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 (1), (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 Noël—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. Noël 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 current 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 κοινωνία of εἴδη which are not αὐτὰ καθ’ αὑτά 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.
Bibliography. F. W. A. Mullach, Fragmenta Philosophorum Graecorum (Paris, 1860), i. 266 seq.; Zeller, Die Philosophie d. Griechen (Leipzig, 1876), i. 534-552; P. Tannery, Pour l'Histoire de la Science Hellène (Paris, 1887), pp. 247-261; H. Diels, Die Fraqmente der Vorsokratiker (Berlin, 1906, 1907). On the mathematical questions raised by certain of Zeno's paradoxes, see G. Noël, Revue de Metaphysique et de Morale, i. 107-125, and Hon. Bertrand
