ZENO OF ELEA, son of Teleutagoras, is supposed to have been born towards the beginning of the 5th century B.C. The pupil and the friend of Parmenides, he sought to recommend his master's doctrine of the existence of the One by contro verting the popular belief in the existence of the Many. In virtue of this method of indirect argumentation he is regarded as the inventor of "dialectic," that is to say, disputation having for its end not victory but the discovery or the transmission of truth. He is said to have been concerned in a plot against a tyrant, and on its detection to have borne with exemplary constancy the tortures to which he was subjected; but authorities differ both as to the name and the residence of the tyrant and as to the circumstances and the issue of the enterprise.
In Plato's Parmenides, Socrates, "then very young," meets Parmenides, "an old man some sixty-five years of age," and Zeno, "a man of about forty, tall and personable," and engages them in philosophical discussion. But it may be doubted whether such a meeting was chronologically possible. Plato's account of Zeno's teaching (Parmenides, 128 seq.) is, however, presumably as accurate as it is precise. In reply to those who thought that Parmenides's theory of the existence of the One involved inconsistencies and absurdities, Zeno tried to show that the assumption of the existence of the Many, that is to say, a plurality of things in time and space, carried with it inconsistencies and absurdities grosser and more numerous. In early youth he collected his arguments in a book, which, according to Plato, was put into circulation without his knowledge.
Of the paradoxes used by Zeno to discredit the belief in plurality and motion, eight survive in the writings of Aristotle and Simplicius. They are commonly stated as follows. (I) If the Existent is Many, it must be at once infinitely small and infinitely great - infinitely small, because the parts of which it consists must be indivisible and therefore without magnitude; infinitely great,, because, that any part having magnitude may be separate from any other part, the intervention of a third part having magnitude is necessary, and that this third part may be separate from the other two the intervention of other parts having magnitude is necessary, and so on ad infinitum. (2) In like manner the Many must be numerically both finite and infinite - numerically finite, because there are as many things as there are, neither more nor less; numerically infinite, because, that any two things may be separate, the intervention of a third thing is necessary, and so on ad infinitum. (3) If all that is is in space, space itself must be in space, and so on ad infinitum. (4) If a bushel of corn turned out upon the floor makes a noise, each grain and each part of each grain must make a noise likewise; but, in fact, it is not so. (5) Before a body in motion can reach a given point, it must first traverse the half of the distance; before it can traverse the half of the distance, it must first traverse the quarter; and so on ad infinitum. Hence, that a body may pass from one point to another, it must traverse an infinite number of divisions. But an infinite distance (which Zeno fails to distinguish from a finite distance infinitely divided) cannot be traversed in a finite time. Consequently, the goal can never be reached. (6) If the tortoise has the start of Achilles, Achilles can never come up with the tortoise; for, while Achilles traverses the distance from his starting-point to the starting-point of the tortoise, the tortoise advances a certain distance, and while Achilles traverses this distance, the tortoise makes a further advance, and so on ad infinitum. Consequently, Achilles may run ad infinitum without overtaking the tortoise. [This paradox is virtually identical with (5), the only difference being that, whereas in (5) there is one body, in (6) there are two bodies, moving towards a limit. The "infinity" of the premise is an infinity of subdivisions of a distance which is finite; the "infinity" of the conclusion is an infinity of distance. Thus Zeno again confounds a finite distance infinitely divided with an infinite distance. If the tortoise has a start of 1000 ft. Achilles, on the supposition that his speed is ten times that of the tortoise, must traverse an infinite number of spaces - 1000 ft., loo ft., to ft., &c. - and the tortoise must traverse an infinite number of spaces - loo ft., to ft., I ft., &c. - before they reach the point, distant from their starting-points III I y ft. and I I I ft. respectively, at which the tortoise is overtaken. Ina word, wood-100+10 &c., in (6) and 2-?-4+a &c., in (5) are convergent series, and I I I I and I are the limits to which they respectively approximate.] (7) So long as anything is in one and the same space, it is at rest. Hence an arrow is at rest at every moment of its flight, and therefore also during the whole of its flight. (8) Two bodies moving with equal speed traverse equal spaces in equal time. But, when two bodies move with equal speed in opposite directions, the one passes the other in half the time in which it passes it when at rest. These propositions appeared to Zeno to be irreconcilable. In short, the ordinary belief in plurality and motion seemed to him to involve fatal inconsistencies, whence he inferred that Parmenides was justified in distinguishing the mutable movable Many from the immutable immovable One, which alone is really existent. In other words, Zeno re-affirmed the dogma, "The Ent is, the Non-ent is not."
If tradition has not misrepresented these paradoxes of time, space and motion, there is in Zeno's reasoning an element of fallacy. It is indeed difficult to understand how so acute a thinker should confound that which is infinitely divisible with that which is infinitely great, as in (I), (2), (5), and (6); that he should identify space and 'magnitude, as in (3); that he should neglect the imperfection of the organs of sense, as in (4); that he should deny the reality of motion, as in (7); and that he should ignore the relativity of speed, as in (8): and of late years it has been thought that the conventional statements of the paradoxes, and in particular of those which are more definitely mathematical, namely (5), (6), (7), (8), do less than justice to Zeno's acumen. Thus, several French writers - notably, Tannery and Noel - regard them as dilemmas advanced, with some measure of success, in refutation of specific doctrines attributed to the Pythagoreans. "One of the most notable victims of posterity's lack of judgment," says Bertrand Russell, "is the Eleatic Zeno. 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. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreamed of any connexion between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an unchanging world, and that the arrow at every moment of its flight is truly at rest." "The interpretation of Zeno's last four paradoxes given by Messrs. Noel and Russell," says G. H. Hardy, "may be briefly stated as follows: The notion of time, which seems at first sight to enter into (5) and (6), should be eliminated. The former should be regarded as asserting that the whole is, not temporally, but logically, subsequent to the part, and that therefore there is an infinite regress in the notion of a whole which is infinitely divisible - a view which at any rate demands a serious refutation. The kernel of the latter lies in the perfectly valid proof which it affords that the tortoise passes through as many positions as Achilles - a view which embodies an accepted doctrine of modern mathematics. The paradox of the arrow (7), says Mr Russell, is a plain statement of a very elementary fact: the arrow is at rest at very moment of its flight: Zeno's only mistake was in inferring (if he did infer) that it was therefore at the same point at one moment as at another. Finally, the last paradox may be interpreted as a valid refutation of the doctrine that space and time are not infinitely divisible. How far this interpretation of Zeno is historically justifiable, may be doubtful. But one may well believe that there was in his mind at any rate a foreshadowing of some of the ideas by which modern mathematicians have finally laid to rest the traditional difficulties connected with infinity and continuity."
Great as was the importance of these paradoxes of plurality and motion in stimulating speculation about space and time, their direct influence upon Greek thought was less considerable than that of another paradox - strangely neglected by historians of philosophy - the paradox of predication. We learn from Plato (Parmenides, 127 D) that "the first hypothesis of the first argument" of Zeno's book above mentioned ran as follows: "If existences are many, they must be both like and unlike [unlike, inasmuch as they are not one and the same, and like, inasmuch as they agree in not being one and the same, Proclus, On the Parmenides, ii. 143]. But this is impossible; for unlike things cannot be like, nor like things unlike. Therefore existences are not many." That is to say, not perceiving that the same thing may be at once like and unlike in different relations, Zeno regarded the attribution to the same thing of likeness and unlikeness as a violation of what was afterwards known as the principle of contradiction; and, finding that plurality entailed these attributions, he inferred its unreality. Now, when without qualification he affirmed that the unlike thing cannot be like, nor the like thing unlike, he was on the high road to the doctrine maintained three-quarters of a century later by the Cynics, that no predication which is not identical is legitimate. He was not indeed aware how deeply he had committed himself; otherwise he would have observed that his argument, if valid against the Many of the vulgar, was valid also against the One of Parmenides, with its plurality of attributes, as well as that, in the absence of a theory of predication, it was useless to speculate about knowledge and being. But others were not slow to draw the obvious conclusions; and it may be conjectured that Gorgias's sceptical development of the Zenonian logic contributed, not less than Protagoras's sceptical development of the Ionian physics, to the diversion of the intellectual energies of Greece from the pursuit of truth to the pursuit of culture.
For three-quarters of a century, then, philosophy was at a standstill; and, when in the second decade of the 4th century the pursuit of truth was resumed, it was plain that Zeno's paradox of predication must be disposed of before the problems which had occupied the earlier thinkers - the problem of knowledge and the problem of being - could be so much as attempted. Accordingly, in the seventh book of the Republic, where Plato propounds his scheme of Academic education, he directs the attention of studious youth primarily, if not exclusively, to the concurrence of inconsistent attributes; and in the Phaedo, 102 B-103 A, taking as an instance the tallness and the shortness simultaneously discoverable in Simmias, he offers his own theory of the immanent idea as the solution of the paradox. Simmias, he says, has in him the ideas of tall and short. Again, when it presently appeared that the theory of the immanent idea was inconsistent with itself, and moreover inapplicable to explain predication except where the subject was a sensible thing, so that reconstruction became necessary, the Zenonian difficulty continued to demand and to receive Plato's best attention. Thus, in the Parmenides, with the paradox of likeness and unlikeness for his text, he inquires how far the cur14nt theories of being (his own included) are capable of providing, not only for knowledge, but also for predication, and in the concluding sentence he suggests that, as likeness and unlikeness, greatness and smallness, &c., are relations, the initial paradox is no longer paradoxical; while in the Sophist, Zeno's doctrine having been shown to be fatal to reason, thought, speech and utterance, the mutual Koevwvia of Elan which are not abra KaO' abra is elaborately demonstrated. It would seem then that, not to Antisthenes only, but to Plato also, Zeno's paradox of predication was a substantial difficulty; and we shall be disposed to give Zeno credit accordingly for his perception of its importance.
In all probability Zeno did not observe that in his controversial defence of Eleaticism he was interpreting Parmenides's teaching anew. But so it was. For, while Parmenides had recognized, together with the One, which is, and is the object of knowledge, a Many, which is not, and therefore is not known, but nevertheless becomes, and is the object of opinion, Zeno plainly affirmed that plurality, becoming and opinion are one and all inconceivable. In a word, the fundamental dogma, "The Ent is, the Non-ent is not," which with Parmenides had been an assertion of the necessity of distinguishing between the Ent, which is, and the Non-ent, which is not, but becomes, was with Zeno a declaration of the Non-ent's absolute nullity. Thus, just as Empedocles developed Parmenides's theory of the Many to the neglect of his theory of the One, so Zeno developed the theory of the One to the neglect of the theory of the Many. With the severance of its two members Eleaticism proper, the Eleaticism of Parmenides, ceased to exist.
The first effect of Zeno's teaching was to complete the discomfiture of philosophy. For the paradox of predication, which he had used to disprove the existence of plurality, was virtually a denial of all speech and all thought, and thus led to a more comprehensive scepticism than that which sprang from the contemporary theories of sensation. Nevertheless, he left an enduring mark upon Greek speculation, inasmuch as he not only recognized the need of a logic, and grappled, however unsuccessfully, with one of the most obvious of logical. problems, but also by the invention of dialectic provided a new and powerful instrument against the time when the One and the Many should be reunited in the philosophy of Plato.
. - F. W. A. MuIlach, Fragmenta Philosophorum Graecorum (Paris, 1860), i. 266 seq.; Zeller, Die Philosophie d. Griechen (Leipzig, 1876), i. 534-55 2; P. Tannery, Pour l'Histoire de la Science Hellbne (Paris, 1887), pp. 247-261; H. Diels, Die Fragmente der Vorsokratiker (Berlin, 1906, 1907). On the mathematical questions raised by certain of Zeno's paradoxes, see G. Noel,. Revue de Metaphysique et de Morale, i. 107-125, and Hon. Bertrand Russell, Principles of Mathematics (Cambridge, 1903), pp. 346-354. For histories of philosophy and other works upon Eleaticism see Parmenides
. (H. Ja.