Zeno "disproved" motion in different ways. Thus, he argued that a body in motion, before reaching a given point, must first have travelled half the distance. But, before this, it must have travelled half of that half, and so on ad infinitum. Thus, when two bodies are moving in the same direction, and the one behind at a fixed distance from the one in front is moving faster, we assume that it will overtake the other. Not so, says Zeno. "The slower one can never be overtaken by the quicker." This is the famous paradox of Achilles the Swift. Imagine a race between Achilles and a tortoise. Suppose that Achilles can run ten times faster than the tortoise which has 1000 metres start. By the time Achilles has covered 1000 metres, the tortoise will be 100 metres ahead; when Achilles has covered that 100 metres, the tortoise will be one metre ahead; when he covers that distance, the tortoise will be one tenth of a metre ahead, and so on to infinity.

From the standpoint of everyday common sense, this seems absurd. Of course, Achilles will overtake the tortoise! Aristotle remarked that "This proof asserts the same endless divisibility, but it is untrue, for the quick will overtake the slow body if the limits to be traversed be granted to it." Hegel quotes these words, and comments: "This answer is true and contains all that can be said; that is, there are in this representation two periods of time and two distances, which are separated from one another, i.e., they are limited in relation to one another;" but then he adds, "when, on the contrary, we admit that time and space are related to one another as continuous, they are, while being two, not two, but identical." (Hegel, op. cit., p. 273.)

The paradoxes of Zeno do not prove that movement is an illusion, or that Achilles, in practice, will not overtake the tortoise, but they do reveal brilliantly the limitations of the kind of thinking now known as formal logic. The attempt to eliminate all contradiction from reality, as the Eleatics did, inevitably leads to this kind of insoluble paradox, or antimony, as Kant later called it. In order to prove that a line could not consist of an infinite number of points, Zeno claimed that, if it were really so, then Achilles would never overtake the tortoise. There really is a logical problem here. As Alfred Hooper explains:

"This paradox still perplexes even those who know that it is possible to find the sum of an infinite series of numbers forming a geometrical progression whose common ratio is less than 1, and whose terms consequently become smaller and smaller and thus ‘converge’ on some limiting value." (A. Hooper, Makers of Mathematics, p. 237.)

In fact, Zeno had uncovered a contradiction in mathematical thought which would have to wait two thousand years for a solution. The contradiction relates to the use of the infinite. From Pythagoras right up to the discovery of the differential and integral calculus in the 17th century, mathematicians went to great lengths to avoid the use of the concept of infinity. Only the great genius Archimedes approached the subject, but still avoided it by using a roundabout method.

The Pythagoreans stumbled on the fact that the square root of two cannot be expressed as a number. They invented ingenious ways of finding successive approximations for it. But, no matter how far the process is taken, you never get an exact answer. The result is always midway between two numbers. The further down the list you go, the closer you get to the value of the square root of two. But the process of successive approximation may be continued forever, without getting a precise result that can be expressed in a whole number.

The Pythagoreans thus had to abandon the idea of a line made up of a finite number of very small points, and accept that a line is made up of an infinite number of points with no dimension. Parmenides approached the issue from a different angle, arguing that a line was indivisible. In order to prove the point, Zeno tried to show the absurd consequences that would follow from the concept of infinite divisibility. For centuries after, mathematicians steered clear of the idea of infinity, until Kepler in the 17th century simply swept aside all logical objections and boldly made use of the infinite in his calculations, to achieve epoch making results.

Ultimately, all these paradoxes are derived from the problem of the continuum. All the attempts to resolve them by means of mathematical theorems, such as the theory of convergent series and the theory of sets have only given rise to new contradictions. In the end, Zeno’s arguments have not been refuted, because they are based on a real contradiction which, from the standpoint of formal logic, cannot be answered. "Even the abstruse arguments put forward by Dedekind (1831-1916), Cantor (1845-1918) and Russell (1872-1970) in their mighty efforts to straighten out the paradoxical problems of infinity into which we are led by our concept of ‘numbers,’ have resulted in the creation of still further paradoxes." (Hooper, op. cit., p. 238.) The breakthrough came in the 17th and 18th centuries, when men like Kepler, Cavalieri, Pascal, Wallis, Newton and Leibniz decided to ignore the numerous difficulties raised by formal logic, and deal with infinitesimal quantities. Without the use of infinity, the whole of modern mathematics, and with it physics, would be unable to function.

The essential problem, highlighted by Zeno’s paradoxes, is the inability of formal logic to grasp movement. Zeno’s paradox of the Arrow takes as an example of movement the parabola traced by an arrow in flight. At any given point in this trajectory, the arrow is considered to be still. But since, by definition, a line consists of a series of points, at each of which the arrow is still, movement is an illusion. The answer to this paradox was given by Hegel.

The notion of movement necessarily involves a contradiction. Consider the movement of a body, Zeno’s arrow for example, from one point to another. When it starts to move, it is no longer at point A. At the same time, it is not yet at point B. Where is it, then? To say that it is "in the middle" conveys nothing, for then it would still be at a point, and therefore at rest. "But," says Hegel, "movement means to be in this place and not to be in it, and thus to be in both alike; and this is the continuity of space and time which first makes motion possible." (Hegel, op. cit., Vol. 1, p. 273.) As Aristotle shrewdly observed, "It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow." But what is this "now"? If we say the arrow is "here," "now," it has already gone.

Engels writes:

"Motion itself is a contradiction: even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." (Engels, Anti-D¸hring, p. 152.)

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