COLLOQUIUM 1

THE DOGMAS OF INDIVISIBILITY: ON THE ORIGINS OF ANCIENT ATOMISM

in Proceedings of the Boston Area Colloquium in Ancient Philosophy
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References

I As an astute a commentator on the Atomists as J. Barnes claims, "'atomic' means 'indivisible'" (Barnes, 1982, p. 54). For a clear account of this ambiguity see C. J. F. Williams' comments to his translation of De Generatione et Corruptione, (Williams, 1982), esp. pp. 67, 128. 1 thank the anonymous referee for directing my attention to a early use of ATOIAOT in Sophocles, Trach. 200, there applied to a sacred meadow. Here it seems that something stronger than "uncut"

may be intended. However, I do not think it need mean uncuttable. It seems closer to an imperative usage "This meadow is not to be cut, for it is sacred." The modal notion at play here (if any) is that since the meadow is not to be cut, it never was, is, nor will be cut. This may be the manner in which someone wants to unpack the meaning of "uncuttable," but, then again, it may not. That is to say, one may "reduce" the modal "uncuttable" to the non-modal "eternally uncut," but one may also (more plausibly, I think for this passage in Sophocles), believe that the meadow will likely never be cut, but were a sacrilegious madman to enter the field with a lawnmower, the plants would be cut. The meadow is uncut, but cuttable. 2 Simplicius in de Caelo 242,20. See my "When Worlds Collide" (Lewis, 1990), pp. 244-45 for a discussion of this argument. S. Makin, in his important Indifference Arguments, (Makin, 1993), interprets this argument differently. I am very sympathetic with much that he has to say. Where we essentially differ is with regards to whether divisibility, as opposed to division, threatens a post-Eleatic pluralist. Makin argues that assorted Eleatic dilemmas, as interpreted by the Atomists, do require indivisible atoms in order to offer a response, I do not, as shall become clear. This admittedly crucial difference aside, his is by far the most sophisticated account of the motivations behind the development of atomism, and well worth reading. There are many places where my account can be strengthened by taking a leaf from Makin's work. 3 Void can be contained in a body in two ways. Either bubbles of void or channels of void may be in a body (or a combination of the two). Yet the former in no way effects the putative divisibility of the body, while the latter has it that the body is already divided. See my "When Worlds Collide" for more on this. In fact, there are precedents for taking Statpeo-tv as division, and not divisibility. For example, Aristotle states, "Again, a point of contact is always a

single point of contact between two things, which implies that there is something else besides the point of contact or division (8�aipeonv)." (GC 1.2, 316b6-8) And, "what is there, then, besides the divisions (8iaipc6 Furley, 1987, 126-127.

S. Makin is in sympathy with this line of reasoning. See also my "When Worlds Collide."

8 Makin, 1989, pp. 143-4. 9 In De Anima 35.12

p. 352. I say here 'physical object,' as opposed simply to 'object' because I take it that one cannot physically divide a non-physical object.

For a lucid and illuminating investigation of some of these issues see S. Makin, "The Indivisibility of the Atom," Archiv fur Geschichte der Philosophie, 1989. See Treatise of Human Nature II, 334ff. This seems to be a misinterpretation of an Epicurean argument. See also (Bames, 1982) for a similar analysis.

14 Hume himself seems to subsume conceiving under imagination, where it seems more plausible to subsume imagining under conceiving, if one does not want to keep them separate. Given Hume's views, since we cannot imagine a 1000 sided polygon, such a polygon is inconceivable, and so cannot exist. Although Democritus would certainly not! 16 The close connection between divisibility and the actually divided is brought out by considering the claim that the atoms are meant to be essentially indivisible. Given any standard modal system where transitivity of possibility holds the claim that something is essentially indivisible is the same as the claim that this thing is essentially undivided. If one wants to claim that the atoms are so indivisible, and call this a type of conceptual or theoretical indivisibility, one should note two

things. First, that this sense of conceptually indivisible has nothing to do with our concepts, our imagination, or any other mental act. It is a claim about the relative accessibility of certain worlds to this one. Mere epistemic possibility is not at issue. Second, one needs to worry about whether dispositional properties, like (in)divisibility, can be essential, which leads one quickly into the thicket of issues involving deducing what is possible if the laws of nature should be other than they are. And, of course, one moves quite far from any notion that Democritus might have countenanced.

One might have the following worries. One gets no univocal answer as to what a body is composed of, given this principle. What a body is composed of will depend upon what set of parts one generates. This is not in and of itself a problem; we do not expect a univocal answer to such questions. I am composed equally by all my organs and limbs, and by all my cells. But there is a problem in basing facts about what something is composed of, on what it can be divided into—on the ways in which it is divisible. This is a problem because divisibility is a disposition and (so) a modal notion, while composition does not seem to be. In so far as divisibility is a modal notion we get different answers to the question of how something is divisible depending on how we unpack the appropriate counterfactuals. Yet it may very well be mistaken to suppose the composition of something to be contingent upon counterfactuals. Something has the composition it has, tout court What this composition is may depend on many factors, but surely not only dispositions. For one thing, the composition something has seems to be a good candidate for being an essential feature of that thing, at least in some cases. And there are notorious problems with dispositional essential properties.

18 Actually a slightly stronger claim is needed. Zeno's argument requires one to privilege the set of parts one generates by dividing everywhere over all others, since only that set of parts yields the dilemma. The results of such a division is being taken to be the discovery of the ultimate, and, for the purposes of the dilemma, the real, constituents of that being divided. 19 This realization is codified in modem point-set and measure theory. Makin seems to disagree with the above, since he states, "Whatever a thing is ultimately divided into is what it is composed out of...." (Makin, 1993, p. 25). 21 It is instructive to read Bolzano's Paradoxien des Unendlichen of 1851 to see both how sophisticated a pre-Cantorian recognition of the "wine-glass fallacy" can be, and just how problematic the issues it raises remain prior to Cantors's work in set theory. See in particular section 4. For a very clear statement of one of the central issues at play here see Simplicius, who tells us disapprovingly, while commenting on Aristotle's claim that a line cannot be composed out of points, that most have assumed that any magnitude "is composed out of those parts into which it is also divided" (in Phys., 925.5f.).

22 de Mix. 222.17f. Whether or not Aristotle also makes this same point is a bit unclear. He does deny that a line can be composed out of points (Phys. 6.1, 231a24f.), but his reasoning is based on his own account of continuity, contact and succession. In addition, his argument applies to indivisibles in general, including extended indivisibles. If the analysis here offered is correct, the Presocratic Atomists are not guilty of the modal fallacy which J. Barnes hoists upon them. He claims (The Presocratic Philosphers, Vol. 2, p. 58) that the atomists move from 1. It cannot be the case that everything has been divided, to: 2. There are some things which cannot be divided.

On the reading here advanced, they move to 2: there are some things that are not divided. Again, for a subtle interpretation of these texts see (Makin 1993). There are many points of contact between our accounts (and also many differences). Most importantly, Makin realizes that the problem in GC 1.8 is directed against pluralists who generate the many by means of division. There is a sense in which the Atomists agree with this. Plutarch (Reply to Colotes 1109A) tells us that Democritus called body TO BEV, and void IA?78iv, which Furley renders as 'hing,' and 'not-hing,' respectively. What separates the units cannot be the same as the units (as the arguments goes on to make clear), and so it must be 'not-being,' or nothing. Via premise one this rules out anything playing the role of that which separates beings. The Atomists, by claiming that nothing in fact is (i.e., the void), agree with the Eleatics that nothing must separate the units of being.

This is, in effect, to postulate a theory of motion not unlike Aristotle's, and very like that of Descartes. Lucretius, attacking the Peripatetics, raises objections to such a theory of motion (one without void) (de rerum natura, I.334f.) Again, this view is close to that of Aristotle. It is the fact that these options are not exhaustive that the Atomists will go on to exploit. The unstated third option, "it is divided in no places," will be the option the Atomists accept. However, as far as this argument goes, the premise as found in the text is to be expected, since the opponents' assumption is that "the universe . .. consists of divided pieces." One should perhaps note the equivocation in premises (5) and (9) between 'divided everywhere' and 'infinitely divided.' Division everywhere would consist in cardinality C divisions, as opposed to a merely countably infinite number of divisions.

31 One might think that the best way to interpret this section of the argument is to assume it is based on the premises "no motion without void," and, "there is no void." Yet this argument is not in fact suggested by this passage. The initial conclusion in (8) is that all is void, and nowhere is it argued that the void does not in fact exist. It is true that premise (1), on my reconstruction is "that there is no void." Yet here we are assuming a different premise, that "the universe ... consists of divided pieces in contact with each other," a model which implies that there is not any void. 32 The argument seems to presuppose something like the principle of sufficient reason. If an initially homogeneous mass of being is divided, why divide it here rather than there? This raises interesting questions about whom this Eleatic argument is directed against. Might it be, among others, Anaxagoras, who postulates an initial state of total homogeneous mixture? See Makin's lndifference Arguments for more on the homogeneity of being, and for an opposed view as to whether the principle of sufficient reason is at play here.

33 This has worried many. Kirk, Raven and Schofield think the argument must apply to "the one" and envisage a Zeno who was both aware of this, and perhaps pleased. Furley offers a less than satisfactory account of why Zeno's argument does not so apply, based on "being" lacking bulk. I agree with D. Held that in any case Parmenides B8 22-25 is best read as an argument as to why being is undivided, not indivisible. A being "all alike" is not divided. Yet such "all alikedness" cannot ground indivisibility. It can, at best, be used to show that being is either indivisible or divisible everywhere. I thank Prof. Held for bringing Parmenides into the discussion.

One might raise the alarm that I am misrepresenting Aristotle's argument, for he is concerned not just with body, but with magnitude. It is true that at 316al4 he starts by asking "suppose that there is a body and (Kai) magnitude divisible everywhere." Yet the "and magnitude" then drops out of the discussion, which centers on the division of body only. When Aristotle finally reaches the resolution of the argument at 316bl4 he states, 'Hence if it is impossible for magnitudes to consist of points of contact, there must be indivisible bodies and magnitudes." Aristotle then hastily returns to talk of the division of "perceptible body" (316bl9). I conclude that there are two plausible ways of explaining this. First is that the addition of talk of dividing magnitudes is Aristotle's, and that it plays no role in the arguments of the Atomists. Second, Aristotle is merely equating bodies with magnitudes (note 316a34f., where Aristotle glosses the escape of a piece of sawdust from a body as a body escaping from a magnitude). This makes good sense, since the whole argument is powered by the fact that all bodies are extended, that they are, in fact, magnitudes. The only magnitudes which Aristotle is here concerned with are the magnitudes of bodies, corporeal magnitudes. I am therefore tempted to read, contra Sorabji, 316al4 as "body and magnitude," not "body or magnitude," this locution indicating to the reader that the argument hinges on the fact that bodies are extended, and so 'are,' magnitudes, or 'have' magnitude. In any case, "and" is, of course, the usual translation of the "Kai" found here. See (Furley 1987) for an alternative reading. 35 Actually, it is difficult to know how much of this argument is Aristotle's own. As presented, it moves from the divisibility of a body, to its division everywhere, and from this to aporia. Clearly, the aporia is avoided by denying that body is so divided. One may go on and ground this fact by claiming that body is indivisible, but of course one need not. It is clear that Aristotle has the modal argument in mind, since his own solution (presented immediately after the passage at hand) involves the notions of potential versus actual divisions. Aristotle nowhere claims that Democritus also had such a model reading in mind. Of course, so far as this passage goes, he may have, but what is important is that he need not have in order to respond to the aporia as presented. For an interestingly similar argument see Aristotle DC 299a24f, where Aristotle argues that bodies cannot be composed out of planes, which cannot be composed out of lines, which cannot be composed out of points, since a point is weightless, and so anything composed out of them, no matter how many, must be weightless, the final conclusion being that all bodies would be weightless. This passage is discussed in J. J. Cleary's very interesting Aristotle and Mathematics, Brill, 1995.

36 CN 1079E. The surrounding texts are extremely pertinent for tracing the further development of many of the ideas discussed in this paper. In particular, the Stoics seem to realize that one should not privilege one set of parts over any other as being those that something is composed out of. Indeed, they seem to deny that there is such a thing as the ultimate parts of any object. The Stoic position on these matters is a topic for another day.

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