The Nature of Zeno's Argument against Plurality in DK 29 B 1Author(s): William E. AbrahamSource: Phronesis, Vol. 17, No. 1 (1972), pp. 40-52Published by: BRILLStable URL: http://www.jstor.org/stable/4181872 .

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The Nature of Zeno's Argument Against Plurality in DK 29 B I

WILLIAM E. ABRAHAM

Simplicius has preserved (Phys. 140, 34) a Zenonian argument pur- porting to show that if an object of positive magnitude has parts from which it derives its size, then any such object must be at once of infinite magnitude and zero magnitude. This surprising consequence is based upon a construction which Zeno makes, but his argument is widely thought to be grossly fallacious. Most often he is supposed to have misunderstood the arithmetic of his own construction. Evidently, any such charge must be premised on some view of the particular nature of the sequence to which Zeno's construction gives rise. I seek to develop a view that Zeno's argument is in fact free from fallacy, and offer reason to fear that his real argument has usually been missed.

For ease of reference, I reproduce a translation of Simplicius in DK 29 B 1.

(He showed the infinity of) size earlier by means of the same reasoning, for having first established that if a being did not have a size, it would not even be, he proceeds: if then it is, each must have some definite size and bulk, and have one (part) of it extend from another. Concerning a' projecting part, the same principle holds, for that too will have size, and part of it2 will project. Indeed, to say this once is equal in force to saying it forever; for no such part of it will be the last, and there will not be a part not extending from another. Thus, if it consists of many parts, it must be both small and huge - so small as not to have size, so huge as to be infinite.

In order to impute the usual fallacy to Zeno, it is held that his con- struction is of a series of continuously decreasing quantities. The charge then is that Zeno mistakenly thought that the sum of such a series approaches infinity.

It is irrelevant to this charge whether the terms decrease geometri- cally or not. Vlastos3 has however offered persuasive considerations,

1 For a discussion of the grammar, see p. 43. 2 W. A. Heidel (followed by Hermann Frankel) took aoiu Tt to be a partitive genitive, rightly. See Proceedings of the American Academy of Arts and Sciences 48 (1913), p. 724. 3 G. Vlastos in "Note on B 1" (p. 3), one of several papers offered to the Institute in Greek Philosophy and Science meeting at Colorado Springs, Summer 1970.

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which indicate that if Zeno's sequence is of continuously decreasing terms, he probably intended them to decrease geometrically. Vlastos cites our knowledge "of the severe norms of symmetry in archaic thought", the fact that the operative ratio in the paradox of the stadium is 1: 2, and the fact that the Porphyry text (apud Simpl. Phys. 139, 27-32) relates the use of the method of dichotomy. Indeed, the use of t-epov in the present text would, as DK surmised,4 suggest a division into two parts, and not three as Frinkel prefers.5 At any rate, given that the sequence is of continuously decreasing quantities, then whether the terms decrease geometrically or not, it would be a blatant fallacy to suppose that their sum, if there is one, could be infinitely large.

That Zeno should have committed the above fallacy is regarded as inexcusable, since, even if he did not know that the sum of the arithmetical progression E (n) does not exceed 1, only wilful fatuity

n- 1, 2,3,...

could blind him to the fact that all the quantities in his own sequence are supposed to be parts of the one given finitely large object, and so could not in sum exceed their matrix. And yet it hardly seems that Zeno had any such amnesia concerning the source of the parts; on the contrary, he makes it a corner-stone of his argument that the sum of all the derived parts would not differ from the original finite existent. This indeed is why he makes the original existent assume an infinite size if the sum of its parts is an infinite magnitude, and zero size if the sum of the parts is of zero magnitude. What is lacking is the proof that the parts would add up to an infinite magnitude and to zero magnitude. As to the equivalence of the parts with the whole, Zeno's B 3 paradox may even seem to formulate precisely this.6

The standard exposition varies as to the ratio in which the given finite existent is continuously divided. While most accounts suggest a bipartite division, some (e.g. Frankel's) have proposed a tripartite division. All of them, however, make certain suppositions which give rise to the imputed fallacy. For the purpose of the following discussion, it is simpler to refer to the bipartite division.

4 Diels/Kranz, Die Fragmente der Vorsohratiker (Dublin/Zurich 1966) I 255, footnote to line 16. b Hermann Frankel, "Zeno of Elea's Attacks on Plurality" in American Journal of Philology, 63 (1942), 1-25, 193-206. 6 DK I 257 f: "If it consists of many parts, they (the parts) must be just as many as they are, neither more nor fewer, and if they are just as many as they are, they must be limited".

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It is assumed in the first place that Zeno gives a particular direction to the repeated division, that it proceeds from left to right, and there- fore that the division is reiterated at each step, only upon the right hand term resulting from the preceeding division. Thus, let the original object be A, let A be divided into two parts, a, and a2; let a2be divided into b, and b2; and similarly ad infinitum. Zeno's sequence would therefore supposedly be represented by the expression (a, + b, + cl + ... ad inf.). This reading of Zeno's construction yields an infinite number of parts not further divided themselves. Obviously, such an operation will produce continuously decreasing terms. One thing to note here is: even if there is a direction or sense to Zeno's sequence, that the direction would be from left to right would only be an assump- tion made natural by our own direction of reading. A Japanese trans- lation may suggest a downward direction, and a Hebrew translation, a right to left direction. Indeed, Simplicius himself may even seem to have assumed that Zeno's sequence moved not from left to right, but from right to left (npo' ro5 Xaotuovo~tievou .eL rL elvoaL 3C -'V E' &7rtpoV

'ropV1nv)7 for the part before would seem to be a left-hand part. A second thing to note is that it is, in fact, only an assumption that

there is a particular direction or sense at all. What Simplicius does report (DK 29 B 1) is that given an existent has a positive size and bulk, a part of it must stretch from (CX&exsLv) another part (meaning, doubtless, that it is divided into two). Zeno adds that this same principle is true of a projecting part (7tepl 'oi 7pouxov'oq). What the standard exposition has done is both to translate this last phrase to mean "concerning the projecting part", and to identify this projecting part with the right hand segment. And yet, at the first division, neither part can be described as the projecting part to the exclusion of the other, for either part stretches away from the other. The relation expressed by &.7XeLV is symmetrical, rendering each part a projecting part. The contrast with "ro npo?'Cov", if there is one, would thus be, not the left-hand part, but rather the whole object, especially as Zeno's stated reason for reiterating the partition is that resulting parts too will have a size (xax yap 'xsZvo tere 0'y?oq XMl npo6ei autoiu x) whether this clause is taken to be epexegetic or argumentative. Since either part in the bipartite division would have a size, it would follow that the principle is to apply to both parts, and, indeed, any part with positive magnitude resulting from the reiterated subdivision. The force of rxe-vo above is therefore to introduce a general rather than a singular 7 DK I 257 2,3.

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proposition. That the division is indeed envisaged for every resulting part would appear to be confirmed by Zeno's remark that there will not be a part which does not abut another (ou'e &TEpOV 7pOq gTepov

oux eacrL)8 for this does not seem to make any exception of left-hand or right-hand parts. If this is so, the form of Zeno's sequence cannot be represented by the expression E (a, + b, + cl + . . .) where the terms are successive undivided left-hand (or right-hand) parts. Ac- cordingly, in the remark "7ZpO TO Xa4 vo[ivou a L EttvacL 8atr FfvV s &7?cLpov -opiv" (DK I 257 2-3), Simplicius is not choosing between left-hand and right-hand, but is saying that "by reason of the infinite division, there is always something (jutting) from any part you take".

The construction I am suggesting will be seen to depend on a reading of 7eP?. TOU 7rP0UX0vT0q in B 1 which denies the article its individuating role; but this is not contrary to Greek grammar. In Greek, the article with the participle can have a generic or a particular force. When it assumes a generic force, the substantive which governs the participle with the article may be suppressed; and just as 'o r3oux6,svoq would mean "anyone who wishes," so sLo 7rpoC'Xov could mean "any projecting part." This interpretation is required by the logic of the argument, and is now seen to be consonant with the grammar of Greek (e.g., Smyth, Greek Grammar 1124, 2050, 2052).

One thing about the rendering suggested here is that it makes it easier to reconcile both the Philoponus passage and the 'Porphyry text' with that of Simplicius. In the Philoponus passage (Phys. 43, 11-13), the following sentence occurs:

s6 yap atvx EX t eLV 7V ?Ntay' 't'o auvexq &'Ct 8Laxpv.ov ?a'tv Ciai 'To

aLMLpeRV EC; t6ptO eateeV 7tXeCoVX.

It would be hard to assume Philoponus to be restricting the further sub-division to some but not others of the products of the division. The Porphyry text (apud Simpl. 139, 27) turns out to be even more felicitous in this respect, for Porphyry's understanding of the argu- ment he is reporting is that the process of division is through and through, and is reiterated for every product of each application: 7tcavtfl OLLo6)&a' M<r4rax L;ttpeov &X' ov rn Fev, 'r] 8a oV (15-16). Second, he indicates the same paradox as in DK 29 B 1, for the thorough division yields either an infinite number of mutually equal parts which are minima of positive magnitude, or are of zero magnitude. The original object would therefore now either be of infinite magnitude or be of zero magnitude. As to Porphyry's attribution of his version to Parme- s DK I 255 20,21.

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nides, it is indeed possible that Parmenides should have used such an argument. Vlastos9 has even drawn attention to the stylistic parallel between Porphyry's "(X?X' o'u 7n ptv, T- 8? ov" and Parmenides' own

"rn n" (DK I 239, 45). It is not necessary to deny the possibility of the attribution; what must be emphasized, however, is that such an attribution does not preclude Zeno from having pressed the argument home. Indeed, the close relationship between the two men claimed by Plato (Parm. 127 e) and repeated by Eudemus (Simpl. 99.7 8-11), would make this natural.

One objection may seem to stand against the suggested inter- pretation of Zeno's premises. In the concordance on Zeno, the verb 7rpoeyXLv occurs only three times, twice in Simplicius (DK 29 B 1, 17, 18) and once in Aristotle (Phys. Z 9, 239 b 14). In the Aristotelian passage, the verb, it would appear, cannot carry the same construction as that here suggested for it in Simplicius, whereas according to the inter- pretation questioned in the foregoing, the verb has the same meaning in all three occurrences. To this, it can be said that the construe of the verb cannot just depend on homonymy but must be guided by the context. Now, the basis of the suggested interpretation is that TO

tpo0uo,v results from the relation &7tsrxzev which is symmetrical. Hence, each product resulting from &7tXScLV would answer to 7repO r7rGupoXovTo0. On the other hand, in the Achilles, the pursuit is not symmetrical. The quicker runner is pursuing the slower runner, the latter does not in turn pursue the former; hence Aristotle can only say:

Me[ TL 7rP0eZXEV xvCyxaouV ro ppot&YSpov.

If we thus construe Zeno's premises, we can see that in this argument against plurality, he did not seek just any infinite division but only one which is through and through. It is obvious that given such a division of a being with a size, Zeno can proceed to develop his paradox. The products of the thorough division, it is assumed, would be parts of the given being. None of these parts can however be larger than any other, or else the division would not be thorough. Hence the parts must be equal to one another. On the one hand each of these parts has positive size, otherwise they could not constitute an object with positive size, and so would not be the parts of such an object. Zeno's argument for this thesis is essentially stated in Simplicius (DK 29 B 2 10-15). If then each of the equal parts has positive size, since there is an infinite number of them, their sum would be infinitely * G. Vlastos, "Note on the 'Porphyry Text'," (p. 2), paper offered to the Insti- tute in Greek Philosophy and Science, meeting at Colorado Springs, 1970.

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large. As they must at the same time be equal in sum to the given being, that being would now be infinite in size. On the other hand, these parts cannot possibly have positive size, or else the division is not through and through. Hence since the given being is thoroughly divided, these parts, while remaining mutually equal, must now have zero size each. In sum, they will still have zero size. Since in sum they must be equal to the given being whose parts they are, it follows now that that being would have zero size.

Even though Zeno's argument does not commit the fallacy of making a decreasing geometrical progression infinite in sum, there still are features of it which may not have commanded acquiescence in his day, and certainly merit discussion in ours. The following propositions are important to his argument:

i) a being with positive magnitude is divided through and through; ii) a through and through division as in (1) is an infinite division;

iii) the products of any division of a being with positive magnitude are parts of that being;

iv) the products of a through and through division of a being with positive magnitude cannot have positive magnitude;

v) the parts of a being are together equivalent to that being (formulated by Zeno himself in B 3).

Evidently, this set of propositions can yield Zeno's contradictory conclusions. Those of them referring to a through and through division have found much disfavour since ancient times, for it is said to be impossible either on the supposition that it evaporates a being into non-existence (e.g., Aristotle, De Gen. et Corr. 316 a; and Epicurus, Letter to Herodotus 56, 5-8) or on the supposition that an infinite division simply cannot be supposed to be complete (i.e., to have been completed).

For Zeno's argument, however, it is still essential to insist that he envisaged a through and through division. Indeed, unless this is done, it would become a wonder that a crude arithmetical fallacy concerning the sum of a geometrically decreasing progression could be committed by Zeno and sustained by thinkers of similar acumen, like Eudemus, Sextus, Simplicius, and Epicurus. In the case of Eudemus, it would be particularly flabbergasting since this man appears to have compiled a History of Geometry and composed an essay on The Angle. It is probable that in complaining that we would be forced in our conception of totals to make things go to waste in the non-existent if infinite division were permitted, Epicurus had in mind the Zenonian argument in

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which a being may well do precisely that if it were divided through and through in the stipulated sense. Clearly unless Epicurus had a through and through division in mind, his fear would be blatantly groundless, for otherwise an indefinite number of sizable chunks of the being would survive the division. Epicurus' further thought (57, 1-5) seems to be that if nevertheless someone obstinately proposes the thorough division into parts (which, as was seen with Zeno, have to be equal, whether they be of zero or of positive size), then however small he may make these parts, the divided being will now be infinitely large, taken (as before) in total. Epicurus' own way out appears to have been to exclude infinite divisibility, that is, to postulate that a being could be thoroughly divided without being infinitely divided, thereby yielding atomic parts.10

If this reading of Epicurus is correct, then it would be futile to have drawn his attention to properties of convergent series, for these are not in question. It is therefore unnecessary to regret that Aristotle failed to convince Epicurus, as there is a temptation to do. Indeed, it is probably too generous to say that the Aristotelian passage in question (Phys. 206 b 7-9) really formulates a theorem concerning decreasing geometrical progressions. What he seems to do is instead to call in question the notion of a complete infinite division ("we shall not traverse the given magnitude").

As for Eudemus, he is reported by Simplicius (Phys. 459, 25-26) to have argued: "to assert of something that it is an infinitely numerous

0OpaL8ek comes to the same as asserting that it is infinite as to size". Evidently 000?takL only needs to be taken to mean congruence with respect to positive size for Eudemus' mathematics to be vindicated. The connection with Zeno is obvious. It would now appear that the suggested basic reading of the Zenonian fragment similarly restores the logic of Zeno, Eudemus, Epicurus, and others. It would seem besides that any attempt to find a background for the atomists in Eleatic thinking must regard infinite divisibility in Zeno as infinite through and through divisibility.'1

The atomist answer to Zeno, as illustrated in Epicurus, is to deny

10 Letter to Herodotus ? 56. 11 Compare Heidel (loc. cit. in n. 2, p. 723) who affirms his belief that the Epicurean doctrine of partes minimae owes its origin to Zeno's argument, and that (724) Zeno's criticism "made it necessary that there should be a limit to the number and the divisibility of the parts of which a revised atomism might concede that it was composed".

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that a through and through division of a finite being is an infinite division, and also to deny that the products of a through and through division would yield indivisible parts. This atomist view would seem to be behind Owen's complaint that Zeno assumed without argument that the conjunction of size with theoretical indivisibility would be a contradiction.'2

It may possibly help to remark that Zeno's arguments involving the notion of infinity may be grouped according as they involve a static infinity or a dynamic infinity. The Achilles would be said to involve a dynamic infinity, for there is a chase which never ends. Similarly, the Stadium involves a dynamic infinity, for there is a race which cannot be completed (apparently can't even be started); and it would seem that in each case the hub of the argument is that each program calls for the undertaking of an infinite number of tasks which must be complete in an ordinal fashion - something that Zeno argues cannot be achieved. It would be piquant if by a volte face Zeno should assume in other arguments that an infinite number of tasks could be after all completed ordinally. The argument against plurality in Simplicius does not in fact make any such assumption. The objection to the effect that Zeno assumes an infinite division could be completed appears to be unjustified. Neither the race across the stadium nor Achilles' pursuit is given as a completed whole, while the divided being is. When Zeno uses the argument of the dichotomy as he in principle does in the present argument against plurality, he does not propose the use of a hatchet, but rather draws attention to a relation that is hereditary in the sense that its terms themselves contain the same relation. To assume the division to be complete is here not to assume it to have been completed. Thus having said that a given being (since it must have size and bulk) has two complementary parts, he asserts that this would be true of the complementary parts themselves and so on without end, for no complementary part could be final or could terminate the relation. Indeed the intransitive verbs o&7r&e LV,

7rpoeXyev themselves suggest that no task or activity is proposed. Zeno never assumes that an infinite task can be ordinally completed,

and whenever the question arises, he is anxious to argue that it cannot, as can be seen in the case of Achilles and the Stadium. If his argument against plurality required the ordinal completion of an infinite task, he would have no argument, for there would now be exactly the same reasons as at any other time against the requirement. The fact that 12 G. E. L. Owen in Proceedings of the Aristotelian Society 1957/58 p. 210.

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Zeno postulates products of the through and through division can hardly show that he envisages the ordinal completion of an infinite task. Consider as follows.

The objection that Zeno assumes the completion of an infinite task assumes that, when he postulates that the being is divided through and through and so infinitely, he is introducing end-products which logi- cally cannot be further divided, or he is assuming a last division beyond which there is no other, and so, a last part. Now even though there is an actual infinite number of points in a line at any of which points the line may be divided, the finite line does have terminal points, both a first and a last point. (This, of course, is made possible by the fact that there is no point next to either terminal point). But it would be a howler, committed by Johann Bernoulli, and decried by Leibniz, that a terminal point would be "the infinitieth point" on the line. Of course, if there were an infinitieth point on the line, there would be an infinite number of "infinitieth points" on the same line. Leibniz's observation was that points are not elements of lines, and if this means that points are not themselves members of line intervals, he is right. Zeno, of course, regarded the products of through and through divisions as members of the aggregate which is the given being, indeed the only parts which are members of the aggregate without being included in it, i.e., without themselves having members. This is the logical import of the thoroughness of the division. On the other hand, parts which are not end-products, i.e., all other parts, are included in the aggregate. That there must be end-products is truly a consequence of the supposition of through and through division, and no less truly through and through division is a consequence of the tenet that every being with positive size is divisible, for, as even Aristotle appreciated, every such being must be divisible at every point.'3 To say that it is infinitely divided is no more than to say that it actually has an infinite number of points at every one of which it is divisible. The point to note is that the infinite divisibility means not an infinite number of points of alternative division (such that the alternatives are inexhaustible), but rather an infinite number of points of simultaneous division. The points of division, being points on the being, belong to it not alter- natively, but simultaneously. It is this simultaneity (and not a process) which is articulated by the postulate of the complete division. It is clear that if Zeno's complete division is thus a cardinal completion rather than an ordinal completion, the infinite division of the given 18 DeGen. etCorr. I 2, 316 a 15-317 a 17.

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being does not imply a last division or last part, any more than the simultaneity of the points on a line imply an infinitieth point. That there are end-products implies neither that there is a last division not that there is a last product. That division which involves the end- product is distinguished from all the other possible divisions in that it is the only division of which it is true that every one of its products is related to the given being (now regarded as an aggregate) not by inclusion but by membership.

It should be evident now that any part with positive magnitude would be included in the aggregate. To say that such a part could be theoretically indivisible would mean at most that it is non-fissile in relation to physical theory, not that it has no segments. It is not necessary to Zeno's argument against plurality that the parts of a being should be fissile parts. The verb 47eXeLv does not require this; it calls only for segments. Solomon Luria14 accurately perceived the nature of the aggregation involved in Zeno's argument against plurality when he saw that Zeno rested on an addition principle for an infinite number of equal magnitudes. Two cases are distinguished:

i) if the addends are of positive magnitude their sum is of infinite magnitude; ii) if the addends are of zero magnitude their sum is of zero magnitude.

Griinbaum calls the enunciation for zero magnitude in question, and yet the analogy whereby he hopes to confute Zeno is in fact only helpful in drawing attention to a crucial difference between Zeno's principles and those of dimension theory, but fails entirely to disclose any fallacy in Zeno's reasoning; for the crucial difference is that whereas in dimension theory the unextended entities belonging to aggregates which make up a line are included in the aggregate but are not members of it, those parts of zero magnitude (the end-products of a through and through division) are members of, and are not included in, the aggregate constituting the being. It is only upon a reversal of this crucial relation (which Gruinbaum himself notes)15 that it seems to Griinbaum that Zeno has at last been refuted.

I wish to pass on now to Aristotle's strictures. In De Generatione et

14 S. Luria, "Die Infinitesimallehre der antiken Atomisten", in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, vol. 2 (1932-33) pp. 106-185. 16 A. Griinbaum, Modern Science and Zeno's Paradoxes (Wesleyan University Press, 1967), e.g. 119-23, 129f.

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Corruitione (315-317 fin), Aristotle states and examines an argument of the genre of Zeno's. In it, he points out that its proponents have misunderstood the concept of infinite divisibility. His opinion is that an infinitely divisible magnitude is only potentially so divisible, and it makes no sense to say: let it have been been (so) divided (wAT[JaT aEXa, or even worse 8 'npa& 8n mitvqt). Aristotle rests his caveat on his belief that even though a magnitude may be divisible anywhere within it, its points of division are alternative and not simultaneous points of division. The magnitude is infinitely divisible only because these alternative points of division are everywhere within it. It appeared to Aristotle that if one were to treat these points as though they were points of division simultaneously, then given any one such point there would have to be another point of division next to it. Since this is an impossibility the points could not simultaneously be points of division. It should be observed however that if this reasoning has any force at all, its force would be against the concept of a through and through division, and not that of a simple infinite division, for with the latter a next point of division need not be an immediate successor point in a dense manifold.

It is not clear what should be made of Aristotle's comment to the effect that should a through and through division be supposed to be complete, any point of division would have another next to it. That there is no such thing as a point "immediately next" to a given point was well known to Aristotle (De Gen. et Corr. 317 a 1-14); he fails to make it clear however why the supposition of a through and through division would dissect the given magnitude at every point conjointly, and so at every point along with its neighbors with which it would now be given simultaneously. Aristotle appears to have believed that the only way to avoid the howler of juxtaposed points would be to insist that at any one time there could be at most only one point (op. cit. 317 a 8) anywhere within a given magnitude. In this there is a misapprehension for the simultaneity of all points within a given magnitude would not make them consecutively well-ordered, such that each point has an immediate successor.

There is however another objection which Aristotle might well have considered. Two assumptions are associated with his reasoning con- cerning infinite divisibility, namely:

i) that every positive magnitude is divisible; ii) that every dividend part of a positive magnitude is itself a positive magni-

tude.

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Aristotle links these two assumptions with two other beliefs:

iii) that dividend parts are discrete parts; iv) that discrete dividend parts are well-ordered consecutively.

Under such assumptions, a complete through and through division would certainly involve dividend parts articulated seriatim and whose points of articulation are everywhere. At the same time, if each dividend part should have positive magnitude, then given any one point of articulation, the body could not be said to have been divided at any point next to this, for there would be other points within the magnitude of each dividend part. He might consequently also say that no con- ceivable division into parts could be a division at every possible point conjointly. The truth seems to be that a complete through and through division is non-intuitive, and there is hardly any reason to suppose that whatever it yields could be regarded as parts in a homoeomerous manner. Zeno, at least, was well aware that it would yield only degenerate "parts".

There is one question concerning Aristotle's treatment which raises the possibility that he was committed to an actual infinite. Aristotle, as can be seen from the foregoing, was against dense manifolds, and preferred to hold that the points at which a positive magnitude is infinitely divisible are alternative points of division, rather than simultaneous points of division. How, however, does Aristotle know that a positive magnitude is infinitely divisible? The answer which is available in his present discussion, seems to be that there is a point of division anywhere within the magnitude, and its points are everywhere within it (op. cit. 317 a). Evidently he supposes for one thing that there is an identical cardinality for the points of possible division as for the points everywhere within the magnitude; for another he must suppose that the latter points are infinitely many. Since, however, the magnitude is itself given as a complete whole, it would seem that the points everywhere within it are themselves given as a simultaneous manifold, for if they only arise in consequence of being points of alternative division, they could not as such make the points of alter- native division infinitely many. In other words, Aristotle claims that the alternative points of division, by reason of being everywhere within the magnitude, make the points of alternative division poten- tially infinite. But they themselves cannot be potentially infnite merely or this would mean that they arise in consequence of the alternative divisions, and so could not guarantee the potential infinitude of the

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divisions which give rise to them. It would seem that they would have to be actually infinitely many.16

Macalester College, St. Paul, Minnesota

"6 The Greek texts have been verified by Jeremiah Reedy, of the Department of Classies in Macalester College.

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