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MathematicsandPractice:Diderot andd'AlembertArgueProbability JEFFLOVELAND Around1660,fromthedisreputabledarknessofintuition,legaljargon, andeverydayhabit,thereroseanewbranchof"mixedmathematics" whichwouldtroubleandanimateintellectualcultureforcenturiestocome— probabilitytheory.1Fromamodernperspective,theearlyformofthistheory, theclassicaltheory,saidtohavelastedupuntilroughlythemid-nineteenth century,wasanunhappymedleyofseveraldisciplines,especiallymathematics, economics,andinformalpsychology,andegregiouslyconfoundedobjective probability(arepresentationofidealbelief,ascalculatedthroughobserved eventfrequencies)withsubjectiveprobability(ameasureofthefirmnessof actualbelief).2Thislastambiguity,notfullyrecognizedtillthemid-nineteenth century,generatedparadoxbutalso,conversely,heightenedprobability's perceivedinterestandvaluebymakingitapromisingfoundationfor psychology,sociology,andotherbuddingsocialsciences.Objective probabilityallowedmathematicianstospeakofrightmodesofthinking,while subjectiveprobabilitygavethemacoverincasetheywereaccusedofbeing toonormativeandnotrealistic.Howmuchdescriptionandhowmuch prescriptiontheEnlightenment'snascentscienceshumainesweremeantto containwascamouflaged,asinsomanyEnlightenmentschemes,bythe epoch'sconfusionoftheidealandactual,ofthesub-andobjective,butwhen probability'senthusiasts,whetherDanielBernoulliortheMarquisdeCon- 99 100/LOVELAND dorcet,turnedtheircalculationstothewholeofsociety,onecouldbesure thatatleastaniotaofreformativezealfueledtheirmathematics. Thearticle"Probabilité,"writtenbyDenisDiderotfortheEncyclopédie, illustratesboththedeepambiguitiesandtheambitionsofearlyprobability theory.Theopeningparagraphistypicalofearlyprobabilists'effortstodis- countinadvanceanyobjectiverealitytheirsubsequentdiscussionsmayseem togranttotheideaofprobability:"Toutepropositionconsidéréeenelle- mêmeestvraieoufausse;maisrelativementÃnous,ellepeutêtrecertaineou incertaine."3Probability,wearetold,isamentalillusion,anunfortunate consequenceofourlimitedknowledge.Inevitably,however,asthearticle moveson,itbecomesclearthatprobabilityistobejudgednotwithrespectto actualsubjectivecertaintyoruncertaintybutwithrespecttoanidealizedset ofbeliefs,thoseoflespersonnessagesetprudentes.Furthermore(although thisisunstatedtilllateinthearticle),conformitytoobservedeventfrequen- ciesiswhatmakesajudgmentofprobabilityareasonablejudgment.Probability ,itseems,hasbeensnatchedfromthepsycheandfoistedonnature. Likeallearlyadeptsofprobabilitytheory,Diderotappearstobemostly oblivioustothetwodifferentsensesofthewordprobability.Midwaythrough thearticle,followingJacquesBernoulli,4hehandilydistinguishesdeductive probability,basedoncausesandprinciples,fromprobabilitiesknownfrom experiencewithnature—aroughcharacterization,itmightseem,oftheob- jectiveandsubjectivefacetsofprobability—butindefininghistwoperspec- tivesDiderotmakesitclearthatbothhavemoretodowithmathematics, logic,andobservedeventsfrequenciesthanwithactualsubjectivedegreesof belief.Openinglinesexcluded,probability,forDiderot,ismainlyobjective. Thesevenrulesgoverningthecalculationofprobabilitieswhicharelisted inDiderot's"Probabilité"signaltheextenttowhichprobability,likeFrancis Bacon'sinductionorDescartes'sdeduction,couldbeconceivedasacenter forallvalidknowledge.Ruleonegivespriorityto"evidence"(logicalcer- tainty,inthemedievalsense)inallcasesinwhichthisisattainable.Rules twothroughfivespecifynon-mathematicallywhatseparatesreasonablefrom unreasonableprobabilityjudgments.Reasonablepeopleconsiderthepar- ticularratherthanthegenericfeaturesofeventsunderscrutiny,theyweigh possiblegainsagainstrisksoflosses,theyalwaysattendtotheirneighbors' opinions,andtheyproportiontheirbeliefstocomputedprobabilities.We have,tothispoint,aquitegeneralmethodfordiscoveringknowledge,equally applicable,Diderotstateslater,topoliticaleconomy,thestudyofnature,and everydaylife.Thelasttworulesgofarther,linkingtheknowledgeideally acquiredinrulesonethroughfivetoimmediatepracticeinanimperfectworld: 6°.Puisqu'iln'estpaspossibledefixeraveccetteprécisionquiseraità désirerlesdegrésdeprobabilité,contentons-nousdesÃ-peu-prèsquenous MathematicsandPracticeI101 pouvonsobtenir.Quelquefois,parunedélicatessemalentendue,l'on s'exposesoi-même,&lasociété,Ãdesmauxpiresqueceuxqu'onvoudroit éviter;c'estunartquedesavoirs'éloignerdelaperfectionencertains articles,pours'enapprocherdavantageend'autresplusessentiels&plus intéressans. 7°.Enfinilsembleinutiled'ajoutericiquedansl'incertitudeondoit suspendreÃsedéterminer&Ãagirjusqu'Ãcequ'onaitplusdelumière, maisquesilecasesttelqu'ilnepermetteaucundélai,ilfauts'arrêterÃce quiparaîtraleplusprobable;&unefoislepartiquenousavonsjugéle plussageétantpris,ilnefautpass'enrepentir.("Probabilité,"394). Pendingcompletionoftheperfectlycertainphilosophicaldomicileopti- misticallyevokedinhisDiscoursdelaméthode,Descartes,muchearlier, haddecidedtoliveinaprovisionaldwelling,ofwhichthefoundationswere fourrulesofaction.5WhilereminiscentincontentofDescartes'sfouraxi- oms,Diderot'slastrulesarequitedifferentinspirit,refusingastheydoto suggestanydefinitetermtotheperiodofuncertaintytheytrytoremedy. Theirpragmaticvitiationofthepreviouslyoutlinedtheoryofidealjudgment willbeimportanttomydiscussionofDiderot'sideasonsmallpoxinocula- tion.Forthemoment,however,Imentiontheserulesonlytoindicatethe perceivedbreadthandimportanceofEnlightenmentprobabilitytheory,of whichDiderot'svisionisquiterepresentative. Themanysimultaneousinterpretationsclassicalprobabilityhadtostand uptomadeparadoxunavoidable,andtheenormousepistemologicalandso- ciologicalpotentialthatprobabilitytheorywasthoughttoembodyguaran- teedthattheseparadoxeswouldreceivemeticulousattentioninintellectual circles.TheSaintPetersburgParadoxandtheparadoxesattendantonsmallpoxinoculation ,bothdiscussedbelow,werethemostcelebratedandthemost controvertedissuesineighteenth-centuryprobabilitytheory,althoughmy presentationdeliberatelydownplaystheirubiquityandnotorietyinEnlight- enmentEuropetoconcentratemorefullyonhowDiderotandJeand'Alembert triedtoresolvethem.Ichoosesuchafocusfortwomajorreasons—first,for thecontrastbetweentheopinionsofDiderotandd'Alembert,twotowering figuresinEnlightenmenthistory,and,second,tohighlightDiderot'smuch neglectedmathematicalcorpus,whichsuggestssomedeficienciesinthestan- darddescriptiondrawnofthisphilosophe.Itmustbenoted,however,that juxtaposingDiderotandd'Alembertinastudyofprobabilityconstitutesa violationofhistoricrealism,sinceDiderot'sprincipalcommentaryonprob- abilitytheory,"Surdeuxmémoiresded'Alembert,"wasunpublishedexcept inGrimm'sCorrespondancelittéraireandremainedalmostunknownuntil afterhisdeath,6whiled'Alembert'sworkshadasignificantimpactonhis 102/LOVELAND intellectualcontemporaries.Itrustthatthecogencyofthecontrastwillmake upforthispartialanachronism.7 TheSaintPetersburgParadox Firstpublicizedin1713afterbeingproposedinaletterbyNicholasBer- noullitoPierreMontmort,theSaintPetersburgParadoxwassodubbedby d'AlembertwhentheImperialAcademyofSciencesatSaintPetersburgpub- lishedanalysesofitin1738.Thegamegivingrisetoit,originallybasedon dicerolls,soontookthefollowingdefinitiveform.AandBtossacoinuntil headsappears.Achievedonthefirsttoss,headsearnsplayerAoneunitof money.Achievedonthesecondtoss(butnotonthefirst),headsnowearnsAtwounitsofmoney.Andsoon.Ingeneral,tossingheadsforthefirsttimeon tossnumberηearnsAprecisely2nlunitsofmoney.WhatisA'sexpectation inplayingthisgame?Inotherwords,howmuchwillA'swinningsbeonthe average,andhowmuchwouldareasonableApayplayerBfortherightto participateinthislopsidedgame? Computingexpectationasthesumoftheproductsofeachpossible outcome'sprobabilityandvaluewasalreadycommonplacein1713,8andat firstglanceitseemedthatA'sexpectationwas (1/2)(1)+(1/4)(2)+(1/8)(4)+...+(1/2")(2nl)+... sincetheprobabilityoftossingheadsforthefirsttimeontossnumberηis 1/2".Thissumisinfinite,however,andalthoughinfiniteexpectationdidnot initselfdiscomfitthetheory(Pascal'sfamouswagergaveinfiniteexpecta- tiontobelieversinGod),commonsensesuggestedthatonlyafoolwould payanylargesumofmoney,letaloneaninfinity,toplaytheSaintPetersburg game.Heads,itwasreasoned,wouldhavetoturnupinthefirstseveral tosses,limitingpossiblewinningstoapittanceofmoney.Howwereintuition andmathematicalexpectationtobebroughtbacktogether? Thosewhowroteonthesubjectallwanted,indifferingdegrees,tomake mathematicsdefertointuition.Threestrategieswerefollowed.9Certainmath- ematicianssuchasDanielBernoulliproposedintroducingaunitofvalue distinctfrommonetaryunitstoreflectbeggars'andrichpeople'sdiffering perceptionsofthesamequantityofmoney.Disentangledfrommixedmath- ematicsfiftyyearslater,thisstrandofclassicalprobabilitytheorybecame thebeginningsofmathematicaleconomics.Othermathematicians,d'Alembert amongthem,soughttoalterthecalculationofsmallprobabilitiessoasto vindicatethecommonintuitionthattheprobabilityoftossingtwentyheads inarowisclosertozerothan1/220.Georges-LouisLederedeBuffon,the MathematicsandPracticeI103 eminentnaturalist,triedbothtacksatonce.10ThentherewasDiderot.Little distressedbytheparadox—tojudgebyhisdefenseoftheclassicaltheory— henonethelessnotedthattruncatingtheinfiniteseriesrepresentedabovemade itbettersuitedformodelingmortals. Throughouthislife,d'Alembertmistrustedprobabilitytheory.Thehand- fulofEncyclopédiearticleshedevotedtoitsuggestonlyoverblowncaution, buttheopinionsrecordedinhisOpusculesmathématiques(1761-80)are pessimisticandhostilewithrespecttothesubject.Theacceptedfoundations ofprobabilitytheory—itsmeansofdemarcatingequiprobablecases,itstreat- mentofconsecutivecointossesandrollsofadieascausallyindependent, anditseasytranslationofactualexperienceintomathematics—areallproved tobequestionableinthecourseoftheOpuscules,andalthoughd'Alembert speculatesonvariousschemesforrectifyingtheanalysis,nothinghepro- posesissufficientlyspecifictohaveimmediatebearingoncalculationsof probability.Furtherexperienceisregularlycitedasthekeytoestablishinga truescienceofprobability,butparticularexperimentsarerarelyproposed, andd'Alembertoffersmorequestionsthananswers.11Diderot,readingthe probabilisticdiscussioncontainedinvolumetwooftheOpuscules,foundits blithedeconstructionofallextantprobabilitytheoryanditsfailuretopro- poseanymoreusefulreplacementthan/en'ensaisriensufficientlyexasper-atingtowriteanunpublishedrebuttal.Asmentionedabove,d'Alemberthopedtoeludetheseemingcontradic-tionoftheSaintPetersburgParadoxbyattackingtheassumptionofequiprob- ableoutcomesonwhichitwasbased.Thisistosay,hebelievedthetossing ofηheadsinarowtobeperceptiblylessprobablethan1/2"andthetossing ofanyparticularsequenceofηmixedheadsandtailstobeperceptiblymore probablethan1/2"—whereasDiderotandothertraditionalistsinsistedthatallofthe2"distinctwaysoftossingacoinηtimesinarowwereequally probable.Inhiswritingsonthematter,d'Alembertattacksequiprobability byappealingtointuitioninseveralways.Heimagines,atonepoint,anex-travagantthoughtexperiment.If2100peopleeachtossacoinonehundred times,isitconceivable,asDiderotwouldhaveit,thatsomeoneofthemwill tossonehundredheadsoronehundredtails?Obviouslynot,d'Alembertan- swers(Opuscules,2:9-10).Elsewhere,heappealstoexperiencemoredi- rectly,andeveninvokesgamblers'wisdom,claimingthisbeliestheaccepted resultsofprobabilitytheory.12Heinvitesreaderstolookatspecificexamples wheretheequiprobabilityassumptionappearscontrarytocommonintuition. Istossingfourheadsinarowjustasprobableastossingtwoheadsandtwo tailsinsomespecifiedorder?Apparentlynot(Opuscules,2:12-13).We must,saysd'Alembert"distinguerentrecequiestmétaphysiquementpos- sible,&cequiestpossiblephysiquement.Danslapremièreclassesonttoutes 104/LOVELAND leschosesdontl'existencen'ariend'absurde;danslasecondesonttoutes cellesdontl'existencenonseulementn'ariend'absurde;maismêmeriende tropextraordinaire,&quinesontdanslecoursjournalierdesévénemens" (Opuscules,2:10).Intuitionandexperience,itwouldseem,aretheonly goodguidestoaphysicaltruththatiseminentlyordinary. D'Alembert'sidentificationofhisownpositionwithintuitionandexperi- enceisleftmostlyunchallengedinDiderot's"Surdeuxmémoiresde d'Alembert."Indeed,Diderotstrengthensthisidentification,bypresenting d'Alembertasanovercautiousandevennescientempiricist:"M.d'Alembert renvoielasolutiondecesdifficultésÃlaconnaissancedescasrareset fréquents,c'est-Ã-direÃl'expérience.Iln'yauradoncquelqueexactitude dansl'analysedeshasardsqu'aprèsdessièclesd'observations?Ilestvrai, répondM.d'Alembert"("Surdeuxmémoires,"343).Andwhen,afterpara- phrasingd'Alembert'ssmugskepticismforseveralpages,Diderotgetsaround torevindicatingprobabilitytheory(andthusrestoringtheconflictbetween intuitionandmathematicsintheSaintPetersburgParadox),hisweaponof choiceisdemonstrativelogic.TheconditionsoftheSaintPetersburggame arelaboriouslyandrepetitivelytranslatedintologicalandmathematicalterms, andanargumentresemblingaformalsyllogismmarchessteadilyforwardto prove,intheend,thatthegame'sexpectationisinfactinfinite.Onlytoward theendofhispaperdoesDiderotciteexperience—andthenvaguelyand briefly—tocombatd'Alembert'sefforttomakeextraordinaryeventsphysi- callyimpossible. BeforeturningtotheimplicationsofDiderot'sneglectoftherhetoricof experience,Ineedtodescribeinsomewhatmoredetailhisresolutionofthe SaintPetersburgParadox,whichtakesplacebelatedlyinthelastpartofhis paper.Probability,saysDiderot,canbeanalyzedintwoways—inutterab- stractionorasaphysico-mathematicalscience.Studiedabstractly,likege- ometryandalgebra,probabilitywillnodoubtproduceparadoxicalresults, justasgeometryandalgebrahavealreadygivenusparadoxicalobjectssuch asincommensurablequantities.D'Alemberthasonlyhimselftoupbraidif hisincessantabstractionawayfromtheconcreteproducesmerenonsense: "ToutelasciencemathématiqueestpleinedecesfaussetésqueM.d'Alembert reprocheÃl'analysedesprobabilités"("Surdeuxmémoires,"351).Inthe physico-mathematicalstudyofprobabilities,however,limitationsonthenum- berofpossiblegamesmustbeimposedtomaintainarelevancetohuman reality.Specifically,AandBarepermittedonlythefinitenumberofgames consistentwiththeirfinancialresources,lifespans,andplayingspeeds.What isconspicuouslyabsentfromDiderot'sphysico-mathematicalmodelisany- thingresemblingd'Alembert'sproposedempiricalsamplingofsupposedly equiprobablerandomevents. MathematicsandPracticeI105 WhileDiderot'sabstractprobabilitybearsaconsiderablelikenessto d'Alembert'smetaphysicalprobability—bothbeingformalizedandnon-an- thropomorphic—Diderot'sphysico-mathematicalprobabilityisdifferent enoughfromd'Alembert'sphysicalprobabilitytomakeacomparisonvaluable...

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