ListofIllustrations
1.1.EdwinAbbott, Flatland:ARomanceofManyDimensions (London:Seeley&Co.,1884),Figures1and2.53
1.2.EdwinAbbott, Flatland:ARomanceofManyDimensions (London:Seeley&Co,1884), finalpage.56
1.3.EdwinAbbott, Flatland:ARomanceofManyDimensions (London:Seeley&Co.,1884),frontcover.57
2.1.FromGeorgeBoole, AnInvestigationoftheLawsofThought (Mineola,N.Y.:Dover,1958),224.71
2.2.FromAugustusDeMorgan, OntheSyllogism,in Onthe SyllogismandOtherLogicalWritings,ed.PeterHeath (NewHaven:YaleUniversityPress,1966),206.71
2.3.FromJohnVenn, SymbolicLogic (London:Macmillan,1881), 106,107.71
2.4.FromLewisCarroll, SymbolicLogic (London:Macmillan,1896), frontispiece.72
3.1.OscarGustaveRejlander(British,bornSweden,1813‒75). TheInfantPhotographyGivingthePainteranAdditionalBrush, c.1856.Albumensilverprint6 7.1cm(23/8 213/16in.).
TheJ.PaulGettyMuseum,LosAngeles.112
3.2.LewisCarroll, TheMissesLutwidgePlayingChess, c.1858. NationalMuseumofPhotography,Film,and Television,Bradford.113
3.3.Clementina,LadyHawarden,1862‒3;©Victoriaand AlbertMuseum,London.114
3.4.Clementina,LadyHawarden,1862;©VictoriaandAlbert Museum,London.116
3.5.Clementina,LadyHawarden,1861;©VictoriaandAlbert Museum,London.118
3.6.Clementina,LadyHawarden,1862‒3;©Victoriaand AlbertMuseum,London.119
3.7.Clementina,LadyHawarden,1862‒3;©Victoriaand AlbertMuseum,London.119
3.8.Clementina,LadyHawarden,1863‒4;©Victoriaand AlbertMuseum,London.119
3.9.Clementina,LadyHawarden,1863‒4;©Victoriaand AlbertMuseum,London.119
3.10.Clementina,LadyHawarden,1863‒4;©Victoriaand AlbertMuseum,London.120
3.11.Clementina,LadyHawarden,1864;©Victoriaand AlbertMuseum,London.122
3.12.Clementina,LadyHawarden,1863‒4;©Victoriaand AlbertMuseum,London.123
3.13.WilliamNotman, MissStevenson,as “Photography,” Montréal,QC,1865;©McCordMuseum,I-14647.1.125
3.14.Clementina,LadyHawarden,1863‒4;©Victoriaand AlbertMuseum,London.127
4.1.IsaacNewton,Plate20fromAndrewMotte’stranslationof Newton’ s TheMathematicalPrinciplesofNaturalPhilosophy (London:BenjaminMotte,1729).142
4.2.MichaelFaraday,Plate1,Figure25from Experimental ResearchesinElectricity,vol.1,(London:Richard andJohnTaylor,1839).143
4.3.JamesClerkMaxwell, “LinesofForceandEquipotential Surfaces,” FigureIVfrom TreatiseonElectricity andMagnetism,vol.1,3rded.,editedbyJ.J.Thomson (Oxford:ClarendonPress,1892).151
4.4.JamesClerkMaxwell,Figure2from “OnPhysicalLines ofForce,” PhilosophicalMagazine,Vol.XXI(1861‒2).152
4.5.JamesAbbottMcNeillWhistler, SymphonyinWhite,no.2, 1864.CourtesyofTateImages.158
4.6.Titian, VenuswithaMirror, c.1555.Oiloncanvas. NationalGalleryofArt,AndrewW.MellonCollection.159
4.7.GeorgesdelaTour, TheRepentantMagdalen, c.1635‒40. Oiloncanvas.NationalGalleryofArt,AilsaMellonBruceFund.160
Introduction
[M]athematicalscience[is]notmerely...avastbodyofabstractand immutabletruths,whoseintrinsicbeauty,symmetryandlogical completeness...entitlethemtoaprominentplaceintheinterest ofallprofoundandlogicalminds,but...possess[es]ayetdeeper interestforthehumanrace,whenitisrememberedthatthisscience constitutesthelanguagethroughwhichalonewecanadequately expressthegreatfactsofthenaturalworld.
AdaLovelace(1843)1
Thetraditionalstoryoftheinfluenceofscienceonnineteenth-centuryart isamongthemostfamiliarinliterarycriticism:industrializingVictorian culture,positivisticandpracticalevenasitsometimesresistedthebroader implicationsofDarwinism,rejectedRomanticidealisminfavorofrealist aestheticsthatfocusedontheconcreteandthequotidian GeorgeEliot’ s “definite,substantialreality.”2 TheascendancyofGradgrindianfactsis memorablydramatizedintheopeningsceneof HardTimes,inwhich childrenaretaughtthathorsesaregraminivorousquadrupedsthatdonot belongonwallpaperbecausetheydon’ t “[walk]upanddownthesidesof roomsinreality.”3 Thisaccountremainsapowerfulone:withtheirrichly detaileddescriptions, finediscriminations,andfamilialandhereditary fixations,manyVictorianworksdoindeedbeartheimprintoftheempiricist mode,classificatorydrive,andfascinationwithevolutionarydynamicsthat characterizedthenaturalsciencesoftheperiod.Buttherelationshipbetween scienceandVictorianaestheticsismorethanjustadramaofrealismandits discontents.ForinitsmoreabstractandtheoreticalincarnationsVictorian scienceoftenprivilegeddeductionoverempiricalfact-findingandvaluedthe internalcoherenceofrepresentationalsystemsmorethantheircapacity
1 AdaLovelace,noteAtohertranslationof “SketchoftheAnalyticalEngine” by L.F.Menebrea.PrintedinTaylor’ s ScientificMemoirs,vol.III(London:Richardand JohnTaylor,1843),696.
2 FromEliot’sreviewofRuskin’ s ModernPainters, WestminsterReview 66(April1856): 626.
3 CharlesDickens, HardTimes (Oxford:OxfordUniversityPress,1989),6.
faithfullytodescribethenaturalandsocialworld.If,initsinductive manifestations,Victoriansciencesponsoredfamiliarvarietiesofrealism,in itsmoretheoreticalruminationsonmethoditsponsoredtheproductionof strikinglyabstractandevenfantasticworksofart.
Theprotagonistofthislessfamiliarstoryis,onthefaceofit,anunlikely one:mathematics.Incontrasttothelifesciences,littlehasbeenwritten abouttheinfluenceofVictorianmathematicsonVictorianaesthetics, althoughinthepastdecadethishasbeguntochangewiththeappearance ofbookssuchasAliceJenkins’ Spaceandthe “marchofmind” andDaniel Brown’ s ThePoetryofVictorianScientists. 4 Thisrelativeneglectisunfortunate,becausenoVictoriandiscourse,withthepossibleexceptionof painting,wasmoredeeplyconcernedwiththeworkingsandlimitsof representation.Mathematiciansexplicitlysetthemselvesthetaskofunderstandingtheroleofimaginationintheproductionofknowledgeandthe meansbywhichwords,images,andsymbolssignify.Historiansofmathematicsspeakof “thecriticalmovement” ofthesecondhalfofthenineteenthcentury,amovementthatledtoafundamentalreconceptionofthe natureandvalueofmathematicalknowledge:mathematicalrepresentations werenolongerpresumedtobedescriptive,andratherthan “relianceupon truththerewastobelogicalcompatibilityorconsistency.”5 Arithmeticitself wasredefinedsuchthat “numbers[were]nolongerinterpretedasobjects, butaspuresymbols,as ‘marks,’ asameansofobjectifyingmathematical thought–i.e.asalanguage.”6 Nineteenth-centurymathematicianHermann vonHelmholtzremarkedthatheregardedarithmeticasamethod “toteach theconsistentapplicationofasystemofsigns(i.e.numbers)ofunlimited extentandunlimitedopportunitiesofsophistication.”7 AshistorianAdrian Ricepointsout, “Britishmathematicsexperiencedadramaticrenaissance
4 AliceJenkins, Space (Oxford:OxfordUniversityPress,2007);DanielBrown, Poetry (Cambridge:CambridgeUniversityPress,2015).Anumberofcriticsarecurrentlycompletingprojectsthatexplorethepertinenceofmathematicalsciencestonineteenth-century culture,fromseniorscholarssuchasMarjorieLevinson,workingonCantorand field theory,toyoungerscholarslikeJeffreyBlevins,AnnaKornbluh,andJessicaKuskey.See alsoCarlaMazzioonmathematicsinRenassainceculture(ed., Shakespeare&Science, specialdoubleissueof SouthCentralReview [WinterandSpring,2009]),Matthew Wickmanonmathematicalconceptsintheeighteenthcentury(LiteratureAfterEuclid: TheGeometricImaginationintheLongScottishEnlightenment [Philadelphia:Universityof PennsylvaniaPress,2016]),andMirandaHickmanonVorticism(TheGeometryofModernism [Austin:UniversityofTexasPress,2005]).
5 MorrisKline, Mathematics:TheLossofCertainty (NewYork:FallRiverPress,1980),270.
6 HansNielsJahnkeandMichaelOtte, “OriginsoftheProgramof ‘Arithmetizationof Mathematics,’” in SocialHistoryofNineteenth-CenturyMathematics,ed.HerbertMehrtens, HenkBos,andIvoSchneider(Boston:Birkhäuser,1981),29.
7 Quotedinibid.,30.
duringQueenVictoria’sreign,” andBritishmathematicianswereparticularlyinterestedinthe “masteryofsymbolicmanipulation.”8
Becauseoftheculturalprestigeofmathematics,itscentralroleinthe explorationofthecharacterofknowledgeandthereferentialcapacitiesof representationswasapparenttoeducatedcontemporariesevenwhenthey werenotmathematiciansthemselves.Thiscentralitywasepitomizedinthe idealof “theCambridgeeducation,” which “culminatedinthemathematical Tripos,[but]wasnotintendedtoproducemathematicalspecialists....The studyofmathematicsformedthemajorpartoftheeducationalenterprise becauseitwouldmostdirectlyenablestudentstocometoknowthenatureof truth.”9 Howthat “truth” wasdefinedchangedoverthecourseofthe century:before1840Euclideangeometryservedasthemodelforrepresentationalreliability;from1840totheendofthecenturymodernalgebraand non-Euclideangeometrywouldstandattheheartofdebatesregardingthe possibilityofrepresentationaltransparency;andbytheturnofthecenturyit seemedthatmathematicallogiccouldserveastheultimatephilosophical language.Theepistemologyofmathematicschangedoverthecourseof thecentury,buttherenownofitsrepresentationalparadigmsdidnot.In theearlyyearsofthenineteenthcenturyWordsworthrhapsodizedoverthe powerofgeometrytofusetheidealwiththereal: “WithIndianaweand wonder/...didImeditate/Upontheallianceofthosesimple,pure/ Proportionsandrelations,withtheframe/AndlawsofNature.”10 Inthe earlyyearsofthetwentiethcenturyT.S.EliotpraisedBertrandRussellfor helpingtomakeEnglishalanguage “inwhichitispossibletothinkclearly andexactlyonanysubject.The PrincipiaMathematica areperhapsagreater contributiontoourlanguagethantheyaretomathematics.”11 Forus denizensofthetwenty-firstcentury,whotendtotaketheriftbetweenthe “twocultures” asagiven,itiseasytounderestimatethecentralityof mathematicalthinkingtoart,butthenineteenthcenturywasatimewhen mathematicianJ.J.Sylvesterpublishedavolumeonprosodytowhichhe appendedalectureonmathematics;physicistJamesClerkMaxwellwrote poetryonsuchtopicsasnon-Euclideangeometry;andpoetGerardManley Hopkinsplannedtowriteabookonwavetheory.12 Eccentricthoughhe
8 AdrianRice, “Introduction,” MathematicsinVictorianBritain,ed.RaymondFlood, AdrianRice,andRobinWilson(Oxford:OxfordUniversityPress,2011),2,6.
9 JoanRichards, “TheArtandtheScienceofBritishAlgebra:AStudyinthePerception ofMathematicalTruth,” HistoriaMathematica 7(1980):363.
10 WilliamWordsworth, ThePrelude:1799,1805,1850,ed.JonathanWordsworth, M.H.Abrams,andStephenGill(NewYork:W.W.Norton&Co.,1979),p.192(BookVI, ll.142–7).
11 T.S.Eliot, “Commentary,” TheCriterion 6:5(October1927):291.
12 IamgratefultoDanielBrownfortellingmeofthis;Hopkinsplannedtotitlethe book “LightandOpticks.”
was,mathematicianLewisCarroll’sfantasticnovelsandpoems,inwhich eventsareoftenshaped(andmisshaped)bytheprinciplesofmathematical logic,arebynomeansasanomalousastheymightat firstappear.
MATHEMATICALFORMALISM
Innumerableinstancesmightbegiveninillustrationoftheimmediateconnectionofthephysicalsciences,mostofwhichareunited stillmorecloselybythecommonbondofanalysis,whichisdaily extendingitsempire,andwillultimatelyembracealmostevery subjectinnatureinitsformulae.Theseformulae,emblematicof Omniscience,condenseintoafewsymbolstheimmutablelawsof theuniverse.
MarySomerville(1846)13
Theeasewithwhichmathematicalprinciplesfoundaplaceinartandart theoryduringthenineteenthcenturyreflectsnotjustthecentralityof mathematicstobourgeoisandeliteeducation onerecallsTomTulliver’ s struggleswithEuclidin TheMillontheFloss butalsotheincreasing interestinpure,asopposedtoapplied,mathematics.Thesalientfeatureof Britishmathematicsofthelatternineteenthcenturywasitspreoccupation withformalstructuresandproceduresperse,quiteapartfromtheir practicalapplications.Putanotherway,insofaras “form” cametobe valuedinandofitself,itbecamedesirabletodiscriminateitfromparticular “ contents ” (themathematicallyconservativeWilliamFrendtellingly describedhisopposedviewas “contentual”).14 Bryon’sdaughterAda Lovelaceunderscoresthesignificanceofformalprocedureinheraccount ofCharlesBabbage’sanalyticalengine;shearguesthatitiscrucialto distinguishthe “results ofoperations” fromoperationsthemselves,by which “ wemean anyprocesswhichaltersthemutualrelationoftwoor morethings.”15 Thepracticalvalueoftheresultofanoperationisclear enough,butLovelacepointsourattentiontotheformulasthatyieldthat result,formulasthatdealinrelationshipsratherthanessences.Shegoeson toinsistthatthe “scienceofoperations,asderivedfrommathematics...is ascienceofitself,andhasitsownabstracttruthandvalue” (693).
13 MarySomerville, OntheConnectionofthePhysicalSciences (NewYork:Harperand Brothers,1846),390.
14 FromaletterofJune22,1836toAugustusDeMorgan;seeHelenaPycior, “Early CriticismoftheSymbolicalApproachtoAlgebra,” HistoriaMathematica 9(1982):396.
15 Lovelace, “Sketch,” 693.
Indeed,formsandformulaswereincreasinglyregardednotsimplyasa meanstoanendbutasimportantinthemselves.Thisviewregistersa crucialmid-centuryshift;priortothe1840sgeometrywas “queenofthe sciences” preciselybecauseitbridgedthegapbetweenidealizedformand real-worlddescription itwassimultaneouslyabstractandconcrete.But withtheadventofnon-Euclideangeometriesaroundmid-century,internal formalcoherence theconsistentmanagementofrelationships gradually cametoseemmoreimportantthandescriptiveness.Aslongastheywere formallycoherent,even “imaginarygeometries” hadvalue.Thestudyof algebrawastransformedalongsimilarlines.InEnglandalgebrahadtraditionallybeenregardedwithsuspicionbecauseitssymbolswerepalpably arbitrary;theywere “symbolsbewitched,andrunningabouttheworldin searchofmeaning.”16 Thisprejudicewouldbegintochangewiththe publicationin1830ofGeorgePeacock’ s TreatiseonAlgebra.Peacock distinguished “arithmetical” algebra,theworkingsofwhichweretransparentbecauseitoperatedonlyonpositiverealnumbers,from “symbolical” algebra,which,becauseitoperatedonnegativeandimaginarynumbers, seemedtorestonuncertainfoundations.Hesalvagedsymbolicalalgebraby proposingthat “thougharbitraryintheauthorityofitsprinciples,” it achievedatleastaformalcoherenceinthatit “[was]notarbitrary” inits useofthoseprinciples,whichwere,moreover,consistentwiththeprinciplesofarithmeticalalgebra.17 OverthecourseoftwovolumesPeacock elaborates “thegenerallawoftransitionfromtheresultsofarithmeticalto thoseofsymbolicalalgebra,” whichlawhedenominates “the ‘principleof thepermanenceofequivalentforms. ’”18
Thistoleranceforungroundedsymbolismmetwithsomeresistance: theeminentmathematicianWilliamRowanHamiltoncomplainedthat Peacock’sbookwas “designedtoreducealgebratoameresystemof symbols,and nothingmore;anaffairofpothooksandhangers,ofblack strokesuponwhitepaper,tobemadeaccordingtoa fixedbutarbitraryset ofrules.”19 TheFrenchanalysisthatPeacockandhisreformistpeershoped ultimatelytopopularizeposedanevengreaterthreat: “Lagrange’salgebraic calculuswasperceivedasdangerousbyconservativeAnglicans[inpart because]itwasassociatedwiththementalityaccompanyingtheFrench
16 AugustusDeMorgan,reviewofGeorgePeacock, ATreatiseonAlgebra,in Quarterly JournalofEducation 9(1835):311.
17 GeorgePeacock, “ReportontheRecentProgressandPresentStatesofCertain BranchesofAnalysis,” in ReportontheThirdMeetingoftheBritishAssociationforthe AdvancementofScience (London:JohnMurray,1834),195.
18 GeorgePeacock, TreatiseonAlgebra,2vols.(Mineola,N.Y.:Dover,2004),vol.1,vii.
19 Froman1846lettertoPeacock.ReproducedinRobertGraves, LifeofSirWilliam RowanHamilton,vol.II(Dublin:Hodges,Figgis&Co.,1882),528.
Revolution....Theabstractnatureofapurealgebraiccalculusseemingly allowedthemindtowanderintofantasythroughthemeaninglessmanipulationofsymbols.”20 ForaconservativeBritishmathematicalestablishment,Newton’swasthepreferredversionofthecalculus:ithadthedual advantagesofbeingarithmeticallyorientedandBritish.TheLagrangian analysisthatreformerssoughttoimportnotonlyinvolvedunfamiliar symbolsbutwasformalratherthanquantitativeinitsnature: “Whenthe infinitesimalcalculusbecamemoreandmorealgebraic,itsontologicalbasis changedaswell. ...someauthorscametoholdtheviewthattheobjectsof theformulaeofalgebraandanalysisarenotquantitiesbutratherrelations betweenalgebraicandanalyticoperations;inthisviewthelettersrepresentingvariablesaremerely ‘bearersoftheoperations.’”21 Acceptanceof thenewanalysis andtheformalaccountofalgebraonwhichitrelied cameonlygradually;aslateas1850aRoyalCommissiontaskedwith assessingthestateofmathematicalstudyatCambridgeraisedthequestion ofwhether “thecourseofMathematicalstudy ...is tendingtobecome moreandmoreexclusivelyanalyticalandsymbolic” andrecommended thatinappliedmathematicsatleast “allunnecessaryexuberanceofan analyticalcalculationberepressed.”22 Changedid,however,come,and onceformalcoherenceratherthanreferentialitybecamethekeycriterion fortheevaluationofalgebra,theproductionofnewalgebras amongthem quaternions,determinants,matrices,andinvariants becamesomething ofacottageindustryinBritain.23
Indeed,algebrabecameamodelforsignsystemsgenerallydespiteits opacityvis-à-visitsparticularobjectsbecauseoftheclarityandprecision withwhichitrevealedrelationshipsamongthoseobjects.24 ThusGeorge Booleandhisfellowlogicianswouldrevolutionizelogicbyrenderingit algebraically;StanleyJevonswouldfoundmarginalisteconomicsbyconceivingofvaluenotasanessentialgoodbutasavariablerelationshipbest understoodusingthetechniquesofanalysis;JamesClerkMaxwellwould
20 WilliamAshworth, “Memory,Efficiency,andSymbolicAnalysis:CharlesBabbage, JohnHerschel,andtheIndustrialMind,” Isis 87:4(December1996):634.
21 HansNielsJahnke, “AlgebraicAnalysisintheEighteenthCentury,” in AHistoryof Analysis,ed.HansNielsJahnke(AmericanMathematicalSociety,2003),107.
22 Fromthe ReportofHerMajesty’scommissioners,appointedtoinquireintothestate, discipline,studiesandrevenuesoftheuniversityandcollegesofCambridge (London: W.ClowesandSons,1852),229,112.
23 OnthepopularityofalgebrasinVictorianEngland,seeI.Grattan-Guinness, “VictorianLogic,” p.374,andKarenHungerParshall, “VictorianAlgebra,” in Mathematics inVictorianBritain,ed.RaymondFlood,AdrianRice,andRobinWilson(Oxford:Oxford UniversityPress,2011).
24 Inhis VictorianRelativity ChristopherHerbertarguesthat “Relativitynamesitselfas thegreattransformativeideaoftheagebeforeEvolutiondoesso” ([Chicago:Universityof ChicagoPress,2001],45.)
deviseequationsthatshowedthatelectricity,magnetism,andlight,however differenttheymightappearinsubstance,wereformallycomparable,and wouldarguethatthiscomparabilitywastheirsalientfeature.Algebrawasthe preeminentlanguageofformalistabstractioninacultureincreasingly devotedtosuchabstraction.HistorianofmathDoronSwadesumsupthe change: “asilentpremiseofcontemporarymathematicsandphilosophy wasthatexamplewasinferiortogeneralization,inductioninferiortodeduction,empiricaltruthsinferiortoanalyticaltruths,thesyntheticinferiorto theanalytic.”25
ThisemphasisonformsandrelationshipsunderwrotethenewintellectualandrepresentationalidealLorraineDastonandPeterGalisoncall “structuralobjectivity.” AsGalisonandDastonexplain,inthelatenineteenthcentury, “especiallyinthe fieldsoflogicandmathematics...the word ‘ structure ’ acquirednewmeaningsandintellectualglamour,” a glamourthatledtoaredefinitionofobjectivity: “objectivitylayinthe invariablerelationsamongsensations,readliketheabstractsignsofa languageratherthanasimagesoftheworld.”26 Thisepistemologicalideal was,inessence,analgebraicone,abstractandformalist: “[forsome]these structureswerelawlikesequencesofsigns;forothers,theyweredifferential equations;forstillothers,logicalrelationships.”27 Underthisdispensation,allknowledgehadadifferentialcharacter;HenriPoincaréremarked that “allthatisobjectiveisdevoidofallqualityandisonlypurerelation. Certainly,Ishallnotgosofarastosaythatobjectivityisonlypure quantity ...butweunderstandhowsomeonecouldhavebeencarried awayintosayingthattheworldisonlyadifferentialequation.”28 When Saussureproclaimedthatlanguage “isaformandnotasubstance, ”29 hewas,then,rehearsinganalready-familiartheme.Indeed,Saussurean linguisticshasmuchincommonwithVictorianalgebra;GeorgeBoole, drawingon “thepresentstateofthetheoryofSymbolicalAlgebra,” hadlongsincedefined “Asign[as]anarbitrarymark,havinga fixed
25 DoronD.Swade, “CalculatingEngines:Machines,Mathematics,andMisconceptions,” in MathematicsinVictorianBritain,ed.RaymondFlood,AdrianRice,andRobin Wilson(Oxford:OxfordUniversityPress,2011),251.
26 LorraineDastonandPeterGalison, Objectivity (NewYork:MITPress,2007),255,253.
27 Ibid.,254.DastonandGalisonexplainthat “manyvoicesspokeoutforstructural objectivityintheperiodbetweenroughly1880and1930” (261)andthat “[t]hosewhodid identify ‘ structures ’ asthecoreofobjectivityunderstoodagreatvarietyofthingsunderthat rubric:logic,orderedsequencesofsensations,someofmathematics,allofmathematics,syntax, entitiesthatremaininvariantundertransformations,anyandallformalrelationships” (254).
28 HenriPoincaré, TheValueofScience:EssentialWritingsofHenriPoincaré (NewYork: ModernLibrary,2001),345.
29 FerdinanddeSaussure, CourseinGeneralLinguistics (1916),trans.WadeBaskin, ed.CharlesBally,AlbertSechehaye,andAlbertRiedlinger(NewYork:McGrawHill, 1966),122.
interpretation,andsusceptibleofcombinationwithothersignsinsubjection to fixedlawsdependentupontheirmutualinterpretation.”30
GeorgLukácswouldlaterdescribethisostensiblyobjectiveformalism asoneoftheessentialfeaturesofbourgeoiscapitalism: “themoreintricate amodernsciencebecomesandthebetteritunderstandsitselfmethodologically,themoreresolutelyitwillturnitsbackontheontological problemsofitsownsphereofinfluenceandeliminatethemfromthe realmwhereithasachievedsomeinsight.Themorehighlydevelopedit becomesandthemorescientific,themoreitwillbecomeaformallyclosed systemofpartiallaws.”31 Certainly,Victoriansthemselvesarguedthat formalconsistencywasthe sinequanon ofanyscientificinquiry: “Any factsare fitted,inthemselves,tobeasubjectofscience,whichfollowone anotheraccordingtoconstantlaws.”32 Overthecourseofthecentury, moreandmore(increasinglyprofessionalized)disciplinesaspiredto “scientific ” status,andtheysoughttoachieveitbyfocusingonmethod, protocol,andthediscoveryofformallaws,fromthelawofdiminishing returnstoGrimm’sLaw.
Victorianmathematicsthusnotonlyredefineditselfasa fieldbutalso allthe fieldsaroundit:inquirycenteredonfactsanddatawasincreasingly distinguishedfromthestudyofthelaws or “operations”—thatstructuredthosefacts.Inher GenresoftheCreditEconomy MaryPooveytracks thisbifurcation,focusingprimarilyonthe fieldsofeconomicsandliterature, fieldsthatwereparadigmaticintheireffortsatself-definitionandtheir privilegingoftheoretical,formalknowledge.AsPooveyexplainsofwriters ontheeconomy, “Economictheoristsofferedavarietyofwaystoexplain thedisjunctionbetween(unreliablebecauseparticular) ‘facts’ and(reliable becausegeneral) ‘principles.’”33 Thisdividegroundedbothgenericand professionaldistinctions: “Inthecourseofthecentury ...onegroupof writers ...collectedfactualinformation,andanothergroupofwriters, whoseworkwasheldtobemoreprestigious,usedthesedatatogenerate modelsthat,especiallyafterthe1870s,aspiredtothelogicalrigorand formalabstractionofmathematics” (221).AsPooveypointsout,even novelistswhohadaninvestmentinpresentingtheirworkasempirically
30 GeorgeBoole, TheMathematicalAnalysisofLogic:BeinganEssaytowardsaCalculusof DeductiveReasoning (1847)(Bristol:ThoemmesPress,1998),3;Boole, AnInvestigationof theLawsofThoughtonWhichAreFoundedtheMathematicalTheoriesofLogicand Probabilities (1854)(Mineola,N.Y.:Dover,1958),25.
31 GeorgLukács, HistoryandClassConsciousness,trans.RodneyLivingstone(Cambridge, Mass.:MITPress,1971),104.
32 JohnStuartMill, ASystemofLogic,RatiocinativeandInductive,vol.8of Collected WorksofJohnStuartMill,ed.J.M.Robson(Indianapolis:LibertyFund,2006),844.
33 MaryPoovey, GenresoftheCreditEconomy (Chicago:UniversityofChicagoPress, 2008),222.
richrecognizedthisdistinctionandoftenprivileged “reliablebecause generalprinciples.” ThusMargaretOliphantarguedthat “Truthisthat grandgeneralruleofhumanity,theharmoniouslawwhichrunsthrough everything. ...Amanwho followsfactinartattheexpenseoftruth,is accordinglytakingthelawless[path].”34 Itisnotbychancethatprofessionalizationcametobeunderstoodlessintermsofagradualtrajectory fromapprenticeshiptomasterythanintermsofknowledgeofand submissiontothelawsofadiscipline.Thiscommonstructurewithin andacross fieldsmadethemamenabletoinstitutionalization,typicallyin theuniversity.AsPooveyremarksofeconomics, “[by]the1870s...the practitionersofpoliticaleconomyhadadoptedamethodologythatresembledthoseofothersubjectstaughtinuniversities,especiallythephysical sciencesandmathematics” (229).
Thiscoordinationof fieldswenthandinhandwiththeestablishmentof theirautonomy.In TheRulesofArt,hisstudyofthenineteenth-century originsofthemodernFrenchliterary field,PierreBourdieuarguesthat while “structuralandfunctionalhomologiesexistbetweenallthe fields,” all fields,evenartisticones,scrupulouslydefinedandguardedtheirown terrain: “Theinventionofthepureaestheticisinseparablefrom...the greatprofessionalartistwhocombines ...a senseoftransgression ...with therigorofanextremelystrictdiscipline.”35 Pooveymakesasimilar argumentforBritishliterature.LikeBourdieu,shebeginsbyunderscoring theroleofgenericdistinctionintheestablishmentofprofessional fields: formalgenericpurityservedasaguarantorofautonomyof fieldand proprietyofmethod.36 Butthisemphasisonformhelpedtoguarantee thatevenas fieldsbecameincreasinglyautonomousandself-referential
34 MargaretOliphant, “NewBooks,” Blackwood’sEdinburghMagazine 108(August 1870):185.
35 PierreBourdieu, TheRulesofArt:GenesisandStructureoftheLiteraryField,trans. SusanEmanuel(Stanford:StanfordUniversityPress,1995),182,111.Tellingly,Bourdieu himselfusesthelanguageof(physical) fieldtheorytomakethepointthattheautonomyof (metaphorical)professional fieldsisonlyapparent,describingcommerceandbohemianart notasexclusivebutasexistinginamutuallydefiningtension.
36 LikePoovey,Bourdieuinsiststhatgenericdistinction(inthiscasebetweendrama,the novel,andpoetry)goeshandinhandwith “theprogressoftheliterary fieldtowards autonomy ” (Bourdieu, Rules,115).Eachofthesegenres,inturn,ispolarizedintoa “researchsector,” withitspretentionstotheoreticalvalue,anda “commercialsector” (120).UnlikeBourdieu,however,Pooveyconcludesherstudybyarguingthatthemathematicalmodelingthatservedasthebasisforeconomictheoryservedtodistinguishitfrom the “versionofformalism” developedbyliterarywriters.Iwouldargue,however,thatitis notbychancethatbothdevelopmentsservedadisciplinaryfunction,forbotheconomic andimaginativewriterswere,ineffect,mathematicizingtheirdiscipline.If,atthebeginning ofthecentury,thatmeanttryingtoproduceorganic,referentialsymbolsinthemannerof geometry,bymidandlatecenturythatmeantproducingself-referentialsystemsinthe mannerofnon-Euclideangeometryoralgebra.
theywerepresumedtobestructuredanalogously.Justastheartist consideredhimselfakindofscientist,scientistsconceivedtheirworkas akintotheworkofprofessionalartists.PhilosopherofscienceWilliam Whewell’scoinageoftheword “scientist” reflectsthisfact: “Aswecannot usephysicianforacultivatorofphysics,Ihavecalledhima physicist.We needverymuchanametodescribeacultivatorofscienceingeneral. Ishouldinclinetocallhima Scientist.Thuswemightsay,thatasanArtist isaMusician,Painter,orPoet,aScientistisaMathematician,Physicist,or Naturalist.”37
ALGEBRAICART
Theincongruitybetweenadvancedmathematicsandversecompositionismoreapparentthanreal Mathematiccommencingasa practicalart,thencepassingintotheformofascience,havingagain emergedintoanartofahigherorder a fineart plasticinthe handsoftheMathematician,obedienttoandtakingshapefromhis will,andalmostadmittingofthefreeplayoffancyuponit,thanksto thedeeperprinciplesevolvedinthenewGeometry,thehigher Algebra,andthecalculusoftheContinuousRiemonn(sic).
JamesJosephSylvester(1876)38
Ofcourse,formhadlongbeenregardedascentraltoartandaesthetic judgment,anditsimportancetoVictorianartistsandthinkersowesmuch totheinfluenceofKantinparticular.IntheKantianaccount,formisdefined inoppositiontofeaturesconsideredsecondary(intheLockeansense)or ornamental.Inart-historicalterms,formis disegno ratherthan colore:
Inpainting,sculpture,andinalltheformativearts inarchitecture,and horticulture,sofarastheyarebeautifularts the delineation istheessential thing;andhereitisnotwhatgratifiesinsensationbutwhatpleasesbymeans ofitsformthatisfundamentalfortaste.39
Andagain,
Everyformoftheobjectsofsense...iseither figure or play.Inthelattercase itiseitherplayof figures(inspace,viz.pantomimeanddancing),orthemere
37 WilliamWhewell, ThePhilosophyoftheInductiveSciences,2vols.(London:John Parker,1840),vol.1,cxiii.
38 JamesJosephSylvester, FliegendeBlätter:SupplementtoTheLawsofVerse (London: Grant,1876),5.
39 ImmanuelKant, Kant’sCritiqueofJudgement,trans.anded.J.H.Bernard(London: MacmillanandCo.,1914),75.
playofsensations(intime).The charm ofcoloursorofthepleasanttonesof aninstrumentmaybeadded;butthe delineation inthe firstcaseandthe compositioninthesecondconstitutetheproperobjectofthepurejudgementoftaste....Evenwhatwecall ‘ ornaments ’ ... i.e. thosethingswhichdo notbelongtothecompleterepresentationoftheobjectinternallyaselements,butonlyexternallyascomplements,andwhichaugmentthesatisfactionoftaste,dosoonlybytheirform.(75‒6)
Thisdefinitionlinksformtomeaningandpurposiveness.Theobjectof aestheticjudgment “canbenothingelsethanthesubjectivepurposiveness intherepresentationofanobjectwithoutanypurpose ...andthusitisthe mereformofpurposivenessintherepresentationbywhichanobjectis given tous...which ...isthe determininggroundofthejudgementof taste ” (69‒70).Asthevehicleofpurposivenesswithoutpurpose,the aesthetichasthepeculiarvirtueofallowingforpurposivenesswithoutthe taintofinterest,which,forKant,wouldcompromisethe “purity” of aestheticpleasure: “Everypurpose,ifitberegardedasagroundofsatisfaction,alwayscarrieswithitaninterest” (69).Aestheticformexercisesthe cognitivefacultiesinawaythatissimultaneouslynon-instrumentaland non-frivolous.
Thusdescribed,Kant’saestheticformsoundsremarkablylikethe Victorianidealof “puremathematics.” ThatidealhaditsrootsinGreek speculativemathematics,butthedistinctionbetweenpureandapplied mathematicswasconsolidatedinthenineteenthcentury.InEngland,this consolidationwasinstitutionalizedinthe1860establishmentofthe SadleirianChairinPureMathematicsatCambridge.Unlike “speculative,” theword “ pure ” expressesthepresumptionthatabstraction one historianofmathcallspuremathematics “formalaxiomatics”40 isthe veryessenceofmathematicalstudy.Butitalsosignalsthebeliefthat abstractionisfreeofthetaintofworldliness.LiketheKantianartwork, puremathematicsembodiespurposivenesswithoutpurpose.
Infact,VictorianmathematicianswouldpressKantianformalismina directionthatwouldprovidethecategoryoftheaestheticwithadditional ideologicalpowerandamuch-expandeddisciplinaryreach.Theydidthis, firstofall,byattackingKant’sownaccountofthevirtuesofEuclidean geometryandbyunderminingtheprestigeofsyntheticknowledge. HermannVonHelmholtzputsthecaseveryclearly: “Theassumptionof aknowledgeofaxiomsbytranscendentalintuitionapartfromallexperience is(a)anunprovedhypothesis,and(b)anunnecessaryhypothesis...also,as
40 HowardWhitleyEves, FoundationsandFundamentalConceptsofMathematics (Mineola,N.Y.:Dover,1990),150.
regardsourobjectiveknowledge,(c)awhollyirrelevanthypothesis.”41 Helmholtzclaimsthat “theaxiomsofgeometryarenotsyntheticpropositions,” andthatEuclidean “space-intuitions” arethereforeintuitions “ofthe kindtheartistpossessesoftheobjectsheistorepresent,andbymeansof whichhedecidessurelyandaccuratelywhetheranewcombinationwhich hetrieswillcorrespondornottotheirnature.”42 InthewordsofArthur Cayley,the firstoccupantoftheSadleirianChair, “Asforeverythingelse,so foramathematicaltheory:beautycanbeperceivedbutnotexplained.”43
Underthisdispensationmathematicsandartcometosharethesame virtues.Therepresentationsofbothare “merely” analytic,intheKantian sense;theycannotclaimtooffertruthsabouttheworlditself.Buttheyare bothpurposive,andtellaformaltruth.Victoriansinvestedthistruthwith transcendentalandevenspiritualvalue.Formathematicianandnovelist EdwinAbbottmathematicalforms,suchasastraightlineoraperfect circle,although “non-existent,” are “realandtrue.”44 ForAbbott,mathis “visionary” (31)and “illusive,” but “alsoleadsustotruth” (47);themathematician’simaginationsimplyworksinthesamewayanartist’sdoes.In effect,mathematicsandartarebothstructuredsymbolicsystemsthetruth ofwhichinheresintheirinternalcoherence.InHelmholtz’swords, “We are...justifiedintakingourspace-perceptionsassignsofcertainotherwise unknownrelationsintheworldofreality,thoughwemaynotassumeany sortofsimilaritybetweenthesignandwhatissignified.Butifonlysomuch standsfast thattounlikesignstherecorrespondunlikeobjectsandtolike signstherecorrespondobjectsthatarelikeinacertainrelationorcomplex ofrelations...thiswillsufficetoyieldusarealcontent.”45
Nineteenth-centuryartistswouldprofitfromthisnewaccountof knowledge,forunderitsauspiceseven fictionalrepresentations,likethe new “imaginarygeometries,” couldbeconceivedasknowledgeofthemost prestigiouskind, “analytic” knowledge thistimeinthemathematical senseoftheword.Initscapacitytoembodystructural,formaltruths,art couldclaim,likemodernmathematics,torelayamoreprofoundknowledge thananempiricallyorientedmimesiscouldprovide.AsBourdieuremarksof Baudelaire,a “pureworkonpureform” providesaccessto “arealmorereal
41 HermannvonHelmholtz, “TheOriginandMeaningofGeometricalAxioms(II),” Mind 3(1878):225.
42 HermannvonHelmholtz, “TheOriginandMeaningofGeometricalAxioms,” Mind 1(July1876):320.
43 ArthurCayley, “PresidentialAddress,” in ReportoftheFifty-thirdMeetingoftheBAAS heldatSouthportinSeptember1883 (London:JohnMurray,1884),7.
44 EdwinA.Abbott, TheKernelandtheHusk:LettersonSpiritualChristianity (London: Macmillan,1886),32.
45 Helmholtz, “AxiomsII,” 224.
thanthatwhichisoffereddirectlytothesenses,” a “realistformalism” ; “[w]ith noreferentotherthanitself,thepoemisacreationindependentofcreation, andneverthelessunitedwithitbyprofoundtiesthatnopositivistscience perceives.”46 ManyBritishwriters,too,understoodtheirworkinformalist terms,asaformalscienceinitsownrightandonethatrequiredtrainingto understand:ThomasHardycelebratedthe “beautyofshape” ofthenovel, andclaimedthatfewnon-professionalreaderscould “enjoyandappreciate” thenovel’ s “artisticsenseofform”“withoutsomekindofpreliminary direction.”47 RobertLouisStevensonarguedthat “[t]hearts,likearithmetic andgeometry,turnawaytheireyesfromthegross,colouredandmobile natureatourfeet,andregardinsteadacertain figmentaryabstraction. Geometrywilltellusofacircle,athingneverseeninnature;askedabout agreencircleoranironcircle,itlaysitshanduponitsmouth.”48
Meanwhile,theprioritymathematiciansplacedoninternalcoherence asopposedtoreal-worldreference,harmoniousorganizationratherthan densityofdetail,wouldleadthemtospeakoftheirownworkintermsthat makethemsoundalmostlike fin-de-siècle aesthetes.AdaLovelacerhapsodizedthatBabbage’sanalyticalenginecouldintheoryoperateon pitchedsoundsratherthanquantities,composing “elaborateandscientific piecesofmusic”;49 WilliamHamiltonclaimedthathisvector-likequaternionshadaquaternionofparents, “geometry,algebra,metaphysicsand poetry ”;50 andJ.J.Sylvestercontendedthattherewas “anexacthomology”“betweenpaintingandpoetryontheonehandandmodern chemistryandmodernalgebraontheother.... ”51 Indeed,Sylvester wouldnotonlyspeakofmathematicsasthequintessenceandevolutionary perfectionoflanguagebutwouldwriteanaesthetictreatise, TheLawsof Verse,inwhichheturnstothemathematicaltheoryofpermutationsto discusslinelengthandwordgroupingsinadditiontousingsquare
46 Bourdieu, Rules,107.
47 ThomasHardy, “TheProfitableReadingofFiction,” in LifeandArt:Essays,Notes, andLettersCollectedfortheFirstTime,ed.ErnestBrenneckeJr.(NewYork:Greenberg, 1925),68,70.
48 RobertLouisStevenson, “AHumbleRemonstrance,” in VictorianCriticismofthe Novel,ed.EdwinEignerandGeorgeWorth(Cambridge:CambridgeUniversityPress, 1985),216.
49 Lovelace, “Sketch,” 694.Similarly,shenotesthattheenginenotonlyusespunched cardslikeaJacquardloom,butthat “wemaysaymostaptlythattheAnalyticalEngine weavesalgebraicalpatterns justastheJacquard-loomweaves flowersandleaves” (696).
50 HamiltontotheRev.RichardTownsend,May14,1855;TrinityCollegeMS notebook1492–126,14–15.
51 JamesJosephSylvester, “Onanapplicationofthenewatomictheorytothegraphical representationoftheinvariantsandcovariantsofbinaryquantics,” in TheCollected MathematicalPapersofJamesJosephSylvester,4vols.(Cambridge:CambridgeUniversity Press,1909),vol.3,190.
