The Meaning of Δυναμiς at Timaeus 31c

7/30/2019 The Meaning of i at Timaeus 31c

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The Meaning of i at Timaeus 31c

PAUL PRITCHARD

1. The inadequacy of current interpretations

The argument from 31b4-32c2purports to prove, byan elaborate deduction,that the world of becoming is composed of just four simple bodies which standto one another in continuousproportion, therebyforming a complete andindissoluble whole. This 'deduction'is in fact anextraordinary tissue of mathe-matical sophistry. Or, to be fairer toPlato, we should recallthat he hasalreadyannounced (29c) that in matters suchas these, the account tobe given will bebut a likeness of a scientific'account, justas its subject matter is a mere likenessof those things which can be thesubjects of a scientific account. Here the

imitation becomes aparody of scientificdiscourse, so perhaps we should allowfor an element of playfulness. _Part of thisargument is a rather convolutedstatement about the termsof a

continuous proportion, which can be rendered as follows:

For if there is a meanamongthreethings,such that as the first is to themean,so themeanis to thelast,andagaininreverse,as the lastis tothemean,so the mean is tothefirst;then the meanbecomes both first andlast,and the last and first alsoappearin the middle.

The point is in fact asimple one. In a three-term continuedproportion we haveF :M : : M :L(e.g. 2:6 ::6:18),

and in reverse

L:M :: M:F(18 :6 : :6:2).

It will also be true that

M:L :: F:M(6:18 :: 2:6).

This ispresumably what is meantby "the mean becomes bothfirst andlast, andthe last and first alsoappear in the middle." Thismuch isuncontroversial. Whatis problematic is the nature of the terms mentionedby Plato in the phrase.


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onozav yap aQL8f..lwvTQLMVFtTc oyxwv EL'tE8vva?.?wv Evlwwvobv fi TOgoov ... That this is a matter fordispute will be clear fromthe followingattempted translations.1. (Archer-Hind 1886)'For when of threenumbers, whether expressing three

or two dimensions, one is a mean...'2. (Taylor 1929) 'For when of threeintegers, or volumes, or characters, the

midmost is...'z3. (Cornford 1937)'For whenever, of three numbers, the middleone between

any two that are either solids(cubes?) or squares ...'4. (Bury 1942)'For whenever the middle term of any three numbers, cubic or

square ...'3

The root of theproblem lies in the meaning of the term variouslyrendered as'two-dimensional number' (Archer-Hind), 'character' (Taylor),and 'square number' (Cornford, Bury).We also have threeviews as tothesyntax. These are:(a) 'Whenever there is a mean among three X's, which are either Y's or Z's'

(Archer-Hind, Bury),(b) 'Whenever of three X's, there is a mean between two which areY's orZ's'

(Cornford).(c) 'Whenever there is a mean among three X's or Y'sor Z's' (Taylor).

2. Thefirst syntacticalalternative

Whichever view wetake of thesyntax, we are here dealing with a three-termgeometrical proportion, the middle term of which is thegeometric mean of theother two. Forexample, each of thefollowing is a geometric proportion:

A(1) 4:6 :: 6:9

(2) 6:12 :: 12:24(3) BD:AD :: AD:DC(see figure)

The thirdexample showsthat theterms need not be numerical. Butif we adoptviews(a) or (b) of the syntax, then the terms must be numerical.Further, underview (a) all three terms must be either6YXOLor 8UVdt[LF-Lg.Consequently these


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words must refer to kinds of arithmoi.What might these be? Archer-Hind'snote on thepassage reads:

oyxogis a solidbody,here a numbercomposed

of threefactors,soas torepresentthreedimensions.6bvapigis the technical term for asquare,or sometimes asquare

root; cf. Theaetetus148A;and herestandsfor a numbercomposedof two factorsandrepresentingtwo dimensions.Thisinterpretationof theterms seemsto me theonlyone atall appositeto thepresent passage.

What we are being asked tobelieve is that Plato is hereruling out arithmoiwhichare not two- orthree-dimensional - but Archer-Hind does not tell uswhyPlato should want to dothis, or why this interpretation is 'apposite to thepresent passage.'

We can, however, fill outthe argument byconsulting SirThomas Heath, whowas in fact aslightly youngermember of Archer-Hind'scollege, and presum-ably discussed these matters with him. Liddell & Scottacknowledge Heath'scontribution to the lexicon. It must have been Heathwho wasresponsible forthe following entryunder 'V . I . b :square number PI Ti 32a.' Theonlyreference given for this meaning of the term isprecisely that which we areconsidering. Elsewhere Heath makes his case as follows:

It is true thatsimilarplaneand solid numbers have the sameproperty [sc.of havinga meanproportionalnumber](Eucl.VIII.18,19);but, if Plato had meant similarplaneandsolid numbersgenerally,I think it would have beennecessaryto specifythat theywere'similar', whereas,seeingthat the Timaeus is as a whole concernedwithregular figures,there isnothingunnatural inallowing 'regular'or 'equilateral'to be understood. FurtherPlatospeaksfirst of 6vvMpEigand 6y%otand then of 'planes' (ejtLJte6a)and 'solids'(oleped)in sucha wayas tosuggestthat 6vv6pEigcorrespondto aiaE6a andoyxovto aiepea. Now theregularmeaningof 6Ovapigis square (orsometimessquare root), and I think it is hereused in the sense of square,notwithstandingthat Plato seems tospeakof threesquaresin continuedproportion,whereas,ingeneral,the meanbetween twosquaresas extremes wouldnot besquarebut oblong.And, if 6vvMpEigare

squares,it is reasonable tosuppose

that oyxoLare alsoequilateral,i.e. the'solids' are cubes.'

We note that Heath's interpretation of oyxog is entirely determined by hisinterpretation of He is led to this view because he thinks that 'theallusion must be to the theorems established inEucl. VIII . l l , 12,that betweenany two square numbers there are two meanproportional numbers."

Now Heath himself notices twodifficulties with this:(i) It is clear that, though the extreme termsmight be square arithmoi, themiddle term need not besquare (see myexample (1) )6 .


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(ii) Further, it is notactually necessarythat any of the termsbe square - all thatis required is that the extreme terms be similar arithmoi(i.e. that their factorsshould be in the sameratio). This caseis illustrated by myexample (2). So whyshould Plato mention square arithmoi at all?

And there is anotherobjection: The existence of a meanproportional arithmosbetween two square arithmoi might excuse themention of squares; but still thereference to cube arithmoi would bequite without motivation. Heath is un-worried by this; he is satisfied by the fact that between any two cube arithmoithere are two meanproportionals. But Plato istalking about a three-termproportion - the later mention at 32a7ff. of thenecessity of finding two meanproportionals to 'bind' twosolids isreally not to the point here. Why shouldPlato gratuitously confuse the readerby expectinghim toguesswhat he isgoingto say next? Further, if in the passage under consideration referenceis beingmade solely to arithmoi, the transition toexamples in the realmof continuousmagnitude is not made more smoothby its inappropriate anticipationsevenlines above. Platois surely a better writer thanthis wouldsuggest. Perhaps theintroduction of cubes is an(unfortunate) afterthought -but then Plato wouldsurelyhave written 'threearithmoi, whether square or cube' andnot, as he does'three arithmoiwhether cubeor square'.

3. The secondsyntactical alternative

Cornford adopts the secondsyntactical alternative to deal with the firstof theseobjections. He says:

The objection ... can be obviatedby construingthe genitivesEL1:E6yxwvEirE8vvawEwvciwzvvwvovvnot(asiscommonly done)as inappositionto butdependingon 1:0J.1(Jov.7

He also mentionsthe third syntactical alternative, sayingGrammatically,the words can. beconstrued:(1) 'Whenever of threenumbers,whether solid orsquare,the middle one issuch...' or (2) 'Whenever of anythreenumbers or solids orsquaresthe middle one issuch...' taking 'numbers'to meannumbers that are neithersquaresnor solids.g

But Heath's interpretation of oyxo5and 6uvauic;leads him toreject the secondpossibility:

This interpretationof the ambiguouswordsoyxoLand 6vvMpEigas 'cubes' and

'squares'seems to bebetter supportedthan anyother. It rules out thenotion thatyxmand 6uv6t4F-Lgare alternativesto arithmoi.9


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So Cornford rejects this interpretation on account of hisfavoured interpreta-tion of 6-6vaRLgand oyxog. His argument for the latter term'smeaning cubenumber relieson a commentby Simpliciusthat Eudemussubstituted xupog for

oyxogin his accountof Zeno's `moving rows' argument.This hardly amounts toproof, and it is clear that his real motive is the same as Heath's -if b6vapigmeans square arithmos, then 6yxog must mean cubearithmos.1OIf it can beshown that b6vapig cannot meanthis, we need not bother with thesuggestionthat oyxogcan mean cube arithmos.

4. dvva,ucs as a mathematical term

Aristotle gives us an account of this term in his'philosophical lexicon' the fifthbook of the Metaphysics. There is no mention of any use to mean squarearithmos, though he does tell us that'66vapig used ingeometry is so calledbyan extension of meaning.'1' The only mathematical sense of b6vapig whichAristotle acknowledges is applicable to geometry.

We find anexample of this geometrical use at Tim. 54b5:

The other[triangle] havingits longersidetriplethe other in respectof

This means that thetwo lineshave the same ratio asthe sidesof twosquares

oneof which has three timesthe area of the other.

5. dvva?ccs in Diophantus

First, the term arithmos inDiophantus is unique to him anddoes not matchEuclid's use.Diophantus uses this term to denote the 'unknownquantity',defining it ashaving in it an indeterminate multitude of units(J[?80uovd6


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WithDiophantus,however,it is notany square [sc.whichis called6uvauig],butonlythesquareof theunknown;where hespeaksof any particular squarenumber,it is isz?aywvo5

This isprecisely the Euclidean usage. So Diophantus does not use theterm8vva?.t5 in a way which could provide a parallel for its use at Tim. 31.

6. The thirdsyntactical alternative

The failure tofind adequate parallelsfor the useof b6vapig to mean squarearithmos directs us back to the source of thisputative interpretation. For it wasthe adoption of one of the first twosyntactical alternatives which led to thesearch for an arithmetical referencefor the terms

b6vapigand

oyxog.If instead

we adopt the third alternative, then the terms need not refer tokinds of arithmoi. Indeed, they need not be mathematical terms at all. All that isrequired is that they should refer toquantities which can standin a continuousproportion. And thatwill betrue of any sets of quantities of the same kind -forexample time intervals, weights, line segments etc.

We can be certain that itispossible to construe in thisway, for this is thewayProclus construes - indeed, he considers no other alternative.In his com-mentary on this passage he writes:

Sincethere are three means(orproportions), 13arithmetic,geometricandharmon-ic,andthesebeingsuch as we havesaid,it is reasonablefor Plato totakethese threeto be the terms ineach case - 6yxoiand For the arithmeticalmeanisin QL8!!OL,the geometricalmean rather incontinuousmagnitude,and theharmonic meanis in

6vvMpEig


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Clearly he takes oyxog and b6vapig to be alternatives to notpartic-ular kinds of Taylor adopts this syntax, and provides us with goodPlatonic parallels. He argues:

The EvEis suppressedbefore thefirst of the alternatives. Forparallels cf.Sophistes217el &noR71x-6vFtvh6yov auxvov xai' ilAaUT6V,EIIExai 3TQ6gl;zEeov,ib. 224e2xaaqhix6v Elzs aiJ1:oJt(oLXV.The effect of thesuppressionisto throwspecialstress onthe firstalternativeas thatwhichischiefly contempated,'three integers,or - forthe matter of that, three 6yxoLor 'to deliver alongdiscourse tomyself,or, as itmaybe, to a companion', 'retailingthe wares of others,or possiblysellinghis own manufactures'."

I believeTaylor isright to follow Proclus thus far.Unfortunately he goes furtherand endorses Proclus'

suggestion"that Plato hasin mind three different kinds

of mean:

But what areyxm and 6vvMpEig?The explanationof Proclus ispretty clearlyright.He takes6yxoito mean,as itusuallydoes, 'bulks','volumes'(soHeraclidesPonticus is said to have called the 'molecules' of hiscorpuscular theory6vac)[totyxm 'uncompoundedvolumes'. As Heraclides is known to havegiven someaccountof the theoriesof thePythagoreanEcphantus,it is notimpossiblethat thephrase may actually belongto Ecphantusand so bePythagorean.... ) What the6vvMpEigmean Proclusexplains by sayingthat the'high' and 'low' inmusical

pitchare an instance of such a and that the interval of a fourth is a

'mean' (paov)betweenthe 'high'and 'low' extremes of the octave. ThusTimaeuswill beillustratinghisgeneral propositionabout 'means'byanexampletaken from


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each of the threePythagoreanstudies,arithmetic geometry (oyxwv),- music(8uv6tREwv).

But the three means arefoundequally

inarithmetic, geometry

andmusic,

so toconnect each mean with its homonymous studywould be misleading andgratuitous. Also, there is no reason to believe that Plato hasany but thegeometric mean in mind here, in spite of the use of all three means in theconstruction of the world-soul a little later.For, immediately after the sentencewe are discussing, he continues:

If thebodyof the universe needed to beplane, havingnothirddimension,thenonemean would have beenenoughto bindtogetheritself and those with it- but as itisthe universe must bethree-dimensional,and solid bodies are joined togetheralwaysby twomeans,neverby one.

The reference here must beto the discovery by Hippocratesof Chios that theproblem of doubling the cube reduces to thediscovery of twomean proportion-als between sides of length one and two units. This is the three-dimensionalanalogue of the doubling of the square, which is solvedby finding one mean

proportional between sides of length one and two units. Thesemeans are, of course, geometric means.

Other Platonic pronouncements lead to the same conclusion.In the Protag-oras we are toldthat communities of men will not holdtogether unless each manhas asense of justice and respect for others, which 'bring order into ourcitiesand create a bond of friendship and union. What is true of thecity also appliesto the universe as a whole, as we are toldin the Gorgias:

Thesagessay,Callicles,that heaven and earth andgodsandmen areheldtogether

bypartnershipand

friendshipand moderation and

justice,and because of this

theycall the whole 'order'(cosmos),myfriend,not disorder and licentiousness.Youseem tome notto attend to thesematters,thoughyouare clever-ithasescapedyouthat geometricalproportionhasgreat poweramongthe godsandamongmen,andyouthinkone musttrytogetmore than a fair share. Thisis becauseyouareignorantof geometry.'8

We find that Platonic justice is clearly a matter of geometric proportion;influence inthe state is to be allotted so thatpower is exercised inproportion tothe intellectual and moralworth of each person or class, while the numbers in

each class are ininverse proportion to their power. In such a state we shall findthe unity of order and friendship, and it is the same withany stable whole,including the entire sensible universe.19


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If this isnot enough, consider Plato's words as hecontinues theargument inthe Timaeus - the foursimple bodies are to one another in thesame ratiozov affrav X6yov). There isabsolutely no reason why any reference toany

other mean than thegeometric should be seen here. With thisout of theway,why should we look fora musical meaning for b6vapig? As ithappens, a muchsimpler and more plausible interpretation is available.

7. What bvva,ucs reallymeans

We are looking for a reference tothings which (i) are magnitudes of the samekind (since they must becapable of appearing in a continuousproportion, andhence havesome ratio with oneanother),2and which (ii) Plato would requireto be inproportion in his cosmos.In his discussion of the infinite Aristotle writes:

For if thepower in onesimplebodyisexceededbythat of anotherbyasmuchas you like - for instanceif fire is finitein quantityandair infinite,and anequalamount of fire is in itspower anymultiplewhatever(so longasit isa finite

multiple)of thepowerof anequalamount of air, nevertheless it is clear that theinfinitequantitywill overcome anddestroythe finite.2'

We see that the'powers' of the simple bodies can be said to haveratios with oneanother. Thusthey satisfyour first criterion. As for thesecond, we find that thismeaning for b6vapig is in fact the mostfrequent in the Timaeus. Threeparticular cases will show that it would beappropriate in this sense at 32al also.At 33a3we find

When heatandcold andanythingwhich hasstrong powers (6vvMpEig)surroundsacomposite bodyand fallsuponit, thisdissolves thecompositebefore itstime,andbybringingon disease and oldageit causes it todecay.


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This iswhy no powers are to be left outside the cosmos.But how are thesepowers to be prevented from tearing apart the cosmos from the inside? Thefollowing is part of Plato's description of the pre-cosmic state, before the

demiurge has put his plan into effect (52el):

[thenurse of becoming]seemed to take on allkinds of forms,andbecauseshewasfilled withpowers (6vvMpEig)whichwere neitheruniform norbalanced,shewas inno sort of self-equilibrium,but, buffetedunevenlyin every part,she was shakenbythese [powers]and, movingherself,she shookthesein turn.

We see how this is dealt withat 56c3:

Andwe must alsosupposethat, withregardto the proportionswhichholdamongthe multitudes,the motionsand the otherpowers [of the simple bodies],that thegod, insofar as the natureof necessity yielded willinglyto persuasionand [theseproportions]werebroughtto accurateperfection byhim,articulated these[viz.themultitudes,motions andpowers] accordingto geometric proportion.

So this sense for b6vapig at 32a would beentirely appropriate. 22We do nothave to accusePlato of sloppy writing,as we would haveto if it were takenin thesense 'square arithmos'. Further, we do not have topostulate a unique use of this term. (This should alwaysbe alast, desperate resort.)The meaning 'power'makes perfectly good sense here.23


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8. Themeaning of oyxosOnce the interpretation of b6vapig as 'square arithmos' isabandoned, the term

6y%og presentsno

problem.Consider this

parallelat Theaet. 155a3:

Nothingcan everbecomelargeror smaller either in bulk(oyxw)or in numberas longas it remainsequalto itself.

Here wefind oyxog as an alternative to Wecan see oyxog used incontrast tob6vapig in Aristotle:"

For, althoughsmall inbulk(oyxw),inpower and worthit farexceeds theothers.

Our text shouldtherefore be read:'Whenever among three arithmoi, or (forthat matter) three bulks orthree powers, one isa mean ...' Oneadvantage of this reading is that weneed notpostulate a unique sense for any of its terms.Neither need we find Platoguilty of incompetent writing. Thirdly,this in-terpretation is entirely appropriate to Plato's wider concerns both in the Tim-aeus itself andelsewhere. The fact that theargument in which thisphrase isembedded is apiece of (playful) nonsense should not lead usto suppose that it isunskilfully handled. It would be madnessnot to findmethod in it. 25


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Works Cited

Burnyeat, M.F., 'The Philosophical Sense of Theaetetus' Mathematics', Isis 69

(1978)489-513.

Cornford, F.M., Timaeus. Plato'sCosmology (London1937).Heath, T.L. (1), The Thirteen Booksof Euclid's Elements(Dover 1956)(2) Diophantus of Alexandria:A Study in theHistory Of Greek Algebra (Dover

1964) .Lee, H.D.P., Plato: Timaeus andCritias (London 1956).Mugler, C., Dictionnaire Historique de la Terminologie Geometrique des Grecs

(Klincksieck 1958-9).Rivaux, A.,Plato OeuvresCompltes Vol. X (1925) (ed.J. Souilh).

Souilh, J., Etude sur le Term i (Paris 1919).Taylor, A.E., A Commentary on Plato's Timaeus (Oxford 1928).Tracy, T.J., Physiological Theoryand the Doctrineof the Meanin Plato and

Aristotle (The Hague 1969).Vlastos, G., 'Isonomia',American Journalof Philology LXXIV (1953)337-66.

Centrefor Ancient PhilosophyUniversityof Bristol.

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The Meaning of Δυναμiς at Timaeus 31c

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