Zeno of Elea · Philosophy
Beneath the famous paradoxes of motion lie Zeno’s arguments against <em>plurality</em> - against the belief that there are many things at all. If reality is divided into many, he argued, each thing must be both infinitely large and infinitely small, both limited and unlimited in number - contradictions that, he claimed, leave only Parmenides’ undivided One.
The paradoxes of motion - Achilles, the Arrow, the Dichotomy - are Zeno’s most famous arguments, but they may not be his most fundamental. Beneath them lies a deeper assault on an even more basic belief: the belief in plurality, the belief that there are many things at all. This was, in fact, the primary target of Zeno’s defence of Parmenides, for Parmenides’ central claim was that reality is One - a single, undivided whole - and the most obvious objection to this was simply that there are plainly many things: this rock, that tree, you, me, countless distinct objects. Zeno’s task was to show that this ‘obvious’ plurality is in fact incoherent, that the assumption of many things leads to contradictions just as severe as anything in the doctrine of the One. And notice the logical priority: motion presupposes plurality. To move is to go from one place to another - but that requires a plurality of distinct places; and it takes a plurality of distinct moments. If plurality itself is impossible, then motion, which depends on it, is impossible too.
So the arguments against the many are, in a sense, the foundation; the arguments against motion are a special case.
Zeno’s strategy against plurality is the same reductio he uses everywhere: assume there are many things, and derive a contradiction. The contradictions he derives are striking and fall into pairs of opposites. From the assumption of plurality, he argues, it follows both that things are infinitely large and that things are so small as to have no size at all; and both that the things are limited in number (just as many as they are) and that they are unlimited in number (infinitely many). Since a thing cannot be both infinitely large and sizeless, and the many cannot be both limited and unlimited in number, the assumption of plurality must be false. There are not many things. Only the One remains. These arguments are less immediately gripping than a footrace, but they cut to the philosophical bone, for they concern the very possibility of a divided, plural reality - and, like the motion paradoxes, they turn on the treacherous notion of infinite divisibility, the problem of how a magnitude can be composed of parts.
Zeno’s first great argument against plurality concerns the size of the supposed many things, and it springs the same trap on either answer to the question of divisibility. Suppose there are many things. Each of them must have some magnitude, some size - for if a thing had no size at all, it would be nothing, and could add nothing to anything; a collection of sizeless things is still sizeless, still nothing. So each of the many must have size. But now, anything with size has parts - it can be divided, for it has a near side and a far side, a this-part and a that-part. And each of those parts, having size, can itself be divided, and each of those parts again, without end - for at every stage the part still has some size and so still has further parts. So each thing contains infinitely many parts, each part itself having some positive size. But infinitely many parts, each with some positive size, however small, must add up to an infinite total size - for any positive size, added to itself infinitely often, grows without bound. So each of the many things is infinitely large.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Zeno’s arguments against plurality (which underlie the motion paradoxes, since motion needs many places and times) try to show that ‘the many’ is contradictory: if there are many things, each must have parts; if those parts have size and are infinitely many, each thing is infinitely large; if the part…
Leads to Democritus.
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