What Zeno is really arguing against here is the attempt to distinguish space from the body that occupies it. If we insist that body must be in space, then we must go on to ask what space itself is in. This is a “reinforcement” of the Parmenidean denial of the void. Possibly the argument that everything must be “in” something, or must have something beyond it, had been used against the Parmenidean theory of a finite sphere with nothing outside it.
Motion.
163. Zeno’s arguments on the subject of motion have been preserved by Aristotle himself. The system of Parmenides made all motion impossible, and his successors had been driven to abandon the monistic hypothesis in order to avoid this very consequence. Zeno does not bring any fresh proofs of the impossibility of motion; all he does is to show that a pluralist theory, such as the Pythagorean, is just as unable to explain it as was that of Parmenides. Looked at in this way, Zeno’s arguments are no mere quibbles, but mark a great advance in the conception of quantity. They are as follows:—
(1) You cannot get to the end of a race-course.[[890]] You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum, so that there are an infinite number of points in any given space, and you cannot touch an infinite number one by one in a finite time.[[891]]
(2) Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it.[[892]]
The “hypothesis” of the second argument is the same as that in the first, namely, that the line is a series of points; but the reasoning is complicated by the introduction of another moving object. The difference, accordingly, is not a half every time, but diminishes in a constant ratio. Again, the first argument shows that no moving object can ever traverse any distance at all, however fast it may move; the second emphasises the fact that, however slowly it moves, it will traverse an infinite distance.
(3) The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move.[[893]]
Here a further complication is introduced. The moving object itself has length, and its successive positions are not points but lines. The successive moments in which it occupies them are still, however, points of time. It may help to make this clear if we remember that the flight of the arrow as represented by the cinematograph would be exactly of this nature.
(4) Half the time may be equal to double the time. Let us suppose three rows of bodies,[[894]] one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions (Fig. 1). By the time they are all in the same part of the course, B will have passed twice as many of the bodies in C as in A (Fig. 2).
| Fig. 1 | ||||||||
| A. | ● | ● | ● | ● | ||||
| B. | ● | ● | ● | ● | → | |||
| C. | ← | ● | ● | ● | ● | |||