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ON THE NATURE OF THE UNIVERSE |
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APPENDIX C Smallest Parts The atomic theory as developed by Epicurus postulates, as we have seen, that matter can be split into tiny invisible indivisible particles. These atoms cannot be divided any further, but they, like everything else, have parts, in that one could (with sufficient magnification) point to 'the part nearest the right' or 'the top half' of an atom, especially in view of the fact that they are three-dimensional. These 'parts' are obviously not perceptible to the naked eye, and can thus only be logically inferred. It may be wondered why Lucretius bothers to mention them at all (I.599-634), and to what extent Aristotle was right to see them as a surrender to the arguments of Zeno of Elea. Let us begin with Zeno. The famous paradoxes that bear his name seek to prove such apparently absurd statements as 'motion is impossible' or that Achilles will never overtake the tortoise in his race if the tortoise is given a start, however small. Most of them rest on the notion of infinite divisibility or dichotomy. So, for instance, if I wish to travel from A to B, I must move along the whole distance AB. But the line AB consists of an infinite range of points {p1, p2, p3 ... pn} and to reach B I must pass each of these points. Thus I need to perform an infinite number of tasks to reach B, as each point on the way is logically a separate task, and an infinite series of tasks will take me an infinite amount of time. I do not have an infinite amount of time. I will therefore never reach B. This sort of dichotomy becomes more relevant to our inquiries when applied to matter itself. A point, we are told, has location but no magnitude: but Epicurus' atoms are not points for the very reason that the aggregate of atoms makes matter but the aggregate of points without magnitude would add up to nothing. Now if an amount of matter has magnitude, it can be divided, as there is no number that cannot be halved. What, then, is to prevent us splitting the atoms themselves - with mathematics if not with a knife? Just as space, in the first paradox, is imagined to be continuous and Euclidean, so in this case matter is also seen as continuous. Now a primary argument of Epicurus is that if matter is infinitely divisible, then nothing would exist as the destruction of atomic compounds is always easier than building them up, and so destruction would always have the upper hand (Letter to Herodotus 40-41): furthermore (in Zenonian terms) it would take an infinite number of tasks to make anything, and the time since the world began has not been infinite. Furthermore, an infinite array of parts each of which possesses magnitude would add up to an infinite magnitude, which would mean that each atom was the size of the whole universe itself (1.615-26 and Epicurus, Letter to Herodotus 57), which is patently not the case. Therefore there must be a minimum magnitude, and hence matter and space must be in some sense 'granular' - like, say, the squares on a chess board - and not continuous. This theory, however, has grave consequences for geometry. For if, say, a cone is bisected and the surface planes on the underside of the top half and the top of the bottom half are compared, they must be of different magnitudes, or else the cone would be a cylinder. But if this is true, then our apparently perfect cone is in fact corrugated and not geometrically continuous. If our geometry is thus shown to be fatally flawed, how can we formulate a system that will be true to reality? The reasons why Epicurus developed this idea of notional smallest parts in an indivisible whole atom are thus several. One reason is to be found in Aristotle's Physics 231a21-b10, where he demonstrates that partless magnitudes cannot touch each other since touching is of either whole to whole, whole to part, or part to part. If it were the first of these then the two magnitudes would be totally co-extensive - which would be impossible for three-dimensional atoms to achieve - and the other two are ruled out because partless things have no parts. The atoms undoubtedly do touch, and so they must have parts: but the atoms cannot be splintered into these parts without surrendering their indivisibility and so running out of the frying pan of Aristotle into the fire of Zeno. Hence the need to work out a balance between these two positions, and hence the doctrine of smallest parts. A more basic reason is simply this: atoms do not differ in substance, but in size and shape. It was the rejected homoeomeria theory of Anaxagoras, which argued that atoms of hair make up hairs etc., and Lucretius laughs this out of court at 1.830- 920. Instead he posits differences in size and shape that demand the existence of a finite number of minimal parts in each atom, whose number and configuration will cause the size and shape of the atom (1.631-3, 2.481-99). Again, the atom is indivisible and so these minimal parts have no motion of their own but only as parts of the whole (on the reasoning that a smallest part cannot move by itself without incurring the possibility of crossing a boundary between places, and this would involve it having one part on each side of the boundary, which, being partless, it could not do). What is more, he also posits minimal quanta of time, the smallest unit of time being that required for an atom to hop from one minimal quantum of space into the next. These theories surely show that Epicurus believed that if he did not postulate the non-continuous, 'granular' nature of matter, time and space, then he would be hauled over the coals of Zeno's infinite divisibility. He could have perhaps argued that the fact that a magnitude can be mathematically subdivided does not alter the fact that space and matter in fact cannot be divided to infinity - not that that is any answer in logic. He could also have suggested that a finite space/magnitude can only be likely to be divided into an 'infinite' number of parts if these parts are unequal and thus form a convergent series such as: {1/2 + 1/4 + 1/8 + ... ∞} whose sum will never be greater than 1: thus fears that gnats will be the size of elephants are somewhat premature; except that that particular line of reasoning, while valid in its own terms, is no answer to the skilled Zenonian, whose infinite parts are not a convergent series but an array where each part is simultaneously divided into an infinite number of equal parts ... The argument goes on, and the curious and the brave are referred especially to Furley, Two Studies in the Greek Atomists and Barnes, The Presocratic Philosophers pp. 23 1-95, 35 2-60, and other further reading suggested in the bibliography. The significance of the theory of smallest parts cannot be overstated, as it brings up the question of the fundamental nature of matter, space and time - whether they are continuous or granular - and demands an answer on which the mathematician, the logician and the theoretical physicist could agree. I doubt that we will find one.
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