. The surviving paradoxes of Zeno of Elea (see ELEATICS) fall mainly into two groups, concerning plurality and motion, though these groups are related. Their interpretation and significance is to some extent controversial.
The idea behind the former group seems to be as follows: to have no size is to be nothing, while to have size is to be divisible (whether in reality or only in principle is left unclear). But the parts resulting from division must themselves either lack size, and so be nothing, or have size, and so be further divisible. Therefore we must end with nothing or with infinitely many parts. If these infinitely many parts lack size they cannot contribute to the whole, which will itself lack size; but if they have size, however small, the whole they form will be infinitely large.
The paradoxes of motion seem intended to argue that space and time can be neither atomic (made of indivisible points and moments) nor continuous. The Moving Rows paradox seems to argue that if both space and time are atomic there is a maximum velocity, namely one point per moment—but anything moving at this velocity relative to one object can always be shown to be moving faster relative to some other object, so there is no maximum velocity. The argument can be made to cover the cases where only one of space and time is atomic. Aristotle, however, who is our source for this paradox, treats it as simply confusing relative and absolute motion. The above version, whether or not historically accurate, is stronger than Aristotle’s. There are other versions.
The Achilles and the Tortoise paradox argues that if space and time are both continuous, then if Achilles allows the tortoise a start in a race he can never overtake it. He takes at least some time to reach the tortoise’s start, during which the tortoise moves at least some distance. While Achilles covers this distance the tortoise moves some more. While Achilles covers this ‘more’ the tortoise moves again. Clearly the argument can be repeated indefinitely: even though the successive stages get shorter and are covered ever more quickly, at the end of any given stage Achilles is still behind the tortoise. How can he reach the end of an endless series of stages? The Dichotomy is a variant of the Achilles, saying that one can never cover a distance, because one must first cover the first half, then the third quarter, and so on, constantly bisecting the remaining distance. In a more general form the paradox claims that if two objects are separated by a certain distance at a certain time they can never be separated by a different distance at any other time. The name Stadium is ambiguous, sometimes meaning the Dichotomy, sometimes the Moving Rows.
The Flying Arrow argues that, since at any moment an arrow occupies a definite position, and since between two moments there is nothing but other moments, the arrow can only be in positions and never move from one to another.
The Grain of Millet, on a different topic, argues that a single grain in falling makes no sound, but a thousand grains make a sound, so a thousand nothings become something, which is absurd. Cf. HEAP.
Modern discussions centre on the Achilles, of which many variants have been developed. Its full solution is still disputed.
