ZENO OF ELEA, son of Teleutagoras, born probably towards the beginning of the 5th century B.C. The pupil and the friend of Parmenides, he sought to recommend his master's doc trine of the existence of the One by controverting the popular belief in the existence of the Many. In virtue of this method of indirect argumentation he is regarded as the inventor of "dialec tic," that is to say, disputation having for its end not victory but the discovery of truth.
In Plato's Parmenides, Socrates, "then very young," discusses with Parmenides and Zeno, "a man of about forty." But it may be doubted whether such a meeting was chronologically possible.
Plato's account of Zeno's teaching (Parmenides, 128 seq.), how ever, is presumably accurate. In reply to those who thought that Parmenides's theory of the existence of the One involved incon sistencies, Zeno tried to show that the assumption of the existence of a plurality of things in time and space, carried with it more serious inconsistencies. In early youth he collected his arguments in a book, which, according to Plato, was put into circulation with out his knowledge.
Of the paradoxes used by Zeno to discredit the belief in plurality and motion, eight survive in the writings of Aristotle and Sim plicius. They are commonly stated as follows : If the Exist ent is Many, it must be at once infinitely small and infinitely great—infinitely small, because its parts 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 inter vention 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 inter vention 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 the paradox does not 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 frcfn his starting-point to the start ing-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 para dox 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.) (7) So long as any thing 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 paradoxes are probably properly regarded as dilemmas advanced in refuta tion of specific doctrines attributed to the Pythagoreans.