Zeno on PluralityAuthor(s): Stephen MakinSource: Phronesis, Vol. 27, No. 3 (1982), pp. 223-238Published by: BRILLStable URL: http://www.jstor.org/stable/4182154 .Accessed: 25/04/2011 21:50

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Zeno on Plurality

STEPHEN MAKIN

In de Generatione et Corruptione A8 Aristotle presents the Atomic theory as due to reflection on the Eleatic view that what is is necessarily one and immovable (325a2: T'O 'EV 't &vyxTjs E'?V ElVaL xati &xLvtTov).1 We want to discuss some Eleatic arguments against plurality,2 which are of interest both in themselves and as precursors of Atomist thought. The arguments to be considered are from Zeno.

We will have two guides in interpreting the arguments. First, they should be such that Atomist theory provides a plausible response to them; second, they should pose no threat to the Eleatic theory. Both these guides embody assumptions that might be challenged. As to the first, it might be doubted whether the Atomic theory was a reaction to Eleaticism; but for the pur- poses of this paper the first guide is the less important, and we need not go into the matter here. As to the second, it might be denied that Zeno's arguments are intended as support for the Eleatic position; if not thus intended, then seeking consistency with the Eleatic position will place an unnecessary constraint on the interpretation of the arguments. The Platonic picture of Zeno as defender of Parmenides3 has come under some attack,4 but has subsequently been defended5 in a convincing way. For details of that debate the reader is referred to the original papers; the Platonic view seems sufficiently defended to justify proceeding with the guide of seeking Eleatic consistency.6

Two substantial passages from Simplicius - in Phys. 139.7-19, 140.27-141.87- are our sole source for what purport to be the actual words of Zeno concerning plurality. Our interest is not primarily in these passages. The interpretation of Zeno that we will put forward will focus mainly on doxographical reports of Zeno's arguments. It will be consistent with the Zenonian arguments preserved by Simplicius at in Phys. 140.27-141.8.

Our discussion will focus on what we find the most fruitful of the doxographical reports, from Simplicius at in Phys. 139.27 - 140.6. For ease of reference we give the passage in two parts, each numbered as it will be discussed, and with line references to pp. 139f of Simplicius' text indicated.

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27 (1) E? yap ?;c, W l, 8&aLpCTOV, TETiiao &;Xa, (2) x&iLa ETv T ppv

EXTEpov &XXa, xOi To'&ov &ei yrvoRivov, (3) 8AXOv, q'nouv, Ws iroL

U'rrOp?EVEL TLVa ?XaaT P.EYE1 aXaXILTa XvL aTo%L, 'nTXOEL 8 &'neLpa, (4)

30 xoLi TO'XOV i Xa)(W'Tov, rTXA0EL 8E 'aTE'LpWV OVU1-qOETaL (5) ii wpoDov

EoTaL xaL ELS O6E'V ETL LacXXOGETaL XaL EX TOv &8evois aGaTrETaI (6)

&Wrep ONoTra. (7) oivx &pa &LOLPefhlETaL, &AXX& VEVi ?V. XiL y&p 8 (1)

1 ~?&is 'rrivrT 6 LOL6v I kUTLv, (2) EL Ep 8&aLpETOV PVt&tpXE, LrTxOVT' 61LOL(ws

EUTrnl &EXLpETOV, &XX O'V 'r EV, Tr 8? o". (3) 8pi- f a 6X r&vr (4) 8AXOV OVV IFTXLV WS OVUeV UVTO I.VE, (5) &XX ?ETaL qfpoubov, XOL EtVTEp

VJTOETal, 'TXLV ?X TOD p4lEVOS UGT1oEL. (6) ai ycap ?OVE Tl,

5 OU8e Ir ) EV1'V OflvTT &11p?LEVOV. (7) "XE xvi bx TOUTwV

pavEpOv, (qGLV, WS &8LaLpETOv TF XXti &ILEpEs xa;L V E"UTaL TO OV.8

We will make references to other passages in the course of our discussion.

Simplicius quotes the passage from Porphyry, who attributes the argument

to Parmenides; but many commentators - Simplicius, Alexander, Philo-

ponus; Lee, Owen9 - hold that the argument is Zeno's. It is an important

question whether the argument actually is from Zeno, rather than later

elaboration of Zenonian material or new argument in the Zenonian style.10

After we have made clear what the argument comes to, we will bring

together a number of points and see that their cumulative weight makes a

Zenonian attribution the most sensible. For the present, let us agree that

the argument is Zeno's; the light that such agreement casts on Parmenides,

Zeno, later Atomism and otherwise opaque doxographical passages will

itself be evidence in favour of a Zenonian origin.

We will set our interpretation out in opposition to that offered by Lee,

Lee's interpretation just being typical of one that takes Zeno's arguments

against plurality to proceedfrom some premiss(es) concerning divisibility.

We want to show that the argument starts from just one premiss, and that

that premiss makes no assumptions about the divisibility of extended

magnitudes.1" This has consequences for our understanding of the

Atomists' theory. For if the link between divisibility and possession of

magnitude be broken, then indivisibility must be given a new grounding.

In discussion of Zeno we shall see that for the Eleatic the indivisibility of

what there is is based on its homogeneity; and if similar arguments are

available for the Atomist, the indivisibility of each atom will be grounded

in its solidity (i.e. in each atom's being homogeneous). Clarity gained

concerning the indivisibility of the atom will be of benefit in consideration

of other aspects of the Atomic theory - for example, consideration of

questions about atomic size, or atomic interaction by collision. In short, to

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get straight on the Zenonian argument is both of value in itself, and a necessary preparation for seeing the Atomic theory right.

Lee says that12 P: the process of division is infinite is a presupposition of the argument, and that (0) any magnitude is by definition divisible is a hidden premiss. But if Zeno holds positively to (0), then he cannot be supporting the Eleatic view that To ov is &&aLdprTov and a R?yt0OO.13 Nor can (0) be the subject of a reductio ad absurdum, for any conclusion of such an argument may then as well be a denial of (0) as the Eleatic view that TO OV iS

)-v and &xLat,peTov. We will see the argument proceeding to this conclusion from a single premiss.

We discuss the later report first (in Phys. 140.1-6). The introductory xai y&tp 8' shows this to be an emphatic form of the first report (139.27-32).14 We have (1) VITFEi 'IaTVTn OLOLOV ?aTLV.

(I) is not hypothetical. Note the use of ku'r and ITV. (1) provides a premiss common to Zeno and his opponents. The parallel with Parmenides' remark (DK 28 B8.22), F?io 'rr&v hTTLV O'iolov, is plain and significant. Parmenides' point is that it is everywhere the same;15 for the properties of the whole follow from its character as To 'ov (thus the deductions of DK 28 B8), and none of the whole can be other than TO OV, since it cannot be that TO [iii OV is.

We may expect Zeno's opponents to agree to (1) - it was the innovation of the Atomists to deny it. The Atomists grant differentiations within the whole, for example concerning divisibility, by allowing that TO >ri1 OV is - i.e. distinguishing those parts which are full and those which are empty, as at DK 67 A 1: Tb p?V 'r&v 'pITELpOv pq(LV (A6xomos). . .TOVTOV 8e TO pL 1V TXipES,

To b' xEv6v. From (1) the text continues (140. If)

(2) ei?i?p &tlapeTbv U?rpXpEL, ITSVf LOLW@ WTXl &LaLpETOV, &\\' Ov TY JEV, Tf

8O, OiV. E'ITEp 8LaLpCToV virapXFL gives the supposition to be reduced to absurdity. Following (1) a trilemma is assumed16: either (a) it is indivisible, or (b) it is divisible here and not there (finitely divisible), or (c) it is divisible every- where (divisible ad infinitum). In that they oppose Parmenides, Zeno's opponents will not hold (a). So they are forced to (b) or (c). (2) is a compressed rejection of (b) along the lines (2) If it is divisible - viz. if either (b) or (c) - then it will be divisible

everywhere, and not divisible here but not there - viz. (b) rejected. So, it is everywhere divisible, if divisible at all - raOvrn 6pRows E'OTaL

225

SLpvrETOV. So, (b) is argued against and rejected without need of either P: the process of division is infinite as a presupposition, or (0) any magnitude is by definition divisible as a suppressed premiss, as Lee holds.

140.2 takes up the consequence of (2) that 'rr&v'rVT oLOLS FU'rT 8aLperTov,

with (3) 81'Pf OW &' stVrn. The use of the perfect imperative 8tLp'aOw indicates that we are to focus our attention not on the process of dividing, but on the products of div- ision.17 Notice that the argument proceeds without any explicit reference to infinity. While it may be true that to divide everywhere is to perform a division to infinity, in the argument as it stands that remains to be shown. We do not, then, need anything like P: the process of division is infinite as a presupposition.

We now have a consequence of the supposition of (3), (140.20) (4) 8iXov oZ,v IT&XLV 'WS OVi8C'V VITOlEvEt,

where .r&aXLV is used temporally. The comparison is with a similar move in the first statement of the argument. The temptation is to read (4) in terms of Zeno's argument (in Phys. 139.9-15 = DK 29 B 2.6-14) that what has no magnitude is nothing, and so non-existent: viz. to suppose that what remains after the division in (3) has no magnitude because (3) gives a complete division, and then to take (0) any magnitude is by definition divisible as a hidden premiss of the argument. But (4) need not be read in this way. The argument continues with (140.3f, 140.4f) (5) 'XXs EUFTatL Cpovov, XviL F';lp voTrlETraOLL, Ir4XLV 'EX TOV ji1&VOS vUG-

and (6) E'L yap VUTOILEV? TL, O, 8' T EVUETOLL 16VT & 81LT VOV.

(5) adds little to (4), but (6) is highly significant. The sense is: 'for if anything does remain (Vi'rOfl?VEL, future), there will not yet be produced

(YEViT,ETOL, future) what has been divided (8&npv'pvov, perfect) eve- where'. (6) gives the ground of (4) 8bXov oiVv 1T6ALV WS ov8Ev vo.ved, and we may suppose VnO,ro?Vet TL in (6) to be compared with o V"v V'fOR?VEL

in (4). (4) follows from the supposition of (3) 8LnYi 'aOw sravrM (3) being intended to close off the alternative that it is 1TvfTn &XLpET6v. The

226

original trilemma is reduced to this alternative by

(I) EAT&L 'OV'IVTn OLOLOV COTLV,

and (1) reflects the Eleatic view that TO OV is homogeneous. So we see the relation of oVi?v lesopvdt in (4) and i(rOrP.VE TL in (6) in terms of the thesis (l) about Being. OvoEv i"ro[eV?t because what results from the division in (3) cannot be, and that for the reason given in (6).

The argument goes thus. To ov is homogeneous, according to (1). Therefore, (2), neither To ov nor any part of TOb y - that is, anything that is - is differentiable as here divisible, there not. So, if divisible at all, To ov is

1T&VTq 8MlpETO?V. But the opponents of Eleaticism hold that TO OV is divisible. Now, what remains when the division of (3) is completed is not divisible, since it has been divided; and so it cannot be, since what is is divisible - because To ov is divisible and homogeneous. Then (4) and (5) follow from (3); TO OV 'vanishes' (ppoi)ov: 'gone', 'vanished') because what remains cannot be. So, the argument proceeds without any need for (0) any magnitude is by definition divisible as a suppressed premiss. It is no longer puzzling why tuiyEOos is not even mentioned in 140.1-6. Further, (6) does provide, as the opening yap sugg- ests, an explanation for (4) and (5). For, if VITOREVrt TL, then the something (T.) remaining is something that is, and so To oy, contrary to the supposition of (3), is not everywhere divided. For, as T0 yv is homogeneous and divisible, anything that is is divisible, so that it cannot be that V'rroivrL TL -that something (that is) will remain after the division is complete. So, since it cannot be that vifro[tEvet Ti then, as (4), oVO?'v VesoiRFvr, with the absurd consequences specified by (5).

So, the argument has just one premiss,

(1) 'rr(VTq OJIOLOV ECOTLV8.

Basically, the argument draws absurd consequences from the conjunction of (I) and 8LaLpET0'TV vIIXPXEL -the opponents' view which is assumed for a reductio, and then discharged as a premiss. So, the conclusion at 140.5f (7) CaTc 'EX TOU'TWV ftvrp6v, fq(L, WS &L0poETOv TE XiL &LpEs XaL EV ECTaL TO

Sy ov

rests solely on (1). Then, since the argument does not involve premisses concerning the divisibility of magnitudes, it cannot pose a threat to the Eleatic view that TO OV is both &a8LpaETov and a tE-yr0os.

A similar interpretation may be given for other doxographical reports of Zenonian arguments. At in Phys. 139.19-22 Simplicius quotes a report from Themistius. The relevant fragment of Themistius has survived, and reads:19 (Z'vwv) 'X TOV aVVE)ES TE ELVOXL XOL a18bL0pET0oV LV ELVLL To Ov XaT-

rOXVcttE, XEYWV "WS Ei 8LlpLETtl, OUME? LOTOL EOMLC 'V, &t& T'V ?EIT ?IELpOV

227

TOLiV Ti.V '

avi&Twv. Zeno's argument is stated to proceed from (Ex) the indivisibility and continuity of TO OV. That need not mean that (i) TO OV is avvrXis and 08tatpeTov features as a premiss in the argument as a whole, which would make for confused logical structure, but may rather mean that the conclusion that Tb

OV is ?v is reached via the claim that it is avvexEs and a'8aPTOV. So, mention of i Ur' a&rSLpov Toi 'v OWxLaTxv cannot be taken to show that any body is divisible ad infinitum, for Zeno is here concerned precisely with something auveXEs which is also &8La;,o?Tv and ?Ev. Then why is avvEXts brought into the argument at all? Simplicius quotes Parmenides thus (DK B8.25): T4O

VVFXE'S r'ITV EUTLV. ?0v yap i6VTL TErX&E&l - T O V is avvEXrs because T OV

'clings close' (?reXaEL) to To ov. This may be taken as a reference to the homogeneity of TO OV, affirmed three lines earlier at B8.22. So we may expect that when use of avvExys is reported in the context of a Zenonian argument, reference is (reported as) being made to the homogeneity of the subject. So, ' as' aTr?ELpOV TOI.Lq TWV OcpaTrV will be what follows from its being the case that T'o ov is divisible at all; i.e. the use of avvEXis indicates that the possibility of there being a finite division is ruled out. NU' E`rTaL

pealws '?v then rules out the possibility of an infinite division, which infinite division would follow from the infinite divisibility signalled by oUVEX.ES - OVU? 'ET'XL OF-otiLwS EV, 8&c T vv ?X' 'X1TELpOV TO[LTqV TWV GWRaT(V. So the only alternative will be that To ov is &BLOpTOV, and from that - ?X TOD

OVVEXES TE eLvaL xai & LpETov - it follows that To ov is ?V, the stated conclusion of the argument.

We see the same moves in an argument reported by Philoponus.20 The clue to the argument is in Philoponus' parenthetical comment (Lee 14.2f): al yAp &UXLpOlTO TO UVVEX.ES, 1'X a"?TELpOV &V EL'T 8LOUpETOV. The use of El is important. At Lee 12.23 Philoponus says that the argument proceeds ix rr s u'r asELpOV TVOV 0VVrx XoTo.LL'aS. Clearly, it cannot be that Zeno holds positively to ij Es' a`&impov TC.EV uveyXC,V8 xoTo>xa, and relies on that as a weapon against his opponents; for the Eleatic doctrine is that To ov is both avnXjs and ?Ev and &8mipefTov. Nor can the story simply be that bn ' a&rrr pov T&v civvEXCv 8LoXoLa gives the view of Zeno's opponents, which is to be reduced to absurdity. For in that case the argument would just show that if there is a plurality, then things are not divisible ad infinitum, and this is precisely what Zeno's opponents want. Philoponus' parenthetical com- ment makes the counterfactual nature of Zeno's argument clear - if To UVVrXiS &MLpolvTo then it would be (av Ei'i) ?Xrr' a&ELpov &atLpETov. This move will be forced by Zeno on his opponents, for neither will admit taTL TO > bV,

this being the move made later by the Atomists, so that the absence of any

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differentiation within To ov will rule out the possibility of a finitely divisible plurality. But, the argument proceeds, neither will an infinite division be possible (Lee 14.1-3, oVSEv 'E'UTrtu XVp;WS iv. . .E, &'E pVqUV i &UTL XVp(WS EV,

o0,b IToXX&); so no division into a plurality will be possible (Lee 14.4f, &8MvrTov apa es -rro 'XXX 8LapdOOM to Tov). There is no mention of the homogeneity doctrine as a block to finite divisibility in the second part of Philoponus' report (Lee 14.5-14). But we shall see that the possibility of a finite division is wholly omitted there, so that there is no need for the homogeneity doctrine. The second part of the report takes off from a dilemma set up thus (Lee 14.7f): lxa'rLT ovv iv&s 1`JOL ,U. ECYTr' xai a8cotpFros,

" xvi o&rVi ds iroXX& 8LapELtTaL. It is significant that the dis- junction is between the adjectival &taxpL-Eo0 and the verbal bL8LpEIxML; that is, not between'... is divisible' and '. . . is indivisible', but between '. . . is indivisible' and '... is divided'. So we do not have a dilemma, 'either finitely divisible or infinitely divisible', with absurd consequences being drawn from finite divisibility to give 'if divisible at all, then infinitely divisible'. Rather, the disjunction is between two ways of considering

ixaarl evas; either (a) as &BMape?os, that is as units, which must, however, as the units of an extended plurality, themselves have magnitude; but, as possessing magnitude, they can be considered not only as units, as &bLaM'pETOS, but also, (b) as &toLpdrotr, that is as divided pluralities, in which case the pluralists still have to provide an account of the units which go to compose their plurality. If this argument is to have any force, i sr' 1rTELpOV

T(&V OVVEy(CV &XoTOrLLca must be assumed in the latter part of Philoponus'

report; while in the earlier part b Ei' &OlTEipOV T(V uVVEXy V 8tI XOTOrLL is argued to follow from divisibility simpliciter, and this most plausibly on the basis of the homogeneity doctrine.

We return now to the passage from which we started, Simplicius, in Phys. 139.27-32. We can pass over the opening steps with little comment, (139.27f, 139.28f), (I) Et yap ?;w, wt, 1AXLpETOV, TET 'COW &ixx' and (2) xafLX?ELTaX T)V REpCaV iX&TEPOV 8L;'X, XaL TOVTOV &Ei YEvoLEvov.

The interesting step is at 139.29f, (3) iX6ov CPqav I(as ijTOL 1hO1ITEVd TLVaX 'E'O)(TaX LE4yh kXWXLOT XCL tXTO[a,

?Xrl~Oct 8'c 0x'rEpa.

(3) gives the first limb of the dilemma resulting from the introduction of divisibility in (1). The possibility considered is that TLvaX EaXaa Iiye'Ov will remain after the division. Note the addition of -nX'0rL b s 'rrELpa; what appears (to us) to be a natural alternative - i.e. that To ov is divisible to a

229

finite number of &aTO[a >yiOq - is not considered. What, then, is covered by (3), and why is a finite division not considered as an alternative at all? Again, it cannot be that (3) reflects the use of (0) any magnitude is by definition divisible as a hidden premiss. For, if that were so, the most obvious reply to the suggestion of EaUXT1xa [ty&0 would be: if they are E-yE'Oq then, by (0), the division is not complete, contrary to (I) and (2); but if they are not >EiyeOi,

then the supposition of 'EOXaXT ,EiyE'Oh comes to nothing. But the reply of (3) is nothing like this: instead we have 'if TtVai iEarXrT RyE-& E'X&aXLT xvi aToroa remained, then they would be rTXiOu a"rTL pa'.

The clue to understanding (3) is to be found in the relation of the two reports, 139.27-32 and 140.1-6. The reports are linked by xai yap &9, and so an emphasis is marked.21 We may render the bridging sentences (139.32-140.1): ov'x &piX 8LOUpEO1ETOU, 'XXXt REVEr EV. xvX y&p 8' &rTd IT&vT'fl

O1OL6v l'rV. . .', as follows: 'It shall then not be divided, but will remain one. For certainly, since it is everywhere the same.. .'. The homogeneity doctrine, which apparently makes no appearance in 139.27-32, is in- troduced as an accepted view in an emphatic restatement of that passage. So it would seem reasonable to suppose that doctrine to be working below the surface in 139.27-32, only being made explicit in the more emphatic 140.1-6. This enables us to explain what is otherwise completely puzzling - i.e. why the possibility of a finite division to a`oT.a Rj.ryiOq is not mentioned in 139.27-32. The dilemma of which (3) is the first limb is based on ?i yap

?'iq. . . 8aLpETOv, read as 'for if it were infinitely divisible'- compare &a? in (2) - and the necessity of the division's being infinite is demonstrated later in 140.1-6. The dilemma is of this form: if it is divided (completely) then either (i) what remains has magnitude, or (ii) what remains has no magnitude. (3) takes up (i). Since the division is stated to be complete what remains must be ECxXaTa, EXiaXtcTa and &ToIua; by (i) they are RE-y&6iq; since the division cannot be finite they must be ITX0OEr arELpa. All this adds up to give (3) &jX6v qnav 'S 7 rTOt V'iOToEVd TlVa EX.TtO P.yEO1 tAXtGT Xai a"TO%a,

X'OE L 8& c1 t lpa.

With the following (139.30f) (4) XiL TO oXov et EXaXL'cTWV, `1XTXOCL 8E p &?TQL'(WV OVOT11UETOL

(3) is reduced to absurdity. If the division of (1)/(2) results in an infinite number of E'X&XVrTa, then the whole will be composable from an infinite

230

number of 0To[LU [Wyq0n. The unstated absurdity is then, presumably, that the whole will itself be an infinite REpyeOos.

139.3 If gives the second limb of the dilemma, (5) "

qppOV8OV EOTUTL XvXL ELS OVOEV ETL bLtVXUOIarTtL XLti 'EX TOi >IiiEV'Os ava-

We consider the alternative that the results are ov0Ev. Strictly, the altern- ative to (3) is that what results has no giyE0os. A step - viz. from 'has no magnitude' to 'is nothing' - must have been omitted. We should under- stand this suppressed move by reference to our earlier comments on 140.2f: 8iXov OViV ItXLV "s oV'&V VrTO[LEVE!, and 140.3f: &XX WTOTXl (PpOUDOV, XvXi EI1TEp

OVOT1iOETil, TrcAWV Ex TOv t,iovobs avwrioETrL. The occurrences of rTaXLv here are intended to refer back to (5) above. We argued previously that the view that what remains is oV8?v is not based on (0) any magnitude is by definition divisible, and Zeno's argument reported by Simplicius at in Phys. 139.7-15,22 that what has no magnitude is nothing, but rather on the homogeneity of To OV,

given explicitly at 140. 1. The comments made earlier on 140.3f will apply to (5), and we need not repeat the details of our interpretation.

The argument concludes with (139.32) (6) 0raiEp cXTOITX

and (ibid) (7) ovx xpt BLOMpEO orETOxL, 4AXXa piEVrE EV.

(6) closes the dilemma introduced by (1), stating that each limb - viz. on the one hand (3)/(4), and on the other (5) - leads to absurdity. The precise absurdities are not, however, made explicit. (7) concludes the argument. The dilemma arose from the assumption that the condition given by (1), dE yap Etq. . .Btatpurov, is fulfilled. That assumption leads to absurdity, and so is negated. So the conclusion is that To ov IEV E'E'v, it will remain one. The aim specified at 139.26 is thus fulfilled; the argument is there claimed 8rLXVVVal TO OV ?yV FOvav R6vov Xti TOi3TO &[jEpES Xai &&lipeTov, to show To ov to be one alone, and this both without parts and indivisible.

We can see that the interpretations we have given will be consistent with any sensible account of the arguments against plurality given in the Zenonian B fragments (DK 29 Bl,2,3 = Lee ??10,9,1 1 respectively). What is important is that any reference to infinite divisibility of magnitudes in those fragments is always under the condition that a plurality exists - i.e. that there is some divisibility of the whole of what there is into a plurality. The argument in B2 (Lee ?9) concerns what follows on there being a plurality (B2.4-7): ELToXXO'& EOTL, xviL E-y&AX #UTL XOi JLXpO. JiEyCAXO pEv L.OTE OtELp Tb RE-yeOos ElVOM, FIxp& 8? OVT(rs (oUTE [i0iv `xctv R7yEOos.

231

Reference to 'iT 'aT' 'X ELpOV TOpn at B2. 15ff concerns 'each of the many', and so presupposes some division into a plurality:. . . oTt [LyEOOS eXEL 'ExxaoTov

TC)V IOXXIA)V Xti aL TELpwv TX3 lTpO TOV Xatpavo LVOV ?EL TL ELv L &a TyV ?X'

&'rTrepOV TO%v. In B1 (Lee ? 10) the reasoning to show the infinity of magnitude starts with the quoted words (B 1.4ff): EL 8? 'EUTLV, &Vvyxyj

EXaOTOV g?yE06S Tl EXELV XOXL 'IT4XOS XaXL &ITEXELV a(VTOV TO ?TEpOV OLSo TOV

?TEpOV. Use of 'ExxOTov shows that a plurality, and each of what makes it up, are under consideration. Hence, the quoted conclusion says what follows on there being a plurality (B1.10ff): OVTWS aElTOXX& EOTLV, &VayXO aXTO'

xLpaL TE ELVOL Xa;L VYaX/AC. ti.XpaX pgiV WAwrTE RL1 E`XELV LiyUOoS, R.Ly&Xot \E' WOUTE

aX1lfpa EVXL. Again in B3 (Lee ? 11) what is offered is an argument con- cerning what follows on there being a plurality (B3.2f): . . .iTL El IoTXXt ?OTL,

Ta avUTat 1T1pTExup.v0( WTL XviL XTrELpa, and the proof concerning infinity goes (B3.8ff): ?i 'ToXX&a kaTLV, &'MeiLpx T& OVTaX &OTLV. aEL yap ET^EpE 1LETatkv) TGV

OVTWV CUTL, xaL ITOcXLV bXEL;V)V ETEpa >LETacv. xvl OVT(T@S a'TrELpO: TaX OVTX kITL.

Of course, pointing out that the B fragments concern what follows on there being a plurality is pointing out what is obvious, and doesn't help much towards a clear interpretation of the B fragments. But it is sufficient to show that any sensible interpretation of the B fragments will be consis- tent with the line we have taken above. The crucial point of that interpre- tation was to show that the view that what has magnitude is ipso facto divisible, was no part of Zeno's thought. That view is nowhere relied upon in the B fragments. Since the B fragments are concerned with what follows on there being a plurality (i.e. what follows on some division being pos- sible), the view required for the B fragments is rather that whatever is at all divisible is infinitely divisible (there is no finite divisibility). Not only is our discussion perfectly consistent with Zeno's having held this view, it moreover shows how Zeno could have argued in its favour from the accepted premiss of the homogeneity of what there is.

We must now, as promised, return to the question of the Zenonian origin of this homogeneity argument. Now, we must allow that we cannot pro- duce any fragment of Zenonian writing wherein Zeno states the homogeneity argument we have discussed. But we know that Zeno had far more arguments than the ones that have been preserved,23 and those that have been preserved were preserved in a short space for a particular exegetical purpose.24 So it is to be expected that quite a few Zenonian arguments will not survive in Zenonian statements. Absence of explicit Zenonian statement is not, then, particularly unsettling; the question really is whether, on the balance of evidence, it is more natural to attribute the homogeneity argument to Zeno.

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We argued earlier25 against challenges to the Platonic view of Zeno as defender of Eleaticism. Since we have that defense, it is to the point to cite evidence that the linking of homogeneity and unity was an Eleatic move; for that will show a Zenonian argument to be possible - i.e. not anachronistic - and even give it a primafacie plausibility. In Parmenides, the subject's being ff&v 'optov and its not being &xLpET6v are linked in an explanatory way, by ?Ei'r, at B8.22: o'vE &MLpETov WoTLV, i'ErL 'T&v EUTLV 6toutov. Melissus upheld both Unity and Homogeneity, and mentioned them together, B7. 1: ovrws oiv &LBLOv EUTL XiL &1TELPOV XaL EV XLa O[LOLOV 'rrav. Aristotle speaks of the homogeneity argument as being Eleaticl,44when he reports the following argument at de Gen. et Corr. A8, 325a9-12: i IxV -yap

'TntvVT'fl 8LOLpETOV, OV&EV EIVtU EV, WA'OTE OiVO? roXa&, &AX& X?VOV TO OXov. at &E

Tr [?V Tnj 8E U TEXTXaUI(LEVW TLVL TOVT' EO0LXEVaL. iEXPL 'ITOGOU -yp XtL && TL T O

REV OVTwS EXEL TOV) bXov xvi iTXip's ?iTL, T6 sE bvI pEvov; So we have reason to say that the argument from homogeneity to unity is ancient and Eleatic; and so it is implausible that the basing of indivisibility is just post-Platonic invention.

Further to this, why should we suppose the homogeneity argument Zenonian? There is no impetus to suppose the argument reported by Simplicius at in Phys. 139.27-140.6 to be, as Porphyry, Parmenidean; the type of argument seems out of style with Parmenides. But we have seen that there is reason to take the homogeneity argument as Eleatic. So that gives us some ground for agreeing with Alexander and Simplicius, Lee and Owen,27 that the argument is Zenonian. Further, as we have seen, the homogeneity argument is not only consistent with, but also required for, the Zenonian B fragments - required, that is, to provide support for the principle that something at all divisible is infinitely divisible. Further, since we have accepted the Platonic view of Zeno's relation to the Eleatics, Zeno does need some means of rendering his use of an infinite divisibility principle consistent with the Parmenidean view of a continuous unity with magnitude. Taking the argument of in Phys. 139.27-140.6 as Zenonian would provide Zeno with what he requires in both these areas. Further, as we have seen, certain difficulties in the doxography28 can be cleared up if we take the homogeneity argument as Zenonian.

So, as Alexander and Simplicius saw, there is some plausibility in taking the homogeneity argument to be Zenonian; and since also Zeno so clearly needs the homogeneity argument, and a Zenonian attribution renders many things more clear, the prima facie plausibility of a Zenonian attribution seems sufficient to support attribution of the homogeneity argument to Zeno - especially in view of the fact that the lack of any

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surviving Zenonian statement is so understandable. To conclude then: we have an interpretation of a number of Zenonian

arguments, reported by the ancient commentators, against plurality, whereby Zeno's conclusion that ror ov is 0a'LapETOV -or ?Ev or alipis - is based on just one premiss; namely that stated by Simplicius at in Phys. 140.1 as bf& s'r&vri OILOLV EUTLV. Our interpretation is consistent with those arguments which Simplicius apparently quotes directly from Zeno, but which we have not discussed at any length. Our interpretation has three main advantages. Firstly, we see that Zeno's arguments do not reveal a tension in the Eleatic view that TO 'ov is both &cL0ipETov and avvEx's and a ,icye0os. Secondly, we see Zeno as in positive accord with the Eleatic position, as evidenced by the parallel between his premiss and Parmenides' ETSEL IXV W'TLV 'Ooiov (DK 28 B 8.22). Thirdly, this interpretation will be of great help in any consideration of the Aristotelian account of the Atomists' theory as a reaction to Eleatic arguments - especially in consideration of what the indivisibility of the Democritean atom comes to.29

Gonville and Caius College, Cambridge

NOTES

Compare too Physics A3, 187al-3; though the reference to Atomism is not explicit, see the arguments of W. D. Ross, Aristotle's Physics (Oxford, 1936), pp. 479ff. It is noteworthy that the fragments of Zeno preserved by Simplicius were preserved in con- nection with this passage. 2 Just against plurality; for our limited purpose we need not consider the better known Zenonian paradoxes concerning motion - though these too might be of interest in connection with the Atomist theory. 3 For the Platonic account of the relation of Parmenides and Zeno, see Plato's Par- menides 127A - 128E. 4 Most thoroughly by F. Solmsen, 'The Tradition about Zeno of Elea Re-examined', Phronesis 16, 1971, 116-141. See also N. B. Booth, 'Were Zeno' Arguments a Reply to Attacks upon Parmenides?', Phronesis 2, 1957, 1-9. 5 For a thorough and convincing defense of Plato's testimony that Zeno was 'a personal intimate and, in his book, a philosophical partisan of Parmenides', see G. Vlastos, 'Plato's Testimony concerning Zeno of Elea', Journal of Hellenic Studies 95, 1975, 136-172. 6 Since the Solmsen-Vlastos debate the Platonic view has been further challenged by Jonathan Barnes, The Presocratic Philosophers, Volume I, pp. 231-237 (London, 1979). Barnes makes a number of points about the aim of Zeno's arguments (op.cit. pp. 234f). First, that Zeno was not defending Parmenides against Pythagorean 'unit-point- atomists': we can agree completely with this. Second, that Zeno did not defend Par- menides against any other philosophical attack - rather it is to satire and jeerings that Plato refers. No doubt Parmenides was the object of much contemporary mirth. Plato

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does use the word xw.Ly&tv at Parm. 128D: Zeno was supporting Parmenides against those who mocked him. But such mockery does not exclude philosophical opposition, which would stand in need of philosophical reply. Plato's report of these opponents continues (ibid): . . eWS ??v uTrVu, IToXX& xai yEXoie aUVRalivEL 'TJXELV TX'r X6yX xiVaVTit O5VT(O. This suggests the mockery was more thought-out than mere popular ridicule; and it is, of course, not unknown for philosophers to scorn one another's views - it is, moreover, a typical philosophical move to call an opponent's views absurd.

But whoever was opposing Parmenides, Barnes does not believe it was Zeno's aim to defend Parmenides. First, because Barnes doubts whether Parmenides was a monist at all; second, because what Zeno aimed to show was that pluralism involved even greater absurdities (Parm. 128D) than monism; third, because it is not true that a refutation of pluralism is equivalent to a proof of monism; fourth, because some of Zeno's arguments appear to bear against pluralism and monism.

Hopefully the discussion in the body of this paper will go to answer some of these points. As to the first, all hangs on what we want Parmenidean monism to be. If we take as an object of enquiry 'everything that there is', and work the deductions of Parmenides' B8 on it, then the Parmenidean view includes that it is finite (B8.4249), balanced (B8.4, reading &TXSavrov for aTeorE?Tov, with Barnes and others), continuous (B8.6, 25), homogeneous (B8.4,22f, 44-49), one (B8.6), and undivided because homogeneous (B8.22-25). Now one might suppose that this won't give monism because nothing is said to show that it is indivisible - and so nothing rules out its having many parts, which could hardly amount to monism. This seems to be the thought behind Melissus' B9, where being one necessitates having no parts, and so no body (otue) and bulk (-rrXos). But in fact the description that Parmenides gives of one homogeneous, continuous body does give a more sophisticated monism than one which cannot handle the divisibility that supposedly follows just on corporeality - what this monism is, and the arguments for it, emerge in our discussion of Zeno: it is precisely the monism that Zeno is in tune with, based on homogeneity, and so that Parmenides would have this type of monism will be a point in our support. Barnes' second and third points above go together: since the Eleatic view ruled out the line 'it is not', then a refutation of pluralism would seem to an Eleatic sympathiser to be a defense of monism; and if pluralism and monism were thus exhaustive alternatives, then to show greater absurdities in pluralism than monism would constitute a defense of monism. Still, monism could not really involve any absurdities, and we will have to take Plato's talk of 'greater absurdities' as a figure of speech. His meaning might be, for example, that the absurdities to be derived from pluralism are greater than any of the absurdities claimed to follow from monism - and this could be so even if no absurdities actually are involved in monism. As to Barnes' fourth point, we should hold fire on this: one of the advantages claimed for our interpretation of Zeno's arguments will be that it does precisely show they can bear with force against only pluralism and not monism. In conclusion, then, we should not abandon the Platonic view of Zeno in face of Barnes' interesting recent points. I Reference is to page and line number of the Commentaria in A ristotelem Graeca edition. 8 This passage is also printed by H. D. P. Lee, Zeno of Elea (Cambridge, 1936), as ?2. Lee's text differs from that given by Diels in the Commentaria in Aristotelem Graeca edition. Lee omits 139.28f, XaVITLTa rCov jLepZ,v bx&7rpOV &)xe' xa;L T6rOu &Ei yEVO[LiVOV iX6v qov,. . . Lee also puts present tense 619O,uEVEL at Lee ?12.10, 17 (139.29, 140.3)

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where Diels has the future im%oveVi. We give the text as printed by Diels. Nothing of substance hangs on the differences. 9 See: Simplicius, in Phys. 140.21; Lee, Zeno of Elea p. 22; G. E. L. Owen, 'Zeno and the Mathematicians', in R. E. Allen and D. J. Furley, eds. Studies in Presocratic Philosophy, Volume II (London, 1975), p. 163, n. 10; Philoponus, indirect support due to the similar argument he reports from Zeno at in Phys. 80.23-81.7. 10 I should thank Jonathan Barnes for emphasising the importance of this question to me in private correspondence, and for mentioning these alternatives as possibilities. 1 Compare the remark by Vlastos, 'Raven's Pythagoreans and Eleatics', in Allen and Furley, op.cit. Volume II, p. 176, on Melissus: '.. . the comparison with the Zenonian fragment would suffice to show that for an Eleatic the possession of spatial extension would of itself entail infinite divisibility. How then could Melissus have supposed that his Being was both spatially extended and indivisible?' It is such a view of the relation of spatial magnitude to divisibility that we will oppose. 12 Zeno of Elea, p. 23. 13 According to Melissusr6 T yO os ii.rrWEipOV, DK 30 B 3: &XX' (3u?rEp 9o'TV &El, 0rS xvi TO

gryEOos .ant?pov &Ei yph Etvat. According to Parmenides, like the bulk of a well rounded ball, DK 28 B8.42ff: avtT&p &'TTEL 'rpas iTTUaiov, 'rETEXE%jsvOv Eai 'Tdt vTOOEV, tv,xuXXoV

acpaip-s kvaxiyxtov oyx. 6EOO6V 'ao'TXis 'rrv'r. But whether infinite or not, there is agreement that TO Ov has pk'ye0os and is &&LaiPETOV. 14 See first Liddell, Scott, Jones. They give xot;L yap as 'for also, for in fact'; with strengthening &v] as 'for of a surety'. See then Denniston, The Greek Particles (Oxford, 2nd Edition 1954) on y&p i including notes on ov y&p 8i and xvi y&p 8A. We can see the sense of xcxi 'y&p bi8 as emphatic in Plato: see Theaetetus 203b2, Hippias Major 289c7, 7th Letter 338b5. For xaL yap & in Aristotle, see for emample Physics B2 194a 1 5, Politics r 13 1283b30. Denniston notes an occurrence in Lysias which is emphatic, Lysias 28.3 (Against Ergocles). There seems no reason to think that xvi yap 8A changed its meaning by the Neoplatonic period. The difficulty here with providing textual evidence is in happening upon it. The related ov' yap 8i, as emphatic to make a negative point, is found: Porphyry, Sententiae Ch.33, Teubner 37.21; ibid Ch.40, Teubner 48.8; lamblichus, de Communi Mathematica Scientia Ch.8, Teubner 34.4; lamblichus, Protrepticus Ch.5, Teubner 35.22. oV yap & emphasises a negative point, which reinforces what has gone before. Had the point to be made not been negative, then xai yap &A would surely have been a suitable choice to get over the same force. In view of this, the translation 'for certainly' given later in the paper seems justified. Other possibilities are 'for of course', 'for indeed', 'for without doubt', 'for in fact' and 'for what's more'.

I owe particular thanks to Nicholas Denyer for his help on this matter, although he may disagree with the way 1 have put things. 15 Compare DK 28 B8.5f: klmi vvv h'riv b 6jo ffav, Ev, avvexis. 16 Putting the argument in trilemmatic form is of use in bringing out the logic of the Eleatic/Pluralist debate. As regards the division of any extended thing there are just three possibilities: it can be divided either (a) nowhere, (b) at only some places, (c) at every place. The denial of any two of these entails acceptance of the third; the relation of Eleatics, their contemporary pluralist opponents and later Atomists can be enlighteningly put in these terms. Briefly: the Eleatics denied (b), (c) - hence upheld (a). Their contemporary pluralist opponents sought to deny (a) of the whole, and uphold (b); Eleatic argument against them, on the basis of homogeneity, is that (b) cannot be defended, so that to deny (a) is to uphold (c) - but (c) is taken to be absurd. Later the

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Atomists deny homogeneity by introducing the void. Then of the whole of what there is, or of some arbitrary compound body, the Atomists deny (a) and (c) - and their intro- duction of void allows them to resist Eleatic arguments forcing them to (c); so, they can uphold (b). Of those entities that contain no void, the atoms, they are in the same position as the Eleatics; they deny (b) and (c), and so uphold (a). Notice that common to each approach is the denial of (c), which was thought absurd; it was the problems raised by (c) that were later tackled by Aristotle. 17 An Aristotelian criticism would follow on just this point, the supposition that what is everywhere divisible may actually be divided everywhere (at once). 18 As the Parmenidean parallel at B8.22, oVO? LaLpLET6V EaTLV, i'rEi 'ir&v knTLV po-Lov, might suggest. 19 Themistius, in Aristotelis Physica, 12.1-3; reference is to page and line number of the Commentaria in A ristotelem Graeca edition. 20 in Phys. 80.23-81.7; reprinted by Lee, Zeno of Elea, as ?3. The text is as follows. Reference in the body of the article is to the page and line of Lee's text, which for convenience are indicated here. 12.21 ZAv.v Si 6 TOUTOV FLOnT'S auvIVVopCv 'TCj 8L ax6Xw\XnTECX tEVOVbTLXai EV T7 OV xa;

&XCV'qTOV it &VcyXTS, TODT(a bE X(KTEOFEXatEV EX TfS &TT TiElpOV Tr.:V UJVEY;)V BLXO-

14.1 TOjUaS. Ei yap L1 Ev E'E T7 'ov Xvi &8LaLpEToV, &XX43 8aLpOLTO EiS 'fXELOVa, OV8EV ELTCL

XVpWS 'E'V (El yap 8LUtpOLTO TO GVVEXES, 6T' aITELpOV v eaE &aLpEToV), Et & ,x986v torTt

xvpiws Ev, oVO& r6XXa, E'; yE Tr iroXXc ix 'floXX(v ivct&.v GvyXedTaL. &bUVTOV apa 14.5 ELS noWXX& bLQLpE7O0aL TO ov. p.6vs apa ev ETLV. 1' Ov'TS. EiL pL1 To EV E'L;, ;, XaL

&&LaipETov, ovS& 'oXX&a C`TTXL. Ta y&p SoAX& 'EX 'rOrXXaV iV&6(8V. E'Xc&OrT OZ,V iv&s

ITTOL LOt iLX i xvTi Xa8bLoipETo-s, 1 xaL avcTl ELiS noTXXa 8LcEIpELT(XL. Et ~LEV OVV FLL &YTI XaLi

&8&cEpETos ixaqTI "oVas, kE aToTLLV PLEyEOoV To IT&VE( TaL, Ei be xvi avcaL &LaLpOivTXaL, 14.10 'rrcXLv ffEpL EXtoTIS T3V bL(aLpov1vL v "v6otv cI EVC'rievooji 0(c T 6t.Ta. XL TO7TO EIT'

(lXTrELpOV. (aTE &TTE?LpaXtS O'TTELpOV LOTaM TO ITOv, EL SoXX& Eiu Ta oVT(X. ?L bi ToDro &TOIrOV, [6vc,,s apa( 'V TO 0V, XC(i SAXoXa ELVOL Ta OVT(a OUX OLOV TE. ex0LaTIqv yap "OV&Sa

14.14 &M?ELpXMS TE?LdV &vyryxq, 8IEp &a7TO'OV. 21 On xaLi y&p oi as emphatic, see note 14. 22 Compare Aristotle, on TO Zivwvos &akta at Metaphysics B4, 100 1b7-13; see too DK 29 B 2. 23 Proclus, DK 29 A 15, mentions there being forty Zenonian X&yoL. Elias, ibid, also mentions there being five arguments against motion. Suidas, DK 29 A 2, mentions four titles of books, but this is probably not to be trusted. 24 Of the four Zenonian B passages given by Diels-Kranz, one (B4) concerning motion is mentioned in passing by Diogenes; and Diogenes may well be in error in his attribution. B I-B3, all concerning plurality, are preserved by Simplicius, in under thirty lines of text, spread over less than two pages of the Commentaria in Aristotelem Graeca edition. Another fragment, B5 (Diels-Kranz I p. 498 = Simplicius in Phys. 562. 1ff = Lee ? 15) is taken from Simplicius, and is an argument against the existence of place. Simplicius appears to have had access to a book by Zeno (in Phys. 140.27f). He quotes the passages B I - B3 to confute the Alexander-Eudemus tradition that Zeno argued also against the One. Simplicius is considering all this in the first place because he is commenting on Aristotle's remark at Phys. A4, 187a l: EVLOL be iviboarv TOlS XOoyS &pLopTepots. 25 See especially footnote 6. 26 He attributes it to those who hold TO Ov it &V6(yXjS eV ElVaL XEaL xvXLVTaTov (325a4f); he

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later mentions (325a27) oL Tr Ev xaTaaxEvaCovTFs. Clearly enough, it is the Eleatics who are referred to. 27 See references at note 9. 28 In the passages quoted by Lee as ?1, from Themistius, and ?3 from Philoponus (the text given at note 20). 29 1 would like to thank Professor Timothy Smiley for the suggestion from which this paper grew. I have benefited greatly from discussions with Professor Elizabeth Anscombe, Mr. Myles Burnyeat, Mr. Nicholas Denyer, Dr. Jonathan Lear and Dr. David Sedley, and from the written comments of Mr. Jonathan Barnes and Mr. Myles Burnyeat. I would like to extend my grateful thanks for all their help.

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