The third and last important thinker of the Eleatic School is Zeno who, like Parmenides, was a man of Elea. His birth is placed about 489 B.C. He composed a prose treatise in which he developed his philosophy. Zeno's contribution to Eleaticism is, in a sense, entirely negative. He did not add anything positive to the teachings of Parmenides. He supports Parmenides in the doctrine of Being. But it is not the conclusions of Zeno that are novel, it is rather the reasons which he gave for them. In attempting to support the Parmenidean doctrine from a new point of view he developed certain ideas about the ultimate character of space and time which have since been of the utmost importance in philosophy. Parmenides had taught that the world of sense is illusory and false. The essentials of that world are two-- multiplicity and change. True Being is absolutely one; there is in it no plurality or multiplicity. Being, moreover, is absolutely static and unchangeable. There is in it no motion. Multiplicity and motion are the two characteristics of the false world of sense. Against multiplicity and motion, therefore, Zeno directed his [{53}] arguments, and attempted indirectly to support the conclusions of Parmenides by showing that multiplicity and motion are impossible. He attempted to force multiplicity and motion to refute themselves by showing that, if we assume them as real, contradictory propositions follow from that assumption. Two propositions which contradict each other cannot both be true. Therefore the assumptions from which both follow, namely, multiplicity and motion, cannot be real things.
Zeno's arguments against multiplicity.
(1) If the many is, it must be both infinitely small and infinitely large. The many must be infinitely small. For it is composed of units. This is what we mean by saying that it is many. It is many parts or units. These units must be indivisible. For if they are further divisible, then they are not units. Since they are indivisible they can have no magnitude, for that which has magnitude is divisible. The many, therefore, is composed of units which have no magnitude. But if none of the parts of the many have magnitude, the many as a whole has none. Therefore, the many is infinitely small. But the many must also be infinitely large. For the many has magnitude, and as such, is divisible into parts. These parts still have magnitude, and are therefore further divisible. However far we proceed with the division the parts still have magnitude and are still divisible. Hence the many is divisible ad infinitum. It must therefore be composed of an infinite number of parts, each having magnitude. But the smallest magnitude, multiplied by infinity, becomes an infinite magnitude. Therefore the many is infinitely large. (2) The [{54}] many must be, in number, both limited and unlimited. It must be limited because it is just as many as it is, no more, no less. It is, therefore, a definite number. But a definite number is a finite or limited number. But the many must be also unlimited in number. For it is infinitely divisible, or composed of an infinite number of parts.
Zeno's arguments against motion.
(1) In order to travel a distance, a body must first travel half the distance. There remains half left for it still to travel. It must then travel half the remaining distance. There is still a remainder. This progress proceeds infinitely, but there is always a remainder untravelled. Therefore, it is impossible for a body to travel from one point to another. It can never arrive. (2) Achilles and the tortoise run a race. If the tortoise is given a start, Achilles can never catch it up. For, in the first place, he must run to the point from which the tortoise started. When he gets there, the tortoise will have gone to a point further on. Achilles must then run to that point, and finds then that the tortoise has reached a third point. This will go on for ever, the distance between them continually diminishing, but never being wholly wiped out. Achilles will never catch up the tortoise. (3) This is the story of the flying arrow. An object cannot be in two places at the same time. Therefore, at any particular moment in its flight the arrow is in one place and not in two. But to be in one place is to be at rest. Therefore in each and every moment of its flight it is at rest. It is thus at rest throughout. Motion is impossible.
This type of argument is, in modern times, called "antinomy." An antinomy is a proof that, since two contradictory propositions equally follow from a given assumption, that assumption must be false. Zeno is also called by Aristotle the inventor of dialectic. Dialectic originally meant simply discussion, but it has come to be a technical term in philosophy, and is used for that type of reasoning which seeks to develop the truth by making the false refute and contradict itself. The conception of dialectic is especially important in Zeno, Plato, Kant, and Hegel.
All the arguments which Zeno uses against multiplicity and motion are in reality merely variations of one argument. That argument is as follows. It applies equally to space, to time, or to anything which can be quantitatively measured. For simplicity we will consider it only in its spatial significance. Any quantity of space, say the space enclosed within a circle, must either be composed of ultimate indivisible units, or it must be divisible ad infinitum. If it is composed of indivisible units, these must have magnitude, and we are faced with the contradiction of a magnitude which cannot be divided. If it is divisible ad infinitum, we are faced with the contradiction of supposing that an infinite number of parts can be added up and make a finite sum-total. It is thus a great mistake to suppose that Zeno's stories of Achilles and the tortoise, and of the flying arrow, are merely childish puzzles. On the contrary, Zeno was the first, by means of these stories, to bring to light the essential contradictions which lie in our ideas of space and time, and thus to set an important problem for all subsequent philosophy.
All Zeno's arguments are based upon the one argument described above, which may be called the antinomy of infinite divisibility. For example, the story of the flying arrow. At any moment of its flight, says Zeno, it must be in one place, because it cannot be in two places at the same moment. This depends upon the view of time as being infinitely divisible. It is only in an infinitesimal moment, an absolute moment having no duration, that the arrow is at rest. This, however, is not the only antinomy which we find in our conceptions of space and time. Every mathematician is acquainted with the contradictions immanent in our ideas of infinity. For example, the familiar proposition that parallel straight lines meet at infinity, is a contradiction. Again, a decreasing geometrical progression can be added up to infinity, the infinite number of its terms adding up in the sum-total to a finite number. The idea of infinite space itself is a contradiction. You can say of it exactly what Zeno said of the many. There must be in existence as much space as there is, no more. But this means that there must be a definite and limited amount of space. Therefore space is finite. On the other hand, it is impossible to conceive a limit to space. Beyond the limit there must be more space. Therefore space is infinite. Zeno himself gave expression to this antinomy in the form of an argument which I have not so far mentioned. He said that everything which exists is in space. Space itself exists, therefore space must be in space. That space must be in another space and so ad infinitum. This of course is merely a quaint way of saying that to conceive a limit to space is impossible.