Zeno's Paradoxes
A paradox (or antinomy) is a statement that appears self-contradictory or contrary to common sense. Scholars believe that the philosopher Zeno wrote his paradoxes around 465 B.C. He probably wrote about forty paradoxes, but only about two hundred words actually written by Zeno have survived; and these mention only two paradoxes. Thus, almost all of the information we have about Zeno's paradoxes comes from other authors, notably Aristotle and Simplicius. In particular, Aristotle dealt with Zeno's paradoxes of motion in his book Physics; he considered Zeno the father of dialectics, the method of argumentative reasoning in which a stated thesis is argued against by an interlocutor who tries to disprove it by showing that it produces a contradiction.
The point of Zeno's paradoxes is to prove that the idea of continuous motion is self-contradictory: that is, he wanted to first confute the ideas of Eraclitus, who thought that the world was in a state of perpetual change, and then to confirm the thesis of Parmenides, who considered the world to be a compact, homogeneous, motionless, immutable sphere.
Tradition ascribes to Zeno four classical paradoxes:
- Dichotomy paradox: before an object can travel a given distance, it must travel half that distance. In order to travel half that distance, it must first travel one quarter of it, and so on infinitely: the process of halving never reaches an end for there is always a distance to be halved, no matter how small. Despite the fact that a finite distance requires a finite amount of time to be traveled, Zeno inferred from this principle of "halving" that no distance could be traveled in a finite amount of time. Hence the paradox.
- Achilles and the tortoise paradox: in a race between the fleet-of-foot Achilles and the plodding tortoise, the former can never catch the later, if the latter is given a head start. This is because in the time it takes Achilles to arrive at some point formerly occupied by the tortoise, the tortoise has since moved ahead some distance; and while Achilles is closing the gap between this point and some farther one, the tortoise has again moved ahead. Thus, it would appear that, if the tortoise keeps moving, Achilles will never catch up, for he will have to travel an infinite number of finite distances.
- Arrow paradox: an arrow in flight is, at any single instant, indistinguishable from a motionless arrow in the same position. The question then arises: is the arrow moving or at rest in that instant? The paradox is that if you say the arrow is moving, how is it possible to be moving during one instant? And if you say the arrow isn't moving, it must therefore be at rest and thus cannot be in flight.
- Stadium paradox (sometimes called moving rows paradox): this is the most obscure of Zeno's paradoxes, little information about it survives. Its basic point appears to be that speed, usually considered an essential property of motion, is not an objective property but a relative one. The paradox involves parallel rows of seats (as in a stadium); it could also be visualized as three parallel trains, A, B, and C. A and C travel at the same speed in opposite directions; B, in the middle, is motionless. Zeno seems to conclude that A takes both the same amount of time and twice as much time to pass any part of B as C does.
The target of the first two paradoxes was probably the thesis that space and time can vary continuously, while the point of the other two was to confute the idea of discrete space or time. In order to overcome the apparent contradiction arising from the first two paradoxes, we have to consider that a geometric series converges. In other words, we have to accept the proposition that the sum of an infinite amount of finite quantities doesn't produces an infinite result!
For instance, if we assume that Achilles runs at a speed ten times faster than the tortoise's, and that the tortoise starts 100 meters in front of Achilles, Zeno's argument leads to the series: 100 + 10 + 1 + 1/10 + ... = 111 + 1/9. Hence, after exactly 111 + 1/9 meters, Achilles will reach the tortoise.
A similar argument based on the geometric series can deal with the dichotomy paradox.
The arguments needed to overcome the paradox of the arrow are probably more subtle, and depend on our assumptions on the nature of space and time--namely whether space (or time) is built by discrete irreducible atoms. A precise definition of the movement is also needed. A discussion about the problems arising in such definitions is in The Principles of Mathematics, by Bertrand Russell.
The stadium paradox, in the formulation classically ascribed to Zeno, is quite vague, and it can be probably overcome by the concept of relative speed. However, the same concept of relative speed presents some "paradoxes" in modern physics, since Einstein's relativity showed some inconsistency in the common sense definition of relative speed and relative time. Also, several philosophers, such as Russell and G. E. Owen, considered the stadium paradox in new ways, producing new logical paradoxes.
The paradoxes on movement were reconsidered by modern philosophers too. For instance, M. Black pointed out that the real logical problem should be that Achilles performed an infinite series of action in a finite time: this can lead to other paradoxes, such as the Thomson lamp. The Thomson lamp is not a real object, but a mental experiment. It deals with proving that machines that perform an infinite number of actions in a finite time can lead to logical problems and paradoxes. Thomson lamp is turned on for 1/2 minute, off for 1/4 minute, on for 1/8 minute, and so on. At the end of one minute, the lamp switch will be moved infinitely many times. The question (disregarding any objection based on the physical impossibility of such lamp) is whether the lamp is on or off after one minute.
Another paradox ascribed to Zeno deals with the relation between a whole object and its parts: if anything does exist, it must be composed of parts. Since these parts can be divided again and again, everything must be composed of infinite parts: such "elementary" parts should have no extension, otherwise they could be divided again. But if the situation is like this, how is it possible that parts with no extension can produce anything extended?
Mathematically, such a paradox can be translated into the fact that points with no extension can produce a line. However, Zeno's argument was reconsidered by modern philosophers, such as William James and A. N. Whitehead, for metaphysical considerations about the discontinuity of the processes of change.
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