Plato's Description of Division Author(s): A. C. Lloyd Reviewed work(s): Source: The Classical Quarterly, New Series, Vol. 2, No. 1/2 (Jan. - Apr., 1952), pp. 105-112 Published by: Cambridge University Press on behalf of The Classical Association Stable URL: http://www.jstor.org/stable/636866 . Accessed: 12/03/2013 07:25 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Cambridge University Press and The Classical Association are collaborating with JSTOR to digitize, preserve and extend access to The Classical Quarterly. http://www.jstor.org
Author(s): A. C. LloydReviewed work(s):Source: The Classical Quarterly, New Series, Vol. 2, No. 1/2 (Jan. - Apr., 1952), pp. 105-112Published by: Cambridge University Press on behalf of The Classical Association
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Accessed: 12/03/2013 07:25
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THEREare many passagesin Plato which look as if they alluded to well-worn
practices, discussions,or lessons in the Academy. As is naturalwith allusions,they are often markedby a puzzling brevityor oddity of expression.One neednot assume that they are always conscious allusions; for every writer hasmoments of obscuritywhich are due not so much to his conclusionsas to his
reaching them along lines that have long been familiar to him. To appreciatehis whole meaning the readerhas then to infer as best he can the writer'strainof thought. I wish to suggestthat the language in which Dialectic is describedin the laterDialoguespresupposesa particularand probablyfamiliarmethod of
illustrating it. This was a geometrical illustration of the rules of Division bymeans of a divided line. By failing to notice it readershave not been led into
any important misunderstandingof the Academy's rules. But I hope it will
appear that the recognitionof it makes Plato's manner of describingDivision
intelligible to an extent that is otherwise difficult. It is only a tentative sug-gestion, and would perhapsnot have been worth makingbut for the possibilitythat some points of interestmight at the same time emerge for thosewho wereunconvinced by it.
So much only is the direct intention of this article. But if the suggestionis
correct,it has also in my view an indirectimportance.For Plato was fascinated
by the mathematical puzzles of infinite divisibility.And by the time he wrotethe Parmenides e considered (I believe) that Zeno's paradoxes indicated asolution of his own paradoxesabout the One and the Many. I do not want todefend this suggestion here: but its upshot may be put very roughly and
dogmatically in the following equations. The Many = the Indeterminate =
the infinite,.e. indefinite,numberof partsof a whole. The 'Ones' or species =
thefiniteor determinate number of parts into which a whole or genus must bedivisible if it is to be an actual whole. For in mathematics magnitudes are
infinitelydivisible: but such magnitudesare only abstractions;and in realitythere are always indivisibleparts.Andjust as mathematicalobjectsare imagesof the real, so the infinite Many are only appearances (due to inadequatedivision) of the One. Both hornsof Zeno's dilemmaaregrasped 2 one accountsfor the intelligible, the other for the sensible.Being is shot through with Not-
being (or Otherness or Matter): but the firstforms a plurality, the second an
infinity.3Dialectic meant always the discoveryof the One in the Many, andin the later Dialogues this consisted of Collection and Division.
In my opinion, then, the illustrationof Dialectic by the division of a line into
parts would be a natural result of Plato's great imaginative feat-his theorythat there was (as we should say now) identity of logical structurebetweenZeno's continuous magnitude and the world itself as an object of experienceand knowledge. And the choice of illustration would help to confirm the
I This is not to be contradicted byAristotle's statement (Met. A 992a22; cf.
M Io84b I-2) that Plato believed in indivisiblelines. For there is more than one sort ofmathematics according to Plato. Parm.
164 c-165 d, where magnitudes are infinitely
divisible, applies to 'popular' as opposed to
'philosophical' mathematics (v. Phil. 56 d-e;Rep. vii. 525 d-526 a). I hope to offer an
explanation of this on another occasion.
2 Cf. Parm. 142 c 7-145 a 2.
3 Soph. 255 e, 256 e-257 a.
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- aVIy7TELPOV ,V KaOo E770V K( T T8EPOVOOLOUEL TOv
0o"o00V tv, iTrws
avTxwrtf',
KLaL7roAAa2TTOVo
KacL flpacV'-rEpov 7rLO
ctOtTO)
3EOV70S,LE7cE EV~L7TEL3Pa
/O8;a, o E aa aVoS
KOET'YEL ....
We are to imagine, as it might be drawn or composed of pebbles on the
ground, a line AB of unknown length. This is TrodrrELpov,hough not because
it is infinitely divisible but because it is not known of how many divisions it is
capable.
(i) We place provisionally' between A and B a point C, thereby 'finding' in
AB a line CB (the tdav18&av). 2) CB is divided at D, giving us CD and DB
(iEr& lar'vo). (3) CD and DB are similarly divided at E and F respectively.
A C E D F B
(4) The divisions of CE, ED, DF, FB are continued as far as necessary, i.e
until it is seen how many indivisible lines there are in CB(Tr Ka7' apxacs EV,
thegeneric Idea). (This
is not of course derivable from thediagram;
in
Dialectic it involves Collection. We shall suppose the process already com-
pleted.) This is equivalent to refraining from considering CB (Tr3r•AiOo0o)
as
infinitely divisible (T7rvT70o L7TElpOV''av7TdaTpoEpEvw) before the exact number
of its components is known.' 7rr•Aioss a word which Plato uses when he means
it to be undetermined whether a magnitude is a7TrrEpovr 7TE7TEpacJLuEvovcf.i8 b 2). The number of lines, viz. CD, (DB,) CE, ED, DF, will be seen to be
between (tEtra7ef)AC, which is the remaining still divisible part (o70v0rrapov)of the original line, and the last indivisible line, FB (To70 vods).The objection to
this is that To70 VdS6as a different denotation both from that of KaT'dpx3.
Evand from
rE"Vn
(6), which,we shall see
shortly,cannot be the lowest
species.
O tIEvovsrefers not merely to the diagram
but to the fact that in Dialectic all Ideas
start as hypotheses. For, incidentally, Plato
never said there is an Idea corresponding to
every general name, although this is now
attributed to him by writer after writer.
Rep. 596 a says Eld2OauEv -IOEaatU, 507 b
1rtOEvrE.Cf. Phaedr. 237 d i(dfoAoylta
OE'?1Evo
opov) for connexion with Socratic method.2 For the use of 'number' where we should
say 'number of parts' see Theaet. 204 d. But
it has also an esoteric meaning, as is men-
tioned below.
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One may reply (a) that if Plato visualized a diagram the difficulty would not
occur, since the 'one' in question could be pointed to; and (b) that (6) is a
new sentence in which the thought makes it quite natural to return to the
original 'one'. (There is, however, an alternative, which strains the expressiona little, but which would avoid the difficulty: the intermediate lines could be
conceived metaphorically as 'standing between' CB as a unit and the indefinite
divisibility of AB; for Plato is thinking as much of the understanding of the
genus [CB] as of the lowest species [FB].)
(5) The lines CD, CE, ED, DF are then ignored--or, what comes to the same
thing, we attend only to CB, DB, FB.
A C E D F B
CF becomes again indefinitely divisible and we lose CD, CE, ED, DF. Two
things are to be noticed in this. First, we are now left, as can be seen from the
diagram below, with the lowest differentia, the superordinate species and the
genus; and this gives us a definition of the lowest species according to the
Academy's rules. Secondly, this definition, which amounts to ignoring the
left-hand side of a division, amounts also to 'dismissing it into the d~rLpov';for the left-hand side contains those things which the contents of the right-handside (the defining characteristics) are not or are otherthan;' and the former, the
'others', are, in respect of their otherness, dirLpa.2
(6) Plato goes on to say that people erroneously reach the limit of divisiontoo quickly, i.e. do not put enough points between C and B, or too slowly, i.e.
put too many. In the first case they might, for example, omit DB by dividingCB only at F. (Thinking of Dialectic only as definition of the
at-ru•rovand of
7rEmLpanly as individuals, editors have often taken the
Evin (6) to be the
lowest species.3 But (a) tleaa would naturally mean between the Ev and the
11eepa,while if the former is the lowest species there are no such &lea; (b)-more conclusively-it is not 'eristic' but correct 'dialectic' to apply the notion
of dJ-IEpla after the lowest species has been reached. -r "v is therefore the genus,
CB.) Such division would be equivalent to 'bad' definition-in Aristotle's
example, to defining Man as Two-footed Animal instead of Two-footed
Footed Animal as Plato required.' This is at once seen if the scheme is repre-sented thus:
(AB)
AC GCBAnimal)
CD DB (Footed)
CE ED DF FB (Two-footed)
This is the more familiar representation, and no doubt it is the one that
Plato generally had in mind when he was making applications of SLaipEUL-.But when he is more concerned with
theory-andwith the structure of
realityas revealed by Division, rather than our piecemeal discovery of it-his languageis that of the Parmenides;and since the language there is appropriate to the
division of lines it is natural to expect it to be so here. Moreover it was
from this geometrical point of view that mutual implications were first seen
among the concepts of irrationality, indefinability, and infinite divisibility.zAnd the last forms the limiting case of the second possible error, the error of
completing the division too slowly. It is what is only prevented from happening,
according to Plato, by the fact that if we collected merely particulars even
these would have a share in Ideas.3 So Plato refers to the danger on this side-
represented by Protagoras' sensationalism-by describinghis
opponentsas
'crumbling' reality into fragments (6pLTrTELV).4 The Eleatics represented the
extreme case of the alternative error, for they allowed only a single 'One' to
be discovered. The via media, advocated by Plato, is that reality is, rather,
'chopped up' (KEKEptaILatILEVOV,KaTaKEKEptaTLaliEv).5 So that he exactlydescribes the attempt to treat changing particulars as reality by saying,
OpV`7rTEae 377L•CtLKEptaaTL1dtLEVOV
CyKr77trav o ov, o aCvTL•
7-j7r8tavo['
(Parm. 165 b 4-6).There is one point which presents some difficulty if we do not recognize the
suggested illustration. I hope, however, that a discussion may be helpful also
to those who do not accept my 'divided line'. In (5) Tr E'vEKcaaTov 7WV ,rdvust refer to all the unities, i.e. Ideas, which have emerged by Collection and
Division. For to take them as individuals6 is surely inadmissible when Ev is just' Ar. Met. Z 1o38ag9-25; De part. an.
6422b5-9.2 ov0 dpEdr' /rELpov iLalpEait,KELKatL
TObAoyov (Procl. in I Eucl., ed. Friedlein,p. 6o.15).
3 For this lesson in the Parmenidesf.
158 b 2-d 8; 164 c 7-d 8; 165a 5-c 3.4 Cf. Soph. 246 b 9-c I: the Idealists
its having Being that Not-being is 'choppedup'. The metaphor is, of course, explainedby its being used for the division of a genusinto species (Meno 79 a so; c 2; Pol. 266 a 2).
6 As does Stenzel, Studien z. Entw. d. plat.Dialektik2 (Leipzig-Berlin 1931) 104, at least
in Allan's interpretation ([Stenzel] Plato's
method of Dialectic, tr. and ed. D. J. Allan,
Oxford 1940, 146). Mr. Allan, who has very
kindly read my manuscript, suggests thatStenzel 'could say that 'v, which has justbeen used in the dialectician's sense, (4), isthen used as a man in the state of
,-a-r•rswould use it, (5), i.e. "those alleged unities" '.But how many readers would grasp thisfrom Plato's text ?
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what is being distinguished from drTELpov,nd when iw-cv 'v• E iaV( EKaT-rovad
denoted species in the previous sentence. Now we are told to dismiss them
into the drTeLpov.But why? All we should be left with is a genus and the
arithmetical number (say 4 or 6 or 8) of its species. And if it is important toknow how manyspecies there are, this can surely only be because we cannot do
this without knowing what they are. In what circumstances ought we then to
forget this latter knowledge? The purpose of Division here may be one or
more of the following possibilities: (a) to define or understand a genus; (b) to
define a lowest species; (c) to show how a genus (or any divisible Idea) is both
a Many and a One; (d) as part of an argument (e.g. to refute the thesis,'Government is necessary, Athenian democracy is government, therefore
Athenian democracy is necessary').But as for (a) we do not define or understand a genus by saying that it has
m species, where m is a mere number. In (b) to omit the superordinate speciesfrom the definition of a lowest species is un-Platonic, as we have seen; and
secondly, it is not obvious-except on my (first) interpretation of (4)-that we
shall have been left with even a lowest species. (c) at least must be admitted
here, because Plato explicitly says so (16 a-c). He is, for the purpose of this
dialogue, fitting his Dialectic into the Pythagorean formula in which i- v is
the first product ofrrpas and cd7rpov, and in which reality is generated by the
imposition of Number or-'-
durov n the ad'tpov. So once the species have been
enumerated the genus has been shown to be a Many, and we can return to its
unity. And it is to be added that the dismissal of the Many into the d'ITtpov will
show how the One and Many together are also diErrepa-aswe were told in (4)-not simply in respect of the lowest species, but all of them.' This Pythagoreanframework would explain the emphasis on the mere number of species, for 'the'
number has then an esoteric meaning (cf. especially 17 c i I-e 6).z (d) is an
aspect to which Hackforth has drawn attention. It too might explain the
'dismissal'. For in our example it is sufficient to know that democracy is a
species of government; and if one asks, 'Why the emphasis on the numberof
species?' it could be replied that until the whole division is completed it is
impossible to know that any single division was a 'real' one and therefore to be
admitted in a genuine, instead of an eristic, argument.3
(d) is not so important, I think, as (c). Nevertheless all four purposes arepresent to Plato's mind. For it is notable that when he goes on to illustrate the
method there is no hint of the 'dismissal'.4 With the possible exception of the
continuation (18 b 6 ff.) of the first one, it seems essential in the illustrations to
know not only how many the species are, but what they are. The improbabilityof both (c) and (d) as explanations could be supported by Politicus 285 a-b,where knowledge of all the species was necessary to an understanding both of
' Cf. L. Robin, Platon:euvres omplktes,i
(1942), 184 n. 20: 'une fois qu'on est arrive'a l'espace dernikre . . 'impossibilite de
"specifier" davantagenous met en
presence,et de l'individu, avec la multiplicit6 de sescaracteres singuliers, et du nombre infinid'individus auxquels s'6tend la notion de
I'espkce derniere, avec tout ce qu'elle im-
plique et qui constitue la chaine des inter-
m diaires.' But his translationc'est alors qued6sormais on doit abandonner l'infini et luidire adieu', is inexplicable.
2 Cf.J. Stenzel,Zahlu. Gestalt1924), 7,I3-18; A. Preiswerk,DasEinzelneb. Platonund Aristoteles', hilologus,uppl.-Bd.xxxii,Heft
I (1939), 55-56.3 Cf. ot vi3vrv &v corwvo o01 of (6)with the vioSof 15 d-e, who 'at one momentkneads any argument into one ball, thenunrolls it again and chops it into pieces'.Compare also Phaedr.237 d.
4 Nor elsewhere. (Pol. 286 e 6, despiteCampbell's note, has nothing to do with the
present point.)
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the lowest species and of the genus. Still more could it be supportedby thenext page (286 c-287 a). For there, in an apologyfor the length of preliminarydiscussions,?uaKpoAoyland /paXvAoylawere also requiredto be tested against
the rules of Division (rrp3)rovTTv/6OOoov v'Tr7VTt/LV 7TOKaT EL77-8vvaoTOVLvatS&atpErv).ut this criterionwas, in fact, the correctnessof the definition of thelowest species,viz. the 'royal art'. Indeed when a concrete applicationof therules was made (287 b ff.), a numberof specieswere dismissed',namely thoseon the left-hand side of the division.And these werejust the ones which wouldhave been dismissed n our suggestedinterpretationof the Philebus assage,butnot on the normal interpretation.The trouble is, in fact, that (a) and (b) as
explanations are incompatible with (c) and (d). However, this need not ruleout the normal interpretation.For one may concentrateon (c)and possibly(d),and say that both Socrates'examplesand the Politicusllustratea use of Division
beyondthat forwhich it is introducedin the Philebus.This further usewould beits more common one in the sciences,and one which had alreadybeen alludedto by Socratesin 16 c 1-3. An alternative,which I should prefer, is to claimthat Plato is trying to combine (a), (b), (c), and (d). For he has a habit of com-
bining two levels of thought--of teaching an immediately relevant lesson inmethod at the same time as an indirectlessonin metaphysics.And it is a habitwhich is at once idiosyncraticand rathervulnerable to strictlogic. To conclude,then, the 'dismissal into the drrELpovis readily intelligible for any of the
purposesof Division that may be intended, if the 'divided line' is agreed toillustrate it: but it does not contain sufficient difficulty to make it positive
evidence for the illustration.
The secondpassagewhichdescribes hemethodofDialectic is in Sophist 53 d:
isvnTaVt3v alls atis fogEK ntanc o qerEe-qtg EtVcvL; ... OVKOVV 0YE
7rTeVTi) oLntsV w7nVa A'KAVA selfVEc-sucet, KhcL a'oe s ntserepasosjAtwv o ojuic tao
eWOEv ITEPLEXOiLEmVL,2) tKatClava' t c•AthwvoAAC'v 'v•
4vvllusrMvyiqv, at
rroAAa'Xp's nTcav-ri&WptEcLEvas.
Especially since Stenzel's work' it has been widely agreed that the passagerefers to the method of Division. To supposeit merely describesfour differentkindsof relationbetweenIdeas,two of Communionand two ofnon-Communionis not at all satisfactory; for instance E'v
V• vvwit1t-iq[- is takenz to mean'remainingon its own' or 'self-sufficient',which does not seem to dojustice tothe Greek. It is much easier to follow if we think of the same illustrationof the
divided line-though, once again, only as influencingthe expression:it is not
necessaryto it, and if Plato had thought it necessaryhe would have made it
explicit. But this second passage is closer, of course, chronologically, to theParmenides.
1aI(sa (our CB) is drawn or stretchedacrossthe whole of the shorter lines
still units (Xwpi S&twptrlt'vas). 7EPLEXELV is the technical term for the relationofof a magnitude to its parts,' and is contrasted in Parmenides150 a with St' jAov
TETa•aEV7VELvaL,which is coextensiveness.2
wlOOEvalludes to the paradox of the
whole-part relation disclosed by Zeno (v. Parm. 145 b-e, and 149 d 8 ff.,specially 150 e 5-151 a) and well known in the Academy (v. Ar. Top. vi. xiii;cf. Phys. A 2IoaI6-I7).3 In (2) it is possible that tlav denotes the generic Idea
as it did in (I), and El the lowest species. But it is preferable to take them the
other way about. The emphasis of the first half will then be upon Collection,the emphasis of the second (marked by av') upon Division, and the two will
form respective explanations of ?TE-r7a-,r3v
ELoS~ETEPOV 47rarOaecLlprE ETEpov
av carodv. Ev Ev vlVv
qpEvl) is a variation from 63d tudis1S'a IEPtEXoIEV7)and
represents the passive of uvAAaJPWvs it was used at 250 b 9.... But the terms
he is employing belong so much to the logic of geometry that the Stranger
finds it necessary to explain that all this is the same thing as what the otherswill already have understood by 'Dialectic' !4
There is one other passage to be discussed; for unlike the previous two it
does, I believe, provide some positive evidence for our suggested illustration.
The Politicus has a puzzling remark in repeating the rule that dichotomy is to
be preferred but, failing that, division into three or more parts:
The usual view of the second sentence is stated byDies:.
'pour la division dansle nombre le plus proche, [cf.] Philebe 16 d: ,Eral lav (18dav)&)o, E' trTo ELUL,
UKOWTELV,l ts , TpELt&g7rtvaAAovJptOLdv.rincipe d'6conomie, ame detoute m6thode.' But the Philebus passage did not say that we should try the
lower number first. For the tEra t1av &8o had nothing to do with alternatives.
True, Plato regards dichotomy as preferable. But this appears to be for no
better reason than the attractiveness of rO'iErov.6 And once the division of
mankind by races into two parts, like Greeks and Barbarians, is seen to be
merely nominal (Pol. 262 d-e), it is unlikely to be an economical method to
proceed by trying three.
At the same time 'the nearest number' must denote the next number, sc. inthe number-series. (It cannot, for example, mean 'nearest to reality', nor, of
course, 'the nearest number' in the English sense of nearest the right one.) I
suggest that the expression contains a reference to the position of divisions in the
that pl'a 13'a stretches through from begin-ning to end.' Cf. Parm. 145 b 8.
2 This is not to reject C. Ritter's conten-tion (Neue Untersuchungen iiber Platon
(19i•o)57 ff.) that
Sta-E'aa0atrefers to the
/dyta-raydEvr, like Otherness which is ta 7rrwovrwvv
•tLEA•v0v7av,and EprEdLXEaOaLo the speciesof ordinary genera-though 250 b 8 is
against it.Plato's final interest, even in the Parm.
passages, is not (pace Cornford, Plato and
Parmenides, 79 ff.) in infinite divisibility, butin a whole, i.e. genus, which is, and yet ismore than, its parts. Cf. Theaet. 201 e-205 e
Pol. 262 b 6-7; 265 a 4. The reason isnotthat which a nominalist logic would give,viz. the exhaustiveness of a class-concept andits contradictory (although this doubtlessinfluenced him in practice, especially in the
Sophist), for a negative class is likely to be
JvEpov (cf. Pol. 262 d) ; cf. Ar. Met. A990bi3
and Ross ad loc. (Platonists' denial of Ideasof negations).
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