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The temperature of the brain1 Edgeworth’s thermodynamical analogies of utility measurement

ThomasMichaelMuellerUniversitéCatholiquedeLouvainMembreassociéduCentreWalras-ParetoAvenueMauriceMaeterlinck171348Louvain-la-Neuve0032494184257thomasvismara@hotmail.com

Abstract

In1881,Edgeworthattemptedtoprovideasolidpsychologicalbasisfortheutilitymeasurement.Hefacedcriticismthatchallengedhisepistemologicalviewsoneconomicscience. Ishowthatthecriticismsystematicallyreferredtothethermometeranalogy,whichEdgeworthandothersusedtoemphasisetheirviewsontheepistemicvaluesofeconomicscience.Throughcarefulreadingofthisanalogy,IdiscussEdgeworth’sideasontheutilitymeasurement,andtheirevolutionfrominitialenthusiasmforpsychologicalreductionismtothemoreprudentialclaimsofhislateryears. Thethermometeranalogyisimportantfortheinventionoftheconceptsofordinalandcardinalutilityscales.

Keywords: Edgeworth (Francis Ysidro), ordinal utility, cardinal utility, economic thought JEL: B13, B16

1TheauthorisgratefultoIvanMoscati,DanieleBesomi,MaximeDesmarais-Tremblay,AndrewSims,LouisLarue,AnnaDrozdzewskafortheircommentsonpreviousversionsofthispaper.CléoChassonery-Zaïgoucheforreading,comments,andmultipleiterationsofthere-reading,re-commentprocessdeservesallmygratitude.

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The temperature of the brain

InthispaperIwillretracetheintellectualstruggleofF.Y.Edgeworthtounderstandtheconceptofutilitymeasurement,fromhisfirstinsightsinMathematicalPsychics(1881)tohismatureviewsof1922. Edgeworthfamouslyimaginedanidealinstrument,thehedonimeter,thatcouldprovideanidealmeasureofutilityinsomefutureworld.

[L]ettherebegrantedtothescienceofpleasurewhatisgrantedtothescienceofenergy;toimagineanideallyperfectinstrument,apsychologicalmachine,continuallyregisteringtheheightofpleasureexperiencedbyanindividual,exactlyaccordingtotheverdictofconsciousness,(…)Frommomenttomomentthehedonimetervaries;thedelicateindexnowflickeringwiththeflutterofpassions,now(…)momentarilyspringinguptoinfinity.

(Edgeworth2003,101(1881))

Edgeworth’sscaleofmeasurementforutilityisgenerallydescribedasacardinalscale(Schumpeter1981,1028–32;PeartandLevy2005;Mata2007;Chaigneau2014).Ifwefollowthecontemporarymeaningoftheterm(Alchian1953),Edgeworth’sscaleis,however,notacardinalscale,butastrongernotionofmeasurementthatwemaylabel,followingMoscati(Moscati2013b),aratioorclassicalscale. Ratioscalesarescalesinwhichthezeropointhasanactualmeaning.Ratioscalesallowustosay,forinstance,thatonemeasurementistwiceanother;cardinalscalesallowustosaythatanincreaseinagivenvalueistwicetheincreaseinanother.Ordinalscalesaresimplyrankings,withnomeaningassociatedwiththenumbers.2Moscati(2013b)considersthatthemarginalistshadaratiounderstandingofutility,andweborrowthistermfromhim3.Inratioscales,azeroquantityisnoquantityatall:zerolengthisnolength,regardlessofthescaleofmeasurement.Epistemologically,whatdistinguishesratiofromcardinalandordinalscalesissubtler:Iclaimthatthedifferencerests,atleastfortheauthorsIamgoingtodiscuss,ontheepistemicvaluesthattheyassociatewithmeasurement. Themeasurementofsensations,suchasutilityor(tosomeextent)warmth,i.e.thesensationofsomethingbeingdesirableorofbeinghot,hasnoobviousphysicalmeaning;nonetheless,asthequoteshows,Edgeworthbelievedthatsensationscouldbemeasuredinthesamewayasphysicalquantitiescouldbe.Edgeworth’sanalogybetweenenergyandutilityallowstheimaginingofaninstrument,thehedonimeter,thatregistersandthusmeasuresutility. Thehedonimeterdescriptionisclosetothatofathermometer,andEdgeworthcitesMaxwell’sTheoryofHeatinafootnote(aftertheterms“scienceofenergy”inourpreviousquotation).Temperatureisgenerallyassociatedwithacardinalscaleofmeasurement,butEdgeworthherereferstoenergyandclearlyhasinmindaratiomeasurement,aswewilldiscusslater.

2Apossibletechnicalexplanationwouldbetosaythatordinalscalesaredefineduptoanorder-preservingtransformation(ifx<ythenanytransformationsuchthatf(x)<f(y)expressingthesamerankingisorder-preserving).Cardinalscalesaredefineduptoaffinetransformation(anyscaley=ax+b).Ratioscalesaredefineduptoalineartransformation(anyscaley=ax).3Moscati(2013b)usestheterm“classical”,buthehasnowchangedhisviews,andpreferstheterm“ratioscales”(privatecorrespondence,26.1.2017).Wethereforefollowthisuse.

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Edgeworthwasnotanaïvereaderofphysics.Hehadanideaandaconceptualbasisforhisboldclaims:psychophysiology.Psychophysiology,inEdgeworth’smind,allowedtheconnectionbetweenasensation–apsychicalfact–andphysics. Hefacedseveralcriticismsforthisidea,mainlyfromFisher,Voigt,PoincaréandNicholson.PsychologyanditslinkwitheconomicswereatthecoreoftheattacksagainstEdgeworth’sutilitarianism.Theexistinghistoricalanalysisofthesecriticisms(Colander2007;Chaigneau2014)hassomemajorproblems:first,itdealsonlywithEdgeworth’searlywritings(basicallyMathematicalPsychics(2003(1881)),andNewandOldMethods(2003c,1877)),thusforgettingmanylatercrucialsteps;second,itconsidersEdgeworthasacardinalist,somethinghewasnot;and,third,itfailstounderstandwhattheconceptualdifferencewasbetweenFisher’santi-psychologismandEdgeworth’spsychophysiologyfromanepistemologicalpointofview.Inviewoftherecentinterestinneuroeconomicsandthemeasurementof“neuralutility”(Padoa-SchioppaandAssad2006;Stuphorn2006;Webbetal.2016;seeFumagalli2013foranepistemologicalcriticism),thisisatopicthatclearlydeservesattention. OurstorywillshowthatEdgeworthwasconsciousofthemainproblemofaratiounderstandingofutility,namelythatitimpliestheparadoxicalideaof“twiceasuseful”.HetriedtofoundthiscounterintuitiveintuitioninpsychophysiologyinMathematicalPsychics,butwasforcedtoseriouslyreconsideritfollowingVoigt’sanalyses(Voigt1893b;SchmidtandWeber2008).ItwasthusnotFisher’santi-psychologismbutVoigt’santi-reductionismthatdefeatedhisratiomeasureofutility.Wewillshowthat,despitesharingaratiounderstandingofutilitymeasurement,EdgeworthandFisherhaddifferentviewsonutilitymeasurementthatlaterdevelopedintocardinalandordinalutilitymeasurement.Wewillshowthatwhatutilitywassupposedtoaccomplishintheirrespectiveviewswasbasedonadifferentconceptionofwhatpoliticaleconomyshouldbe. Edgeworth’sandFisher’sdifferentepistemicvalues,i.e.adifferentideaofwhatshouldcountasvaluablescientificknowledge,reinforcesMoscati’sclaimthatcardinalismwasinventedduringtheordinalrevolution:cardinalismwastheconsequenceofendorsingFisher’sepistemicvalues,despiteFisher’sratiounderstandingofutilitymeasurement.Edgeworthonhissidewasmoretemptedbyordinalismandalmostendorseditattheendofhislife.NeitherFishernorEdgeworthwasthusacardinalist(Edgeworthusedtheword“cardinal”inadifferentmeaning),buttheepistemicvaluesthattheyadvocateweretheinitialcrossroadsfromwhichtheordinalandthecardinalconceptsofmeasurementweregenerated. OurstorywillbeanaccountofEdgeworth’squest,duringtheperiod1880–1910,andtherolethatthethermometeranalogyplayedasaconceptualtoolinthedebateonutilitymeasurement.

1. Measurement in context: a comparison of uti l i ty and temperature

WhatutilityisandwhetheritcanbemeasuredarequestionsatthecoreofEdgeworth’seconomicalreasoning.Edgeworthhadamajorinfluenceonthedevelopmentofwhatisoftencalledtheordinalrevolution.HefoundhisoriginalmotivationinSidgwick(Kaminitz2013)andmorebroadlyinthemixtureofpsychology,moralissuesandpsychophysiologythatwasamajortopicofinterestintheVictorianscientificcommunity,dealingwithcomplexandcontrastingaims,suchasanincreasedinterestanddesireformeasurement,especiallyforthemeasurementoffeelingsandemotions,and

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forthepossibledescriptionofmentalphenomenaintheformofnaturallaws.Psychophysicsandpsychophysiologyarepronetodescribementalphenomenaas“movementsofparticlesoforganizedmatters[that]arecausesoruniversalconcomitantsofallmentalprocesses”(Sidgwick1880,355);nevertheless,italsoattemptedtoreconcilethisdeterministicandmechanisticviewwithfreewillandmoralresponsibility,anissuewhoseimportancewasacknowledgedevenbythemostradicalfollowersofscientificdeterminism. Thistension,whichhasbeendescribedbydifferentauthorsonseveraloccasions(Daston1978;Maas2005,164–9;Smith2013),wasanimportantissueforEdgeworth’sNewandOldMethods(Edgeworth2003c[1877])andMathematicalPsychics(Edgeworth2003(1881)).Famously,inMathematicalPsychics,Edgeworthdefinedapossiblewaytomeasureutility,followingtheideasofGustavFechner.Thequotationatthebeginningofthispapershowshowconfidentandenthusiastichewasaboutthepossibilityofmeasuringutility.Edgeworth’smainideaswerethatsensationincreaseswithstimulus,butatadecreasingmarginalrate.Despiteourintrospectivequalitativeknowledgeofsensation,itwouldbepossible–throughpsychophysicalmeasurement–togroundthisrelationinasolidphysicalbasisanddescribeitmathematically,thuscalculatingoptimaldistributionsofwealth. Thereisanepisodeinthehistoryofsciencethatcloselyresemblethatofutilitymeasurement:themeasurementanddefinitionoftemperature(Chang2004).Beforethermometers,measurementsandunits,asChangclearlysays,temperaturescalesweregroundedonhumansensation(p.42).Conformitywithourhumansensationistheprimaryjustificationforthetrustworthinessofmeasurement,evenifthemeasurementinstrumentislaterallowedtocorrectandevencontradictoursensation.But,asEdgeworthhimselfwilloftennote,temperatureinthepasthadprimarilybeendefinedasasubjectivefeeling,onlymeasurableonanordinalscale. Followingthisfirstconsideration,twodistinctresearchprogrammesmaybeoutlinedinthequestfortheinventionoftemperatureandanalogouslyfortheinventionofutilitymeasurement:ononeside,thereisthelongstruggletodefineathermometerandameasurementscale.Thermometershadtorelyonfixedpointsasthereferenceforstable,intercomparablepoints;thethermometerwasthenbuiltby“findingaprocedureforassigningnumberstothedegreesofheatbetweenthefixedpointsandbeyondthem”(Chang2004:57).Inproceduralterms,temperatureisnothingelsethan“whatismeasuredbyathermometer”.Thereisnoneedforanontologicalgroundingoraphysicaldeepmeaning.Theinstrumentdefinestheobject.Sincedifferentthermometersdefinedifferentscales,thereisnosuchthingasa“truevalue”oftemperature:eventuallytheremayberulestomovefromonescaletoanother.ThiswasthepaththatwouldbefollowedbyFisherinthemeasurementofutility. Butatthesametime,temperatureisalsotheoreticallydefinedintermsoftheKelvinscale,the“true”temperature.Thisisascalethataimsmoreattheoreticalgroundingandcoherencethanatempiricalmeasurement.Thosewhothinkinthiswaywillconsiderthatthereisanobjectivething–temperature–thatwetrytodiscover.Thisthingisdefinedmorebysomedeeper,morefundamentaltheoryofnaturethanbyinstrumentsandprocedures. Inanabsolutescale,zerotemperaturemeanstheabsenceofit,notemperatureatall.Whilethisimagemaylookcounterintuitivetothemoderneconomist,thiswaspreciselythekindofconceptualgroundthatEdgeworthassociatedwithutility.Utilitywasneitheraconventionnoraconceptualtoolintheeconomist’stoolbox;utilitywasreal,somethingthatexistedinthemindandcouldbemeasured.

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Thetemperatureanalogy,willguidethequestforthemeasurementofutility.Theconstantlinktotemperaturewhendiscussingutilityanditsmeasurementismorethanafancymetaphor.Onewordwillappearquiteoften,sidebysidewiththermometers,hedonimetersandmeasurementscales:objectivity. Objectivityisavalue,andacomplexone.Itis,tosomeextent,anepistemicvalue(Kuhn1977[1968]),sinceitdefineswhatscienceoughttobe:whattherelevantcriteriaareonwhichscientificknowledgestandsandisdefined.But,ashadbeenstressedbyDastonandGalison(2007),objectivityhasahistoryandissubjecttochange.Tothatextent,wemayconsiderthatobjectivityisacovering,multifacetedvaluethatcontainsamixtureofseveralepistemic(sub)values. Whatthoseepistemicvaluesaredepends,ofcourse,onthehistoricalperiodunderscrutiny;asKuhnhadalreadynoticed,epistemicvaluesmaysometimesbeinconflict,eitherbecausethereisachangeinimportancebetweenthemorbecausenewonesarecreatedandenterthedebateaslegitimatepartsofwhatagoodscientificpracticeissupposedtoachieve.Periodsofinnovation,change,andthusdoubtarealsoperiodsinwhichtherelevantsetofepistemicvalueswillchange.Valuesaregenerallytacitlyadmittedtotherobustcoreofwell-definedscientificpractices,buttheyaresubjecttomuchmorestressattheboundariesofthosedisciplines.Unsurprisingly,whenthoseboundariesmove,conflictingvaluesmaycomeintocontactandscientistswillfeelcompelledtostresstheirownvalues,whentheycomeunderscrutinyandunderattack. Utilitytheorywas,andstillis,builtoverthegeologicalfaultseparatingpsychology,ethicsandeconomics,aterriblyunstablegroundatEdgeworth’stime,whenneitherpsychologynoreconomicswasclearlydefined.Edgeworthwasamanofsolidvalues,epistemologicallyspeaking.Hestronglybelievedintheimportanceofquantification,whatKuhnlabels“precision”,but,inviewoftheabsenceofeffectivemeasurement,mightbebetterdescribedas“innumberswetrust”.Trustinnumbers4characteriseswellthenumerousreferencesbyEdgeworthtomathematics,fromthemostvaluablecontributions,suchastheintroductionuseofLagrangianmutlipliersinMathematicalPsychics,tothemorebizarreandfancifulanalysisofthestochasticdistributionofdigitsinthenumberπ(Edgeworth1994c(1911)).Healsovalued“coherence”,toquoteKuhnagain,notonlywithinagiventheory,butalsowithotherneighbouringtheories,especiallythosethatweresupportedbyleadingauthoritiessuchasinphysics.Edgeworth’sunderstandingofcoherencewasfullyorientedinthedirectionof“reductionism”and“theunityofscience”.Reducingeconomicstopsychologyandpsychologytopsychophysicswasanattempttodescribealmostanyphenomenonthroughaunifiedscientificlanguage. Trustinnumbers,reductionismandtheunityofsciencearevaluesthatconstantlysuggestwhatgoodeconomicsshouldlooklike.Theyareepistemicvalues,sincetheyconstrainwhatknowledgeisvalued;theyarealso,tosomeextent,valuestoutcourt.Itisonlywhentheyareinterpretedfromtheinternalpointofviewofascientificpracticethattheyareepistemic;atthehistoricalmomentinwhichtheyshapeadiscipline,theyarejuststand-alonevalues.Therefore,epistemicvaluesarealwaysretrospective:itisonlywhentheyhavecreatedasetofknowledgethattheybecameepistemic. Edgeworthreliesheavilyonreductionismandontheunityofscience,andtrustsnumbersmorethananything:thiswilldefinewhatwillcountforhimandforhisfollowersasscientificobjectivity.Edgeworthwasgoingtofaceaconcurrentsetofvaluesthatwilldefineaparallelresearch

4IdonotclaimanydirectlinktoPorter’sfamousandgreatbookwiththesametitle(Porter1995)thatdiffersinbothcontentandtimeperiodfromthispaper.

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programme.IrvingFisher,withhisanti-psychologicalfoundationofeconomics,carriedtheirbanner.Fisherstressedthatobjectivityisprocedural:itisbydefiningasetofrulesforpractisingeconomics,acommonprocedure,thatwearetrulyobjective.Proceduralobjectivityimpliestheexistenceofwell-definedsettings.Scientistsshouldagreeonascientificprotocol;theycanthusavoidpersonalideasandmoralvaluesenteringthescientificdiscourse.Proceduralobjectivityhasnoneed,andnotime,forpsychologicalfoundations.ThusFisherwouldrejectreductionismandtheunityofscienceastherelevantvaluesdefiningobjectivity,infavourofproceduralism,without“thehazesofethics,psychology,biologyandmetaphysics”,asFisherclearlystated(1892:23)withanexplicitreferencetoEdgeworth. Thequestforthemeasurementofutilitythenstartedtofollowtwodistinctdirections.Ontheonehand,proceduralobjectivityledtotheabandoningofanylinkwithpsychology.Itdidnotbotheraboutmoralissues,orperception.Thissidewillfinallymovefromtheratiounderstandingofutilitytothecardinaloneintheperiodbetween1910and1940(Moscati2013a).Mechanicalproceduresareeasytoformaliseandtocompute,henceprovidinggroundforcalculationbasedonthehighlyidealisedhomooecomomicusasaunit:thereisnoplaceforindividualfreedomandmoralvalues,notevenforanindividualinfact.Politicaleconomywasamoralscience,buttherewasnoplace,inthenewscienceofeconomics,forsubjectivity,emotionsandthusforpsychology.Thenewscienceofeconomicshadnointerestinmoralvaluesandintheconnectionwiththemind. Ontheotherhand,Edgeworth’spsychologismwillfacemajorissuesonitsown,andhewillmovehisconceptualframeworkmoreandmoreinthedirectionofordinalism.Edgeworthlosthisbattleforaratiomeasurementofutility:ordinalismwasmorecredibleandappealing,whileratiomeasurementhadtofacedeadlyconundrums,aswewillsee.

2. Psychophysics: A chemical bond between pol it ical economy and the mind

NewandOldMethodsofEthicsisEdgeworth’sfirstbook(Edgeworth2003c[1877]);itpresentsmostofthegreatestaccomplishmentsoftheIrisheconomist,suchastheuseofcalculusofvariationsandthestressonutilitytheorybasedonFechner’slaw. Edgeworth’sambitionisstatedinclearterms,attheverybeginningofhistreatise.HecomparesBarratt’sPhysicalEthics(Barratt1869)andSidgwick’sMethodsofEthics(Sidgwick1907[1874])andcontraststhephysicalscienceswithintrospectionasalternativemethodsforunderstandingthemind. ThetensionthatEdgeworthacknowledgesisanimportantandwidespreadoneinnineteenth-centuryBritishpsychology:theattempt,ontheonehand,toexpressthelawsofthemindinthelanguageofphysics,thusincludingtheminadeterministiclaw-likeframework,andontheotherhandtosolvethemoraldilemmathatdeterminismimpliesforfreewillandfreechoices(Chaigneau2002). Edgeworthbelievesinthepossibilityofdescribingthemindinphysicaltermsandbyphysicallaws,andespeciallyreliesonFechner’spsychophysicsforthatpurpose.Thereis“interdependenceofbodyandmind”(2003c:189)andthusthemindactivityimpliesaphysicalactivity.Butitremainstobeprovedthatphysicalenquiryisthebestwaytodeepentheunderstandingofourpsychicalsensations.

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Forthispurpose,Edgeworthreliesheavilyonthemathematicalscienceofwhichhehasprovedtobeamaster.Afterall“[m]athematicaljudgementsarethesameforallpersons,thedatabeingthesame”(p.188)andthusprovide,forEdgeworth,afirmbasisforobjectiveknowledge. InNewandOldMethods,healsosuggestsananalogythathewilllaterstateexplicitly:pleasureistheanalogueofenergy.Thereisanequivalenttothisquestinthehistoryoftemperaturetoo:theabsolutescaleoftemperature,Kelvindegrees,measuresan“objective”temperature,inthesenseofreductionism,theunityofscience,andtrustinnumbers.Kelvindegreeslinktemperaturetothemorefundamentalconceptofmatterinmotionandenergy:itisareductionistmovethatanchorstheconceptof“coldandhot”intomechanics.Italsoallowstheassociationofanumericalscalewithtemperature,withameaningfulzeropoint. Thisanalogywillrisetoprominencefouryearslater,with“MathematicalPsychics”(Edgeworth2003,89).

Thecomparisonbetweenpleasureandenergymaybeviewed[…as]adeepandrealanalogy,themaximumofpleasureinpsychicsbeingtheeffectoraconcomitantofamaximumphysicalenergy.

Whilethisisonlyananalogy,itisnotaninnocentone:thermodynamicsanditstheoreticalunderstandingwerefacingatthetimeanimportanttransformation,anchoringmacrovariables,suchastemperature,pressureandvolume,intomicrovariables,suchasaveragespeed,positionand…energy.Thus,thermodynamicswasmoreandmoreconfrontedwithareductionistprocessthatanchoreditslawsinthedeterministicparadigmofmatterinmotion5andenergyconservation.Bysecuringutilityinasimilarmathematicalframework,Edgeworthwasacceptingtheepistemicvaluesofasuccessfulresearchprogramme. EdgeworthcitesseveralofMaxwell’sbooksandmentionshisnameninetimes,Kelvin’sfour,andTait’sfivetimesinMathematicalPsychics.Andwhenitcomestotheastonishingimageofthehedonimeterthatisreferredtoatthebeginningofthispaper,Maxwell’stheoryofheatisquotedinafootnote.Thefootnotereferstopage139oftheTheoryofHeat,whereMaxwelldiscussestheCarnotcycle(Maxwell1871).AsChang(2004:168–86)stresses,Carnotcyclesareidealcyclesandtheconceptualtoolthatallowsthedefinitionofanabsolutescaleoftemperature.Byanalogy,thehedonimetershouldhavebeentheconceptualtoolthatallowsthedefinitionofan“absolute”scaleforutility(aratioscale). WhatEdgeworthseeminglyhadinmindwastointroduceanaturalunitytomeasureutility,aunitythatwasbasedonpsychophysicalassumptionsandwasrootedintheenergeticmetaphor.Thehedonimeterwastheinstrumentthatwasgoingtoreduceperceptionandfeelingstophysicalmeasurement,thusjustifyingtheuseofmathematicalinstruments.

Theconceivability[ofmeasurement]atleastmaybethoughtnecessarytomathematicalreasoning.Wemustthencarefullyconsiderthispossibility,or,whatismuchthesamething,theexistenceandnatureofaunitofpleasure

(Edgeworth(1881),98)

Theenergeticmetaphorwasplayingtheinstrumentalroleofareductionisthope:reducingtemperaturetoenergyhadpermittedtheexplanationofthermodynamicsasmatterinmotion,thusgivingamechanicalexplanationtoheat;reducingethicstoutilityshouldhavereducedthemindtoapleasuremachine,anchoringthehigherconceptoffreedomandtheselfinamechanisticframework.

5Notwithoutresistance:seee.g.(Porter1981;Hacking1983;Mueller2015).

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Thekeytothisreductionrelies,however,onaboldclaim:thatpleasure,asenergy,canbemeasured.Buthowisitpossibletomeasurepleasure?Whatistheunit?Edgeworthmadeanoutstandingproposition,basedonFechner’sideas.Theunitofpleasureisthe“justperceivableincrement”ofpleasure.“Justperceivableincrementsofpleasureareequitable”(Edgeworth(1881):99),andnotonlyforthesameindividual:“Anyjustperceivablepleasure-incrementexperiencedbyanysentientatanytimehasthesamevalue”(ibid:101). Thuswehavearegularunitandanunequivocalzeropoint(forwhennothingisfelt):itisenoughtostatethatpleasurecanbemeasuredfollowingaratioscale.Thereisevenmoretoitthanthat.Unitsoflength,ofelectriccurrentoroftimeareconventional.Thereisnoparticularreasontoprefermeterstoparsecs,inchesorstadions.ButEdgeworth’sperceivablepleasuresarenotconventional.Theyarerootedinthevery(biological)natureofthehumanbeing.Thus,theyanchorthescienceofpoliticaleconomyinapsychological,unconventionalnotion:justperceivableincrementsofpleasure. Justperceivableincrements,Edgeworthadmits,leave“theterrafermaofphysicalanalogy”(2003(1881):99).But,headds,“theminimasensibiliabeingequated,whatisfeltis”(p.99).Heisthusidentifyingourperceptionofnature(whatisfelt)withhowthingsreallyare,thusmergingtheepistemologicalandtheontologicallevels.Theparadigmaticrealmofsubjectivity,perceptionisidentifiedwithameasurableobjectthatisconsideredtobeidenticalbetweentimesandindividuals.Subjectivitybecomesobjective:thelinkbetweenpoliticaleconomyandthemindcomesatahugeprice. AndEdgeworthalsoacknowledgesthatthoseminimasensibiliacouldultimatelynotbemeasurableatall:theycouldjustberanked,since“weseemtobecapableofobservingthatthereishereagreater,therealess,[…]andthatisenough”(p.9).Ifthisisafirsthintofanordinalscaleofutility,itremainsquiteclearthatEdgeworthbelievesinratiomeasurement.Hisaimistoshowthatamathematicaltreatmentoftheproblemispossibleandthatmathematicscanapplytoordinalmeasurementaswellbuthedoesnotgoasfarasclaimingthatutilitymaybeordinal.Whathewantsistoclaimthatpleasurecanbemeasured,thusallowingthemathematisationandquantificationofsubjectivity:trustinnumbers. Ofcourse,asEdgeworthperfectlywellknew(p.85),ordinalityisnotenough:theformandshapeofafunction,itssecondderivativeandthereforeconcavity,aremeaninglessifutilityisordinal.Edgeworthexpendsconsistenteffortinavoidingtheproblem:ononeside,hereplacesFechner’slogarithmicalfunctionwithageneralfunctionhavingnopreciseform.Butdespitehisgainingenerality,hehastosupposeanegativesecondderivative(pp.61–6).Whilethiscertainlygainsinabstraction,Edgeworth’suseofsecondderivativesoftheutilityfunction,impliesmorethantheordinalscale.Edgeworth’sutilityfunctionisacontinuous,derivablefunctionwithashape,evenifitisasgeneralaspossible.Edgeworthrealisesthatitmaybeproblematictogiveameaningto“twiceasuseful”:nevertheless,hedefendsthepointthatitactuallyhasameaning,throughtheintroductionofhisminimasensibilia. Edgeworthjustifieshishypothesisofequalminimasensibiliaforeveryindividual,onanalogywithprobabilitytheory,where“casesaboutwhichweareequallyundecided,betweenwhichweperceivenomaterialdifference,countasequal”(p.99). Edgeworthsuggeststhatthehypothesisthatminimasensibiliaareidenticalforagivenindividualandbetweenindividualsmaybejustifiedanalogouslytothehypothesisthatgivesequalchancestoa

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setofeventsaboutwhichweknownothing6(letuscallittheprincipleofindifference).Again,itisnotanaïveanalogy.Probabilitiesarethekeymathematicalinstrument,allowingustomovefromtheleveloftemperatureandpressuretotheatomiclevelofmatterinmotion.Itisbecauseweassignuniform(equal)probabilitiestothedirectionofmoleculesinmotionthatwecanderiveaGaussiandistributionforspeeds,asMaxwellhadwellunderstood(Maxwell1867). WhenMaxwell(Maxwell1860)derivesinhis1860papertheGaussiandistributionforthemolecularvelocities,heusesHerschel’sderivationoftheGaussian(Porter1981).OneofHerschel’shypotheseswasthatwehaveanequalknowledgeofthecauseofeveryevent(molecularmovementforMaxwell),adirectuseoftheprincipleofindifference.Thisderivationanchorsthenotionoftemperatureinthenotionofenergy.EdgeworthprobablyknewofMaxwell’sreadinganduseofHerschel.7Thus,thelinkthatEdgeworthsuggestsbetweentheprincipleofindifferenceandtheprincipleofminimasensibiliaisanalogoustotheuseofthesameprinciplebyMaxwell. Edgeworth’saimistoprovideasolid,physicalbasistothemeasurementofpleasure.Pleasurecomesinsmallgrains,likeatoms,andgrainscanbeaddedonetotheothertogiveatotalamountofpleasure,astheenergyofatomsgivesthetotalenergy.“‘MécaniqueSociale’mayonedaytakeherplacealongwith‘MécaniqueCeleste’,throneeachuponthedouble-sidedheightofonemaximumprinciple”(2003(1881):12).Averagingoverlargegroupsofatomsallowstherelationofmacrovariablessuchastemperaturetomicrovariablessuchastheenergyofsingleatoms.Analogously,averagingoverlargegroupsofpleasureatomsallowstherelationofmicrovariablessuchaspleasure-units,tomacrovariablessuchasindividualpreferencesandsocialwelfare.Andthusindividualpreferencesthatarerelatedtotheutilityfunctionaremeasurableinthemostcompletesenseoftheterm:thatis,ratiomeasurement. Edgeworth’stourdeforceusesalltheingredientsofthesuccessfulresearchprogrammeofstatisticalmechanics–probabilities(jointlywiththeirmostcriticisedassumptions),energyandtemperature–allinconjunctiontograntthescienceofthemindafirmfoundationinphysicalscience.Withtheresearchprogrammeofstatisticalmechanics,healsoinheritsepistemicvalues:reductionismandtheunityofscience. Edgeworthwascertainlyconsciousofthecomplextaskthathadtobeaccomplishedinordertotransformaboldclaimintoacrediblescientifictheory.Hewas,nonetheless,atruebeliever:“[it]isnotmoreparadoxical[than]thatthedemonstratoroftheatom-theoryshouldforeseetheremotepossibilityofitsapplication,nolessapossibilitythanthetriumphoverthesecondlawofThermodynamics”(2003(1881):80). Thermodynamicshadbeensuccessfulbyusingaspecificreductionistprogram.Whynoteconomics?Afterall,thermodynamicshadbeengroundedforatimelongonepistemicshifting6Ifwetossafaircoin,thereisa50:50chancethatitwilllandoneachside.Butifwetossacointhatwedonotknowtobefairorunfair,itisarguablewhetherweshouldstillassociatea50:50chancewithitorifweshouldrefrainfromassociatinganychanceatall.Thisquestion,whichhadbeenatthecoreofanimportantdebateatthetime,isreferredtoasthefirstprincipleofprobabilitiesbyEdgeworth.Italsoprovidessomejustificationsandisgenerallyknownastheprincipleofindifference.7In1911Edgeworthstates(Edgeworth1994c(1911):275)thatin“molecularchaos[…]itmaybepresumedthaterrorsinalldirectionsareequallyprobable”andthatthisideainMaxwellisdirectlyderivedfromHerschel(whohimselfusedQuetelet).Whilethisexplicitquotearrivesmuchlater(30yearslaterthanMathematicalPsychics),EdgeworthcitesMaxwellsofrequentlyduringhiscareer,andnotablyinMathematicalPsychics,thatitseemshighlyprobablethatheknewofMaxwell’spaperindetailasearlyas1881.

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sands:soascienceofthemindcouldsimilarlybebasedonpsychophysics.“[T]hecorrespondencebetweenthetendencyofamechanicalsystemtomaximum[…]energyandthetendencyofvolitiontomaximumpleasure[…]seemstobethephysicalbasisofvolition,andifso,ofbelief”(Edgeworth1994d(1883):148);thiswas,forEdgeworth,enoughforanup-and-comingresearchprogramme.

3. Heating up probabil ity and uti l i ty to their melt ing point

Edgeworthwascertainlyawarethatunitsofminimasensibiliamayhavebeenconceptuallyproblematic.Whilehedefendstheidea,heshowshimselftobeverycautious,especiallyinviewoftheavailablecapabilityofempiricalmeasurement.Wewillseethatthesuggestedanalogybetweenprobabilitiesandutilitieswasgoingtoriseinimportanceinhismind,andwasoftenthoughtofalongsidetheenergeticmetaphor. In1884hewroteapaperinMind,dealingwiththephilosophicalunderpinningsofchance(Edgeworth1884b).EdgeworthwasparticularlyinterestedinJohnVenn’sLogicofChance(Venn1888[1866])andwiththequestionofwhether“gradationsofbelief[are]asubjectofscience”(Edgeworth1884b:224),aquestionthatEdgeworthrelatedtothemoregeneralmeasurabilityofsensations.Vennhaddoubtsconcerningthemeasurabilityofbeliefs;Edgeworthwascautiousbutmoreopentoit.Notably,hestatedthat:

Themeasurementofasubjectivefeelingislikethemeasurementoffeltheatbythethermometer.ItisveryliketheFechnerianmeasurementsofsensation.

(ibid:224)FeltheatisamatterofperceptionthatEdgeworthcomparestothethermometerandthentotheFechnerianmeasurementofsensation.Here,Edgeworth’sintentionwasprobablytostressthepossibilityofgivinganobjectivemeaning(ameasure)toasubjectivefeeling.Buttheperceptionofhotandcoldisnotameasuringscaleinthesamesenseasthethermometer:Edgeworthisherecontrastingacardinalandanordinalscale.ThattheFechnerianmeasurementofutilityexistsinEdgeworth’smindasalogicalanalogueofthetemperaturemeasurementandasanalmostautomaticassociationisafirststepinrealisingtheinnercomplexityofutilitymeasurements. Edgeworththinksthatthemeasurementofprobabilitiesfacesasimilarproblemtothemeasurementofutility.ThepostulateofequalchancesisagainevokedbyEdgeworth:itisbased“uponasortofunconsciousinduction”(Edgeworth1994a(1884):168). Thesameyear,inOntheReductionofObservations(Edgeworth1884a),Edgeworthstatesthat“inthemoralmeasurementsthereneverisanobjectiverealvalue”tobeevaluated“bytheirdifferencefromanobjectivepoint”(p.141).Edgeworthishereacknowledgingtheintrinsicdifferencebetweenthemeasurementoffeelingsandthatofphysicalquantities.Thedistancefromanobjectivepointsuggestsaratiounderstandingofmeasurement:Edgeworthmayberealisingthatfeelingsdifferfromphysicalquantitiesontheveryquestionofthemeasurementscale.Hethentriestosavewhateverispossible.Edgeworthisprobablyrealisingthatthemeasurementofpleasureisnotonlypracticallyunfeasibleonaratioscale:itcouldevenbemeaningless.Ordinalmeasurementseemstobeintuitivelymoreappealingandepistemicallylessdemanding.Nonetheless,Edgeworthmaintainsafirmbeliefinthepsychologicalgroundingofutility:heacceptslessertrustinnumbers,butnotinreductionismortheunityofscience.

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Utilityandprobabilitiesshareasetofcommonproblemscomparedtophysicaldimensions:theybothrelatetopsychologyandlackanynotionofan“objectivepoint”fortheirnumericalevaluation.Butprobabilities,whenrelatingtotemperature,hadbeentamed,despitehavingbeenstronglyconnected,atleasthistorically,withproblemssuchasfreewill(Porter1981)andmoralissues(Hacking1983).Edgeworthprobablyhopedforasimilarfelicitousdestinyfortheutilitytheory:theconnectionbetweenprobabilityandutilitywouldreachitsclimaxin1887,whenEdgeworthwroteoneofhismostcryptictexts,Metretike. Metretike’sboldinitialclaimwasthat“TherelationoftheCalculusofProbabilitiestothePhilosophyofUtilityispartly(i)ofSimilarity;partly(ii)ofIdentity”(p.81,Edgeworth’scapitalletters)and“thatthemeasurementoffeelingmayonedayceasetobeparadoxical”(p.82).Bothprobabilitiesandutilityareanattempttomeasuresensationsandfeelings,andtheconceptualbarriertopassisasimilarone.Theoriginalinterestofmechanicallydescribingfeelingsfindsananswerinprobabilitytheory:thereisnot“anenormouschasmwhichseparatesBelieffromVolition”(p.82).Thus,volitioncanbedescribedbyutilitytheoryanalogouslytothermodynamics.Thereductionoffeelingstophysicalquantitiesandtheirnumericaltreatmentwouldthenbepossibletoachieve. Nonetheless,Edgeworthshowsmoreandmorecareanddoubtsconcerningthemeasurement“inthehumansciences”where“[q]uestionsofdegreeespeciallydefymethod”(p.100). AfterstatingthatutilityismeasurableinMathematicalPsychics,Edgeworthmentionsmoreandmoreoftenthepossibilitythatmeasurementscouldbeexpressedintermofmoreandless,withoutanydefinitenumericalvalue(Edgeworth1994a(1884):167,1994b(1887):105).Whileconsciousofthedifferencebetweenmentalandphysicalphenomena,heisstillhopefulofbeingabletomeasurementalphenomenainaratiosense:whileherecognisesthatheisunabletodoit,hestillbelievesthatitmaybefeasibleinprinciple.

4. Rival measurement scales for r ival epistemic values: rat io cardinal and ordinal mixed phase

AfterMetretike,Edgeworthstartstowriteagainonpoliticaleconomy,especiallyonthetheoryofbanking,withoutforthisveryreasonabandoninghisnewinterestinstatisticsandprobabilitytheory.In1892,IrvingFisherpublisheshisPhDThesisMathematicalInvestigationsintheTheoryofValueandPrices(Fisher1892).Inthepreface,hestronglycriticisesEdgeworth’stheoryofutility,focusingontheproblemofmeasurementunits,andmoregenerallyonitspsychologicalbasis. Fisherconsidersthat“utilitymustbecapableofadefinitionwhichshallconnectitwithitspositiveorobjectivecommodityrelations”(ibid:5).Heleavesnodoubtabouthisdisregardforpsychologysince“[t]hisfoistingofpsychologyoneconomicsseemstomeinappropriateandvicious”(ibid.).Fisherisquiteambiguousaboutthenotionofmeasurabilitythatheuses.Ononeside,utilitycanbemeasuredas“anyothermathematicalmagnitude”(p.12).Whatthismeansismadeclearbyareferencetotemperature:“Foralongtimephilosopherscoulddefineanddeterminewhentwobodieswereequallyorunequallyhot.Butnottillthemiddleofthecenturydidphysicistsattachameaningtothephrase‘twiceashot‘”(p.14).Fisherthenaddsafootnotestatingthat:“[t]hefirstthermodynamicdefinitionofonetemperatureasamultipleofanotherwasmadebyW.Thomsonin1848”(ibid.).ThisisareferencetoThomson(LordKelvin)’spaper“Onanabsolutethermometricscale”(1848),andthustoKelvindegrees.OnemaybetemptedtoconcludethatFisherunderstands

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measurementintheratiosense.Hecontrastssensationoftemperature–anordinalscale–withtheKelvinscale.Fisher,JosiahWillardGibb’sPhDstudent,probablyjustinheritshismaster’sknowledgeaboutthermodynamics. ButthereisanimportantdifferencewithEdgeworth:Fisher’sfocusisontheindividualbehaviour,notonsensation.Itlinksutilitywith“commodities”andnot–asEdgeworthdoes–withfeelings.Sensationsandfeelingsarewhatutilityisabout,butcommodities,despitebeingonlyatbestaproxyofourutilityfeelings,canbeobserved,andtheirknowledgecanbeshared.Commoditiesconcernutilityinalessdirectway,buttheyarealsolesspronetointerpretation,intuitionandfuzziness.“Thelawsofeconomicsareframedtoexplainfacts.Theconceptionofutilityhasitsorigininthefactsofhumanpreferenceordecisionasobservedinproducing”(Fisher1892,11). Thisisnotveryfar,intermofepistemicvalues,fromthekindofproceduralobjectivitythatcharacterisesthethermometer.Proceduralobjectivitydefinesastandard,findsfixedpoints,eitheratthemarginofautilityfunctionorattheboilingpointofwater,andthendefineswhattherelevantpropertiestobemeasuredare. Fisher’sdefinitionofutilityiscarefullystatedinaproceduralwaythatcloselyresemblesthecalibrationofalaboratoryinstrument:Fisherdefinespreferencebycarefullystatingwhatkindofobservationcountsasa“largerthan”orasanequality.Later,Fisherwilldefineutilityratiosinaprudentialway:theutilityofthenthandmthunitofagivengood(loaf)areinanumericalrelation,thussuggestingalessstrongrequirementthanratiomeasurement;acardinalscalewouldbe

compatiblewithFisher’srequirement.TheutilityofthenthloafmeansU n( )−U n−1( ) ,i.e.anincrementinutility.Sayingthattheincrementfromm-1tomistwiceasbigastheincrementbetweennandn-1isacardinalstatement. IamnotclaimingherethatFisherhadaclearunderstandingofthekindofscalehewasusing,orthathehadinfactunderstoodcardinalityantelitteram,andIwarnmyreadersagainstsuchaWhiggishstatement.ItishardtosaywhetherFisherwasconsciousoftheactualdifferencebetweenratioandcardinalmeasurement:hewasprobablynot. WhatclearlydistinguishFisherfromEdgewortharetheepistemicvaluesthattheyemphasise.ForFisher,trustinnumbersismorerelevantthantheunityofscience,andthereforeheprefersapositivisticproceduralismtoEdgeworth’spsychologism(Chaigneau2014). FisherclearlydistinguisheshisunderstandingofutilityfromEdgeworth’sonthebasisofasubjectivity/objectivitydifference:

Beforemechanicswasascience,“force”stoodfor[…]muscularsensationfelt[…]Buttoconstructapositivescience,forcemustbedefinedwithrespecttoitsconnectionwithspace,timeandmass.Soalso,whileutilityhasa[n][…]meaningrelatingtofeelings,wheneconomicsattemptstobeapositivescience,itmustseekadefinitionwhichconnectsitwithobjectivecommodity.

(Fisher1892:17)Afootnotestressingagainstsubjectivefeelingsversusobjectivephenomenaandmeasurementsconcludesthat“ineconomicsthephenomenaareobjectiveandlikewisetheirmeasure”(ibid.). Utils,theunitofmeasurementforutility,arethendefinedfollowingablack-boxpostulate:“eachindividualactsashedesires”,withoutinvolvingtheeconomist:“incontroversyastothelawsofthesubjectivestatesofpleasureandpains”(ibid:23).Psychologyhasbeendefinitelybannedfromtheeconomictheatre. OneshouldnotthinkthatFisher’sproceduralismbansphysicalmetaphorsfromthescene;itisquitetheopposite(Cot2007).ButmetaphorsinFisherwillalwaysbedirectedatmechanisms

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describingconsumersandgoods,andalwaysremainattheeconomiclevel.Theaimofthoseexamplesispedagogical,notanalogical. EdgeworthwillreactverypositivelytoFisher’sthesis,despitethesharpcriticismthathereceived(Edgeworth1893).HeacknowledgesthatFisher“impartsnewclearnesstotheideaofmarginalutilitybyintroducinga‘unitofutility’”(p.108)despitethefactthat:

DrFisher,whohasajustconceptionofthegreatgulfwhichseparateseconomicsfrommoralphilosophy,regardscomparisonsbetweenthepleasuresofdifferentindividualsas“mysterious”(p.99)which“donotbelonghere”(p.87).Atthesametimehethrowsoutsomehintswhichwillbevaluabletotheutilitarian.

(p.112)EdgeworthsawtheprobutalsothecontraofFisher’santi-psychologism.DespitehistasteforanalogieswithmathematicsandphysicsthatwerecertainlysatiatedbyFisher’streatise,Edgeworthwasawareofthemajoradvantagesthatinterpersonalcomparisonwouldhaverepresented.And,aswesaw,Edgeworthwasalreadyinthewayofself-criticismonhisnotionofutility,buthewasnotreadytogiveupthemainepistemicvaluesthatheassociatedwithit. TherewasasecondsourceofdoubtthatshapedEdgeworth’sunderstandingofutilityandmovedhiminthedirectionofanordinalscaleofmeasurement.Aswehaveseen,hewasalreadyconsideringhimselfthatsometimesutilitycouldonlybemeasuredordinally:in1893heencounteredtheordinalviewinapaperbyAndreasVoigt(1893b)dealingwiththeon-goingdebateonordinalnumbers. Voigt,aneconomistandamathematician,wasdirectlyconcernedbothwithmathematicalideasonordinalnumbersandwithdirectphysicalandeconomicapplications(SchmidtandWeber2008). Heusedthethermometeranalogyinaverycleverway:

Themeasurementoftemperaturebyathermometerisnotofamuchhigherlevel(thanthehardnessofanobject).Itisalsojustanorderthatasourceofheattransmitsalongamercurycolumnthatgrowswithtemperature.Thethermometergradesdonotgivethetemperaturerate.Asimilarordercouldalsobedirectlyestablished,throughthesensationofheat.Weshouldinfactbesatisfiedwithfewdifferencesofgrades,andeventuallywiththosethatmaybeeasilyenumeratedbyoralcounting.Nonethelessthisorderhasadeficiencycomparedwiththethermometer.Itissubjective,i.e.itdependsonthepersonaltemporalandspatialsensibilitytoheat,whiletheotherpossessesanobjectivevalidity:thedependenceoftheheightofthemercurycolumnwithrespecttotemperature.Everymeasurementinpsychophysicsisasubjectiveorderofsensationsrespectivetointensity,wheredifferentdegreesexpressthejustperceivabledifferences.8

8DasMessenderTemperaturmitHilfedesThermometersstehtaufkeinervielhöhrenStufe.AuchdiesesistnureinOrdnenderWärmequellenvermittelstdesLängeeinerQuecksilbersäule,diemitderTemperaturwächst.DieThermometergradegebennichtdasVerhältnisderTemperaturenan.EineähnlicheOrdnungkönntemanauchunmittelbar,mittelst(sic!)derWärmeempfindungherstellen.ManmüsstesichdannzwarmitwenigenunterscheidbarenGradenbegnügen,vielleichtmitdenjenigen,diesichohneHilfevonZahlenleichtsprachlichbezeichnenlassen.DochdieseOrdnunghatgegenüberdermittelstdesThermometershergestelltennocheinenMangel.Sieistreinsubjektivd.h.siehängtvonderpersönlichen,zeitlichenundörtlichenEmpfindlichkeitfürWärmeab,währendjeneobjectiveGültigkeitfürallehat,diedieAbhängigkeitderLängederQuecksilbersäulevonderTemperaturanerkennen.AllesMessenderPsychophysikisteinsubjektivesOrdnender

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Voigt’spaperis,seemingly,thefirstplacewherethewords“ordinal”and“cardinal”appearinrelationtoutilitymeasurement(SchmidtandWeber2008),althoughwithadifferentmeaningthantheonethatcontemporaryeconomistsuse. Whilehisintentionisclearlytodefendtheideathatutilityisonlymeasurableordinally,notonlyforpracticalpurposesbutbecausethenotionof“twiceasuseful”ismeaningless(Voigtexplicitlystatesthispoint–(Voigt1893a,citedinSchmidtandWeber2008)–andcitesFisherasdefendingtheoppositeview),theuseoftheworld“cardinal”hasnodirectlinktothemeaningthatmoderneconomistsassociatewithit.“Cardinal”and“ordinal”justmeannumbersandrankingsinVoigt’spaper. Voigt’suseofsubjectivity,ascomparedtoobjectivity,focusesonadifferencebetweenthethermometer,andtemperature:betweentheinstrument,andwhatwewanttomeasure.Theinstrumentisamediatingobject,standinginbetweentheobserverandthequantitytobeobserved,thusimplyingalessdirectaccesstoit.Butitisalsoawaytoincreasetherepeatabilityandcomparabilityofmeasurements.Voigtunderstandsthattemperatureisnotaratiomeasure,butnonethelessitallowsastrongernotionofmeasurementthanpsycophysics:heat“transmits”anorder,itsorder,tothethermometer.Buttheorderthatisactuallymeasuredisnotheat(ortemperature)butsomethingelse,namelytheheightofamercurycolumn.Finally,measurementbysensationcanonlybeordinal,whileatemperaturescalepossessesanobjectivevalidity.Whatisatstakehereistheindependenceoftheobserverfromtheobserved.Whilethethermometerallowsthisseparationinquiteaclearway,thesubjectiveperceptionofutilitydoesnot. Sayingthatagiventemperatureistwiceashotasanotherhasnomeaning,butsayingthatanincreaseof6degreesistwicethatofanincreaseof3degreeswouldbecompatiblewithcardinality.Measuringthetemperaturebymeansofathermometercausesalosswithrespecttoratiomeasurement:asaconsequencethattemperaturecanbemeasuredonlycardinally.9Thus,Voigt’ssuggestionmaybeinterpretedasstressingthreeconceptuallydifferentnotionsofmeasurement:ratio,cardinalandordinal.Althoughtheyarenotyetstatedexplicitly,andareprobablynotfullymasteredbyVoigt,thethreenotionsarestartingtoseparateconceptually. EdgeworthreadVoigt’spaper,probablyduetohisinvolvementintheeditorialboardoftheEconomicJournal(SchmidtandWeber2008:17).Hemayhavebeenquiteimpressedbytheargument,sincehequoteditseveraltimeswhilechanginghisownideasonutilitymeasurement. TheshortcommunicationintheEconomicJournalthatsummarisesVoigt’spaperisanonymous(“Recentperiodicalsandnewbooks”1894),butwasprobablywrittenbyEdgeworthhimself10anditisthereforehisfirstreferencetoVoigt.Itstates:

EmpfindungennachihrerIntensität,wobeidieeinzelnenGradedenebenmerklichenUnterschiedenentsprechen(Voigt1893b:583).9Here,IamofcoursetakingaretrospectiveviewofVoigt’sclaim.Heisnotawareofthedifferencebetweencardinalandratioscales.Manyphysicistssimplywouldnotcareeventoday:itisaproblemfortheeconomistonly.ThisclaimmakesenseonlyinthecontextofunderstandingtheevolutionofEdgeworth’sthinking,andmoregenerallyoftheinventionofcardinalandordinalmeasurementsofutility.Again,thereisnoreasontobelievethatVoigthadanythingofthiskindinhismind.Hecertainlydidnot.Itisonlyinretrospectthatithasaninterest:butretrospectionislegitimate,andevennecessarytosomeextent,inestablishingthehistoryofaconcept.10Edgeworth’sparticularstyleofprosecanhardlybemistakenforsomeoneelse’s.Weagreewiththisjudgement(SchmidtandWeber2008:27).

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Economicsdealwithquantities,suchasutility,which,notbeingexpressibleinunits,aremeasuredonlybyordinalnumbers.Aswemaysayofseveralsensationsofheatthatthesecondismoreintensethanthefirst,thethirdthanthesecond,sothedegreesofadvantage…

(myemphasis,p.202)Heretheauthorunderlinesthedifferencebetweensensation(thussubjectivemeasurement)andunit-basedmeasurement,butdoesnotstressthedifference,asVoigtdoes,betweenthethermometerandtemperature.Edgeworthwasprobablyreadytorecogniseordinalmeasurement,sincehehadalreadydevelopedhisownthoughtsaboutit,aswesaw,butnotthemoresubtledifferencebetweenratioandcardinal. Inthesameissue,EdgeworthcommentsonapaperbyJosephShieldNicholson(Edgeworth1894).NicholsoninhisPrinciplesofPoliticalEconomy(Nicholson1893)discussthemeasurabilityofutilitybymoney,andbasicallyrejectedit.EdgeworthwasfavourablyinclinedtothisparticularrejectionandcitedextensivelyNicholson’sanalogybetweentheutilitymeasurementandthethermometer:

“Thepennydoesnotmeasureaccuratelyeithertheutilityorthedisutility.Itthrowsnolightwhateveronthequestionofhowmuchpleasurehefeelsinsmokingorhowmuchpaininworking”(prof.N.p.57).“Aclinicalthermometerwillmeasureaccuratelytheheatofthebody,butitsaysnothingofthecorrespondingfeeling.Heatisonethingasamodeofmotion–anotherasasensation”(prof.N.p.64).

Iadmitthatapennydoesnotmeasureutilityinthesamesenseasayardmeasureslength;andIaccepttheanalogyofthethermometerasfair,andevenfavourable.

(Edgeworth1894:154–5)Thisparagraphendswithafootnoteon“DrVoigt’sproposaltouseonlyordinal–notcardinal–numbers”. Theparagraphisveryinteresting:Nicholsoncomparesathermometermeasurementwithasensation;Edgeworthinterpretshisclaimasthedifferencebetweenthemeasurementoflengthandthemeasurementofsensations,andcallsthefirst“cardinal”(probablyinfluencedbyVoigt,andthusreferringtoordinalandcardinalnumbers,andnottomeasurementscales). Edgeworthrecognisesthatthethermometerdoesnotdirectlymeasuresensation,butstressesthefollowingfact:athermometercanbeusedtomeasurethetemperatureofalibrary,andsuchameasurementwillsufficeasaproxyofhumansensation.Itmaybetruethat“heatisonethingasamodeofmotion,anotherasasensation”(Edgeworth,citingNicholson,1894:155),neverthelessEdgeworthstressesthattheobjectivemeasurement–temperature–isagoodproxyforthesubjectivefeeling.Nicholson,whoconsidersthatutilityandexchangevaluesaredistinct,andthatmoneymeasuresonlythesecond,defendsanindependenceofeconomicsfrompsychology,butEdgeworth,despiteagreeingwithNicholsononthematterofmeasurability,perseveresindefendingpsychologism. Sixyearslater,EdgeworthwillrememberVoigt’spaperintheEconomicJournal(Edgeworth1900:178).HereEdgeworthrefers,indiscussingUtilitarism,to“theproblemoftheunit,whichsomeregardasinsuperable”(ibid.),andcallsforattentionto“DrVoigt’sreflectiononthemeasurement(…)byordinalnumbers,degreesofutilitybeingdistinguishedasfirst,second,&c.,(…)butnotasmultiplesofaunit”(ibid.).

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Sevenyearslater,EdgeworthwillcommentagainonVoigt’spaperinareviewarticleonmathematicaltheories(Edgeworth1907:222).

Howeverwedefinethe“satisfactions”whicharemeasuredbythemethodunderconsideration,thequestionmayarise:Withwhatunitaretheymeasured?ProfessorIrvingFisher’sunit,“thedesireofanarbitrarilychosenindividualAforB,asmallincrementofagivencommodityundergivencircumstances,”ispreferredbyMrPigoutoa“justperceivableincrementofpleasure”.Perhapsitisbettertosay,withProfessorA.Voigt,thatnounitisrequired:quantitieslikeutilityaretobemeasuredonlybyordinalnumbers.

Interestingly,Edgeworthagainreferstotheunitofmeasurementofutility,butnowclearlydistinguisheshisownpreferredchoice–“justperceivableincrementofpleasure”–fromFisher’schoices. Edgeworthwillneverdisownhisbeliefinapsychologicalfoundationforeconomics,but,throughhisconstantstruggletomaintainit,hewillhelpthesubtledifferencebetweenordinal,cardinalandratioscalestoemerge.

5. Edgeworth’s mature views on uti l i ty measurement Edgeworth’sideas,eveninthelateryears,wereprobablystillmorefavourabletohisinitialpsychophysicalintuitionthantoordinalism;nevertheless,thequestionwasonethatEdgeworthseriouslyandhonestlyconfrontedwithanopenmind. In1911hewroteanarticlefortheeleventheditionoftheEncyclopediaBritannica(Edgeworth1994c(1911))onprobabilityandexpectation.ItistheoccasionforEdgeworthtodiscussoncemorethepossibilityofquantitativemeasurementofcredibility,and,oncemore,herelatescredibilitytotemperature.Edgeworthrecognisesthatthemeasurementofafeelingmaybeonlyinpartialagreementwiththeindividualperception:nonetheless,eventhispartialagreementmaybeofsomeuse:“[T]heindicationsofathermometermaynotcorrespondtothesensationsofafeveredpatient,buttheyservetoregulatethetemperatureofapubliclibrary”(p.268).Edgeworthdeducesfromthisfuzzinessofmeasurementthat“twoorthreeroughlydistinguisheddegreesofcredibility”(ibid.)maysufficeforcalculations.Healsoacknowledgesthepossibilityofnonquantifiablescales,suchasnominalscales,(probablyinfluencedbyPearson1900). Edgeworth’smindismovingawayfromhisinitialbeliefinthenecessityofaratioscale,probablybecauseherealisesthatmathematicaltreatmentandstatisticsarepossibleeveninnon-numericalsituations.Hemaintainsnonethelessanimportantanalogybetweenthehumanandthephysicalsciences,bothbecause“theprincipleofprobabilitiesisessentiallythesameinthesetworegions”(Mirowski1994:292)andmorebroadlybecausethekindofknowledgethatispossibleinbothsciencesisthesame:statisticalknowledge.Evenifmeasurabilityisnotgrantedforeveryphysicalorsociologicalphenomenon,theepistemicvaluesandmoregenerallytheepistemologytobeassociatedwithbotharethesame,becausetheyrely,finally,onbeliefandcredibility. In1915EdgeworthwroteapaperfortheEconomicJournal,dealingwith“Recentcontributionstomathematicaleconomics”(Edgeworth1915)inwhichhecomesback,oncemore,tothequestionofthemeasurementofutility.HequotestothispurposealetterthatPoincaréwrotetoWalrasaboutthisveryquestion:

“Cansatisfactionbemeasured?Imaysaythatonesatisfactionisgreaterthananother,becauseIpreferonetotheother;butIcannotsaythatoneistwoorthreetimesgreaterthan

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another…Satisfactionthenisamagnitude,butnotameasurablemagnitude.Nowisamagnitudethatisnotmeasurablethereforenotamenabletomathematicaltheory[…]?Bynomeans.Temperature,forinstance(atanyratebeforetheterm‘absolutetemperature’hadacquiredasignificationwiththeriseofthermodynamics)wasanon-measurablemagnitude.Itwasarbitrarilydefinedandmeasuredbytheexpansionofmercury.Itmightquiteaslegitimatelyhavebeendefinedbytheexpansionofanyothersubstanceandmeasuredbyanyfunctionofthatexpansion,providedthatitwasacontinuallyincreasingfunction.Likewise,inthepresentcase,youmaydefinesatisfactionbyanarbitraryfunction,providedthatthefunctioncontinuallyincreasesalongwiththesatisfactionwhichitrepresents.”

(EdgeworthquotingPoincaré1915:57–8)Poincaréwasherestrikinglyclear.Hewasdistinguishingbetweenanordinalscale–satisfaction–andaratioone(absolutetemperature).Absolutetemperatureacquiresameaningwiththeriseofthermodynamics,i.e.byareductionistlinktoaphysicalzeropoint.Buttemperature,previoustothatlink,wasinstrumentally:“definedandmeasuredbytheexpansionofmercury”.Aswesaw,itwasinsuchawaythatthescalewasnotonlyincreasing,butwaslinearlyincreasingwithtemperature,i.e.itwasacardinalscale,evenifthisnotionhadnointerestandnomeaninginthehistoricalcontextoftheinventionoftemperatureandwasnotyetentirelydeveloped.EdgeworthcontinuesinhispapersbyendorsingordinalityandagaincitesVoigt’spaper,henceforthafriendlynemesis. In1922anoldEdgeworthwillwriteforthelasttimeaboututilitymeasurement,inapaperagainentitled“Thephilosophyofchance”,butdevotedthistimetocommentonJohnMaynardKeynes’streatiseonprobability(Edgeworth1922).Poincaré’slettertoWalrasiscitedagain,andEdgeworthinsistsontheordinalscaleforutility,butalsocomesbacktohisoriginalFechnerianideaofjustperceivableincrementsofpainandpleasureinMathematicalPsychics,wonderingwhetheritwouldnotbeanacceptableconcept.

6. Conclusions Whatisthemeaningof“twiceasuseful”?Aswesaw,suchaclaimwouldbemeaninglessbothinacardinalandinanordinalunderstandingofmeasurement.Itisasmeaninglessas“twiceashot”ifwemeanbyitthemeasurementofathermometer.Togiveameaningtosuchaclaimwouldrequireanchoringtheutilitymeasurementintoaphysicalunderstandingofpreferences,feelingsandbeliefs.Itwouldrequiretheobjectivemeasurementofsensation. Theattempttoanchortheutilitymeasurementfacedtwodistinctconceptualproblems.Ononeside,therewasanontologicalproblem:itcouldbesimplymeaninglesstoassociatenumericalvalueswithutility.ThiswasthecoremessageofVoigtcriticism,andthemainstruggleofEdgeworth.Ontheotherside,therewasanepistemicproblem:itmighthavebeenunnecessarytogivetheutilitymeasurementdeeppsychologicalfoundations. Ifnoinstrumentcanpossiblyassociateameaningtoautilitymeasurement,weareconfrontedwithanordinalunderstandingofit;butifsuchanumericalvalueexists,despitenotbeingaccessibletodirectmeasurement,thanaproxyofitcouldsuffice.Theorderthatsuchaproxywouldshowwillbelessthanratiomeasurement,butmorethanasimpleranking;itwillbeanalogoustothemercurycolumnofathermometer:acardinalmeasurement. WehaveshownthattemperatureandthethermometerwerethepolarstarbeyondEdgeworth,FisherandVoigt’sunderstandingofutilitymeasurement.Theyhelpedthembygivingaclear

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groundinginexistingknowledge.Thedivergentepistemicvaluesthattheyassociatedwithmeasurementwereatthecoreoftheirconflict,andthemainmotivationfortheirdifferentanswers.Ourauthors,Edgeworth,Fisher,PoincaréandVoigt,werefullyconsciousoftheanalogybetweenwhattheyweretryingtodo–measuringutility–andwhathadbeenaccomplishedthroughthequestofmeasuringtemperature.Theymadeconstantandsystematicreferencetothepropertiesofthethermometer,andthevicesandvirtuesofpastmeasurement,andwereseriouslyattemptingtounderstandthemeaningofutilitybyusingpastresultsonthemeaningoftemperature.Ibelieve,inviewofthenumerousreferencesthatIhaveshown,thatthiswasmorethanrhetoricalbombast.Temperaturewasaseriousconceptualtoolfortheunderstandingofutility.Itwasnotjustphysicsenvy(Mirowski1989):itwasadeepanalogythathadbothpedagogicalandconceptualvirtues. Ofcourse,noneofthoseauthorswereprofessionalhistoriansorphilosophersofscience(withthepossibleexceptionofPoincaré).Buttemperaturewasakeyinterpretationalconceptforourauthors,helpingtheminunderstandingwhattheyweredoing.Anditisforusavaluablehistoriographicalinstrumenttounderstandwhatthoseauthorsdid. WhileEdgeworthwastemptedbetweenratioandordinalmeasurement,becausebothforhimwereabletoconnectpoliticaleconomytopsychologyandthustrulycreateascienceofmind,itwascardinalversusordinalmeasurementthatbecamethecentreofinterestofeconomics.Edgeworthwasthelasteconomistofhisgenerationwhoexplicitlyaimedtogroundpoliticaleconomyinpsychology:hehadtorealisethatutilitymaysimplymean,inpsychologicalterms,moreorlessuseful,withoutanysensebeingassociatedwithtwiceasuseful.Thatwasastrongintuitionabouthowweunderstandutility. Edgeworthnevercompletelyabandonedhisfirmbeliefinpsychologismanditshopeforaratiomeasurementforutility.Lakatos(1978:20)famouslystatedthat“weliveanddieintheprisonofour‘conceptualframeworks’”.Nevertheless,wemustatleastacknowledgetotheOxfordeconomistthat,asintheproverbialeyeofthecyclone,thegustingwindsoftheordinalrevolutionweretwistingandturningaroundthepeacefulIrishintellectual.

References Alchian,ArmenA.1953.“TheMeaningofUtilityMeasurement.”TheAmericanEconomicReview43(1):26–50.

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