Menn Plato Analysis

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Plato and the Method of Analysis

Author(s): Stephen MennSource: Phronesis, Vol. 47, No. 3 (2002), pp. 193-223Published by: BRILLStable URL: http://www.jstor.org/stable/4182698

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Plato and the Methodof AnalysisSTEPHENMENN

ABSTRACrLate ancientPlatonists and Aristotelians describe the methodof reasoningto firstprinciplesas "analysis."This is a metaphorfrom geometricalpractice.How farback were philosophers akinggeometricanalysis as a model for philosophy,andwhat work did they mean this model to do? After giving a logical description ofanalysis in geometry, and arguing that the standard(not entirely accurate) lateancient logical descriptionof analysis was already familiar in the time of Platoand Aristotle, I argue that Plato, in the second geometricalpassage of the Meno(86e4-87b2), is taking analysis as a model for one kind of philosophicalreason-ing, and I explore the advantages and limits of this model for philosophicaldiscovery, and in particular or how first principles can be discovered, withoutcircularity,by argument.

Aristotle cites Plato as asking whether, at any given stage in an argument,"we areon the way to the principles or from the principles"(NE 1,4 1095a32-3). The word"principle,"&pXq,meansliterally"beginning."Greek philo-sophersuse the word to meanwhatever s priorto everythingelse. Thismight be straightforwardemporalpriority- for the pre-Socratics, he&pXai re whatever herewas before the ordereduniversewas formedoutof them- but Plato andAristotleextend theword to more abstract ensesof priority.Philosophersbefore Plato had assumed that the beginningofthingsis also the rightbeginning or ourargument r discourseabout thethings;Plato andAristotlearesaying, by contrast, hatthereare two stagesof argument, irst to the cpXai (contrary to the "natural" rder of thethings)andthenfromthedCpXaifollowingthe natural rder).Plato'spointis that,when we begin an argument,we are not immediately n a posi-tion to grasp the &pXai,but must somehow reason back to them fromsomethingmoreimmediately vident.As Aristotleputsit, we mustbeginwith "the things that are betterknown to us" and reason to "thethingsthat are better knownby nature"and are priorby nature: he goal is tomakethe thingsthat arebetterknownby naturealso betterknownto us,so that we can use them as a starting-point f argument o gain scientificknowledgeof the thingsthatare derivedfromthese apXai.? Koninklijke Brill NV, Leiden, 2002 Phronesis XLVII13Also available online - www.brill.nl


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Late ancient philosophers middle-and neo-Platonists,and the Peri-pateticAlexanderof Aphrodisias)are very interested n this processofreasoningback to firstprinciples.They call it "analysis,"and contrast twith "synthesis" s reasoningfrom he principles; nd something ike thisdescription of analytical reasoning remains familiar to us.' But thisdescriptions very problematic, ndmy goal here is to begin to get clearon the problems hatit involves.To describePlato's or Aristotle'sprocedure f reasoning o principlesas "analysis" s to apply a metaphor rom geometry:"analysis" s thename for a definite geometricalprocedure,and the neo-Platonistsareconscious thatthey are speakingmetaphoricallyn extendingthe term tophilosophy.The neo-Platonistswere (at their best) mathematicallywell-educatedpeople,and the studyof analysis(whatPappuscalls the &vaXvo-ievo0 t6iOo;) is the key to the non-elementary partof geometry, in Pappus'words "a special resourcethatwas prepared,after the discoveryof thecommonElements,for those who want to acquire n geometrya powerof solving problems et to them".2The neo-Platonistswantto claima con-nection betweenthis geometricalprocedure nd the philosophicalmethodof arguing o firstprinciples.Proclusactuallysays thatPlato "taught" eo-metrical analysis to the geometer Leodamas(In Euclidemp. 211, cp.p. 66), apparentlymplying hat Platoinvented he methodof analysisandpassedit on to the mathematicians.3ertainlyPlatodid not inventanaly-sis; there s good reasonto thinkthat the methodwas usedby Hippocratesof Chios as early as 430 BC; ProclusroutinelycreditsPlatowith invent-ing any mathematicaldea or proposition hat is mentioned n a Platonicdialogue,4and we need not take these ascriptions oo seriously.But it isworth thinkingabout why Proclus and other Platonists would want toclaim the methodof analysisfor Plato. The methodof analysishadenor-mousprestige, n antiquityand down to thedaysof DescartesandFermat,

AcceptedNovember 2001' Many of the relevant texts of late ancient philosophers(from Alcinous throughthe sixth centuryAD) are collected and discussed by Donald Morrisonin a work-in-progress,which I have used with profit.2 From the beginning of Pappus' Collection VII, Jones' translationmodified (textand translation rom Pappusof Alexandria,Book 7 of the Collection, editedwith trans-lationandcommentaryby AlexanderJones,2 vols., New York-Berlin-Heidelberg,986).3 Diogenes Laertius in his Life of Plato (DL III,24) attributesthe same reporttothe early second centuryAD Academic Favorinus;it must have become a common-place of the Platonic school.I As noted by Wilbur Knorr,The Evolution of the EuclideanElements(Dordrecht,1975), p. 6.


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PLATO AND THE METHOD OF ANALYSIS 195

because it was seen as the basic method of mathematicaldiscovery:notsimplya way for a student o discover and assimilatefor himself propo-sitionsalreadyknownto his teachers,but also a way for a maturegeome-ter to discover previously unknown propositions.While analysis is amethodwith clearrules for step-by-stepwork (though t is not a mecha-nical method the geometermustapplythe rulesintelligently n ordertosucceed), t terminateswhensomethingunpredictablyclicks";hen,if andwhen this happens, he geometermustagain proceedmethodically again,not mechanically)by the method of synthesis, to confirmwhat has beendiscoveredby analysis; f this succeeds,thenthe newly discoveredpropo-sitionmay be presentedwith a demonstrationn the usualhighly stylizedform given in the classic Greek mathematical exts. Since it is obviousthat the propositionsof, for instance,Euclid's Elements or Apollonius'Conics were not first discoveredby means of the demonstrationshat arenow used to justify them,it is natural o ask how else they were discov-ered;and in manycases it is natural o suspectthat they were first dis-coveredby analysis.Indeed,Archimedesactuallygives us the analysisaswell as the syntheticdemonstrationor several propositionsof On theSphereand the CylinderII, and so does Apolloniusfor severalproposi-tions of the Conics, and for all the propositionsof the Cutting-offof aRatio(extant n an Arabictranslation):hese areby far the earliestextantexamplesof analysis,and they show that muchlater accounts of analy-sis, such as Pappus',do faithfullyreflectgeometricalpracticeat least asfar back as the thirdcenturyBC.s So it was natural or the philosophers

I Archimedes On the Sphereand the CylinderBook HI,1 nd 3-7; ApolloniusConics11,44-47and 49-51; Jones discusses the Cutting-offof a Ratio, op. cit., pp. 510-12 andtranslatessome sections of it pp. 606-19. All of these analyses are of problems ratherthan theorems. There are theoretical analyses in the manuscriptsof Euclid ElementsXIII,1-5, but these are agreed to be post-Euclidean,and their origin and date are uncer-tain (the text is printedin Heiberg's Teubner Elements in an appendix,v. 4 pp. 364-77). On the history of analysis, as opposed to the history of descriptionsof analysisby mathematiciansor philosophers, there is useful material in Wilbur Knorr, TheAncient Tradition of Geometric Problems (Boston, 1986) and in Richard Ferrier'sintroduction o TheData of Euclid, translatedby George McDowell and Merle Sokolik(Baltimore, 1993); McDowell and Sokolik also translate one of the extant analysesfromApollonius' Conics, showing how it makes use of the Data. Also JaakkoHintikkaand Unto Remes in The Method of Analysis (Dordrecht, 1974), in addition to dis-cussing the logical structureof analysis from a modernpointof view, give useful anno-tated versions of two analyses in Pappus, pp. 22-6 and pp. 52-3. Finally, AlexanderJones' translation and commentary on Pappus Collection Book VII, cited above, arevery useful.


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to regardanalysis as the living core of Greekmathematicalhought,whatthe geometersdid by themselvesand taughtto their students,while mostof theworks they publishandmakeavailable o people outsidethe schoolare only the dead husks.If analysisis such a powerfulsource of insightin mathematics, t is natural or the philosophers o hope to find some-thing like it in philosophy.At least from the second centuryAD on, the philosopherswere inter-ested in taking geometricalanalysis as a model for philosophy: he hopewas that the proved success of the method n mathematicswould rub offon the philosophers, r, morecynically, thatthe prestigeof the method nmathematicswould help to justify what the philosopherswere alreadydoing.6WhatI aminterestedn is whetherPlato,in the fourthcenturyBC,was already akingthis kindof interest n the methodof analysis.Procluslikes to think of Plato the philosopheras giving directions o the mathe-maticians about what problemsto work on, and even as teaching themmathematicalmethods, but it is more fruitfulto ask what philosophicalinspirationPlatomay have taken from the mathematicians:or it is cer-tainlyclearthatPlatowas an enthusiastormathematicalrainingand thathe "everywhereriesto arouseadmirationormathematics mongstudentsof philosophy" Proclus n Euclidemp. 66).7Platoand his contemporariesdo not use "analysis," s the neo-Platonistswill, as a generaltermfor allreasoning o firstprinciplesn philosophyas well as mathematics.ndeed,Plato never uses the word "analysis" or the verb&vakX_tv)t all. Butthat does not show that Plato and othersin the Academy may not havethoughtaboutor alluded to geometricalanalysis,or takenit as a modelin theirown reasoning.I will arguethat Platodoes, at least once, alludeto geometricalanalysis,and that he at least experimentedwith takingitas a model for philosophical easoning.First I should say what the method of analysis was. It is standardto startby commentingon the late ancient definitionsor descriptionsof analysis, since we have nothinglike a definitionof analysis beforeabout the firstcenturyAD (probably he earliestareHeronandAlcinous;the only extended descriptionis in Pappus, probablythird century).8Unfortunately,hese descriptionsof analysis are unclearand sometimes

6 This history will be traced in Don Morrison's work-in-progress ited above.7 For a sceptical treatmentof the image of Plato as "researchdirector"of mathe-matics in the Academy, see now Leonid Zhmud, "Plato as 'Architect of Science,"' inPhronesis, v. 43 (1998), pp. 211-44.K The Heron text, short and probablynot improved n translation, s cited by Nairizi


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PLATO AND THE METHOD OF ANALYSIS 197misleadingat crucial points, so they need a fair amountof commentary.Fortunately,we are not dependenton these "official"descriptions,and(despitethe impressionone mightget from the scholarly iterature) hereis no real doubt about what analysis was - there are the extant texts ofArchimedes,ApolloniusandPappuspracticinganalysis,and this is not alost art,but somethingone can easily train oneself to do on the ancientmodel;and once we master the practice,we can understand he official

("Anaritius"); medieval Latintranslation s printed n the TeubnerEuclid, v. 5, p. 89,lines 13-21. Alcinous discusses analysis in Didaskalikos chapter 5, sections 4-6 (his"secondtype of analysis"is the relevantone); also the interpolatedanalyses of EuclidElementsXIII,1-5,cited in a previousnote, begin with brief definitions of analysis andsynthesis (thoughthe definition of synthesis is corrupt),which may conceivably be theearliest extant. The only account that might actually be helpful is Pappus': "What iscalled the Domain of Analysis, my son Hermodorus, s, in sum, a special resource thatwas prepared, after the discovery of the common Elements, for those who want toacquire in geometrya power of solving problems set to them; and it is useful for thisalone. It was written by three men- Euclid the Elementarist,Apollonius of Perga,andAristaeus the elder - and it proceeds by the methods of analysis and synthesis. Nowanalysis is the path from what one is seeking, as if it were established [or agreed],throughthe things that follow, to something that is established by synthesis. For inanalysis we assume what is sought as if it has been achieved, and look for the thingfrom which it comes about, and again what comes before that, until by regressing inthis way we come upon some one of the things that are already known, or thatoccupythe rank of a first principle. We call such a method 'analysis,' that is, ava'icktv Xi'at;['solution backwards']. In synthesis, by reversal,we assume what was obtained lastin the analysis to have been achieved already,and by arranging n their natural orderas antecedentswhat were consequents in the analysis, and by putting them together,we reach the goal of the constructionof what was sought; and we call this 'synthe-sis.' There are two kinds of analysis: one of them seeks aftertruth,and is called 'the-oretic' [or 'theorematic'], while the other tries to furnish something that has beenprescribed [for us to construct], and is called 'problematic.'In the theoretickind, weassume what is sought as being, i.e. as true, and then proceed, throughwhat followsas true-and-beingaccordingto the assumption, to somethingthat is [already] agreed:if it is agreed to be true, what was soughtwill also be true,and its proof is the reverseof the analysis;but if we encountersomething agreed to befalse, then what was soughtwill also be false. In the problematickind, we assume the thing prescribedas if known,and then, proceedthroughwhat follows as true, to somethingthat is [already] agreed:if it is agreed to be possible and furnishable(what the mathematicianscall 'given'),the thing prescribedwill also be possible, and again the proof will be the reverse ofthe analysis; but if we encountersomething agreed to be impossible, the problemtoowill be impossible. Atoptag6o is adding a condition [reading npoa5txaroX'i fornpo8taatoXi] on when, how, and in how many ways the problem will be possible.So much, then, on analysis and synthesis."(Pappus, Collection VII,1-2, Jones' trans-lation modified).


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198 STEPHEN MENNdescriptions,while recognizingtheirimprecisions.'While analysis is notterriblymysterious, t is difficultto give a precise logical account of it;and given the generalfailureof the Greeksat describing he logical struc-ture of mathematical easoning, t is not surprising hat their descriptionsof the logic of analysis can be improvedon. A basically correct ogicaldescriptionhas been given by Hintikkaand Remes (along with, unfortu-nately, much that is incorrect),and I will make use of theirwork, as wellas of the extant ancient examples and descriptions f analysis.The most commonaccountof analysis, bothin the ancientsourcesandin modernreconstructions,goes somethinglike this. We are trying toprove a propositionP, the Clqtoiu'evovr thing-sought.As a heuristictoward indinga proof,we assume the 4iyrol4tevov as if it were knownto be true, and then draw inferences fromthis assumption; he analysisterminates andsomething"clicks" when we derive eithera propositionknown to be true(fromthe principlesof geometryand from theoremswehave alreadyproved),or else a propositionknown to be false. If we caninferfromP to a propositionknown to be false, then we have proved1 Pby reductioad absurdum. f we have inferred rom P to a propositionRthatwe know to be true, henwe cantryto reverseeachstepof the deriva-tion of R from P: if this succeeds,then the proofof R, togetherwith thederivationof P fromR, give us a proofof P. Of course,there s no guar-antee that the analysis (i.e. the derivationof R fromP) is reversible:butit does very frequentlyhappen that steps of geometricalargumentsarereversible i.e. thatif a step P-+Q is legitimate,so is Q-+P;e.g. "if trian-gle ABC is isosceles, it has equalbase angles",and also "if triangleABChas equal base angles, it is isosceles"),and in carryingout the analysisintelligentlywe will try to avoid obviouslynon-reversibleteps (e.g. "iftriangleABC is isosceles, its angles are equal to two right angles").Soalthoughan analysis leadingto a positiveresult does not guarantee hatthe 4Toi4tEVOV can be proved, t may still be heuristicallyuseful,since itconstructsa plausibleoutline for a proof,and we can then try to fill inthe steps.Some ancient(and modem) writers,botheredby the logical gapbetweena successfulanalysisof P and a successfulproofof P, say insteadthat analysis begins from the 4iyrou'4evov P and proceeds (not to proposi-tions that follow from P but)to "propositionsrom which the iiTo'gevov

9 Nineteenth-centurygeometry texts often gave instructionsfor how to carry outanalyses and syntheses - these instructions are logically inexact, but show that theauthors, and the teachers and studentswho used their books, did habitually carryoutthe practice of analysis.


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PLATO AND THE METHOD OF ANALYSIS 199would follow," so that as soon as we reach a propositionknown to betrue,we would have a guaranteedproof of P. But it is historicallyclearthatanalysiswas alwaysa deductiveprocedure tartingwith the 4itouie-vov, and this is also muchmore practicableand heuristicallyuseful. Theonly legitimatesense in which in analysiswe are looking for "proposi-tions from which the 4To{jievov would follow" rather han"propositionswhich follow from the 4TfloU{gevov" s that we are looking for propositionswhich, in the completeddemonstration,will be priorto P, and so which,in some vague "causal" ense, may be seen as "naturally" riorto P.

However, as Hintikkaand Remesrecognized, his standard ccountofanalysisis logically very imprecise,and its imprecisionsmake it hardtosee why analysiswould be heuristically aluable.Whileanalysis ooks fora proof of a proposition by assuming the 4Tyou'evov as if it were knownand drawing inferencesfrom it, it is a serious mistake to identify the

tol4icEVOV with the propositionwe are tryingto prove.To begin with,Greekmathematicalexts contain wo kinds of propositions,heoremsandproblems, and analysismay be seeking a proofof either.Only theoremsare what we would call propositions: theorem s a statementassertingthat all figuresof a given class have some particularproperty,while aproblem s a challenge o constructa figurehaving certainprescribed rop-erties (and/orcertainprescribedrelations to a given figure).While theenunciationof a theorem is a complete sentence, the enunciationof aproblem s an infinitivephrase(e.g. "to inscribe n a given circle a trian-gle similarto a given triangle"or "to constructan equilateral nd equian-gular pentagon").Pappus distinguishesaccordinglybetween "theoretic"analysis analysisof theorems) nd"problematic"nalysis analysisof prob-lems). In problematic analysis the 7n?ob'gevovis not a proposition at all,but ratheran object,the figurewe are tryingto construct:we assume thedesiredfigureas if its size, shape and positionwere known, and makeconstructions ut of it (anddraw nferencesabout he figureswe construct,from the assumption hat the 4nto{vtevovhad the prescribedproperties),until"something licks" and we constructa figurewhose size, shapeandpositionwe recognizethat we can determine rom the givens of the prob-lem alone (togetherwith the principlesof geometryand with propositionswe have already proved), independentlyof our assumptionabout the411tro1)I.tvov.Having reached this point, we then try to reverse the con-structionand the accompanying nferences to producea constructionofthe 4ntouigtvov and a proof that it does indeed have the prescribed prop-erties. The ancientgeneralaccountsof analysis which speakof assumingthe 4nTou'evovas if it wereknown,inferring o something ndependently


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known, and then reversing he reasoning o demonstrate he IJTo{iPEvov,are deliberatelyvague enoughto include this case; we miss problematicanalysis,and thus badly misrepresenthe analyticmethod, f we identifythe 4toU0gEVOV with the propositionwe are tryingto prove.In fact, even in the case of theoretical analysis, the il?ou1gevov is gen-erally not the proposition o be proved.A typical theoremcan be repre-sented as Vx (Px-+Qx)'? e.g. "forany triangleABC, if AB = AC, thenZABC = ZACB"). In analyzing this theorem, the 411o-14EVOV will not bethe propositionVx (Px-+Qx),but rather he propositionQx. We take asgivenan arbitrary suchthat Px; andthenwe also assume he 4fltOu'eVOVQx as if it were known to be true;we then draw inferencesfrom the4nrobVg?vovQx, makinguse of the "given"Px as well as of the principlesof geometryand of otherpropositionsalreadyproved. The analysis ter-minateswhen we infer a propositionRx whose truthvalue we can deter-mineindependentlyf the 4itou'evov Qx. If Rx is knownto be true,fromPx as well as the principles of geometry and other propositions alreadyproved,then we can try to reversethe analysis.We could describe thisreversalas turning he derivationof Rx from Qx into a derivationof Qxfrom Rx; but since we used the fact that Px in derivingRx from Qx, andsince we will also have to use Px in reversing he analysisto deriveQxfrom Rx, it is more accurate o say that we are turninga derivationofRxrPx fromQxrPx into a derivationof QxrPx fromRxroPx. f we cando this, we have a proof of the theorem Vx (Px-+Qx):first assume anarbitrary such thatPx, then infer from Px to Rx, then infer fromPxnRxto Qx, then concludethat Vx (Px-+Qx)."Or the analysiscould have anegativeresult: eitherwe could infer from QxnPx to a propositionRxthatwe knowto contradict x(inwhichcase we haveprovedVx(Px-_--Qx),so Vx (Px-*Qx)is false unless Vx -1Px);or we might recognizethat Rxis true for some but not all x such thatPx, in which case we know thatVx (Px-+Qx) s false, but we can conjecture hatthe analysis s reversibleand that Vx (PxrRx-+Qx), with the extra conditionadded,will be true.As Hintikka and Remes point out, the analysis is not simply working"backward" from the lirrou'u?vov Qx to infer the "beginning" Px: rather,

t0 When I say "Vx", this means a block of universal quantifiers,possibly more thanone (but it is important hat there are no existential quantifiers n Greek theorems);similarly, "Px"may really be a relational expression "Pxyz"." The argumentwill depend on a natural-deduction tep, but so do all proofs inGreekgeometry,since they all work by provingthe instanceof the propositionset outin the c0e'At; and toptag'S;, and then inferringto the universalproposition.


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PLATO AND THE METHOD OF ANALYSIS 201theanalysisstartsby assumingbothends together,and this additionalog-ical powerhelps to explainwhy we are likely to be able to infer some-thing thatwe can recognizeas true(or as false): whereas, f we were sim-ply arguing"backwards"romQx to Px, it is hardto see why this shouldbe heuristicallyany moreuseful thanarguing"forwards."'2We can try to give a similarlogical descriptionof problemsand ofproblematic nalysis.This shouldcomewith a warning abel, since, in thecase of problemsmore clearly than in the case of theorems,we are inconflictwith the Greek conceptionof these propositionswhen we repre-sent themin the notationof the predicatecalculus.To every Greekprob-lem there s a corresponding ropositionwith the logical form Vx (Px-+3yQxy) [or in prenexnormalformVx3y (Px-+Qxy)]- so the problem"toinscribein a given circle a trianglesimilarto a given triangle"can berewritten s "foranycirclex andanytrianglex', there s a triangley suchthaty is inscribed n x andy is similar to x"'. Any demonstration f theproblemalso demonstrateshis universal-existential roposition;but theonly acceptabledemonstrations f the problemare constructivedemon-strations, hatis, procedureshatshow how to construct he Ctycol,uevovfromany given x suchthatPx, accompanied y a proofthatQxy.'3Greektheorems,as opposed to problems,never involve existentialquantifiers;universal-existentialropositions ppear n Greekmathematics nly in theguise of problems.'4 am not saying thatGreekgeometers dentified he

12 With the above account compare Hintikkaand Remes, esp. pp. 31-9.13 However, the proof might not satisfy the scruples of a modem constructivist,since it might rely on the law of the excluded middle.14 This needs some qualification, since there are some propositions, sometimescalled "porisms"rather han"problems,"which are phrasedas a challenge, not to con-struct something,but tofind something (because it is a point or an abstractmagnitudeor number,none of which are properly"constructed," r because it has alreadyimplic-itly been constructed n the ?W6eat; r in the demonstrationof a previousproposition).But these, like problems,are infinitive phrasesratherthan what we would call propo-sitions. There are also some propositions in Euclid's arithmeticalbooks (the groupbeginning VIII,8, on how many numbers"fall" in continued proportionbetween twogiven numbers,and IX,20, "prime numbersare more than any assigned multitudeofprime numbers"),as well as X,1 ("two unequalmagnitudesbeing set out, if from thegreaterthere be subtracteda magnitudegreaterthan its half, and from that which is

left a magnitude greaterthan its half, and if this process be repeatedcontinually,therewill be left some magnitudewhich will be less than the lesser magnitudeset out"),which are phrasedas theoremsbut might be stated in modernterms as universal-exis-tentialpropositions,with the existentialquantifier angingover finite sequences of arbi-trary ength. But Euclid certainlydoes not understand he propositions n this way, and


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problemwith the universal-existentialroposition,but insisted that suchpropositionscould only be provedconstructively;nor am I saying thatthey distinguished he problem romthe universal-existentialroposition,and said thata narrower ange of demonstrations ould be acceptable orthe problem han for the theoreticalproposition.The truth s thatneitherGreekmathematicians or Greekphilosophershad a conceptionof a uni-versal-existential ropositionat all, and thatGreekgeometersphrased omanyof theirpropositionsas problems,as challengesto do or constructsomething,partlyas an attempt o compensate or the lack of a logic thatcouldhandlemultiplyquantifiedpropositions.With these caveats,we can try to give a logical descriptionof prob-lematicanalysis. In the problemVx (Px-+]y Qxy), on the Greekunder-standing,the 41iTo0visEvovs not a propositionbut an objecty such thatQxy. In analysis,we assumea given x suchthatPx, and we also assumethe illToievov y such that Qxy. We then make constructionsand in-ferences from x and y, using the given Px and the assumptionQxy.Eventuallywe constructan object z havingsome particularelation o x,such that we can determinez merelyfromknowing what x is and fromknowingthatPx, withoutrelying on y or on the assumptionQxy. Using"4p"tc. assymbols orconstruction-procedures,hishappensf fromPxrQxywe can deriveRxq(x,y), and if fromPxnRxz we can derive z = y(x). Ifthis succeeds, we try to reverse the analysis, first by provingthat PximpliesRxNI(x), nd thenby reversing he construction f z = (p(x,y) togive a construction = x(x, z), andproving hatPxr)RxzimplyQxX(x, ).If this can be done, then setting y = x(x, Ny(x)) ives us a construction-procedure nd a prooffor Vx (Px-+3yQxy).SOr, withproblematic s with

indeed the awkwardnessof these propositionsreflects his lack of the concept of asequence as the kindof object thatcan be quantifiedover. This kindof difficultyseemsnot to arise in the geometricalbooks.Is It is worth stressingthat there is a close formalparallelism between problematicand theoretic analysis, since this may be disguised by the symbolism I have used,which is considerably more complicated for problematicthan for theoretic analysis.But we could insteaddescribeproblematicanalysis by mimicking the simplerdescrip-tion of theoretic analysis, by saying "assume the tnirobUgvov 3y Qxy as well as thegiven Px, then infer a proposition3z Rxz which we know to be true (or false) on thebasis of the given Px, then (if the result was positive) try to reverse by showing that3z Rxz implies the 4qtoi'*evov 3y Qxy".Here,by treating3y Qxy as if it werea monadicpredicateof x, we are putting the problematicanalysis into the form of a theoreticanalysis. And this is in fact a logically correctdescriptionof problematicanalysis, aslong as we require proofs of existential propositions to be constructive:thus proving


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PLATO AND THE METHODOF ANALYSIS 203theoreticanalysis,we mightreacha negativeresult, discoveringthattheassumptionQxy contradictshe given Px: thiswould happen f (as before)we construct ome(p(x,y) satisfyinga relationRx(p(x, ), andthen,insteadof (as in the positivecase) recognizing hat [PxrRxz],[Pxn(z = (x))],we recognizethatPxrRxz implya contradiction, nd thatthe problem sthereforeunsolvable. But more often than reaching a purely negativeresult,we will discoverthat the problem s solvable only undersome con-ditions: hiswill happenwhenwe recognizethat PxrRxz implysome fur-ther conditionSx; we can then tryto reverse he analysisby showingthat,under the hypothesisSx, z is given by some construction V(x), nd thatPxrSx imply RxNI(x), nd so on, using z = V(x) to constructy and todeduce Qxy. In this case analysiswill have revealedthe StoptaRgo,henecessaryandsufficient onditions or theproblem o be solvable,and thiswas indeedone majoruse of problematicanalysis in Greekgeometry. tmay also happenthat we do not know how to lead the analysisto eithera positiveor a negativeresult,butthatwe can reducethe problem"givenx such thatPx, to constructy such thatQxy" to an easier or more fun-damentalproblem,"givenx such thatPx, to constructz such thatRxz":thiswill happen f, as before,we find a construction-procedurep(x,y) andprovethatPxriQxyimplyRx(p(x, ), andif we canthenreverse hisanaly-sis by findinga construction-procedure(x, z) and provingthatPxnRxzimplyQxx(x,z). Once again,this was a majoruse of problematic naly-sis: indeed,it seems reasonable o describeHippocrates f Chios' reduc-tion of the problemof doublingthe cube to the problemof findingtwomeanproportionals s an earlyapplicationof problematicanalysis.Problematic nd theoreticanalysisareformallysimilarenoughthat(inboth ancient and modem accounts) they are often coveredby the samegeneraldescription:hese descriptions end to applymoreimmediately oAied etic analysis,which is logically simpler, eaving problematicanaly-sis is an awkwardcomplication.Nonetheless,it is clear both thatprob-lematic analysis is historicallyolder, and that it was heuristicallymorefruitful.That problematicanalysis is older is naturalenough, since it isan older and more basic taskof geometry o constructor find objectssat-isfyinggiven descriptionsandthe mostbasic task,thatof measuring .g.

[Pxr)3y Qxy]-+[3z Rxz] means finding a construction-procedurep(x, y) and provingVy ([Pxn3y Qxy]-+Rxp(x, y)); proving Px[3z Rxz] means finding a construction-procedure4i(x) and provingPx-+Rxxy(x); nd proving [Pxr3z Rxz]-*[3y Qxyl meansfinding a construction-procedure(x, z) and proving Vz ([PxnRxz]-+Qxx (x, z)).


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an area,was construedas the problem"to constructa squareequal to agiven area"),while clearstandards f proofandjustificationdeveloponlyover time. And while theoreticalanalysisis essentiallya methodfor dis-coveringa proof of a given proposition,problematic nalysisis originallya methodfor discoveringa construction-procedure,lthough t can alsohelp us discover a proofthat this proceduredoes what it is supposed o.This also helps to explainwhy problematic nalysiswas heuristicallymoreimportant:to apply theoretical analysis to a propositionP, we mustalreadyhave come to suspect somehowthatthe proposition s true(onecommonuse would be in teaching,where the studentbelieves P becausethe teacher[or a book] says so, and thentries to findthe prooffor him-self usinganalysis).'6 n problematic nalysis,on theotherhand,whilewemustsuspectthattheproblemhassomesolution(andoftenit is intuitivelyobvious[e.g. by a vriat;] that it does, at leastsubject o somebtoptapo6;),we mayhave no clue at all aboutwhatthe solutionwill be. If we aretry-ing to prove(and proveconstructively) propositionVx (Px-ey Qxy),problematicanalysis can suggest a function(pand suggest that we tryprovingVx (Px-+Qxp(x)),and this is likelyto get us over thebiggesthur-dle towardsfindingthe proof;and in this way problematic nalysismaylead us to theoremsas well as to solutionsof problems.All these reasonshelp to explainwhy, in the Greektexts, we find mostlydescriptionsofwhat seems to be theoreticalanalysis,but mostly examples of problem-atic analysis.

IIGiven this descriptionof geometricalanalysis, as it was practicedinPlato'stime andafter,we are in a positionto ask: does Platorefer to themethod of analysis, and does he (like late ancient Platonists andPeripatetics) ake it as a model for philosophicalreasoningtoward firstprinciples?As I have alreadysaid, Platonever uses the word"analysis";but thisis compatiblewith his being aware of analysisas a distinctivegeometri-cal practice,and with his alludingto this practicewithoutusingthe name"analysis." n fact it is tolerablycertain,not only that Plato was awareofanalysis as a distinctivegeometricalpractice,but also that he knew itunder he name"analysis," nd thathe was familiarwithroughly hesame

16 Compare Diogenes Laertius VII,179, where Chrysippus tells Cleanthes that hewants only to be taught the doctrines,and he will find the demonstrations or himself.


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PLATO AND THE METHOD OF ANALYSIS 205(inadequate) ogical descriptionof analysis that we find in laterwriterssuch as AlcinousandPappus.The reason I think Plato must have knownthe name"analysis," ndtheancient ogical description f analysis, s thatwe find the name and the description n three passages of Aristotle(twoof them plausibly written in Plato's lifetime). Aristotle had no specialexpertise n mathematics oing beyondhis Academiccolleagues, and thetexts show that he is referring o a methodthat he expects his students,not merelyto have heardof, but to be accustomed o practice hemselves(the texts are in fact unintelligible xceptto someone who alreadyknowswhat analysis is, and have oftenbeen misunderstood y both ancient andmodemreaders).Aristotle s reflectingmathematical nowledgethat wascurrentin the Academy, and using it to make his own philosophicalpoints;and the same mathematical nowledgewas available for Plato touse in makinghis philosophicalpoints, if this is what he wanted to do.I will firstbrieflygo through he texts of Aristotle, o show whatknowl-edge of analysiscouldbe presupposedn the Academy; hen I will arguethat in at least one extant text - the Meno - Plato does refer to the methodof analysis, althoughnot by name; then I will commenton the harderquestionof what philosophical points Plato thoughtthis mathematicalpracticecould illustrate.Two of theAristotlepassages arein logical contexts,and makeroughlythe same point: Posterior Analytics I,12 78a6-13 and Sophistici Elenchi16 175a26-28.The PosteriorAnalyticspassage says: "If it were impossi-ble to show [86t4ai = deduce] something ruefromsomething alse, thenanalysis [6o avakXUelv]ould be easy: for [the analysis]would necessar-ily convert[avtLop?(ppEtv].or let A be true [ov]; but if this is true,thesethings (say, B) are true[i.e. I can deduceB fromA], which [sc. B] I [infact] know to be true.Then from these thingsI will show that that [sc.A] is true.'7Now mathematicalarguments] onvert moreoften, becausethey do not assumesomethingaccidental as a premiss](andin this theydiffer fromdialectical arguments),but rather they assume as premisses]

17 I am translatingov throughoutas "true",which is the easiest way to take thepassage (but NB "true" in the first sentence is the unambiguousaikXiOk);f this isright, Aristotle is talking about theoreticalanalysis. But it is just possible that "A isov means "A exists"; e.g., if "A" stands for "equilateraland equiangular pentagon,"then "A is ov" means "there is an equilateraland equiangular pentagon,"in whichcase Aristotle is giving a - ratherless logically precise - descriptionof problematicanalysis. Nothing much hangs on this; either (as I will assume) Aristotle is talkingabout theoretical analysis, or he is assimilating problematicand theoretical analysis soclosely that it is impossible to tell them apart.


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206 STEPHENMENN

definitions."The passageis very condensed,but Aristotle'sbasic pointisclear enough.'8Aristotlehas been saying that thereare valid argumentswith (some or all) false premissesbut true conclusions.He then illustratesthis with an appeal to his readers' or hearers')experience: ife wouldbemucheasierin geometry f thiswere not the case, i.e. if every valid argu-ment to a true conclusion also had (all) true premisses, since then ananalysis could always be converted nto a syntheticdemonstration. amtryingto demonstrate 4ntouigivovA;'9 so, for purposesof analysis, Iassume A as if it were known to be true, and deduce B, which I in factknow to be true;so the analysis terminates. f every valid argument o atrueconclusionhadall its premisses rue,then, since there s a validargu-ment from A to B, and since B is true, necessarilyA wouldalso be true;so we could automatically onvertthe argumentrom A to B into a validargumentromB to A; and since we know thatB is true,this would givea demonstration f A. In fact, since some argumentsdo lead fromfalsepremisses o true conclusions,not all analysesconvert;but Aristotleaddsthat,as a matterof mathematical xperience,analysesoftendo convert.Iam not sureexactlywhatto makeof Aristotle'sexplanation,namelythatmathematical nferences ake definitionsrather han accidentalpropertiesas theirpremisses;butone illustrationwouldbe that,if an inference say,thata certainequalityholds) dependson the premissthat a certainangleis right, it will use the full strengthof the premiss hatthe angle is right,and not merelythat it is (say) greater han 80 degrees;so it is likely tobe possibleto reversethe inference, o infer thatif the equalityholds theangle is right, since if the angle was slightly more or less than a rightangle, one quantitywould be slightlytoo largeor too small. Something

18 Themistius misunderstands he passage in his paraphrase Analyticorumposte-riorumparaphrasis 26,22-8), apparentlydue to his ignorance of geometrical practice.Aristotle means: "Suppose we are trying, by the method of analysis, to find a proofthat A. We infer from A to B, which we recognize to be true. But this does not yetshow us that A is true, because a false premiss could yield a true conclusion."Themistius takes Aristotle to mean: "Supposewe are trying to find a proof that A. Werealize that B implies A. But that doesn't yet show thatA, because B might be false."The Aristotle text is (like the whole Posterior Analytics) highly elliptical, andThemistius fills in the ellipses incorrectly,apparentlybecause he is unfamiliarwith thepractice of analysis and does not realize that the analyst takes the 4toUicvov (hereA) as a premiss in the analytical stage of his argument. Themistius is apparentlyassuming that analysis must be something like rhetorical inventio, a practice withwhich he is much more familiar.

'9 Assuming that A is a proposition rather han an object. We could rewrite all thisin the case where A is an object.


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PLATO AND THE METHOD OF ANALYSIS 207like this is why theorems ike the Pythagoreanheorem end to have trueconverses,althoughof coursethis does not hold in every case.20This passage rom hePosteriorAnalyticss enough o show thatAristotle,like later Greekwriters,conceives of analysis as a process in which atqxoiLuevovs assumedto be true,and deductionsare made from it untilwe deducesomething ndependently nownto be true;we thentry to con-

vert the argument into a demonstration of the inTo{'EVOV.What may notbe clear is what part of this process is called "analysis."Aristotlesaysthatif all arguments onverted,T6o vaX{Etvwould be easy: this suggeststhat &vaXvitvis not just what later writers call analysis, namely theprocessof deducing rom the 4ito(ugvov a propositionknown to be true,butrather he whole processthat later writerscall analysis-and-synthesis,culminating n a demonstration f the 4irro{wEvov. owever,Aristotle'susage is in fact the same as later writers', as is shown by SophisticiElenchi 16 175a26-28:"it sometimeshappensas in 8taypa'ija-a ['dia-grams' but also 'geometricproofs']: for there too sometimes,after wehave analysed [&vaX{6avte;Iwe are unable to synthesize [ouvOeivat]again."Hereava&kut;ando{vOecn;areclearlytwo successivestages,andthe difficultyof passingfrom&va&XuGt;o aCUv0eoi;s what the PosteriorAnalyticspassage calls the difficultyof "converting." o when Aristotlesays in the PosteriorAnalyticsthat "it would be easy to &vakc6av",emust mean not simply that it would be easy to find an analysis of a4n11?O{UjVOV,but that it would be easy to find a good analysis,where agood analysis is one that can be converted nto a demonstration f the

tnToitevov. (It is always trivial to give some formally legitimate butmathematically seless analysis,and no one would take this as a goal.)In these passages from the Posterior AnalyticsandSophisticiElenchi Aris-

totle is interested n drawinganalogiesbetween the methodsof analysis20 Aristotle'sdescription s inadequate n some of the same ways thatother descrip-tions of analysis, ancientand modem, typically are. He does not distinguishthe 4qo'u-iiFvov from the proposition to be demonstrated. Connected with this, he treats theargumentthat might or might not convert as simply an inference from one proposi-tion to another,when it is actually embedded in a natural deduction context (i.e. weare arguing from Px to Qx, neither of which is properly speakinga proposition,sincethey each contain a free variable).Aristotle also speaks as if therewere a single pre-

miss A and a single conclusion B, when it would be more accurate to say that we candeduce B from the premiss A and auxiliary premisses C,, C2, etc.; the argumentisvery unlikely to be "convertible" o an argument rom B to the conjunctionof the pre-misses Ar-C1qC2, but it might be convertible to an argument from the conjunctionBqCrlC2 to A.


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208 STEPHEN MENNandsynthesis n geometryandprocedures f non-mathematicaleasoning,although t is notclearthat he is willingto use the wordavaXiTetvn non-mathematicalcases. However, in a third passage, Aristotle does use"analysis"metaphoricallyodescribe,notphilosophicalnferences, utprac-tical reasoning romends to means.

Having posited the end, they examine how and by what means it will come about:and if it seems that it can come about in many ways, they also examine whichis the easiest and best, but if it is accomplished [only] in one way, they ex-amine how it will come about through this means and how this means itselfwill come about, until they come to the firstcause, which is last in discovery. Forthe person who deliberates seems to be inquiring and analyzing [4InrEiV0avaXi)elv] in the way that has been described,as if [he were inquiring nto andanalyzing] a 8taypagiga (it seems that not all inquiry [i]t71yt;] is deliberation- mathematicalones are not - but all deliberation s investigation); and the lastthing in the analysis is the first in the coming-to-be.And if they encountersome-thing impossible, they desist, for instance if money is neededand there is no wayto provide this; but if it seems possible, they try to do it. (NE 11,3 1112bl5-27)

This passage is difficult, but it is clear that Aristotle is thinking specificallyof problematic analysis, and using it as a model to describe practicalreasoning:we begin with a specificationof the object we are trying toproduce,and, positing a situation in which this has been achieved,wereasonbackto theway it might havebeenproduced,until we reachsome-thing that is immediatelyn our powerto produce.This last thingcorre-sponds,in a problematic nalysis, to the last thingwe construct romthe4TnOiO.gvov,which we recognize as something that is determined by thedata of the problem, so that we are able to construct it directly fromthe data:so this "last thing in the analysis" s the "first n the coming-to-be"of the4tovi'Pjevov;hen,in reversing heanalysis,we construct achof the subsequenthingsout of this firstthing,in the reverseof the orderin which we found them in the analysis, until we have constructed he4iyoiu'gvov. The analysis has succeeded only when we have inferred fromthe ntrovIrvov ack to a "firstcause"or &pxil,meaningnot a propositionwe know to be true,but an object we know we can construct;whereas,if we inferto an objectrelatedto the givens of the problem n an impos-sible way, we have a reductioad absurdum, nd we give up the problemas unsolvable in the given case.2'

21 Aristotle may well also be thinkingof analysis at NE VI,8 1142a23-30, where(ppovmlat; the ability to deliberate well), which perceives some E,aXaxrovhat isnpa-T v, is compared to an ability to perceive that ro EvToi; gaEPhlTtKOIK; i?xaTovis (for instance) a triangle.To ev Toig actrucoi; ''aGrTovmight mean simply a


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PLATO AND THE METHOD OF ANALYSIS 209These passagesfrom Aristotle show that it was possibleto presupposea familiaritywith the practiceof geometricanalysis in Academic circles(in Plato's lifetimeand in the subsequentdecades);thatthe practicewasknown by the name"analysis," nd thatessentiallythe same (inadequate)logical descriptionof analysisthat we find in Pappuswas alreadyavail-able; and, finally, that Academic philosopherswere interested n usinggeometricalanalysis,so described,as a model forphilosophical and prac-tical) reasoning.So, even thoughPlato never uses the word"analysis,"heand his students n the Academywere familiar with the practice;Plato

could ifhe wanted)allude o thisgeometric ractice expecting isAcademicreaders to fill in the name "analysis"and the logical description),andmake whateverpoint he might want to make about the relationbetweenthis kind of mathematical easoningand reasoning n philosophy.I willnow arguethat Plato does, once, so allude to analysis,in the secondgeo-metricalpassage of the Meno (86e4-87b2); and then I will offer somespeculationsaboutwhat philosophicalpoint Plato wantedto makeby theanalogywith geometry.Socrateshas proposedto examine "from a hypothesis"Meno's ques-tion whethervirtue is teachable.He then says,

I mean "from a hypothesis"in this way, the way the geometers often examine,when someone asks them, for example, about an area, whether it is possible toinscribe this area in this circle as a triangle. [A geometer] might say, "I don'tyet know whether this [area] is such [as to make the constructionpossible], butI think I have as it were a hypothesis that would help towards the question, asfollows: if this area is such that when it is appliedto the given line [sc. the diam-eter of the circle], it falls short by an area similar to the applied area, then onething seems to me to follow, but anotherif it is impossible for this to happen.So afterhypothesizingI am willing to tell you what follows about inscribing[thearea] in the circle, whether it is impossible or not."

Here Plato is consideringa geometricalproblem,"to inscribe in a givencircle a triangleequalto a given area"; n fact the hypothesishe gives isdesignedto solve the morespecificproblem"toinscribe n a given circlean isosceles triangle equal to a given area."22 he hypothesisthat Platomathematical particular, hough it seems odd to posit a special quasi-sensory abilityto recognize (individual, but perfect and hence non-sensible) mathematicaltriangles;but it seems more likely that to ?v toi; iahgartncoi; EGxa'rovs the last thing con-structed in an analysis, which we quasi-perceptually recognize as something wealready know how to construct from the givens.

22 The special problem is equivalent to the general problem in the loose sensethat whenever there is a solution to the generalproblemthere is also a solution to the


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210 STEPHEN MENN

F E

A DB x

Figure 1

mentions,namelythat the given area can be applied to the diameterofthe given circle (in the form of a rectangle) n such a way that it fallsshort by a figuresimilarto the appliedarea,23s in fact a necessaryandsufficientconditionfor the problem o have a solution;furthermore, nysolutionto the application-of-areasroblemcan be straightforwardlyon-verted into a solution of the problem"to inscribe in a given circle anisosceles triangleequal to a given area." For (see Figure 1) let AB bea diameterof the given circleY, and let the rectangleCDBEbe equaltothe given area X, and let the rectangleCDBE fall shortof the line AB bythe rectangleFADC, in such a way that the rectangleCFAD is similarto the rectangleCDBE. Thus the line CD is a meanproportional etweenthe line AD and the line BD. Produce he line CD beyondD to G, so thatGD = CD. Since the rectangleon GD andCD (beingequalto the squareon CD) is equal to the rectangleon AD and BD, it follows (by the con-verse of Euclid111,35)hat the pointsA, B, C and G lie on a circle. Sincethe chord AB perpendicularlyisects the chordCG, AB must be a diam-eter, so the circle on which the pointsA, B, C and G lie is in fact thegiven circle Y. Now the triangleCDB, which is half of the rectangle

special problem. It is not equivalent in a strongersense, since there is no straightfor-ward procedure or converting a solution to the generalproblem into a solution to thespecial problem.23 An area X is applied to the line AB in the form of a rectangle (or parallelogram)

if a rectangle (parallelogram)equal to X is constructedwith AB as base; the appliedfigure exceeds AB if its base is the line AC which extends AB to C lying beyond B(and it exceeds AB by the portion of the figure that lies over BC); itfalls short of ABif its base is AC for C lying in between A and B (and it falls short of AB by the por-tion of the figure that lies over BC).


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PLATO AND THE METHODOF ANALYSIS 211CDBE,is also half of the triangleCGB.So CGB is equalto CDBE,whichis equalto the given areaX. So CGB is an isosceles triangle nscribed nthe given circle Y and equal to the given areaX: which is what was tobe found.Plato explicitlycites this example,not as an exampleof analysis,butonly as an exampleof how a geometermightreasonfrom a hypothesisin answeringa given question.The questionhe cites proposesa problemrather hana theorem,and a full answerwouldbe a solutionto the prob-lem: that is, the aim is not simplyto answer the question"is it possibleto inscribe hisareain this circle as a triangle?"with yes or no, butrather,in the case where the answer is yes, to give a construction-procedureshowing how to inscribethe area in the circle in the formof an [isosce-les] triangle.24When the geometeranswersthe question"froma hypoth-esis," he is taking a step towardansweringthe questionfully, that is,toward giving both a &toptogot6or the problemand a construction-pro-cedurefor solvingit where it can be solved. So when the geometeroffersto answerthe question rom thehypothesis"thegiven area can be appliedto the diameter n such a way that it falls shortby a figuresimilar to theappliedarea,"he is not simplyclaimingthatthe originalproblem s solv-able if and only if the application-of-areasroblem s solvable,but alsoofferinga construction-procedureo convertany solutionof the applica-tion-of-areasproblem nto a solutionto the originalproblem.The solution"froma hypothesis" hus reduces the originalproblem o the application-of-areas problem: the task that remains is to give a Btoptagji; determiningwhether he application-of-areasroblemcan be solvedfor the given areaand the given line, and to give a construction-procedureor solving itwhere it can be solved. WhenPlato recommends he geometers'practiceof answering"froma hypothesis,"he is recommendingacklinga difficultquestionby reducing t step-by-step o more basic questionsuntilwe cananswer t directly:and this is the lesson Socratesdrawswhen, in answer-ing Meno's question"is virtue teachable?"romthe hypothesis"virtue sknowledge,"he reducesMeno'squestion o the question"is virtueknowl-edge?"(87b2-dl), then answersthis question n turnfromthe hypothesis"virtue is good" (87d2-89a7;explicitly called a "hypothesis"at 87d3),which presumablywe can immediatelygraspto be true.So Plato is rec-ommending,not simplythatwe learn how to answera given questionX

24 If the only interestwere in giving a Btoptcjo';, this would be much easier: theanswerwould be "if and only if the given area is less than or equal to an equilateraltriangle inscribed in the given circle."


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212 STEPHEN MENN

froma given hypothesisY, but also that we learn how to tackle a givenquestionX byfindingan appropriatehypothesis"o reduce t to.25 n giv-ing the geometricalexample,Plato leaves it mysterioushow the geome-ter finds the appropriate ypothesis:on a superficial eading, t looks asif the geometer s simplyguessing, or intuitivelydiviningthat the hypoth-esis "thegiven areacan be appliedto the diameter n such a way that itfalls shortby a figuresimilar to the appliedarea,"would be useful forinvestigating he problemat hand;it would thenbe just a lucky coinci-dence, or a confirmation f the geometer'spower of intuition,that thehypothesisturnsout to be necessaryand sufficient or solving the prob-lem. But in fact this hypothesiswas certainlyfound by the methodofanalysis,and is very typicalof the use of analysis in reducinga problemto an easier problem;and since Plato is recommendinga methodforfindingappropriate ypotheses and so reducinghardquestions to easierones, it is analysisthathe is recommending.To see how analysisof the problem"toinscribe n a given circle Y anisosceles triangleequalto a given area X" would lead to Plato'shypoth-esis, assume the problem olved. So (see Figure1) let BCGbe an isosce-les triangle,BC = BG, inscribed n the circle Y andequal to the rectilin-eal area X. Then let BA be a diameterof the circle Y; the diameterBAperpendicularlyisects thechordCG at a pointD. ConnectAC. TheangleZACB is inscribed n a semicircle,and is thereforea rightangle. So thetrianglesADC and CDB are similar,to each other and to the triangleACB. So, completing he rectanglesADCF andCDBE,we see that theserectanglesare similar,and therefore hatthe rectangleCDBE falls shortof the line AB by a figure similar to itself. Since the rectangleCDBE isdouble the triangleCDB, which is half of the triangleBCG, it followsthat CDBE = BCG; but BCG = X, so CDBE = X. So the given area X hasbeen appliedto a diameterof the given circle Y in the formof a rectan-gle, in such a way that it falls shortof the diameterby a figuresimilar othe appliedarea. As we have seen, the analysiscan be reversed,so thehypothesis"the area X can be applied o the diameterof Y in the formof

25 In Prior Analytics11,25,Aristotlegives this passage (withoutciting the Meno byname) as an example of reduction [&1Lay l]: we wish to know whether teachablebelongs to virtue, it is clear that teachable belongs to knowledge, so we reduce thequestion whether teachable belongs to virtue to the (hopefully) simpler questionwhetherknowledge belongs to virtue. Aristotle compares this to a geometrical exam-ple, Hippocratesof Chios' attemptto reduce the problemof squaring he circle to sim-pler problems.


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PLATO AND THE METHOD OF ANALYSIS 213a rectangle n such a way that it falls shortby a figuresimilar to the ap-plied area"gives a sufficientas well as a necessarycondition or solvingthe problem"to inscribe n the circleY an isosceles triangleequalto X."This is not only an easy and straightforwardse of analysis,but alsoa very typicalone. In fact, it seems to be partof a systematicprogramofreducingproblemsof all kinds to problemsof applicationof areas,in thehopethattheseproblems ouldall be solvedin a simple and uniformway.An important xampleis theproblemof constructing regularpentagon,which can be reducedby analysis to the problem"to divide a line inextreme and mean ratio"; his in turn can be reducedby analysis to theproblem"to applya squareto its own side in the form of a rectangle nsuch a way that it exceeds by a square,"and this in turncan be reducedby analysis to the problemof finding a mean proportional,and thussolved. Probably beginning from the analysis of the regularpentagon,earlyGreekgeometersdeveloped techniques or solving a broad class ofproblems f applicationf areas:Proclus InEuclidem 19-20)citesEudemusas attributinghese techniques o "the Muse of the Pythagoreans"i.e. tothe tradition romHippasus o Archytas)and Euclidpresents heir resultsin developed orm in ElementsVI. Theoriginalproblemswould have been"toapplya given area to a given line, in the form of a rectangle, n sucha way that it exceeds [or falls short]by a square,"but the techniques orsolvingtheseproblemscan be generalized o solve "toapplya given areato a given line, in the form of a rectangle, n such a way that it exceeds[or falls short] by a rectanglesimilar to a given rectangle"or even "toapplya given area to a given line, in the form of a parallelogram,n sucha way that it exceeds [orfalls short]by a parallelogramimilarto a givenparallelogram,"which is the problem that Euclid solves in ElementsVI,28-29.Euclidgives a 8toptaji6; or theproblemof falling-short whichcannot be solved in all cases), and gives a constructionwhich worksbyreducingboth problems o the problemof constructing parallelogram fa given shapewith a given area,which in turncan be reduced o the prob-lem of findinga meanproportional; hile Euclid does not explicitlygivethe analysesof his application-of-areas roblems,his expositionmakesitobvious that his constructions (and his toptaFgo) were first discovered byanalysis.26The application-of-areas roblemthat Plato proposes as his

26 In fact the proof of VI,27, giving the S&optajo6for the falling-short problemVI,28, is a disguised analysis (and thus, in a sense, the earliest extant analysis). WhatEuclid is doing in these propositionsseems to arise from a generalizationof the resultsneeded to construct the regular pentagon. Euclid draws from VI,29, the problem of


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214 STEPHENMENN"hypothesis" n the Meno is obviously similarin formulationo Euclid'sproblems, and comes from the same geometricalresearch-program;utPlato's problemis more difficult,and Euclid does not discuss it in theElementsbecause it cannotbe reduced o findinga mean proportional rsolved by ruler-and-compassonstructions.However, thereis a solutionusingconics,which Platomaywell have knownwhen he wrote theMeno,and which was certainlywithin the capacityof Greekgeometersat leastleast by mid-fourth-century.27ut whetherPlatoknew the solutionor not,he would have seen the problemas part of a promising programforfinding&toptagoi of any given construction-problemnd for solving anyproblemwhen it can be solved.28Thus Plato alludes at Meno 86e4-87b2 to the method of analysis,and more specificallyto the programof reducingconstruction-problemsthroughproblematic nalysis;and he holds up the program f analysis asa methodologicalmodel for philosophical nquiry.But Plato does all thiswithoutever using the word "analysis" thoughhe must have known theword), and withoutdescribingclearly either the logic of the method ingeneral or the geometryof the case he describes: he does not explaineitherhow his application-of-areasroblemwas derived romthe originalproblem,or how it would help to solve the original problem,or how itmight itself be solved; indeed, he does not describe either the originalproblem or the "hypothesis"clearly enough for anyone who did notalreadyunderstand he problemPlato is describing o understand im.29

excess, the corollary VI,30, "to divide a line in extreme and mean ratio."Euclid doesnot use VI,30 to construct the regular pentagon, because he has already done it inIV,10-1I without using proportion heory, using II, 1, "to divide a straight ine so thatthe rectangle contained by the whole and one segment is equal to the square on theother segment": but this is simply VI,30 reformulated,and re-proved,in such a wayas to avoid proportions.Euclid is certainly modifying an earlier order of presentationwhich used application of areas to construct the pentagon.

27 See the solutionof the Menoproblemgiven by Heath,HistoryofGreekMathematics(Oxford, 1921), v. 1, pp. 300-301, using the same methods that Menaechmus (a stu-dent of Eudoxus)used to find two mean proportionalsbetween two given lengths (seeHeath, v. 1, pp. 251-5).28 In the passage of Philodemus' Academica (ed. Gaiser, Stuttgart, 1988, p. 152)reportingsome Academic or Peripateticsource on the progressof mathematics n the

Academy, and speaking of Plato as research-director,t is said that "analysisand thetaking of Stoptogoi [il a&va6kutqKalt6 ni p' Soptagolb; ),.tga]" were then broughtforth; the conjunctionis apparentlythe subject of a singularverb.I Hence the gross misunderstandings f this passage e.g. in Jowett's and Grube'stranslations, and by Bluck in the appendix to his edition of the Meno (Cambridge,


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PLATO AND THE METHOD OF ANALYSIS 215This compressedpassage, alluding without explanation to what wouldhave been the forefrontof mathematical esearch, mmediatelyafter anextremelyslow and patientdiscussion of a trivial mathematical act, wasnot meant to be understoodby most of its readers;Meno, who asks noquestions at all at this crucial turningpoint in the argument,obviouslydoes not understandwhat is going on. In fact, the passageis perfectprooffor Gaiser'sthesis that Plato's dialoguesallude to doctrines hat they donot fully explain, in an attemptto rouse Plato's readersto seek furtherenlightenmentn the Academy.30 hose of Plato's readerswho are famil-iar with currentgeometricpracticewill understand is mathematical llu-sions; his other readerswill pick up that Plato is referring o some geo-metric result and to some geometricpracticewhich is supposedto bephilosophicallymportant nd which they would understandf they cameto study geometry n the Academy.3'Butwhat was the philosophicalpay-off supposed o be?

IIIAs I said at the beginning,analysis appealedto philosophersbecause itpresenteda method for discovery. In the Meno, the progressof inquiryhad been frustrated:Socrates cannotanswerMeno's question"is virtueteachable"until Menotells him what virtue s, andMeno,unable o definevirtue afterrepeatedattempts,gives what Socratescalls an "eristicargu-ment"(80e2) to show that one cannot searchfor something f one doesnot alreadyknowwhat it is. So Socrates' mmediate ask is to show Menohow inquiry s possible, by giving him successfulmodels of it: this is thepointof both the first and the secondgeometricalpassages.The firstgeo-metricalpassage, togetherwith the accountof immortalityand recollec-tion which it illustrates,helps to show how we can inquire"what is X"

1964); but the passage is correctly translatede.g. by Guthrie, and by Heath, Historyof Greek Mathematics,v. 1, p. 299, followed in the revised version of Grube's trans-lation in John Cooper and D.S. Hutchinson,eds., Complete Worksof Plato (India-napolis, 1997).30 I am not endorsingGaiser's view of the content of these doctrines.'1 CompareHeath, History of Greek Mathematics,v. 1, p. 302. Benecke had objected

that, on the (correct) interpretation avored by Heath, Socrates would be describingquite a difficult geometrical problem, and that therefore "Plato is unlikely to haveintroduced it in such an abrupt and casual way into the conversation between Soc-rates and Meno"; Heath replies, rightly, that "Plato was fond of dark hints in thingsmathematical."


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(e.g. what is virtue?):we have encounteredX before this life and so havedim memoriesof it, which will help us to recognizethe thingwhen weare confrontedwith it again,and which can be teased out and "tieddown"to becomeknowledge.The secondgeometricalpassageserves a comple-mentary unction n showinghow inquiry s possible.SocrateshadwantedMeno (encouragedby the accountof recollection)to keep on inquiring",whats virtue?" 86c4-6), but Meno wants to go back to his originalquestionwhethervirtue s teachableor acquiredn some otherway (86c7-d2), that is, to ask what virtue is like [0i10v ?aTI] before determining whatit is [ti ?aort].Surprisingly, lthoughSocrates had earlierinsisted that itwas impossibleto inquire n this way, he now immediatelygives in, andoffers to investigateMeno's oiw6vvSTt questionon thebasis of a hypoth-esis about the xi iart. As we have seen, this hypothetical nvestigationmeansusing somethingcomparable o the method of analysis to reducethe lo6ov Es;t question o a ti esm question,and to keepreducing t untilwe reach a question hat we can answerdirectly.In a sense, Socrateshasnot concededmuch on the logical priorityof the ti Eart to the loi6ovc'aTlquestion,since he continues o insist that we cannotknowwhethervirtueis teachable until we can demonstrate he answer from a knowledgeofwhatvirtue is. But as a matterof heuristics,Socrates s concedingthatitmay be usefulto begin with the logically posteriornioi6v9art question, nthe hope of discovering an answer both to the Tiirtt and to the noiov E'atquestions.CertainlyMeno had not beenmakingmuchprogress n his suc-cessive attempts o answer the ri iarrt questiondirectly,so perhaps t isworthtryingan indirectapproach.The geometersare supposed o be able,throughanalysis, to reason from logically posteriorthings to logicallyprior hings, and so to discoverthe appropriate rinciples or demonstrat-ing an answer to a given question;so perhapswe can imitate their suc-cess in philosophy.Thus the firstgeometricalpassage suggeststhatit isin principlepossible come to knowledgeof what virtue s, if someonecandiscover the rightseries of questions o ask;the secondpassagesuggeststhat something ike an analyticinvestigationof whethervirtueis teach-able might be the path that succeeds, in bringingus to knowledgeofwhethervirtueis teachable,and thus also of what virtue is.32

32 If Plato had thought it was worth while, he could also have illustrated he methodof analysis in the first geometrical example, by showing how the line of questioningthat promptsrecollection is an application of the method of analysis. In one sense heis in fact doing this, since part of what prompts recollection is the refutationof theanswers that the side of the eight-foot squareis four or three;and such a reductio ad


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PLATO AND THE METHOD OF ANALYSIS 217What is much less clear is how this is supposedto work. If we startby not knowingthe &pXaihat we need for demonstratinghe answer toour question(here "is virtueteachable?"),how is analysis or its philo-sophicalanalogue going to help us find the demonstration? aterGreekphilosophers dentifyanalysiswith arguing"upward"o the appXai,ndsuggest that we can firstargue"up"from posterior hings to the &pxac,then argue back "down" rom the apxai to the posterior hings.But thiskind of argumentwill not give a demonstration, nd so will not give usknowledgeof the posterior hings,unlesswe have acquiredknowledgeof

the &pXai: ndhow is analysisor its analoguesupposedto help in that?There are a numberof different enses in which analysiscould be saidto lead to knowledgeof &pXac,nd it will help to sortsome of these out.To beginwith,apxii anbe taken as equivalent o iunt0ec;t,as the propo-sition which is "laid down"at the beginningof a discourse, o fix the ref-erence of a termorto give a premiss or a deduction.33n this sense,whenabsurdum s just an analysis with a negative result. However, the positive result thatthe side of the eight-foot square is the diagonal of the four-foot square is not shownas being reached analytically: f Socrateshad not alreadyknown the answer,and askedthe boy the rightstring of questionsbased on knowledge of this answer,the boy mightnever have discovered it. But Plato could instead have shown this result as beingreachedby analysis: suppose a squareof area eight squarefeet has been found; drawthe diagonals, dividing the square into four equal isosceles right triangles, each ofwhich is thus of area two square feet. At this stage, probably,something clicks, andwe recognize that half of the given two-foot-by-two-footsquare, divided by a diago-nal, is also an isosceles right triangleof area two squarefeet. We thus know how toconstructpurely from givens a figure similar and equal to the figure we have con-structed from the lqToi4Levov, namely the isosceles right triangle whose base is theside of the 4toiotrvov square.And we can then reverse the analysis to constructthe41TtOi4LEVOV squareof area eight squarefeet from the isosceles right triangleof areatwo squarefeet, by constructing our equal and similar isosceles right trianglesaroundthe same vertex. The diagramthat would result is the diagram that Socrates in factdraws, and Plato could have represented t as the result of reversingthis analysis. ButPlato probablythought thatthe methodof analysis was too important o waste on sucha trivial example: its power is betterbroughtout by showing how it can contributetoa difficultproblembelonging to currentor recent mathematicalresearch.

3 Carl Huffmangives a useful collection and discussion of evidence on the earlyhistory of the terms apXTiand rno60rat;,specially in Hippocratictexts, in the intro-duction to his Philolaus of Croton: Pythagoreanand Presocratic (Cambridge,1993),pp. 78-92. The noun bn?6rat; is a relatively late developmentfrom the phrase bn7oti-Ora0ct a pXTIv,to lay down a beginning"for a discourse,where it is often assumedthat the appropriate beginning must be something that the listeners will agreeto. The sense "beginning of a discourse" connects with the physical sense of &pXPi,since often the appropriatebeginningfor the discourse will be the "natural"beginning


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problematic analysis discovers the Stoptc06 for a given problem, it isdiscoveringan apx"i or the proposition, hatis, a hypothesis romwhichthe proposition an be proved.But this is notdiscoveringan &p i in sucha way that the apXyi s known to be true: and while I thinkpart ofwhatPlato is interested n is simplydiscoveringan appropriate ypothe-sis for proving a given proposition,he also wants more than that. Thusthe Phaedo speaks of going from a hypothesisto a higherhypothesis"untilyou come to somethingsufficient" t which you can stop (102el);the Republic says that dialectic proceedsfrom a hypothesisC'' apXilvavul6OEtov51Ob6-7),.e. to an apxiiwhich is not a hypothesis,butwhichis somehowimmediatelyknown(and not merelyassumed)to be true. Ifa hypothesis s something ike the Stoptor.o;of a problem,or moregen-erallyanyconditionof a propositionhat can be reachedby analysis,thenwhat is an apX'9hat is not a hypothesis?A look at the Menoexamplesuggestswhythe"hypothesis"here s insufficient,ndwhatamoresufficientapxyjmight look like. Recall that the problemwas posed with regardtoa particular rea and a particularircle, "whethert is possibleto inscribethis area in this circle as a triangle": he geometersays that he can do it"if this area is suchthatwhen it is applied o thegiven line [sc. the diam-eter of the circle], it falls shortby an area similarto the appliedarea,"but he does not know whether his hypothesisholds.This is becausethehypothesis is itself a difficultproblem(or says that a problem can besolved),and there s no directway to verifywhether t holds of the givenarea.By contrast, f the hypothesiswere "this area is smaller hananothergivenarea," herewouldbe a directway to checkwhether t holds(assum-ing bothareas aregiven as rectilineal igures): f it holds, it can be estab-lished,not by a generalproof,but by a construction hatmust be verifiedby directperceptionof this particular iven area. It seems reasonable osay thatwhen a geometricpropositionhas beenreduced o something hatwe can verify by directperceptionof the given figure,then it has beenreducedto an apxi'jhat is not a hypothesis.But this dependson visualperceptionof a figure:this is no help in dialectic,which, unlikegeome-try, makes no use of visual images, and can lead us to knowledgeonlyby reasoning.How can reasoning ead us to knowledgeof an &pXTlhatis not a hypothesis?of the thing. The beginning of the discourse may also be something like a definition,not so much as a startingpoint for deduction as to make sure the speaker and listen-ers are talkingaboutthe same subject (so too in Plato,Phaedrus237b7-d3). Plato canuse &p ' and 'in6eeoa; as equivalent, as with i?n0evoi; at Phaedo 101d7 and &pxjat l0le2.


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PLATO AND THE METHOD OF ANALYSIS 219

I don't have a fully adequateanswerto this question,and I don't thinkPlato did either. But furtherreflectionon geometricalanalysiswill shedsome light on how Plato thought ts philosophicalanaloguemight work.To begin with, there is an obvious sense in which analysis is reasoningback to an apxil - not simply to a hypothesis (or to the 8toptagi6 of aproblem)but to an 'apxi thatis knownto be true.In (say) the theoreticalanalysisof a theoremVx (Px-+Qx), we reasonfrom the Cqro{ievovQx(togetherwith the "given"Px and the principlesof geometry)back tosomethingknown to be true,where this could be eitheran apXiTjf geom-etry absolutely,or an apxilrelativelyto this particular roposition, hatis,one of the givens of the proposition Px or - since Px is typicallya con-junction one of the conjunctsn Px), or something hat has alreadybeendeducedfromsome combinationof these &pXai.n any of these cases itis fair to say that we are reasoning back from the 4qtoiTVevov to an apnthat is known to be true.34 ut, since the argumentbeginsby assuminga4MMoi>evovhich is not (at the outset) known to be true, the inferencefrom the 4irrovuevovo the apXTiannotbe the cause of our knowing the&pxqio be true.Onepossibleway out would be to say thatwe may beginthe"upwardway"from a Cirro{jievovhich we "know" o be true hroughsense-experienceor from authority,but which we do not know scienti-fically, becausewe don't understandwhyit's true. Indeed, n geometryweoften startby believing hat a theorem s true,on the authority f a teacheror of a book, and then apply theoreticalanalysis in order to discover aproof, and so to understandwhy the theorem s true.In such a case, webegin with a 4toV?gVOV which we "know" in a weak sense, reason up toan apXijwhich we know, and reason back down to the inrrov4ievov,ocoming to know it in a strongersense; Plato would call our initial state"trueopinion"rather hanknowledge,and he would describethe wholeprocess as convertingtrue opinion into knowledge by "tying it down"through"reasoningout the cause" (Meno 98a3-4). But again, if we haveno means of recognizing the truth of the &pXijndependentlyof the41toljiEVOV, this processcannotgive us scientificknowledge: t will leaveus with only trueopinionof the &pXij,nd so with only trueopinionof

34 Plato seems not to be interested n the "logical direction"of analysis, i.e. the factthat startingfrom the lntoi?rvov Q, we work back to a principleor a given P suchQ-+P, in the hope of proving, when we reverse the analysis, that P-+Q. Plato speaksas if we just divined P as a plausible startingpoint for proving Q, and establishedthatP-+Q; but Plato thinks that we also examine the consequences of the hypothesis -_P,proving (_P)(_Q), and thus indirectly proving Q-+P.


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220 STEPHEN MENNthe 4iiTouig4vov. This cannot be what Plato means by Socrates' explana-tion of how we can arriveat knowledge n philosophy. ndeed, f we takethe comparisonwith geometricalanalysisseriously, t rules thisout, sincethe successof analysisdependson our nferringosomething hatwe alreadyknow to be true, independently f the analyticalchain of inferences hatlead us to it.But we need to draw a distinction.An analysis terminateswhen wesucceed in inferringsomethingthat we alreadyhabituallyknow, but weneed not actuallyknow it before we infer it from the 4qto{iEvov. Here Iam using Aristotle'sterminologyof actual and habitualknowledge:I amactuallyknowinga theorem f I am currentlyhinkingabout the theoremandunderstanding hy it is true;I have habitualknowledgeof the theo-rem if I am in such a state that, whenever I turnmy attentionto thetheorem, f nothingobstructsme fromthinkingabout t, I will understandwhy the theorem is true. Thus someone who has masteredelementarygeometryalwayshas habitualknowledgeof a large numberof theorems,althoughmost of the time he will not have this particularheoremor itsproof present o his mind. For an analysisto succeed, we must, when wemake the final inference,recognizethat its conclusionis somethingweknow to be true: this means that we must alreadyhave had habitualknowledgeof the conclusion,and so the analytical nference tself cannotbe the causeof our habitualknowledgeof the conclusion,but it may verywell be the cause of our actual knowledge of the conclusion;that is, itmay be the occasion that turnsour attention o thisproposition, ctualizesour habitualknowledge of it, and begins the chain of actualizationsaswe reverse each step of the analysis) which leads to our having actualknowledgeof the theoremwe were tryingto prove.

Indeed, t is in some sense necessarythat,when I am in the processofdoing an analysis,I do not yet have actualknowledgeof the proposition(alreadyhabituallyknown)in which the analysiswill terminate.SupposeI am doing a theoreticalanalysisof the theoremVx (Px-+Qx). Supposethat,in the analyticalchain of inferencesbeginning romthe 'n-ToigeVOvQx (andalso assumingthe given Px), the finalproposition reach is Rx;when I reach the conclusionRx, I recognizethatI alreadyhave habitualknowledgethatVx (Px-+Rx),and therefore,using the given Px, that Rx.Now suppose that the next-to-lastpropositionI reach in the analysis,immediatelybefore Rx, is Sx. If the analysis is step-by-stepreversible,then,as the firststep in reversing he analysis,I will be able to proveVx(PxnRx-+Sx); since I can also prove Vx (Px-+Rx), I can prove Vx(Px-*Sx). But,clearly,I did not have actualknowledgeof the proposition


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PLATO AND THE METHODOF ANALYSIS 221Vx (Px-+Sx) before I drewthe analytic nference romSx to Rx; for, if Ihad, I would have stopped he analysisat Sx, rather hangoing on to Rx.So,even if I hadactual andnotmerelyhabitual) nowledgeof Vx (Px-+Rx),thatknowledgemusthave been somehow"obstructed"nd prevented romproducingactualknowledgeof its consequences.If I had not been thusobstructed omewheredown the line, I would have seen right down thechain of consequences rom Px to Qx, and so I would have known thetheorem immediately,without having to apply analysis to discover aproof. So analysis,if it succeeds,has the psychologicaleffect of remov-ing an obstruction rom some of my habitualknowledge,to allow it tohave its full consequences n actualknowledge.Geometricalanalysis can thus providePlato with a model for philo-sophicaldiscovery, n one sense of "discovery":t does nothing o explaina transition rom nothavinghabitualknowledge o havinghabitualknowl-edge, but it helps to explain the transition rom having merelyhabitualknowledgeto having actualknowledge, that is, the process of removingan obstruction rom our habitualknowledge.But, afterall, this is all wecanexpectfromPlato,sincehe renounces he possibilityof explaining hefirstkind of transition.The point of the account of learningas recollec-tion is just to give up on this, and to say thatwe have alwayshad habit-ual knowledge,but that it has been somehow obstructed,and that we"learn" y removingobstructions ndreawakening he habitualknowledgethat is under the surface of our minds. The two geometricalpassagesofthe Meno servecomplementaryunctions n explaininghow we can cometo have actualknowledge: he firstarguesthat we have alwayshadhabit-ual knowledge,and the second uses the modelof geometricalanalysistoexplain how we can go fromhabitual o actualknowledge.

Analysis infers from a inrrovtEvov, assumed but not known to be true,to some kind of apl'j alreadyhabituallyknown to be true;we come toactuallyknow the apx'il,and thus it becomesavailableas a starting-pointfor demonstratingthe 4nTob'4evov. Analysis is designed to bring some pos-sible &pXiio our attention,and also to bringto our attentiona possibleseries of inferencesfrom this apxil through ntermediatepropositions othe 4Pqo{iEvov; of course, this can occasion our discoveryof an actualproof only if we do have habitualknowledge that the apxl' is true andthat each of the inferential tepsis justified.If we beginfrom a trueopin-ion of the CnTo'U14EVOVbased, perhaps, on the authority of a competentteacher),we havegoodreasonto hopethatwe will reacha trueand usableappi, thus stimulatingrecollection of something we alreadyhabituallyknewbutdid not have present o ourminds;and,as Platosays, "since all


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222 STEPHENMENN

nature is akin, and the soul has learned all things, nothing preventssomeone,once he has recollected ust one thing - and this is what peo-ple call learning fromfindingout all the others" Meno81c9-d3).None-theless,there s an important isanalogybetweengeometricalanalysisandthekindof philosophicalnquiry hat Plato wants it to illustrate. n geom-etry,we are interested n awakeningactualknowledgeof the apXljonlyas a means to discovering a proof of the 4Ptov'4evov; the &p j (in theexampleI havebeenusing, Rx, or Vx (Px-+Rx)) s not in itself somethingespeciallydesirable o know- it is, generally,an obviousfact butone thathad not occurred o us in this connection,or had not seemeduseful as astarting point for proving the 4PTou{jvov.As Plato sees it, the philo-sophicalcase is different:althougha particularnquirer such as Meno)may be more interestedn the posteriorquestion(whethervirtue s teach-able) than in the prior question(what virtueis), so that in a particulardialecticalsituationwe may be led to ask about the apxyilor the sake ofknowing the ,notUoevov, nonetheless Plato thinks that the knowledge ofthe apxil (of what virtueis, and ultimately,of the good) is intrinsicallymuch more desirable hanall the knowledgewe can derive from it. Theknowledge of the &pXnjs a great good, but it is one we alreadyhave,deep within us; but like the food and drink of Tantalus (apparentlyrecalled at Euthydemus 80b-d), it is a possessionthat we are preventedfromusing,and so does not actuallybenefitus. Since this knowledge iesdeeply buriedwithin us, to uncover it and make it availablewould be agreatgood;whereasthe principlesof geometry ie prettyclose to the sur-face, and the greatthingis notto dig themupbut to build somethingwiththem. Despite this difference between the aims of geometry and ofPlatonicphilosophy,Plato finds the method of analysis an encouragingmodel for what he hopes can happenin philosophicaldiscovery.In theSeventhLetterhe says that the knowledgehe aims at "suddenly, ike alight kindled from a leapingfire,comes to be in the soul and then nour-ishes itself' (341c7-d2);but in this same passagehe is warning againstfalse claims of insight,and insistingthat the leapingand kindlingcomeaboutonly "with the maximumof practiceand muchtime"(344b2-3),35

35 There is an untranslatablepun in ptpIi: practice" n the sense of repeatedexer-cise as opposed to theoreticalinstruction(perhapsas a way of leaming to apply theinstruction,but Gorgias 463b2-4 describes rhetoric and relish-makingas "not TExVnbut tigrnpia ca'itp4o "); also just "spending time" (like taurp43sn,he standardwordfor philosophicaleducation throughconversationand companionship);but also "rub-bing," a sense which Plato makes good use of here, with a suggestion of startingafire by friction.


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PLATO AND THE METHODOF ANALYSIS 223throughpatientandrigorous nquiry.WhatPlato is describing s an every-day experience n geometry.Whenanalysis succeeds,something uddenlyhappens,a spark umps,we suddenlyunderstandomethingor see some-thingin a new light, and see how to find what we were lookingfor; butanalysis is also a precisedisciplinethat we can become trained n, andthat must be practicedrigorously and patientlyfor the result to comeabout;and it is accompanied y a rigorousmethodof synthesisforcheck-ing and for discarding alse inspirations.One can only wish there wassomething ike that in philosophy.36Departmentof PhilosophyMcGill University37

36 Long after writing the above, I discovered the following passage in Galen,expressing a similar judgment: "O{5' a&XX1 ctq OEcopixa Ete06v; EI'ppaivEt 'TcIv'rxiiv &vapo'; pVOib;Ti5 &vxiirV ci, 6tav y? n; aV &f cpo?X - wax &p&;ev yap eintiov6o EoTtV, IboEitp K Xiti &kXat aXe86v 'ainacat. cadtot Kav Ei if&gtiavcVxPPOaGVIVtXE,S' a6TO'ry? O XE?IV aup'iaOi ipb; r&yTaa IaX( aveiXeV MWNvat Ta' a1rTiv C'aiperov e`Xoioav, 3; E&prv, T6OaPTxPrEipC0aXtpO;auTwv T&v EbpTEVcV, O6IEpO1K EaTtV ?V Xt015; CCa&tXoaopiv eptUapiopivgo;"(On the errors of the soul, 5,87,14-88,6 Kuehn; repunctuating ollowing Marquardtand De Boer).

37 This paperwas originally read at a conferenceat the University of Chicago hon-oring Bill Tait on the occasion of his retirement.I would like to thank members ofthat audience for useful discussion, especially Bill Tait, Michael Friedman, IanMueller, and Howard Stein; I also received valuable feedback, not in Chicago, fromEmily Carson,Michael Hallett, Rachana Kamtekar,Alison Laywine, a class at McGillUniversity,and an anonymous referee. Years of conversations both with Bill Tait andwith Ian Mueller, and reading their writings, have helped to shape my understandingof Greek mathematics and of its philosophical ramifications.

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