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.
That is the first horn. The second horn is the opposite. Suppose instead that we try to avoid the infinite largeness by saying the ultimate parts have no size - that division terminates in sizeless points or elements. But then we face the opposite disaster: a thing composed of sizeless parts can have no size, for adding sizeless things together, however many, yields nothing with size - zero plus zero plus zero, however many times, is still zero. So if the ultimate parts are sizeless, each thing is sizeless, that is, nothing at all. Thus the dilemma: the many things must be either infinitely large (if their infinitely many parts have size) or sizeless and so nothing (if their parts lack size). Neither is acceptable - nothing is both infinitely large and a real, finite thing, and nothing that exists is sizeless and so nothing. So, Zeno concludes, there cannot be many things. The whole argument turns on the impossibility, as Zeno sees it, of giving a coherent account of how a thing of finite size can be composed of parts - whether those parts are infinitely many and sized (leading to infinite largeness) or sizeless (leading to nothing). The modern reply, foreshadowing the resolution of the motion paradoxes, is that infinitely many parts of diminishing size can sum to a finite total (a convergent series), so that a finite thing can indeed be composed of infinitely many ever-smaller parts without being infinitely large - but this requires the rigorous theory of the infinite that Zeno’s arguments helped provoke.
Zeno’s arguments against plurality matter enormously because they posed a challenge so severe that the entire subsequent development of Greek natural philosophy - and, through it, the foundations of physics - can be read as a series of attempts to answer them. Parmenides and Zeno had argued, with formidable rigour, that a divided, plural, moving reality is incoherent, and that only the changeless One survives logical scrutiny. But the changeless One could not be the world we live in - a world of many things, changing and moving. So the challenge for every philosopher after Parmenides was: how can we save the plural, moving world from the Eleatic arguments? How can there be many things, and change, and motion, in a way that escapes Zeno’s paradoxes? This challenge drove the great pluralist systems of the fifth and fourth centuries. Empedocles posited four eternal ‘roots’ (earth, air, fire, water) that combine and separate. Anaxagoras posited infinitely many kinds of stuff, with ‘a portion of everything in everything.’ And, most consequentially, the atomists Leucippus and Democritus posited indivisible atoms moving in the void - a system designed precisely to grant the Eleatics their point (that being is unchanging and indivisible) at the level of the individual atom, while recovering the plural, moving world at the level of atomic combinations.
The atomist solution is the most important and the most enduring, and it is a direct response to Zeno. The atomists accepted the Eleatic premise that what truly is must be unchanging and indivisible - and then said: very well, but there are many such indivisible beings (atoms), separated by the void (non-being, which the atomists daringly admitted exists), and the changing world is produced by their rearrangement. Each atom is a tiny Parmenidean One - eternal, unchanging, indivisible - but there are countless atoms, and their combinations and separations give us the plural, moving world of appearance. Crucially, the atoms are physically indivisible: division stops at the atom and never reaches the paradoxical infinity Zeno exploited. So the plurality paradox, which depended on infinite divisibility, is escaped by denying that matter is infinitely divisible. This was a momentous move, and it is the conceptual ancestor of all atomic theory in physics and chemistry down to the present day - and it was forged in direct response to Zeno’s assault on the many. Few episodes show more clearly how a great philosophical challenge can drive the development of science: Zeno argued that plurality was impossible, and in answering him, the atomists invented one of the most powerful and lasting ideas in the history of human thought. The paradoxes of the many were not idle puzzles; they were the goad that drove philosophy to ask what matter is made of - and to give the answer that, refined over twenty-four centuries, underlies our physical understanding of the world.
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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