plato and the method of analysis
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Plato and the Method of Analysis
Author(s): Stephen MennReviewed work(s):Source: 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 Analysis
STEPHENMENN
ABSTRACr
Late 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) late
ancient 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 stages
of 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 scientific
knowledgeof the thingsthatare derivedfromthese apXai.
? Koninklijke Brill NV, Leiden, 2002 Phronesis XLVII13Also available online - www.brill.nl
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194 STEPHENMENN
Late ancient philosophers middle-and neo-Platonists,and the Peri-pateticAlexanderof Aphrodisias)are very interested n this processof
reasoningback to firstprinciples.They call it "analysis,"and contrast t
with "synthesis" s reasoningfrom he principles; nd something ike this
description of analytical reasoning remains familiar to us.' But this
descriptions very problematic, ndmy goal here is to begin to get clear
on the problems hatit involves.
To describePlato's or Aristotle'sprocedure f reasoning o principles
as "analysis" s to apply a metaphor rom geometry:"analysis" s the
name for a definite geometricalprocedure,and the neo-Platonistsareconscious thatthey are speakingmetaphoricallyn extendingthe term to
philosophy.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 the
commonElements,for those who want to acquire n geometrya power
of solving problems et to them".2The neo-Platonistswantto claima con-
nection betweenthis geometricalprocedure nd the philosophicalmethod
of arguing o firstprinciples.Proclusactuallysays thatPlato "taught" eo-metrical analysis to the geometer Leodamas(In Euclidemp. 211, cp.
p. 66), apparentlymplying hat Platoinvented he methodof analysisand
passedit on to the mathematicians.3ertainlyPlatodid not inventanaly-
sis; there s good reasonto thinkthat the methodwas usedby Hippocrates
of Chios as early as 430 BC; ProclusroutinelycreditsPlatowith invent-
ing any mathematicaldea or proposition hat is mentioned n a Platonic
dialogue,4and we need not take these ascriptions oo seriously.But it is
worth thinkingabout why Proclus and other Platonists would want to
claim 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 through
the 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 (text
and 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 reportto
the 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 and
when this happens, he geometermustagain proceedmethodically again,not mechanically)by the method of synthesis, to confirmwhat has been
discoveredby 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 obvious
that 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 the
Sphereand 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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196 STEPHENMENN
to regardanalysis as the living core of Greekmathematicalhought,whatthe geometersdid by themselvesand taughtto their students,while most
of theworks they publishandmakeavailable o people outsidethe school
are only the dead husks.If analysisis such a powerfulsource of insight
in 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 hope
was that the proved success of the method n mathematicswould rub off
on the philosophers, r, morecynically, thatthe prestigeof the method nmathematicswould help to justify what the philosopherswere already
doing.6WhatI aminterestedn is whetherPlato,in the fourthcenturyBC,
was already akingthis kindof interest n the methodof analysis.Proclus
likes to think of Plato the philosopheras giving directions o the mathe-
maticians about what problemsto work on, and even as teaching them
mathematicalmethods, but it is more fruitfulto ask what philosophical
inspirationPlatomay have taken from the mathematicians:or it is cer-
tainlyclearthatPlatowas an enthusiastormathematicalrainingand that
he "everywhereriesto arouseadmirationormathematics mongstudentsof philosophy" Proclus n Euclidemp. 66).7Platoand his contemporaries
do not use "analysis," s the neo-Platonistswill, as a generaltermfor all
reasoning o firstprinciplesn philosophyas well as mathematics.ndeed,
Plato never uses the word "analysis" or the verb&vakX_tv)t all. But
that does not show that Plato and othersin the Academy may not have
thoughtaboutor alluded to geometricalanalysis,or takenit as a model
in theirown reasoning.I will arguethat Platodoes, at least once, allude
to geometricalanalysis,and that he at least experimentedwith takingit
as a model for philosophical easoning.First I should say what the method of analysis was. It is standard
to startby commentingon the late ancient definitionsor descriptions
of analysis, since we have nothinglike a definitionof analysis before
about the firstcenturyAD (probably he earliestareHeronandAlcinous;the only extended descriptionis in Pappus, probablythird century).8
Unfortunately,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,"' in
Phronesis, 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 197
misleadingat 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) here
is no real doubt about what analysis was - there are the extant texts of
Archimedes,ApolloniusandPappuspracticinganalysis,and this is not a
lost art,but somethingone can easily train oneself to do on the ancient
model;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 Euclid
ElementsXIII,1-5,cited in a previousnote, begin with brief definitions of analysis and
synthesis (thoughthe definition of synthesis is corrupt),which may conceivably be the
earliest extant. The only account that might actually be helpful is Pappus': "What is
called the Domain of Analysis, my son Hermodorus, s, in sum, a special resource that
was prepared, after the discovery of the common Elements, for those who want to
acquire in geometrya power of solving problems set to them; and it is useful for this
alone. It was written by three men- Euclid the Elementarist,Apollonius of Perga,and
Aristaeus 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 in
analysis we assume what is sought as if it has been achieved, and look for the thing
from which it comes about, and again what comes before that, until by regressing in
this 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 last
in the analysis to have been achieved already,and by arranging n their natural order
as 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 been
prescribed [for us to construct], and is called 'problematic.'In the theoretickind, we
assume what is sought as being, i.e. as true, and then proceed, throughwhat follows
as 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 of
the analysis; but if we encountersomething agreed to be impossible, the problemtoowill be impossible. Atoptag6o is adding a condition [reading npoa5txaroX'i for
npo8taatoXi] 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 MENN
descriptions,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 descriptions
of the logic of analysis can be improvedon. A basically correct ogical
descriptionhas been given by Hintikkaand Remes (along with, unfortu-
nately, much that is incorrect),and I will make use of theirwork, as well
as of the extant ancient examples and descriptions f analysis.
The most commonaccountof analysis, bothin the ancientsourcesand
in modernreconstructions,goes somethinglike this. We are trying toprove a propositionP, the Clqtoiu'evovr thing-sought.As a heuristic
toward indinga proof,we assume the 4iyrol4tevov as if it were known
to be true, and then draw inferences fromthis assumption; he analysis
terminates andsomething"clicks" when we derive eithera proposition
known to be true(fromthe principlesof geometryand from theoremswe
have alreadyproved),or else a propositionknown to be false. If we can
inferfromP to a propositionknown to be false, then we have proved1 P
by reductioad absurdum. f we have inferred rom P to a propositionR
thatwe know to be true, henwe cantryto reverseeachstepof the deriva-tion of R from P: if this succeeds,then the proofof R, togetherwith the
derivationof 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:but
it does very frequentlyhappen that steps of geometricalargumentsare
reversible 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 triangleABC
has equal base angles, it is isosceles"),and in carryingout the analysis
intelligentlywe will try to avoid obviouslynon-reversibleteps (e.g. "if
triangleABC 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 it
constructsa plausibleoutline for a proof,and we can then try to fill in
the steps.Some ancient(and modem) writers,botheredby the logical gapbetweena successfulanalysisof P and a successfulproofof P, say instead
that 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 out
analyses and syntheses - these instructions are logically inexact, but show that the
authors, and the teachers and studentswho used their books, did habitually carryout
the practice of analysis.
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PLATO AND THE METHOD OF ANALYSIS 199
would follow," so that as soon as we reach a propositionknown to betrue,we would have a guaranteedproof of P. But it is historicallyclear
thatanalysiswas alwaysa deductiveprocedure tartingwith the 4itouie-
vov, and this is also muchmore practicableand heuristicallyuseful. The
only 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 propositions
which, 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 hardto
see why analysiswould be heuristically aluable.Whileanalysis ooks for
a proof of a proposition by assuming the 4Tyou'evov as if it were known
and 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 theorems
are what we would call propositions: theorem s a statementasserting
that 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 a
problem 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, shapeand
positionwe recognizethat we can determine rom the givens of the prob-lem alone (togetherwith the principlesof geometryand with propositionswe have already proved), independentlyof our assumptionabout the
411tro1)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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200 STEPHENMENN
known, and then reversing he reasoning o demonstrate he IJTo{iPEvov,
are deliberatelyvague enoughto include this case; we miss problematic
analysis,and thus badly misrepresenthe analyticmethod, f we identify
the 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, then
ZABC = ZACB"). In analyzing this theorem, the 411o-14EVOV will not be
the propositionVx (Px-+Qx),but rather he propositionQx. We take as
givenan arbitrary suchthat Px; andthenwe also assume he 4fltOu'eVOV
Qx as if it were known to be true;we then draw inferencesfrom the
4nrobVg?vovQx, makinguse of the "given"Px as well as of the principles
of 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,from
Px as well as the principles of geometry and other propositions already
proved,then we can try to reversethe analysis.We could describe this
reversalas turning he derivationof Rx from Qx into a derivationof Qx
from 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 deriveQx
from Rx, it is more accurate o say that we are turninga derivationof
RxrPx fromQxrPx into a derivationof QxrPx fromRxroPx. f we can
do this, we have a proof of the theorem Vx (Px-+Qx):first assume an
arbitrary such thatPx, then infer from Px to Rx, then infer fromPxnRx
to Qx, then concludethat Vx (Px-+Qx)."Or the analysiscould have a
negativeresult: eitherwe could infer from QxnPx to a propositionRx
thatwe 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 that
Vx (Px-+Qx) s false, but we can conjecture hatthe analysis s reversible
and 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 than
one (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 in
Greekgeometry,since they all work by provingthe instanceof the propositionset out
in the c0e'At; and toptag'S;, and then inferringto the universalproposition.
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PLATO AND THE METHOD OF ANALYSIS 201
theanalysisstartsby 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 should
be heuristicallyany moreuseful thanarguing"forwards."'2
We can try to give a similarlogical descriptionof problemsand of
problematic nalysis.This shouldcomewith a warning abel, since, in the
case of problemsmore clearly than in the case of theorems,we are in
conflictwith 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-+3y
Qxy) [or in prenexnormalformVx3y (Px-+Qxy)]- so the problem"to
inscribein a given circle a trianglesimilarto a given triangle"can be
rewritten s "foranycirclex andanytrianglex', there s a triangley such
thaty is inscribed n x andy is similar to x"'. Any demonstration f the
problemalso demonstrateshis universal-existential roposition;but the
only acceptabledemonstrations f the problemare constructivedemon-
strations, hatis, procedureshatshow how to construct he Ctycol,uevov
fromany given x suchthatPx, accompanied y a proofthatQxy.'3Greektheorems,as opposed to problems,never involve existentialquantifiers;
universal-existentialropositions ppear n Greekmathematics nly in the
guise 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 abstractmagnitude
or 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 group
beginning VIII,8, on how many numbers"fall" in continued proportionbetween two
given numbers,and IX,20, "prime numbersare more than any assigned multitudeof
prime numbers"),as well as X,1 ("two unequalmagnitudesbeing set out, if from the
greaterthere 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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202 STEPHEN MENN
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 that
couldhandlemultiplyquantifiedpropositions.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 from
knowingthatPx, 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 the
given Px, then infer a proposition3z Rxz which we know to be true (or false) on the
basis 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 203
theoreticanalysis,we mightreacha negativeresult, discoveringthattheassumptionQxy contradictshe given Px: thiswould happen f (as before)
we construct ome(p(x,y) satisfyinga relationRx(p(x, ), andthen,instead
of (as in the positivecase) recognizing hat [PxrRxz],[Pxn(z = (x))],
we recognizethatPxrRxz implya contradiction, nd thatthe problem s
thereforeunsolvable. But more often than reaching a purely negative
result,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 to
deduce Qxy. In this case analysiswill have revealedthe StoptaRgo,he
necessaryandsufficient onditions or theproblem o be solvable,and this
was indeedone majoruse of problematicanalysis in Greekgeometry. t
may also happenthat we do not know how to lead the analysisto either
a positiveor a negativeresult,butthatwe can reducethe problem"given
x 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 provingthatPxnRxz
implyQxx(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 findingtwo
meanproportionals s an earlyapplicationof problematicanalysis.
Problematic nd theoreticanalysisareformallysimilarenoughthat(in
both ancient and modem accounts) they are often coveredby the same
generaldescription: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 heuristicallymore
fruitful.That problematicanalysis is older is naturalenough, since it is
an 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 means
finding a construction-procedure(x, z) and proving Vz ([PxnRxz]-+Qxx (x, z)).
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204 STEPHEN MENN
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(one
commonuse 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.
II
Given this descriptionof geometricalanalysis, as it was practicedin
Plato'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 it
under 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 known
the name"analysis," ndtheancient ogical description f analysis, s that
we find the name and the description n three passages of Aristotle(two
of them plausibly written in Plato's lifetime). Aristotle had no special
expertise n mathematics oing beyondhis Academiccolleagues, and the
texts 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 alreadyknows
what analysis is, and have oftenbeen misunderstood y both ancient andmodemreaders).Aristotle s reflectingmathematical nowledgethat was
currentin the Academy, and using it to make his own philosophical
points;and the same mathematical nowledgewas available for Plato to
use 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 argue
that in at least one extant text - the Meno - Plato does refer to the method
of analysis, althoughnot by name; then I will commenton the harder
questionof 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 Elenchi
16 175a26-28.The PosteriorAnalyticspassage says: "If it were impossi-
ble to show [86t4ai = deduce] something ruefromsomething alse, then
analysis [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,these
things (say, B) are true[i.e. I can deduceB fromA], which [sc. B] I [in
fact] know to be true.Then from these thingsI will show that that [sc.A] is true.'7Now mathematicalarguments] onvert moreoften, because
they 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 the
passage (but NB "true" in the first sentence is the unambiguousaikXiOk);f this is
right, Aristotle is talking about theoreticalanalysis. But it is just possible that "A is
ov 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 problematic
analysis. Nothing much hangs on this; either (as I will assume) Aristotle is talking
about theoretical analysis, or he is assimilating problematicand theoretical analysis so
closely 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 arguments
with (some or all) false premissesbut true conclusions.He then illustrates
this with an appeal to his readers' or hearers')experience: ife wouldbe
mucheasierin geometry f thiswere not the case, i.e. if every valid argu-
ment to a true conclusion also had (all) true premisses, since then an
analysis could always be converted nto a syntheticdemonstration. am
tryingto demonstrate 4ntouigivovA;'9 so, for purposesof analysis, I
assume A as if it were known to be true, and deduce B, which I in fact
know 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 valid
argumentromB to A; and since we know thatB is true,this would give
a demonstration f A. In fact, since some argumentsdo lead fromfalse
premisses o true conclusions,not all analysesconvert;but Aristotleadds
that,as a matterof mathematical xperience,analysesoftendo convert.I
am not sureexactlywhatto makeof Aristotle'sexplanation,namelythat
mathematical nferences ake definitionsrather han accidentalpropertiesas theirpremisses;butone illustrationwouldbe that,if an inference say,
thata certainequalityholds) dependson the premissthat a certainangle
is 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 to
be possibleto reversethe inference, o infer thatif the equalityholds the
angle is right, since if the angle was slightly more or less than a right
angle, 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 proof
that A. We infer from A to B, which we recognize to be true. But this does not yet
show 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. We
realize 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, and
Themistius fills in the ellipses incorrectly,apparentlybecause he is unfamiliarwith the
practice of analysis and does not realize that the analyst takes the 4toUicvov (here
A) as a premiss in the analytical stage of his argument. Themistius is apparentlyassuming that analysis must be something like rhetorical inventio, a practice with
which he is much more familiar.
'9 Assuming that A is a proposition rather han an object. We could rewrite all this
in the case where A is an object.
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PLATO AND THE METHOD OF ANALYSIS 207
like this is why theorems ike the Pythagoreanheorem end to have trueconverses,althoughof coursethis does not hold in every case.20
This passage rom hePosteriorAnalyticss enough o show thatAristotle,
like later Greekwriters,conceives of analysis as a process in which a
tqxoiLuevovs assumedto be true,and deductionsare made from it until
we deducesomething ndependently nownto be true;we thentry to con-
vert the argument into a demonstration of the inTo{'EVOV.What may not
be clear is what part of this process is called "analysis."Aristotlesays
thatif all arguments onverted,T6o vaX{Etvwould be easy: this suggests
that &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's
usage is in fact the same as later writers', as is shown by Sophistici
Elenchi 16 175a26-28:"it sometimeshappensas in 8taypa'ija-a ['dia-
grams' but also 'geometricproofs']: for there too sometimes,after we
have analysed [&vaX{6avte;Iwe are unable to synthesize [ouvOeivat]
again."Hereava&kut;ando{vOecn;areclearlytwo successivestages,and
the difficultyof passingfrom&va&XuGt;o aCUv0eoi;s what the PosteriorAnalyticspassage calls the difficultyof "converting." o when Aristotle
says in the PosteriorAnalyticsthat "it would be easy to &vakc6av",e
must mean not simply that it would be easy to find an analysis of a
4n11?O{UjVOV,but that it would be easy to find a good analysis,where a
good analysis is one that can be converted nto a demonstration f the
tnToitevov. (It is always trivial to give some formally legitimate but
mathematically 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 analysis
20 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 the
argumentthat 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. we
are arguing from Px to Qx, neither of which is properly speakinga proposition,since
they 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 argumentis
very 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 conjunction
BqCrlC2 to A.
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208 STEPHEN MENN
andsynthesis 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 which
is 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 itself
will come about, until they come to the firstcause, which is last in discovery. Forthe person who deliberates seems to be inquiring and analyzing [4InrEiV0
avaXi)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 specifically
of problematic analysis, and using it as a model to describe practical
reasoning: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 romthe
4TnOiO.gvov,which we recognize as something that is determined by the
data 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 the
4tovi'Pjevov;hen,in reversing heanalysis,we construct ach
of the subsequenthingsout of this firstthing,in the reverseof the orderin which we found them in the analysis, until we have constructed he
4iyoiu'gvov. The analysis has succeeded only when we have inferred from
the ntrovIrvov ack to a "firstcause"or &pxil,meaningnot a proposition
we 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?xaTov
is (for instance) a triangle.To ev Toig actrucoi; ''aGrTovmight mean simply a
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PLATO AND THE METHOD OF ANALYSIS 209
These passagesfrom Aristotle show that it was possibleto presupposea familiaritywith the practiceof geometricanalysis in Academic circles
(in Plato's lifetimeand in the subsequentdecades);thatthe practicewas
known by the name"analysis," nd thatessentiallythe same (inadequate)
logical descriptionof analysisthat we find in Pappuswas alreadyavail-
able; and, finally, that Academic philosopherswere interested n using
geometricalanalysis,so described,as a model forphilosophical and prac-
tical) reasoning.So, even thoughPlato never uses the word"analysis,"he
and 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),and
make whateverpoint he might want to make about the relationbetween
this kind of mathematical easoningand reasoning n philosophy.I will
now arguethat Plato does, once, so allude to analysis,in the secondgeo-
metricalpassage of the Meno (86e4-87b2); and then I will offer some
speculationsaboutwhat philosophicalpoint Plato wantedto makeby the
analogywith 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 to
inscribe this area in this circle as a triangle. [A geometer] might say, "I don't
yet know whether this [area] is such [as to make the constructionpossible], but
I think I have as it were a hypothesis that would help towards the question, as
follows: 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 one
thing 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[the
area] in the circle, whether it is impossible or not."
Here Plato is consideringa geometricalproblem,"to inscribe in a given
circle a triangleequalto a given area"; n fact the hypothesishe gives is
designedto solve the morespecificproblem"toinscribe n a given circle
an isosceles triangle equal to a given area."22 he hypothesisthat Plato
mathematical particular, hough it seems odd to posit a special quasi-sensory ability
to 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 we
already know how to construct from the givens.
22 The special problem is equivalent to the general problem in the loose sense
that 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 an
isosceles 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. Since
the 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 AB
if 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 211
CDBE,is also half of the triangleCGB.So CGB is equalto CDBE,whichis equalto the given areaX. So CGB is an isosceles triangle nscribed n
the given circle Y and equal to the given areaX: which is what was to
be found.
Plato explicitlycites this example,not as an exampleof analysis,but
only as an exampleof how a geometermightreasonfrom a hypothesis
in answeringa given question.The questionhe cites proposesa problem
rather hana theorem,and a full answerwouldbe a solutionto the prob-
lem: that is, the aim is not simplyto answer the question"is it possible
to inscribe hisareain this circle as a triangle?"with yes or no, butrather,in the case where the answer is yes, to give a construction-procedure
showing 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 geometeroffers
to answerthe question rom thehypothesis"thegiven area can be applied
to the diameter n such a way that it falls shortby a figuresimilar to the
appliedarea,"he is not simplyclaimingthatthe originalproblem s solv-able if and only if the application-of-areasroblem s solvable,but also
offeringa 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; determining
whether he application-of-areasroblemcan be solvedfor the given area
and the given line, and to give a construction-procedureor solving it
where it can be solved. WhenPlato recommends he geometers'practice
of answering"froma hypothesis,"he is recommendingacklinga difficultquestionby reducing t step-by-step o more basic questionsuntilwe can
answer t directly:and this is the lesson Socratesdrawswhen, in answer-
ing Meno's question"is virtue teachable?"romthe hypothesis"virtue s
knowledge,"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: the
answerwould be "if and only if the given area is less than or equal to an equilateral
triangle 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 the
hypothesisturnsout 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 follows
that 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 teachable
belongs 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 213
a rectangle n such a way that it falls shortby a figuresimilar to the ap-plied area"gives a sufficientas well as a necessarycondition or solving
the problem"to inscribe n the circleY an isosceles triangleequalto X."
This is not only an easy and straightforwardse of analysis,but also
a very typicalone. In fact, it seems to be partof a systematicprogramof
reducingproblemsof all kinds to problemsof applicationof areas,in the
hopethattheseproblems 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 in
extreme and mean ratio"; his in turn can be reducedby analysis to theproblem"to applya squareto its own side in the form of a rectangle n
such a way that it exceeds by a square,"and this in turncan be reduced
by analysis to the problemof finding a mean proportional,and thus
solved. Probably beginning from the analysis of the regularpentagon,
earlyGreekgeometersdeveloped techniques or solving a broad class of
problems f applicationf areas:Proclus InEuclidem 19-20)citesEudemus
as attributinghese techniques o "the Muse of the Pythagoreans"i.e. to
the tradition romHippasus o Archytas)and Euclidpresents heir results
in developed orm in ElementsVI. Theoriginalproblemswould have been"toapplya given area to a given line, in the form of a rectangle, n such
a way that it exceeds [or falls short]by a square,"but the techniques or
solvingtheseproblemscan be generalized o solve "toapplya given area
to 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 "to
applya given area to a given line, in the form of a parallelogram,n sucha way that it exceeds [orfalls short]by a parallelogramimilarto a given
parallelogram,"which is the problem that Euclid solves in Elements
VI,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 f
a 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 expositionmakesit
obvious that his constructions (and his toptaFgo) were first discovered by
analysis.26The application-of-areas roblemthat Plato proposes as his
26 In fact the proof of VI,27, giving the S&optajo6for the falling-short problem
VI,28, is a disguised analysis (and thus, in a sense, the earliest extant analysis). What
Euclid 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;ut
Plato's problemis more difficult,and Euclid does not discuss it in the
Elementsbecause it cannotbe reduced o findinga mean proportional r
solved by ruler-and-compassonstructions.However, thereis a solution
usingconics,which Platomaywell have knownwhen he wrote theMeno,and which was certainlywithin the capacityof Greekgeometersat least
least by mid-fourth-century.27ut whetherPlatoknew the solutionor not,
he would have seen the problemas part of a promising programfor
finding&toptagoi of any given construction-problemnd for solving anyproblemwhen it can be solved.28
Thus Plato alludes at Meno 86e4-87b2 to the method of analysis,
and more specificallyto the programof reducingconstruction-problems
throughproblematic nalysis;and he holds up the program f analysis as
a methodologicalmodel for philosophical nquiry.But Plato does all thiswithoutever using the word "analysis" thoughhe must have known the
word), and withoutdescribingclearly either the logic of the method in
general or the geometryof the case he describes: he does not explain
eitherhow his application-of-areasroblemwas derived romthe originalproblem,or how it would help to solve the original problem,or how it
might itself be solved; indeed, he does not describe either the original
problem or the "hypothesis"clearly enough for anyone who did not
alreadyunderstand he problemPlato is describing o understand im.29
excess, the corollary VI,30, "to divide a line in extreme and mean ratio."Euclid does
not use VI,30 to construct the regular pentagon, because he has already done it in
IV,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 the
other segment": but this is simply VI,30 reformulated,and re-proved,in such a way
as 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 (see
Heath, 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 brought
forth; the conjunctionis apparentlythe subject of a singularverb.I Hence the gross misunderstandings f this passage e.g. in Jowett's and Grube's
translations, and by Bluck in the appendix to his edition of the Meno (Cambridge,
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PLATO AND THE METHOD OF ANALYSIS 215
This compressedpassage, alluding without explanation to what wouldhave been the forefrontof mathematical esearch, mmediatelyafter an
extremelyslow and patientdiscussion of a trivial mathematical act, was
not meant to be understoodby most of its readers;Meno, who asks no
questions at all at this crucial turningpoint in the argument,obviously
does not understandwhat is going on. In fact, the passageis perfectproof
for Gaiser'sthesis that Plato's dialoguesallude to doctrines hat they do
not fully explain, in an attemptto rouse Plato's readersto seek further
enlightenmentn 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 be
philosophicallymportant nd which they would understandf they came
to study geometry n the Academy.3'Butwhat was the philosophicalpay-
off supposed o be?
III
As I said at the beginning,analysis appealedto philosophersbecause itpresenteda method for discovery. In the Meno, the progressof inquiry
had been frustrated:Socrates cannotanswerMeno's question"is virtue
teachable"until Menotells him what virtue s, andMeno,unable o define
virtue afterrepeatedattempts,gives what Socratescalls an "eristicargu-
ment"(80e2) to show that one cannot searchfor something f one does
not alreadyknowwhat it is. So Socrates' mmediate ask is to show Meno
how inquiry s possible, by giving him successfulmodels of it: this is the
pointof 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, History
of 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 have
introduced 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 things
mathematical."
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216 STEPHEN MENN
(e.g. what is virtue?):we have encounteredX before this life and so havedim memoriesof it, which will help us to recognizethe thingwhen we
are 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 original
questionwhethervirtue s teachableor acquiredn some otherway (86c7-d2), that is, to ask what virtue is like [0i10v ?aTI] before determining what
it is [ti ?aort].Surprisingly, lthoughSocrates had earlierinsisted that itwas impossibleto inquire n this way, he now immediatelygives in, and
offers 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 until
we reach a question hat we can answerdirectly.In a sense, Socrateshasnot concededmuch on the logical priorityof the ti Eart to the loi6ovc'aTl
question,since he continues o insist that we cannotknowwhethervirtue
is teachable until we can demonstrate he answer from a knowledgeofwhatvirtue is. But as a matterof heuristics,Socrates s concedingthatit
may be usefulto begin with the logically posteriornioi6v9art question, n
the hope of discovering an answer both to the Tiirtt and to the noiov E'at
questions.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 questioning
that promptsrecollection is an application of the method of analysis. In one sense he
is in fact doing this, since part of what prompts recollection is the refutationof the
answers 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 217
What is much less clear is how this is supposedto work. If we startby not knowingthe &pXaihat we need for demonstratinghe answer to
our question(here "is virtueteachable?"),how is analysis or its philo-
sophicalanalogue going to help us find the demonstration? aterGreek
philosophers dentifyanalysiswith arguing"upward"o the appXai,nd
suggest that we can firstargue"up"from posterior hings to the &pxac,then argue back "down" rom the apxai to the posterior hings.But this
kind of argumentwill not give a demonstration, nd so will not give us
knowledgeof 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 said
to 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,when
absurdum s just an analysis with a negative result. However, the positive result that
the side of the eight-foot square is the diagonal of the four-foot square is not shown
as being reached analytically: f Socrateshad not alreadyknown the answer,and asked
the boy the rightstring of questionsbased on knowledge of this answer,the boy might
never have discovered it. But Plato could instead have shown this result as being
reachedby analysis: suppose a squareof area eight squarefeet has been found; draw
the diagonals, dividing the square into four equal isosceles right triangles, each of
which is thus of area two square feet. At this stage, probably,something clicks, and
we 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 to
constructpurely 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 the
side of the 4toiotrvov square.And we can then reverse the analysis to constructthe41TtOi4LEVOV squareof area eight squarefeet from the isosceles right triangleof area
two squarefeet, by constructing our equal and similar isosceles right trianglesaround
the same vertex. The diagramthat would result is the diagram that Socrates in fact
draws, and Plato could have represented t as the result of reversingthis analysis. But
Plato probablythought thatthe methodof analysis was too important o waste on such
a trivial example: its power is betterbroughtout by showing how it can contributeto
a difficultproblembelonging to currentor recent mathematicalresearch.
3 Carl Huffmangives a useful collection and discussion of evidence on the early
history 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 assumed
that the appropriate beginning must be something that the listeners will agree
to. 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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218 STEPHENMENN
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. Thus
the 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'' apXilv
avul6OEtov51Ob6-7),.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,ndwhatamoresufficient
apxyjmight 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 verified
by 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 &pxj
at 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 shed
some light on how Plato thought ts philosophicalanaloguemight work.
To begin with, there is an obvious sense in which analysis is reasoning
back to an apxil - not simply to a hypothesis (or to the 8toptagi6 of a
problem)but to an 'apxi thatis knownto be true.In (say) the theoretical
analysisof a theoremVx (Px-+Qx), we reasonfrom the Cqro{ievovQx
(togetherwith the "given"Px and the principlesof geometry)back to
somethingknown 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 alreadybeen
deducedfromsome combinationof these &pXai.n any of these cases it
is fair to say that we are reasoning back from the 4qtoiTVevov to an apnthat is known to be true.34 ut, since the argumentbeginsby assuminga
4MMoi>evovhich is not (at the outset) known to be true, the inference
from the 4irrovuevovo the apXTiannotbe the cause of our knowing the
&pxqio be true.Onepossibleway out would be to say thatwe may begin
the"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 geometrywe
often startby believing hat a theorem s true,on the authority f a teacher
or of a book, and then apply theoreticalanalysis in order to discover a
proof, and so to understandwhy the theorem s true.In such a case, we
begin with a 4toV?gVOV which we "know" in a weak sense, reason up to
an apXijwhich we know, and reason back down to the inrrov4ievov,o
coming 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 have
no means of recognizing the truth of the &pXijndependentlyof the
41toljiEVOV, this processcannotgive us scientificknowledge: t will leave
us 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 fact
that startingfrom the lntoi?rvov Q, we work back to a principleor a given P such
Q-+P, in the hope of proving, when we reverse the analysis, that P-+Q. Plato speaks
as if we just divined P as a plausible startingpoint for proving Q, and establishedthat
P-+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 MENN
the 4iiTouig4vov. This cannot be what Plato means by Socrates' explana-tion of how we can arriveat knowledge n philosophy. ndeed, f we take
the comparisonwith geometricalanalysisseriously, t rules thisout, since
the successof analysisdependson our nferringosomething hatwe alreadyknow to be true, independently f the analyticalchain of inferences hat
lead us to it.
But we need to draw a distinction.An analysis terminateswhen wesucceed in inferringsomethingthat we alreadyhabituallyknow, but we
need not actuallyknow it before we infer it from the 4qto{iEvov. Here I
am 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 the
theorem, f nothingobstructsme fromthinkingabout t, I will understandwhy the theorem is true. Thus someone who has masteredelementary
geometryalwayshas 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 we
make the final inference,recognizethat its conclusionis somethingweknow to be true: this means that we must alreadyhave had habitual
knowledgeof the conclusion,and so the analytical nference tself cannot
be the causeof our habitualknowledgeof the conclusion,but it may verywell be the cause of our actual knowledge of the conclusion;that is, it
may 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 actual
knowledgeof 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-ToigeVOv
Qx (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 221
Vx (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 the
theorem 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 we
canexpectfromPlato,sincehe renounces he possibilityof explaining he
firstkind 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 to
actuallyknow 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 usable
appi, 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 isanalogybetweengeometricalanalysisand
thekindof philosophicalnquiry hat Plato wants it to illustrate. n geom-
etry,we are interested n awakeningactualknowledgeof the apXljonly
as a means to discovering a proof of the 4Ptov'4evov; the &p j (in the
exampleI havebeenusing, Rx, or Vx (Px-+Rx)) s not in itself something
especiallydesirable o know- it is, generally,an obviousfact butone that
had 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 particular
dialecticalsituationwe may be led to ask about the apxyilor the sake of
knowing the ,notUoevov, nonetheless Plato thinks that the knowledge of
the apxil (of what virtueis, and ultimately,of the good) is intrinsically
much more desirable hanall the knowledgewe can derive from it. The
knowledge of the &pXnjs a great good, but it is one we alreadyhave,deep within us; but like the food and drink of Tantalus (apparently
recalled at Euthydemus 80b-d), it is a possessionthat we are prevented
fromusing,and so does not actuallybenefitus. Since this knowledge ies
deeply buriedwithin us, to uncover it and make it availablewould be a
greatgood;whereasthe principlesof geometry ie prettyclose to the sur-
face, and the greatthingis notto dig themupbut to build somethingwith
them. Despite this difference between the aims of geometry and of
Platonicphilosophy,Plato finds the method of analysis an encouraging
model for what he hopes can happenin philosophicaldiscovery.In theSeventhLetterhe says that the knowledgehe aims at "suddenly, ike a
light kindled from a leapingfire,comes to be in the soul and then nour-
ishes itself' (341c7-d2);but in this same passagehe is warning against
false claims of insight,and insistingthat the leapingand kindlingcome
aboutonly "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 the
instruction,but Gorgias 463b2-4 describes rhetoric and relish-makingas "not TExVnbut tigrnpia ca'itp4o "); also just "spending time" (like taurp43sn,he standardword
for philosophicaleducation throughconversationand companionship);but also "rub-
bing," a sense which Plato makes good use of here, with a suggestion of startinga
fire by friction.
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PLATO AND THE METHODOF ANALYSIS 223
throughpatientandrigorous nquiry.WhatPlato is describing s an every-day experience n geometry.Whenanalysis succeeds,something uddenly
happens,a spark umps,we suddenlyunderstandomethingor see some-
thingin a new light, and see how to find what we were lookingfor; but
analysis is also a precisedisciplinethat we can become trained n, and
that must be practicedrigorously and patientlyfor the result to come
about;and it is accompanied y a rigorousmethodof synthesisforcheck-
ing and for discarding alse inspirations.One can only wish there was
something ike that in philosophy.36
Departmentof Philosophy
McGill 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>iav
cVxPPOaGVIVtXE,S' a6TO'ry? O XE?IV aup'iaOi ipb; r&yTaa IaX( av
eiXeV 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 Marquardt
and 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 of
that audience for useful discussion, especially Bill Tait, Michael Friedman, Ian
Mueller, and Howard Stein; I also received valuable feedback, not in Chicago, fromEmily Carson,Michael Hallett, Rachana Kamtekar,Alison Laywine, a class at McGill
University,and an anonymous referee. Years of conversations both with Bill Tait and
with Ian Mueller, and reading their writings, have helped to shape my understanding
of Greek mathematics and of its philosophical ramifications.