Aristotle made Alexamenos the first writer of dialogues.[[865]]
Plato gives us a clear idea of what Zeno’s youthful work was like. It contained more than one “discourse,” and these discourses were subdivided into sections, each dealing with some one presupposition of his adversaries.[[866]] We owe the preservation of Zeno’s arguments on the one and many to Simplicius.[[867]] Those relating to motion have been preserved by Aristotle himself;[[868]] but, as usual, he has restated them in his own language.
Dialectic.
157. Aristotle in his Sophist[[869]] called Zeno the inventor of dialectic, and this, no doubt, is substantially true, though the beginnings at least of that method of arguing were contemporary with the foundation of the Eleatic school. Plato[[870]] gives us a spirited account of the style and purpose of Zeno’s book, which he puts into his own mouth:—
In reality, this writing is a sort of reinforcement for the argument of Parmenides against those who try to turn it into ridicule on the ground that, if reality is one, the argument becomes involved in many absurdities and contradictions. This writing argues against those who uphold a Many, and gives them back as good and better than they gave; its aim is to show that their assumption of multiplicity will be involved in still more absurdities than the assumption of unity, if it is sufficiently worked out.
The method of Zeno was, in fact, to take one of his adversaries’ fundamental postulates and deduce from it two contradictory conclusions.[[871]] This is what Aristotle meant by calling him the inventor of dialectic, which is just the art of arguing, not from true premisses, but from premisses admitted by the other side. The theory of Parmenides had led to conclusions which contradicted the evidence of the senses, and Zeno’s object was not to bring fresh proofs of the theory itself, but simply to show that his opponents’ view led to contradictions of a precisely similar nature.
Zeno and Pythagoreanism.
158. That Zeno’s dialectic was mainly directed against the Pythagoreans is certainly suggested by Plato’s statement, that it was addressed to the adversaries of Parmenides, who held that things were “a many.”[[872]] Zeller holds, indeed, that it was merely the popular form of the belief that things are many that Zeno set himself to confute;[[873]] but it is surely not true that ordinary people believe things to be “a many” in the sense required. Plato tells us that the premisses of Zeno’s arguments were the beliefs of the adversaries of Parmenides, and the postulate from which all his contradictions are derived is the view that space, and therefore body, is made up of a number of discrete units, which is just the Pythagorean doctrine. Nor is it at all probable that Anaxagoras is aimed at.[[874]] We know from Plato that Zeno’s book was the work of his youth.[[875]] Suppose even that it was written when he was thirty, that is to say, about 459 B.C., Anaxagoras had just taken up his abode at Athens at that time,[[876]] and it is very unlikely that Zeno had ever heard of him. There is, on the other hand, a great deal to be said for the view that Anaxagoras had read the work of Zeno, and that his emphatic adhesion to the doctrine of infinite divisibility was due to the criticism of his younger contemporary.[[877]]
It will be noted how much clearer the historical position of Zeno becomes if we follow Plato in assigning him to a somewhat later date than is usual. We have first Parmenides, then the pluralists, and then the criticism of Zeno. This, at any rate, seems to have been the view which Aristotle took of the historical development.[[878]]
What is the unit?
159. The polemic of Zeno is clearly directed in the first instance against a certain view of the unit. Eudemos, in his Physics,[[879]] quoted from him the saying that “if any one could tell him what the one was, he would be able to say what things are.” The commentary of Alexander on this, preserved by Simplicius,[[880]] is quite satisfactory. “As Eudemos relates,” he says, “Zeno the disciple of Parmenides tried to show that it was impossible that things could be a many, seeing that there was no unit in things, whereas ‘many’ means a number of units.” Here we have a clear reference to the Pythagorean view that everything may be reduced to a sum of units, which is what Zeno denied.[[881]]
The Fragments.
160. The fragments of Zeno himself also show that this was his line of argument. I give them according to the arrangement of Diels.
If the one had no magnitude, it would not even be.... But, if it is, each one must have a certain magnitude and a certain thickness, and must be at a certain distance from another, and the same may be said of what is in front of it; for it, too, will have magnitude, and something will be in front of it.[[882]] It is all the same to say this once and to say it always; for no such part of it will be the last, nor will one thing not be compared with another.[[883]] So, if things are a many, they must be both small and great, so small as not to have any magnitude at all, and so great as to be infinite. R. P. 134.
For if it were added to any other thing it would not make it any larger; for nothing can gain in magnitude by the addition of what has no magnitude, and thus it follows at once that what was added was nothing.[[884]] But if, when this is taken away from another thing, that thing is no less; and again, if, when it is added to another thing, that does not increase, it is plain that what was added was nothing, and what was taken away was nothing. R. P. 132.
If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.
If things are a many, they will be infinite in number; for there will always be other things between them, and others again between these. And so things are infinite in number. R. P. 133.[[885]]
The unit.
161. If we hold that the unit has no magnitude—and this is required by what Aristotle calls the argument from dichotomy,[[886]]—then everything must be infinitely small. Nothing made up of units without magnitude can itself have any magnitude. On the other hand, if we insist that the units of which things are built up are something and not nothing, we must hold that everything is infinitely great. The line is infinitely divisible; and, according to this view, it will be made up of an infinite number of units, each of which has some magnitude.
That this argument refers to points is proved by an instructive passage from Aristotle’s Metaphysics.[[887]] We read there—
If the unit is indivisible, it will, according to the proposition of Zeno, be nothing. That which neither makes anything larger by its addition to it, nor smaller by its subtraction from it, is not, he says, a real thing at all; for clearly what is real must be a magnitude. And, if it is a magnitude, it is corporeal; for that is corporeal which is in every dimension. The other things, i.e. the plane and the line, if added in one way will make things larger, added in another they will produce no effect; but the point and the unit cannot make things larger in any way.
From all this it seems impossible to draw any other conclusion than that the “one” against which Zeno argued was the “one” of which a number constitute a “many,” and that is just the Pythagorean unit.
Space.
162. Aristotle refers to an argument which seems to be directed against the Pythagorean doctrine of space,[[888]] and Simplicius quotes it in this form:[[889]]
If there is space, it will be in something; for all that is is in something, and what is in something is in space. So space will be in space, and this goes on ad infinitum, therefore there is no space. R. P. 135.
What Zeno is really arguing against here is the attempt to distinguish space from the body that occupies it. If we insist that body must be in space, then we must go on to ask what space itself is in. This is a “reinforcement” of the Parmenidean denial of the void. Possibly the argument that everything must be “in” something, or must have something beyond it, had been used against the Parmenidean theory of a finite sphere with nothing outside it.
Motion.
163. Zeno’s arguments on the subject of motion have been preserved by Aristotle himself. The system of Parmenides made all motion impossible, and his successors had been driven to abandon the monistic hypothesis in order to avoid this very consequence. Zeno does not bring any fresh proofs of the impossibility of motion; all he does is to show that a pluralist theory, such as the Pythagorean, is just as unable to explain it as was that of Parmenides. Looked at in this way, Zeno’s arguments are no mere quibbles, but mark a great advance in the conception of quantity. They are as follows:—
(1) You cannot get to the end of a race-course.[[890]] You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum, so that there are an infinite number of points in any given space, and you cannot touch an infinite number one by one in a finite time.[[891]]
(2) Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it.[[892]]
The “hypothesis” of the second argument is the same as that in the first, namely, that the line is a series of points; but the reasoning is complicated by the introduction of another moving object. The difference, accordingly, is not a half every time, but diminishes in a constant ratio. Again, the first argument shows that no moving object can ever traverse any distance at all, however fast it may move; the second emphasises the fact that, however slowly it moves, it will traverse an infinite distance.
(3) The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move.[[893]]
Here a further complication is introduced. The moving object itself has length, and its successive positions are not points but lines. The successive moments in which it occupies them are still, however, points of time. It may help to make this clear if we remember that the flight of the arrow as represented by the cinematograph would be exactly of this nature.
(4) Half the time may be equal to double the time. Let us suppose three rows of bodies,[[894]] one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions (Fig. 1). By the time they are all in the same part of the course, B will have passed twice as many of the bodies in C as in A (Fig. 2).
| Fig. 1 | ||||||||
| A. | ● | ● | ● | ● | ||||
| B. | ● | ● | ● | ● | → | |||
| C. | ← | ● | ● | ● | ● | |||
| Fig. 2 | ||||
| A. | ● | ● | ● | ● |
| B. | ● | ● | ● | ● |
| C. | ● | ● | ● | ● |
Therefore the time which it takes to pass C is twice as long as the time it takes to pass A. But the time which B and C take to reach the position of A is the same. Therefore double the time is equal to the half.[[895]]
According to Aristotle, the paralogism here depends upon the assumption that an equal magnitude moving with equal velocity must move for an equal time, whether the magnitude with which it is equal is at rest or in motion. That is certainly so, but we are not to suppose that this assumption is Zeno’s own. The fourth argument is, in fact, related to the third just as the second is to the first. The Achilles adds a second moving point to the single moving point of the first argument; this argument adds a second moving line to the single moving line of the arrow in flight. The lines, however, are represented as a series of units, which is just how the Pythagoreans represented them; and it is quite true that, if lines are a sum of discrete units, and time is similarly a series of discrete moments, there is no other measure of motion possible than the number of units which each unit passes.
This argument, like the others, is intended to bring out the absurd conclusions which follow from the assumption that all quantity is discrete, and what Zeno has really done is to establish the conception of continuous quantity by a reductio ad absurdum of the other hypothesis. If we remember that Parmenides had asserted the one to be continuous (fr. [8], 25), we shall see how accurate is the account of Zeno’s method which Plato puts into the mouth of Sokrates.