The Greek philosopher Zeno of Elea was celebrated for his paradoxes. Aristotle called him the ‘founder of dialectic’. He wrote in order to defend the Eleatic metaphysics of his fellow citizen and friend Parmenides, according to whom reality is single, changeless and homogeneous. Zeno’s strength was the production of intriguing arguments which seem to show that apparently straightforward features of the world – most notably plurality and motion – are riddled with contradiction. At the very least he succeeded in establishing that hard thought is required to make sense of plurality and motion. His paradoxes stimulated the atomists, Aristotle and numerous philosophers since to reflect on unity, infinity, continuity and the structure of space and time. Although Zeno wrote a book full of arguments, very few of his actual words have survived. Secondary reports (some from Plato and Aristotle) probably preserve accurately the essence of Zeno’s arguments. Even so, we know only a fraction of the total.
According to Plato the arguments in Zeno’s book were of this form: if there are many things, then the same things are both F and not-F; since the same things cannot be both F and not-F, there cannot be many things. Two instances of this form have been preserved: if there were many things, then the same things would be both limited and unlimited; and the same things would be both large (that is, of infinite size) and small (that is, of no size). Quite how the components of these arguments work is not clear. Things are limited (in number), Zeno says, because they are just so many, rather than more or less, while they are unlimited (in number) because any two of them must have a third between them, which separates them and makes them two. Things are of infinite size because anything that exists must have some size: yet anything that has size is divisible into parts which themselves have some size, so that each and every thing will contain an infinite number of extended parts. On the other hand, each thing has no size: for if there are to be many things there have to be some things which are single, unitary things, and these will have no size since anything with size would be a collection of parts.
Zeno’s arguments concerning motion have a different form. Aristotle reports four arguments. According to the Dichotomy, motion is impossible because in order to cover any distance it is necessary first to cover half the distance, then half the remainder, and so on without limit. The Achilles is a variant of this: the speedy Achilles will never overtake a tortoise once he has allowed it a head start because Achilles has an endless series of tasks to perform, and each time Achilles sets off to catch up with the tortoise it will turn out that, by the time Achilles arrives at where the tortoise was when he set off, the tortoise has moved on slightly. Another argument, the Arrow, purports to show that an arrow apparently in motion is in fact stationary at each instant of its ‘flight’, since at each instant it occupies a region of space equal in size to itself. The Moving Rows describes three rows (or streams) of equal-sized bodies, one stationary and the other two moving at equal speeds in opposite directions. If each body is one metre long, then the time taken for a body to cover two metres equals the time taken for it to cover four metres (since a moving body will pass two stationary bodies while passing four bodies moving in the opposite direction), and that might be thought impossible.
Zeno’s arguments must be resolvable, since the world obviously does contain a plurality of things in motion. There is little agreement, however, on how they should be resolved. Some points can be identified which may have misled Zeno. It is not true, for example, that the sum of an infinite collection of parts, each of which has size, must itself be of an infinite size (it will be false if the parts are of proportionally decreasing size); and something in motion will pass stationary bodies and moving bodies at different velocities. In many other cases, however, there is no general agreement as to the fallacy, if any exists, of Zeno’s argument.
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