Zeno of Elea Born around 489 BC

This Zeno, as opposed to Zeno of Citium, is perhaps most famous for his paradoxes. These strange scenarios are designed to make the think about mathematics and physics in new and different ways. Zeno was opposed to many of the ideas of Pythagoras, supported many of the ideas of Parmenides. His paradoxes have been reconstructed in slightly different ways by modern scientists from his cryptic and fragmentary writings.

• Let us suppose that reality is made up of units. These units are either with magnitude or without magnitude. If the former, then a line for example, as made up of units possessed of magnitude, will be infinitely divisible, since, however far you divide, the units will still have magnitude and so be divisible. But in this case, the line will be made up of an infinite number of units, each of which is possessed of magnitude. The line, then, must be infinitely great, as composed of an infinite number of bodies. Everything in the world, then, must be infinitely great, and a forteriori the world itself must be infinitely great. Suppose, on the other hand, that the units are without magnitude. In this case, the whole universe will also be without magnitude, since, however many units you add together, if none of them has any magnitude, then the whole collection of them will also be without magnitude. But if the universe is without any magnitude, it must be infinitely small. Indeed, everything in the universe must be infinitely small. Either everything in the universe is infinitely great, or everything in the universe is infinitely small. The supposition from which the dilemma flows is an absurd supposition, namely, that the universe and everything in it are composed of units.

o To understand this paradox, think of “magnitude” as “size,” and consider the number line. In geometry, you learned that points are infinitely small; in order for a finite line segment to contain an infinite number of points, they must be infinitely small. Yet an infinite number of infinitely small points would still be infinitely small.

• If there is a many, then we ought to be able to say how many there are. At least, they should be numerable; if they are not numerable, how can they exist? On the other hand, they cannot possibly be numerable, but must be infinite. Why? Because between any two assigned units there will always be other units, just as a line is infinitely divisible. But it is absurd to say that the many are finite in number and infinite in number at the same time.

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o Zeno would like us to conclude, then, that there is not a “many” – i.e., that there is not a plurality of objects in the universe, but rather that there is at most one object in the universe. What we normally consider to be different objects, Zeno would call different phases of one continuous substance “existence” or “being.” The final mathematic resolution of this paradox waited for over 2,000 years, until the German mathematician Georg Cantor showed, in the late 1800’s, that there were both countable and uncountable infinities.

• Does a bushel of corn make a noise when it falls to the ground? Clearly. But what of a grain of corn, or the thousandth part of a grain of corn? It makes no noise. But the bushel of corn is composed only of the grains of corn or of the parts of the grains of corn. If, then, the parts make no sound when they fall, how can the whole make a sound, when the whole is composed only of the parts?

• If an existent were infinitely divisible, no contradiction should arise from the supposition that it has been divided exhaustively. But an exhaustive division would resolve the existent into elements of zero extension. This is impossible, for no extensive magnitude could consist of extension-less elements.

• Zeno denies the existence of the void or empty space, and tries to support this denial by reducing the opposite view to absurdity. Suppose for a moment that there is a space in which things are. If it is nothing, then things cannot be in it. If, however, it is something, it will itself be in space, and so on indefinitely. But this is an absurdity. Things, therefore, are not in space or in an empty void.

• Let us suppose that you want to cross a certain distance. In order to so, the Pythagoreans say that you would have to traverse an infinite number of points. Moreover, you would have to travel the distance in finite time, if you wanted to get to the other side at all. But how can you traverse an infinite number of points, and so an infinite distance, in a finite time? We must conclude that you cannot cross any specified distance. Indeed, we must conclude that no object can traverse any distance whatsoever (for the same difficulty always recurs), and that all motion is consequently impossible.

• Let us suppose that a sprinter and a tortoise are going to have a race. Since the sprinter is a sportsman, he gives the tortoise a head start. Now, by the time that he has reached the place from which the tortoise has started, the latter has again

Zeno Elea – page 2 advanced to another point; and when the sprinter reaches that point, then the tortoise will have advanced still another distance, even if very short. Thus the sprinter is always coming nearer to the tortoise, but never actually overtakes it – and never can do so, on the supposition that a line is made up of an infinite number of points, for then the sprinter would have to traverse an infinite distance.

o On the Pythagorean hypothesis, then, which Zeno opposes, the sprinter will never catch up to the tortoise; and so, although Pythagoras asserts the reality of motion, he makes it impossible on his own doctrine. For it follows that the slower moves as fast as the faster. By pointing out this internal contradiction, Zeno wishes to persuade us to reject two Pythagorean ideas: that a line is made up of an infinite number of points, and that motion is real.

• Suppose a moving arrow. According to the Pythagorean theory, the arrow should occupy a given position in space. But to occupy a given position in space is to be at rest. Therefore the flying arrow is at rest, which is a contradiction.

o The path of the flying arrow can be noted as Cartesian pairs of the form (Pn,Tn), where Pn is the place, and Tn is the time, so the arrow moves through (P1,T1), (P2,T2), (P3,T3), etc., until the flight ends. But to be at (Pn,Tn) is to be at rest.

Scholars point out that the exact number of paradoxes that Zeno wrote, and their exact formulations, are subject to some ambiguity. There is at least one more than those listed above, but the reader has enough here to capture the general flavor as well as the specific issues within mathematics and physics.

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