Version: v1, Published online: 1998

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## 3. Minima

The fifth-century atomists (see Atomism, ancient; Leucippus; Democritus §2) had first introduced atoms at least partly to circumvent the puzzles that Zeno of Elea (§§4–6) had derived from the supposition of infinite divisibility: any magnitude will, as the sum of infinitely many parts, be infinitely large; and motion will be impossible, since it will require the traversal of infinitely many sub-distances. But if atoms, as seems inevitable given their varying shapes and sizes, are not the smallest possible magnitudes but simply indissoluble lumps, it is hard to see that this kind of indivisibility can do anything to thwart Zeno: each atom will consist of infinitely many parts, and threaten to be infinitely large and/or untraversable.

Epicurus (Letter to Herodotus 56–9) starts by making the first clear distinction in ancient thought between (1) things which are physically indivisible, and (2) those which modern scholarship sometimes calls theoretically, conceptually or mathematically indivisible, but which in ancient usage are called either ‘minimal’ (*elachista*) or ‘partless’ (*amerē*). (For Epicurus’ main predecessor in a theory of type (2), see Diodorus Cronus §2.) Not only, Epicurus says (Letter to Herodotus 56), (1) can things not be ‘cut’ to infinity (a reference to atoms, which are physically ‘uncuttable’), as proved earlier, ‘but also (2) we must not consider that in finite bodies there is *traversal* [moving along something part by part] to infinity, not even through smaller and smaller parts [that is, not even in a convergent series such as
${\scriptstyle \frac{1}{2}},{\scriptstyle \frac{1}{4}},{\scriptstyle \frac{1}{8}},\dots $
]’. Otherwise, he warns, Zenonian consequences will ensue.

By showing that a finite body cannot contain an infinite number of parts, Epicurusconsiders that he has established the existence of an absolutely smallest portion of body, which henceforward he calls ‘the minimum (*elachiston*) in the atom’. Clearly it cannot be larger than an atom, or it would not be a minimum, so it must be either an entire atom or part of one. (For reasons which emerge later, it must in fact be the latter.) The idea that an extended magnitude like an atom consists of minimal or partless units is designed to avoid Zenonian paradoxes, but, as Aristotle had pointed out in Physics VI, it generates paradoxes of its own. In particular, it is hard to see how two partless items can be adjacent, since they cannot touch either part to part (they do not have parts) or whole to whole (or they would be co-extensive, not adjacent).Epicurus answers brilliantly with the analogy of the ‘minimum in sensation’ – the smallest magnitude you can see. Any larger visible magnitude consists of a precise number of these minima, which are seen as adjacent without touching in either of the two ways offered by Aristotle, but, as Epicurus puts it, ‘in their own special way’. This provides a model which is readily transferable to the ‘minimum in the atom’.

But it also follows, of course, that *any* extended magnitude will consist of an exact number of minima, since after analysis into minima there could not be a fraction of a minimum left over. This makes all magnitudes commensurable, and conflicts with the geometers’ recognition of incommensurable lengths, for example, the side and diagonal of a square. So much the worse for geometry, was the conclusion of Epicurus and his collaborator Polyaenus, himself formerly a distinguished geometer. They concluded, at least provisionally, that geometry is false.

In one respect the analogy between the sensible and actual minimum fails. Visible minima are capable of independent motion (for example, a falling object viewed at a suitable distance might be a sensible minimum), but actual minima are not, says Epicurus. This perhaps reflects Aristotle’s argument in Physics VI 10 that a partless entity (he is thinking especially of a geometric point) could not move except incidentally to the motion of a larger body. At all events, it follows that, since atoms move, no atom consists of just one minimum.

Atoms must vary widely in shape and size in order to account for the large range of phenomena (Lucretius II 333–477). But they cannot vary indefinitely, since out of a finite set of minima only a finite number of shapes can be formed. Being partless, minima cannot be partly adjacent, so two minima can only be arranged in one shape, three minima in only two shapes, and so on (Lucretius II 478–531).

Sedley, David. Minima. Epicureanism, 1998, doi:10.4324/9780415249126-A049-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/epicureanism/v-1/sections/minima.

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