“The Geometry Behind Poincaré’s Conventionalism” Jeremy ......Poincare’s famous sphere...

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“TheGeometryBehindPoincaré’sConventionalism”

JeremyHeisUniversityofCalifornia,Irvine

UnderReviewJuly2019

HowdoesPoincaréargueforhisconventionalismaboutgeometry?Inparticular,

whatfeaturesofgeometrydoeshisargumentrelyon?Accordingtosomecommon

interpretations,Poincaré’sargumentdependsonfeaturesthatarenotuniqueto

geometry.Forinstance,ononerecentreading(Gimbel2004),Poincare’s

conventionalismamountstonothingmorethantrivialsemanticconventionalism:

thetruthofageometricalsentencesuchas“thedistancefromAtoB>thedistance

fromCtoD”dependsinteraliaonthemeaningoftheword“distance”;butwhich

meaningweassigntotheword“distance”issimplyaconventionalfactaboutour

language;sothetruthofthesentenceisamatterofconvention.Sinceitisclearthat

thisargumentcouldberunforanysentence(thisiswhyitis“trivial”),this

interpretationofPoincaré’sconventionalismturnsonlyonthemundanefactthat

sentencesofgeometryarecomposedofwords.

InterpretationsthatassimilatePoincaré’sargumenttoDuhemianarguments1

basedonunderdeterminationalsoturnonafeaturethatisnotuniquetogeometry:

1.Themetricofspaceisunderdeterminedbyouraprioriandempirical

evidence.

1ForexamplesofinterpreterswhoattributethisargumenttoPoincaré,seeStump1989,note52;BenMenahem2006presentsamoresophisticatedversionofthisreading.

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2.Givensuchunderdetermination,onlyconventioncanbeusedtodecideona

metric.

3.Mattersofconventionarenotmattersoffact.2

So,themetricofspaceisnotamatteroffact,butofconvention.

Poincare’sfamoussphereargumentin“SpaceandGeometry”(1895/1913)3might

becitedinsupportofpremise1.Poincareimaginesbeingslivingintheinteriorofa

spherewithatemperaturefieldthatdecreasestoabsolutezeroattheboundaryof

thesphereaccordingtotheformulaR2-r2,whereRistheradiusofthesphereandr

isthedistanceofapointfromthecenter.Ifthesizesofbodiescontract

proportionatelytotheirtemperature,nobodycouldtravelinfinitetimeandatfinite

speedfromthecenterofthespheretoitsboundary,sincethebodywillgetsmaller

andsmallerasittravelstowardtheboundary.Itwouldbeconsistentwithallofour

aprioriandempiricalevidencetosayofsuchbeingsthattheyliveinaninfinite

worldwherethemetricishyperbolic.Alternately,itwouldbeconsistentwiththe

empiricalandapriorievidencetoclaimthattheyliveinafiniteregionwithina

spacewherethemetricisEuclidean,aslongaswemodifyourusualphysicallawsto

includethetemperaturefieldIdescribedabove.Furthermore,Poincarésupposes

thatlightwithinthediscrefractswithanindexofrefractioninverselyproportional

2AsPoincaréusestheterm“convention,”a“convention”isnotatruthandsoalsonotamatteroffact.See,e.g.,(1891/1913,39).3TheoriginalFrenchversionofthisessayappearedin1895.Itwaslightlyrevised–i.e.,oneparagraphwasremoved–andrepublishedin1902intheFrenchversionofScienceandHypothesis,whichwasthentranslatedintoEnglishin1905,andthenre-translatedintoEnglish(byGeorgeHalsted)inafarsuperiortranslationfrom1913.Icitepagenumbersfromthe1913Englishre-translation.ButIalsoincludethepublicationdateoftheoriginalessaysaswell,sincetheoriginalpublicationdateswillbeimportantinmydiscussionofRussellandPoincarébelow.

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toR2-r2.Insuchaworld,beamsoflightwilltravelincirculararcswithinthesphere

(seefigure1).Again,wecandescribethepathsofthelightbeamsasstraightlines

thatobeyhyperbolicgeometry,orwecandescribethemascirculararcsthatobey

Euclideangeometry–aslongaswemodifyourusualphysicallawstoincludethis

lawofrefraction.Sinceourtotalphysicaltheory–includingourphysicallawsand

ourgeometry--facesexperienceonlyasunit,wegettheusualDuhemianconclusion

thatbothalternativesareconsistentwithallofourevidence,andsoitisonly

conventionandnotmattersoffactthatdeterminewhichgeometrytochoosefor

suchaworld.

OntheDuhemianreading,therelevantfactaboutgeometrythatleadsto

conventionalismisjustthefactthatgeometricalsentencesarepartofourtotal

physicaltheory.ThismakestheDuhemianreadingveryunattractive.4Itwouldseem

thenthateverysentenceineveryphysicaltheorywouldturnoutconventional,

whichwouldjustcollapsethedistinctionbetweenmatteroffactandconventionand

therebyleaveconventionalismwithoutmuchinterest.Moreover,therearepartsof

ourtotalsciencethatPoincaréclearlythinksarenotconventional.Inparticular,

Poincareseemstobelievethatitisnotaconventionthatspaceallowsfor

displacementsthatformagroup,5andhealsobelievesthatallofarithmeticconsists

ofsyntheticaprioritruths,notconventions.

Becauseofthefailuresofthesereadings,BenMenahem2006hasarguedthat

thisDuhemianargumentneedstobesupplementedbyspecificfactsabout

geometry.Onherreading,whatisimportantaboutthespheremodelofhyperbolic 4SeeStump1989andFriedman1999.51895/1913,53.

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geometryisthatitprovidesarecipeforsystematicallyredescribingeveryfactabout

thegeometryofaEuclideanworldinhyperbolictermsandviceversa.Itthusgivesa

convincingcasefortheunderdeterminationofthegeometryofphysicalspacebyall

possibleevidence.Poincare’sargument,onherreading,doesnotemployorrequire

ageneralDuhemianargument(based,say,onglobalconfirmationholismaboutour

totalphysicaltheory),andsodoesnotgeneralizebeyondtheveryspecialcaseofthe

metricofspace.

Onedifficultywiththiskindofreading–adifficultyI’llreturntobelow–is

thatthisargumentforconventionalismdependsonlyontheexistenceofEuclidean

modelsofnon-Euclideanspaces,andthesemodelspredatedPoincare’swritingsin

thephilosophyofgeometry.ThisreadingthusleavesitunexplainedwhyPoincaré

wasledtoconventionalismwhenothergeometers(whoalsounderstoodthese

modelsperfectlywell)werenot.

Areadingthat,ifsuccessful,wouldexplainwhyPoincaréwasledto

conventionalismwhenothergeometerswerenotisprovidedbyMichaelFriedman.

AccordingtoFriedman1999,Poincaréargues(onphilosophicalgrounds)thatwe

canknowapriorithesyntheticclaimthatspacehasagrouptheoreticstructurethat

allowsforfreemobility.BytheHelmholtz-Lietheorem,6thisrequirementrestricts

thepossiblegeometriestothosewithconstantnegative,positive,orzerocurvature,

butdoesnotprivilegeoneoveranother.Moreover,FriedmanarguesthatPoincaré

wascommittedtoahierarchyofthesciences,wherethesciencesareorderedby

leveloffundamentalityinsuchawaythatnofactinamorefundamentalsciencecan

6Stein1977.

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bedeterminedbyafactofalessfundamentalscience.Thishierarchicalpicturethus

rulesoutappealingtoempiricalfactstodecideonthecorrectgeometry.This

readingthusgivesanargumentbyeliminationforconventionalism:

1. Euclid'spostulateiseitherananalytictruth,asyntheticaprioritruth,an

empiricaltruth,oraconvention.

2. Theexistenceofmodelsofnon-EuclideangeometryshowthatEuclid's

postulateisnotanalytic.

3. Euclid’spostulateisnotsyntheticapriori,sincewecanknowapriori

onlythatspacehasagrouptheoreticstructurethatallowsforfree

mobility,which(byHelmholtz-Lie)doesnotdecidethetruthofEuclid’s

postulate.

4. Euclid'spostulatecannotbeanempiricaltruth,sincethesciencesare

arrangedhierarchically.

5. So,itisaconvention.

Therearemanyfeaturesofthisinterpretationthatareattractive.Itexplains

whyPoincaréwasledtohisconventionalismwhenothergeometerswerenot,and

whyPoincarédidnotconsiderallofgeometrytobeconventional,letaloneall

sciences.Moreover,thisreconstructionoftheargumentnicelycapturesthe

argumentativestructureof“Non-EuclideanGeometries”(1891/1913)–Poincaré’s

firstsustaineddefenseofconventionalism–whichclearlyarguesinaneliminative

way.7Onemighttakeissuewithsomefeaturesofthisreconstruction,8butitisnot

7AsimilareliminativeargumentappearsinPoincaré1887,214-6.8Inparticular,Dunlop2016arguesthatthereislittlegroundforattributingahierarchicalpictureofthesciencestoPoincaré.

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mygoalinthispapertodecideonitsfidelity.(Indeed,asI’llclaimbelow,Idonot

believethatthereisauniqueargumentthatPoincaréputsforwardfor

conventionalism.Rather,hesupplementedhisargumentsandaddednewonesover

thenearlytwentyyearsofhiswritingsonthephilosophyofgeometry.)Whatisnot

wellknown,isthattherewasaquitedifferentinterpretationofPoincaréthatshares

thesameinterpretivevirtuesandwasinfactofferedupinPoincaré’slifetimebyone

ofhisforemostcritics.BertrandRussell,inhisearlybookEssayontheFoundations

ofGeometry,arguedthatPoincaréwasledtohisconventionalismbyadistinctive,

mathematicalinterpretationofthemodelsofnon-Euclideangeometry.This

mathematicalinterpretationwascommontoPoincare,Klein,andCayley,but

differedfromtheinterpretationthat,say,Beltramigaveofhismodel.

SincethesesystemsareallobtainedfromaEuclideanplane,byamere

alterationinthedefinitionofdistance,CayleyandKlein[thoughnot

Beltrami]tendtoregardthewholequestionasone,notofthenatureof

space,butofthedefinitionofdistance.Sincethisdefinition,ontheirview,is

perfectlyarbitrary,thephilosophicalproblemvanishes…,andtheonly

problemthatremainsisoneofconventionandmathematicalconvenience.

ThisviewhasbeenforcefullyexpressedbyPoincaré*:"Whatoughtoneto

think,"hesays,"ofthisquestion:IstheEuclideangeometrytrue?The

questionisnonsense."Geometricalaxioms,accordingtohim,aremere

conventions:theyare"definitionsindisguise."9

9Russell1897,§33.RussellisquotingPoincaré1891/1913,39.

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RussellgoesontoarguethatwhatmatteredforPoincaréwasthatdistancewasnot

aprimitivenotionasitwasforBeltrami,butwasmathematicallydefinedusing

intrinsicallynon-metricnotionsdrawnfromadifferentareaofmathematics

(namely,projectivegeometry).AsI’llexplainbelow,RussellclaimsthatPoincaré’s

argumentdependsondefiningthedistancebetweentwopointsintermsofthe

crossratioofthosetwopointsandtwootherarbitrarilychosenpoints.But,Russell

claimed,distanceisinfactaprimitivenotionandsotheargumentfor

conventionalismcollapses.

ThereareafewfeaturesofthisreadingthatIwouldliketohighlight.First,it

assimilatesPoincaré’smathematicalworktoearlierworkbyCayleyandKlein.

Second,itclaimsthatPoincaré’sargumentforconventionalismdependsonvery

specificfeaturesofhismathematicalwork.10Third,thereconstructiondependsin

nowayonDuhemianunderdeterminationarguments,andinfactdoesnotturnon

theapplicationofgeometrytophysics.Rather,theargumentturnsonthe

applicationofonemathematicaltheorytoanother.Insteadofarguingthatthereisa

10RussellisnottheonlyreadertoseePoincaré’sconventionalismasdependentonhisparticularmathematicalwork.Zahar(1997,185)arguesthat"strictlyinternalfactorsconnectedwithhisworkonFuchsianfunctionsgaverisetohisso-called'conventionalism.'"Zahar'sprincipalconclusionisthatPoincarédidnotemployRiemanniangeometryinhisinvestigationsoftheinvariantsinFuchsianfunctions.ThisexplainswhyPoincaréwasnottemptedtoconsiderRiemanniangeometriesofvariablecurvatureaslegitimategeometries(1891/1913,37)–animportantmoveinhisdefenseofconventionalism.IagreewithZahar'sreadingofPoincaré'smathematicalworlkonFuchsianfunctions,astherestofthispaperwillshow.However,IbelievethattheconnectionbetweenPoincaré'smathematicalworkandhiscommitmenttoconventionalismrunsdeeper.Afterall,therewerecontemporariesofPoincaré's(e.g.,HelmholtzandRussell)whoalsodeniedthatRiemanniangeometriesofvariablecurvaturewerelegitimategeometries,andyetdidnottakethefurtherstepandembraceconventionalism.ThereneedstobesomeexplanationforwhyPoincaréinparticulartookthisfurtherstep.

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loosenessoffitintheapplicationofpuremathematicstothephysicalworld–thus

leavingopenadegreeoffreedomthathastoberestrictedbyconvention–itturns

ontheloosenessoffitindefiningdistance(apurelymathematicalnotioninmetric

geometry)usingtermsdrawnfromanotherareaofpuremathematics.

Inthispaper,Iwanttoexplainandcriticallyevaluatethisreading.InSection

I,I’llexplainRussell’sobjectiontoPoincaré,itsphilosophicalmotivations,andwhy

itultimatelyfailsasareadingofPoincaré.InsectionII,I’llarguethatthereisa

successfulmodifiedRussellianreading.Thatis,thereispresentinPoincaréan

argumentforconventionalismfromthepossibilityofalternativedefinitionsof

distancewithinpuremathematics.Thisargumentdoesnotderivefromthe

underdeterminationofphysicalgeometrybyexperience,butbytheapplicationof

onemathematicaltheorytoanother.

SectionI:EarlyRussellversusPoincaré

Russell’scriticismofPoincaré’sconventionalismappearedinhis1897book,Essay

ontheFoundationsofGeometry[EFG].Thisbookappearedfiveyearsbefore

Poincaré’sclassictreatmentofthespaceprobleminPartTwoofScienceand

Hypothesis[SH],whichcollectstogetherandre-arrangespapersthatwerepublished

between1891and1900.InEFG,Russellcitestwopapersthateventually

reappearedinSH:“Non-EuclideanGeometry”((1891/1913),whichwasreprinted

withminor–thoughsignificant–deletionsaschapter3),and“SpaceandGeometry”

((1895/1913),whichwasreprintedaschapter4ofSH,withoneparagraph

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deleted).11ThefirstchapterofEFGistitled“AShorthistoryofMetageometry”and

arguesthatthehistoryofmathematicalworkonnon-Euclideangeometryshouldbe

dividedintothreeperiods,withthesecondperiodcharacterizedbytheuseof

differentialgeometrybyGauss,Riemann,andBeltrami,andthethirdperiod

characterizedbytheuseofprojectivegeometrybyCayleyandKlein.Itisinthis

contextthatRussellarguesthatconventionalistinterpretationsofmodelsofnon-

Euclideangeometrymakesenseonlywhenthesemodelsareconstructedusing

techniquescharacteristicofthethirdperiod,notthesecond.

ToseewhatRussellisgettingat,weneedtoexaminethedifferentroutesthat

BeltramiandKleintooktoconstructingtheirmodelsofhyperbolicgeometry.12

Beltramiusedmethodsfromdifferentialgeometry,and(inhis1868Saggio)

constructedamodelof2-dhyperbolicgeometryinEuclidean3-space.Beltrami

beganwiththestandardwayofprojectingthedistancefunctiondefinedonpoints

onthesouthernhemisphereofasphereontoaplane(figure2).Theinversefunction

11Russellquotes1891/1913at§33(quotedabove).Hementions1895/1913inalongfootnoteto§100,whichgivesalonglistofthe“mostimportantrecentFrenchphilosophicalwritingsonGeometry.” ThatfootnotealsolistsPoincaré1897,whichisPoincaré’sreplytocriticismsofhisphilosophyofgeometryleveledbyLechalasandCouturat.Thispaperisdevotedlargelytoline-by-lineresponsestospecificclaimsofLechalasandCouturat,andcontainsnonewclaimsmateriallydifferentfromthosein1891/1913and1895/1913.Moreover,thereisnoevidencethatRussellengagedwiththispaperatallinEFG,beyondsimplylistingit.SoIwillfollowRussell’sleadinignoringitinthispaper. Twoparagraphsaredeletedfrom1891/1913inch.3ofSH.Thelastparagraphissimplymovedonepageovertobecometheopeningofch.4ofSH.Thepenultimateparagraph–apassagedevotedtothepossibilityofdeterminingthecorrectgeometryempiricallybymeasuringstellarparallaxes–ismovedtoch.5ofSH.Seenote16andpage37below.12MypresentationofBeltramiisderivedfromStillwell1996.

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thatmapsthecoordinateofthepointontheplane(x,-R,z)backontothesphereis

then

𝑥!,𝑦!, 𝑧! =𝑅

(𝑅! + 𝑢! + 𝑣! (𝑥,−𝑅, 𝑧)

Nextwedefineaninduceddistancefunctionds2onpointsontheplanesothatthe

distancebetween(x1,-R,z1)and(x2,-R,z2)isjustthestandardlengthofthegeodesic

onthesurfaceofthesphereconnecting(x1’,y1’,z1’)and(x2’,y2’,z2’).(Thisisthekind

ofinducedfunctionthatmapmakersareinterestedinwhentheyrepresent

distancesonaglobeusingflatmapsinanatlas.)Beltramithennoticedthatthis

induceddistancefunctiondependedonlyonx1,z1,x2,z2,andR2,andsowouldbe

meaningfulifRwasreplacedwithR√-1.BeltramithenshowedthatreplacingRwith

R√-1inducesonaplaneanon-Euclideanmetricthathasconstantnegative

curvature,andthepointsprojectedfromContotheplanearenowallinsideadisk

(figure3).

Klein("Ontheso-calledNon-EuclideanGeometry":Klein1871])arrivedat

thesamemodelutilizingtechniquesforbuildingadistancefunctiononthecomplex

projectiveplaneusingonlyresourcesdrawnfromprojectivegeometry.Kleinstarted

withanideafromVonStaudt.VonStaudtshowedhowtoinduceasetofrational

coordinatesonthepointsonaprojectivelinebyrepeatedapplicationofthe

quadrilateralconstruction,apurelyprojectiveconstructionthatrequiresonlypencil

andstraightedge(figure4).Itwasawell-knowntheoremofprojectivegeometry

thatthefourpointsonalinepickedoutbythequadrilateralconstructionhavethe

samecrossratio:

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𝐶𝑅 𝑃,𝐴,𝐵,𝑄 =𝑃𝐴𝑃𝐵 ×

𝑄𝐵𝑄𝐴

anditwaswellknownthatthecrossratiooffourpointswasinvariantunder

projection.VonStaudtthenshowedthatiftwoofthesepointsarearbitrarilylabeled

nand∞,thenrepeatedapplicationsofthisconstructionwillassignthestandard

Euclideanmetrictothepointsontheline.Ingeneral,ifwetakeaprojectiveline

withtwopointsfixed(suchasPQinfigure5),wecandefineafunctionthatassignsa

distancebetweentwopoints:

𝑑 𝐴,𝐵 = 𝑐[log𝐶𝑅(𝑃,𝐴,𝐵,𝑄)]

Employingthelogofthecrossratioensuresthatthedistancefunctionhastheright

additiveproperty

𝑑 𝐴,𝐵 + 𝑑 𝐵,𝐶 = 𝑐[log𝐶𝑅(𝑃,𝐴,𝐵,𝑄)]+ 𝑐[log𝐶𝑅(𝑃,𝐵,𝐶,𝑄)]

= 𝑐[log𝐶𝑅(𝑃,𝐴,𝐶,𝑄)] = 𝑑(𝐴,𝐶)

sincethecrossratioofalinesegmentistheproduct,notthesum,ofcrossratiosof

itsparts:

𝐶𝑅 𝑃,𝐴,𝐵,𝑄 ×𝐶𝑅 𝑃,𝐵,𝐶,𝑄 = 𝐶𝑅(𝑃,𝐴,𝐶,𝑄)

Adistancefunctionforthewholeplanecouldthusbedefinedpurely

projectively,usingnoundefinednotionsfrommetricgeometry–ifonlywehada

systematicwaytopickouttwoarbitrarypoints,nand∞,onanylineintheplane.

Buteverylineintersectsaconicinthecomplexprojectiveplaneintwopoints,and

sowecandefinedistanceprojectivelyifwejustpickanarbitraryconicinthe

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plane.13Infact,Cayley(“SixthMemoironQuantics”:Cayley1859)hadshownhowto

constructthestandardEuclideandistancefunctioninthisway,inthespecialcase

wheretheconicpickedoutisimaginaryanddegeneratesintoapointpair.Klein's

ideawasthatdifferentkindsofmetricgeometrieswouldariseiftheconic(whichhe

calledthe“fundamentalconic”)werechosendifferently.14Inparticular,heshowed

thatifthefundamentalconicisimaginaryandnon-degenerate,thenwehave

sphericalgeometry(spacesofconstantpositivecurvature),andifthefundamental

conicisrealandnon-degenerate,thenwehavehyperbolicgeometryinthespace

enclosedwithintheconic(spacesofconstantnegativecurvature).Thelattercaseis

depictedinfigure5.ItiseasytoseethatBeltrami’smodelisamodelofthislatter

Kleiniankind,wherethediskaroundtheoriginisaspecialcaseofareal,non-

degenerateconic.

KleinandBeltramithusproducedtheirmodelintwoconceptuallyquite

distinctways.Beltramiusedtechniquesfromdifferentialgeometry,whose

fundamentalobjectsarevariousspaceswithdistancefunctionsonthem;Kleinused

techniquesfromprojectivegeometry,whosefundamentalobjectiscomplex

projectiven-spacewithprojectiverelationsoutofwhichcanbeconstructedvarious

distancefunctions.Russellwasabsolutelycorrect,then,toarguethatoneandthe 13Again,pickingaconicisprojectivelyacceptableaswell,sinceSteinerhadshownthatconicscanbeconstructedpoint-by-pointbytakingtheintersectionpointsoflinesthatstandtooneanotherinconstantcrossratios(Steiner1881,§41.I).14Klein’sadvanceoverCayleywastwofold.First,hegeneralizedCayley’sproceduretodistancefunctionsotherthanEuclideanones.Second,heusedthetechnicalresourcesfromVonStaudttoshowthatthedistancefunctionscouldbedefinedinapurelyprojectiveway.Thisremovedtheworryofcircularityindefiningdistanceintermsofcrossratio,whichwas–afterall–initiallydefinedastheproductofquotientsofstandardEuclideandistances.Athirdadvance,discussedbelow,isKlein’sinterpretationofhisclassificationofgeometriesgroup-theoretically.

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samemodelcanbeunderstoodinmathematicallyquitedistinctways.Russell

furtherassimilatedPoincaré’sapproachtoKlein’s.TounderstandRussell’sclaim,

weneedtounderstandtherelationbetweenPoincaré’sspheremodel(a2-dversion

ofwhichisfigure1)andtheKlein-Beltramimodel.

AsI’llshowinsection2,Poincaréarrivedathismodel(inPoincaré1880a)

whiletryingtorepresentgeometricallytheinversesofquotientsofsolutionsto

secondorderlineardifferentialequations.Butthoughheproducedthismodelusing

techniquesdistinctfrombothKlein’sprojectiveandBeltrami’sdifferentialmethods,

heshowedgeometricallyhowitcouldbeconstructedfromtheKlein-Beltrami

model.BeginbytakingaspherewhoseequatorialdiskisaKlein-Beltramimodel

(figure6).Projecttheequatorialplaneontothesouthernhemispherebyparallel

projectionsorthogonaltotheequatorialplane(figure7).Thestraightlinesinthe

Klein-Beltramimodelarenowprojectedontocirculararcsthatmeettheequatorat

rightangles.Thenprojectthebottomhemisphereontotheplanetangenttothe

southpoleofthesphere,usingasthecenterofprojectionthenorthpoleofthe

sphere(figure8).Thecirculararcsonthesouthernhemispherewillbeprojected

ontocirculararcsofthedisk,andtherightanglesatwhichthecirculararcsonthe

southernhemispheremeetthediskwillbeprojectedontorightangleswhere

circulararcswithinthediskmeetthecircumferenceofthedisk.Theregionofthe

planeboundedbytheprojectionoftheequatorofthespherewillbetheboundaryof

Poincaré’smodel(figure1).

RussellinterpretedPoincaré’sargumentforconventionalisminsuchaway

thatitdependedonKlein’swayofconstructinghismodel.Therearethree

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philosophicallysignificantfeaturesofKlein’sprocedurethatdistinguishitfrom

Beltrami’s.First,inKlein’sconstructiontheEuclidean,hyperbolic,andspherical

metricsarealldefinedononeandthesameunderlyingcomplexprojectiveplane.It

thereforeseemsmorenaturaltodescribethesemetricgeometriesasthreedifferent

waysoftalkingaboutthesameunderlyingreality.Second,inKlein’sconstruction

thedistancefunctionisnotprimitive,butdefined.Itthereforeseemsnaturalto

characterizethechoiceofametricasachoiceofadefinition,and–inasmuchas

definitionsofwordsarearbitrary–asnotamatteroffact.15Third:

Theprojectivegeometer…whenheintroducesthenotionofdistance,he

definesit,intheonlywayprojectiveprinciplesallowhimtodefineit,asa

relationbetweenfourpoints.(EFG,§37)

RussellthereforereconstructsPoincaré’sargumentinthefollowingway.The

Cayley-Kleinmetricdefinesd(A,B)intermsofthecrossratioofAandB,andtwo

otherarbitrarilychosenpointsPandQ.Moreover,theproceduresforpicking

arbitrarypointsP,Qonalinecorrespondtothethreegeometriesofconstant

curvature.So,thechoiceamongthethreegeometriesofconstantcurvatureis

arbitrary.

OnRussell’sreading,Poincaré’sargumentforconventionalismdoesnot

dependonanyfactsaboutphysicalgeometry,norontheunderdeterminationoftotal

theoriesbyphysicalevidence.Thisisnotsurprising,sincePoincaré’sextended

treatmentinScienceandHypothesisofthepurportedempiricalbasisofgeometryis

15Onthesefirsttwopoints,seeEFG,§33,quotedabove.

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ch.5(“ExperienceandGeometry”),whichappearedafterRussell’sbook.16Indeed,as

DavidStump(1989,348)haspointedout,thereislittleindicationinthetextof

1895/1913thatPoincaréintendedthesphereargumenttobemakingapointabout

thewayinwhichphysicalgeometryisunderdeterminedbyourempiricalevidence.

Instead,thesphereargumentissupposedtoshowthatwecanimaginespaces

whereEuclid’saxiomfails.(Andinanycase,Russelldoesnotmentionthesphere

argumentinEFG,despiteciting1895/1905.)Infact,Poincaréintroducesthesphere

thoughtexperimentinthisway:

Ifgeometricspacewereaframeimposedoneachofourrepresentations,consideredindividually,itwouldbeimpossibletorepresenttoourselvesanimagestrippedofthisframe,andwecouldchangenothingofourgeometry.Butthisisnotthecase;geometryisonlytheresumeofthelawsaccordingtowhichtheseimagessucceedeachother.Nothingthenpreventsusfromimaginingaseriesofrepresentations,similarinallpointstoourordinaryrepresentationsbutsucceedingoneanotheraccordingtolawsdifferentfromthosetowhichweareaccustomed.(1895/1913,49)

Sothepointofthespheremodelisthatthestructureofspaceisnotsomethingthat

isintrinsicinanygivenrepresentation,butemergesonlyfromthelawsof

connectionamongrepresentations.AsPoincaréputsthepointafewpageslater

(1895/1913,59),theobjectofgeometryisnotaformofsensibility,butaformof

ourintellect.Andsowecanimagineadifferentgeometrybyimaginingaworld

whereoursensationsrelatetooneanotheraccordingtodifferentlaws.Usingthe

eliminativeargumentforconventionalismPoincaréintroducesin1887and 16Indeed,thebulk–i.e.sections4-7–ofPoincaré1902/1913(titled,“ExperienceandGeometry”)consistsofpassagesfromPoincaré1899and1900,wherePoincarédevelopedamoresustainedargumentagainsttheclaimthatEuclid’spostulateisanempiricaltruth,inresponsetoRussell’sclaiminEFGthatitstruthcouldbedecidedbyexperiment(EFG§140).Theexceptionis§3,aparagraphthatoriginallyappearedattheendofPoincaré1891/1913,butwasremovedwhenthechapterwasrepublishedinSH.Seealsop.37below.

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1891/1913,thespheremodelisthereforeintendedtoshowthatEuclid’spostulate

isnotasyntheticaprioritruth.

RussellrejectsPoincaré’sconventionalismbecauseherejectsthesecondand

thirdfeaturesofKlein’sprocedure.Inparticular,withrespecttothethirdfeature,

Russellargues:

Distance,intheordinarysense,remainsarelationbetweentwopoints,not

betweenfour;anditisthefailuretoperceivethattheprojectivesensediffers

from,andcannotsupersede,theordinarysense,whichhasgivenrisetothe

viewsofKleinandPoincaré.(EFG,§37)

Russellmaintainsthatdistanceisarelationbetweentwopointsbecausehewants

geometrytoplayatranscendentalrole:toprovidetheformofexternality.Geometry

should“permitknowledge,inbeingswithourlawsofthought,ofaworldofdiverse

butinterrelatedthings”(§58).Distance,asaprimitivetwo-placerelation,allowsus

todistinguishbetweentwopointswhilealsointerrelatingthem.Theabilityto

perceivetwothingsatdistancefromoneanotheristhustheabilitytoperceive

identityindifference–themostprimitiveabilitywithoutwhichwecouldnot

cognizeobjectsofperception.Ifdistancewerearelationamongfourpoints,thenit

wouldpresupposesomeperceptualwayofdistinguishingthosefourpointsfrom

oneanother,andwouldthuspresupposetheformofexternalityinsteadof

constitutingit.Andso:"BeltramiremainsjustifiedasagainstKlein"(§33).

Russell’sinterpretationisnotcorrect.InhisreviewofEFG,Poincarépointed

outthatRussellhadmisinterpretedhisargumentbymakingitdependentonthe

17

metricofCayleyandKlein(Poincaré1899,273).17Infact,inanotherpaper,

publishedafterRussell’sbookbutbeforePoincaréknewofit,Poincareexplicitly

deniedthattheproperwaytodefinedistancefromAtoBisintermsofthecross

ratioofA,B,andtwootherpoints:

[VonStaudt]obtain[s]themetricalproperties[by]definingaharmonic

penciloffourstraightlines,takingasdefinitionthewell-knowndescriptive

property.Thentheanharmonic[i.e.cross]ratiooffourpointsisdefined,and

finally,supposingthatoneofthesefourpointshasbeenrelegatedtoinfinity

theratiooftwolengthsisdefined.Thislastistheweakpointoftheforegoing

theory,attractivethoughitbe.Toarriveatthenotionoflengthbyregarding

itmerelyasaparticularcaseoftheanharmonicratioisanartificialand

repugnantdetour.Thisevidentlyisnotthemannerinwhichourgeometric

notionswereformed.(Poincaré1899,§XVII)

DespitethefactthatRussell’sreadingmissesthemark,isthereamore

successfulreadingofPoincaréthatsharesmanyofthefeaturesandvirtuesof

Russell’stheory?SuchareadingwouldmakePoincaré’sargumentdependon

specificfeaturesofhismathematicalworkingeometryandwouldexplainwhy

conventionalismseemednaturaltohimwhenitdidnotforothergeometerswho

understoodthemodelsofnon-Euclideangeometryequallywell.Itwoulddependon

thespecificwayinwhichdistanceisdefinedusingresourcesfromadifferentareaof

mathematics,andnotonthewaymathematicsisappliedinphysicalscience.Inthe

nextsection,I’llarguethatthereissuchanargument.Inordertounderstand 17PoincaréwroteareviewofRussell’sbook(Poincaré1899)andafurtherpiece(Poincaré1900)inresponsetoRussell’sreply(Russell1900).

18

Russell’sreadingofPoincaré,wehadtolookatthedifferentmathematicalroutes

thatBeltramiandKleintooktoarriveattheirmodels.Inthenextsection,we’lldo

thesameforPoincaré.

SectionII:FuchsianFunctions

Poincaré’searliestargumentsforconventionalism,including1891/1913(which

wasthefirstsustainedphilosophicaltreatmentofgeometry,andtheworkthat

Russellcitesanddiscusses),explicitlydrawonhisearlierworkinpure

mathematics.

Ifgeometryisnothingbutthestudyofagroup,onemaysaythatthetruthof

thegeometryofEuclidisnotincompatiblewiththetruthofthegeometryof

Lobachevsky,fortheexistenceofagroupisnotincompatiblewiththatof

anothergroup.(1887,215)

Nothingremainsthenoftheobjection[thattheremaybeahidden

contradictioninLobachevskiangeometry]aboveformulated.Thisisnotall.

Lobachevski'sgeometry,susceptibleofaconcreteinterpretation,ceasesto

beavainlogicalexerciseandiscapableofapplications;Ihavenotthetimeto

speakhereoftheseapplications,noroftheaidthatKleinandIhavegotten

fromthemfortheintegrationoflineardifferentialequations.(1891/1913,

34).

19

Thefirstquotationisfromtheconcludingpagesof“Surleshypothèses

fondamentalesdelagéométrie,”whichisamathematicalpaperthatendswitha

pageandahalfofprogrammaticphilosophicalremarks.HerePoincaréclaimsthat

thechoicetoacceptorrejectEuclid’spostulateisnotamatteroftruth,andhis

argumentclearlydependsonthinkingofthevariousmetricgeometriesasarising

fromselectingamongallthepossiblegroupsofcoordinatetransformationsonone

andthesameunderlyingspace.Thesephilosophicalremarksarethefirststatement

ofPoincaré’sconventionalism,andareexpandeduponinhisfirstpurely

philosophicalessay,1891/1905.Inthelatteressay,Poincaréarguesthatwhat

movesnon-Euclideangeometryfroma“merelogicalcuriosity”tosomethingmore,

isnotitsphysicalapplications,butitsapplicationingrouptheoryandinthetheory

ofdifferentialequations.

TounderstandwhatPoincaréisgettingat,weneedtolookatPoincaré’s

mathematicalworkfromtheearly1880s.Poincaré’sfirstapplicationofhyperbolic

geometrywasinthetheoryofFuchsianfunctions.Fuchsianfunctionsarea

generalizationofellipticfunctions,whichwereoneofthemostwidelystudied

topicsinnineteenthcenturymathematics.18Inordertokeeptrackoftheirorigins

andapplications,weneedtointroduceellipticfunctionsinthecontextofelliptic

integralsandellipticcurves.Anellipticintegralisanintegraloftheform

𝑅 𝑡, 𝑝(𝑡) 𝑑𝑡

18SeeBottazziniandGray2013,Ch.1.Foradiscussionofthephilosophicalsignificanceofellipticfunctions,seeTappenden2006.MypresentationhereisindebtedtoStillwell2010,chapter12.

20

whereRisarationalfunction19andp(t)isapolynomialofdegree3or4.An

especiallysimpleexampleofsuchanintegralisthelemniscaticintegral

𝑑𝑡1− 𝑡!

!

!

whichgivesthearclengthofthelemniscateofBernoulli.TheCartesianequationof

Bernoulli’slemniscateis

𝑥2 + 𝑦! ! = 𝑥2 − 𝑦!

anditsgraphisfigure9.

TheseintegralshavebeenstudiedsinceLeibniz.Theyappearinvery

elementarysettings:e.g.,theintegralthatgivesthearclengthofanellipseisan

ellipticintegral–hencethename.Earlyon,itwasdiscoveredthattheycannotbe

expressedintermsofelementaryfunctions,whichfrustratedtheintuitivenatural

idea(beginningwithLeibnizhimself)thatthesolutionofeveryintegrationproblem

shouldbeexpressedintermsofelementaryfunctions.20Abreakthroughcame

around1800whenGauss(and,later,AbelandJacobi)studiedtheirinversesand

consideredtheirbehaviorinthecomplexplane.Theinverseofanellipticintegralis

calledanellipticfunction.Forexample,theinverseoftheleminscaticintegralGauss

calleda"lemniscaticsinefunction,"standardlyabbreviatedsl(x),onanalogywith

thesinefunction.Recallthatthesinefunctionistheinverseof

sin!! 𝑥 =𝑑𝑡1− 𝑡!

!

!

19Arationalfunctionisaquotientofpolynomials.20Afunctioniselementaryifitcanbedefinedbyarithmeticaloperationsonafinitenumberofexponentials,logarithms,constants,andnthroots.

21

whichistheequationforthearclengthofacircle.21Thesinefunction,togetherwith

itsfirstderivative–thecosinefunction–,canbeusedtoparameterizetheequation

ofacircle

𝑥2 + 𝑦! = 𝑟!

withx=cost=sin'tandy=sint.Justassintcanbeusedtoparameterizethe

equationofacircle,sotoocanellipticfunctionsbeusedtoparameterizethe

equationofcertaincurves.Thatis,iff(x)isanellipticintegral,thentherearecurves

thatcanbeparameterizedsothat

𝑥 = 𝑓!!(𝑢)

and

𝑦 = 𝑓!!′(𝑢)

wheref-1(x)isanellipticfunction.Thosecurvesthatcanbeparameterizedby

ellipticfunctionscametobecalledellipticcurves.22

MathematiciansbeforeRiemannstandardlydefinedellipticfunctionsin

termsoftheirpowerseriesexpansions.AbelandJacobiinthe1820sdiscoveredthat

ellipticfunctionsaredoublyperiodic,andafterRiemannmathematiciansbeganto

definethemsoastohighlighttheirdoubleperiodicity.Consideragainthe

elementarysinefunction.Thesinefunctionissinglyperiodic(seefigure10):itis

invariantundersubstitutions

𝑥 ↦ 𝑥 + 2𝑛𝜋

21Thisiswhysin-1iscalled“arcsin.”22Infact,anycubiccurvecanbeparameterizedbyanellipticfunction–aresultannouncedbySteinerbutfirstprovenbyClebschin1864.

22

Interpretedgeometrically,thismeansthatwecanslidetheentireplane2πunits

alongthexaxisntimeswithoutchangingwhichpointslieonthefunctionsinx.

Gaussnoticedthatthelemniscaticsinefunctionisdoublyperiodicinthecomplex

plane:

𝑓 𝑥 = 𝑓(𝑤 +𝑚𝜔! + 𝑛𝜔!)

withm,nintegersandω1andω2complexnumbers.Thiscanberepresented

geometricallybyaplanetiledbyparallelograms,asinfigure11.Themappingof*-

pointstopointsonthecurvewillbeunaffectedby

𝑥 ↦ 𝑥 +𝑚𝜔! + 𝑛𝜔!

Interpretedgeometrically,thismeansthatwecanslidetheentirecomplexplaneω1

unitsalongoneaxisntimesandω2unitsalongtheotheraxismtimeswithout

changingthevaluesoftheellipticfunction.Thesemappingsarejustrigid

(Euclidean)translationsoftheplanethatkeepthetilingintact.Therigid

translationsthusformagroup,witheachelementofthegroupcorrespondingtoa

tile.Forexample,theparallelogramwhosebottomleftcorneristheorigin

correspondstothegroupidentity,thatis,theoperationthatleavesallpointsonthe

planeunaffected.Theparallelogramwhosebottomleftcorneristhepoint(ω1,0)

correspondstotherigidtranslationthatmoveseverypointtotherightbyω1.And

soon.

Thetheoryofellipticfunctionsprovidedagreatsimplificationand

unificationinthetheoryoffunctions.Forinstance,inlecturesdeliveredin1874-5,

Weiserstrass23showedthatanyanalyticfunctionofasinglecomplexvariablethat

23SeeBottazinniandGray2013,424-9.

23

admittedanalgebraicadditionformulawasexpressibleasarationalfunctionofthe

simplestellipticfunction,whichWeierstrasscalled℘

𝓅 𝑧 =1

𝑧 +𝑚𝜔! + 𝑛𝜔! !

!

!,!!!!

whichparameterizestheellipticcurve

𝑦! = 4𝑥! − 𝑔!𝑥 − 𝑔!.

Butcouldthistheorybefurthergeneralized?Aretherefruitfulgeneralizationsof

ellipticfunctions–generalizationsthatwouldparameterizealargerclassof

algebraiccurvesandfurthersimplifythetheoryofcomplexfunctions?Inpapers

from1880LazarusFuchstriedtodopreciselythis.Fuchsbeganwithsecondorder

lineardifferentialequationswithrationalcoefficients:

𝑦!! + 𝑃 𝑧 𝑦! + 𝑄 𝑧 𝑦 = 0

wherePandQarerationalfunctions.Sincethefunctionissecondorderandlinear,

allofitssolutionscanbeexpressedaslinearcombinationsoftwosolutions,f(z)and

ϕ(z).Fuchsthenclaimed(Fuchs1880)thatthequotientofthesetwosolutions

𝜁(𝑧) =𝑓(𝑧)𝜑 𝑧

couldundercertainconditionsbeinvertedtoformawell-defined,single-valued,

meromorphic24function:

𝐹 = 𝜁!!.

24Ameromorphicfunctionisafunctionthatisholomorphicexceptatisolatedpoints.Aholomorphicfunctionisafunctionthatiscomplexdifferentiableintheneighborhoodofeverypointinthecomplexplane.(AsPoincaréshowed,theisolatedpointsatwhichaFuchsianfunctionisnotholomorphicarethepointsontheboundaryofthedisk.)

24

ItisthisfunctionFthatisanalogoustoanellipticfunction,with𝜁akintoanelliptic

integral.

Inhisprizeessaysubmittedon28May1880,25Poincaré–thenayoungand

unknownmathematician–showedthattheconditionsFuchsidentifiedwereneither

necessarynorsufficientforFtobesingle-valued,well-defined,andmeromorphic

(1880a,331).Underwhatconditions,then,doesFexistwiththespecified

properties?PoincarébeganwiththefactthatanyFwouldbeinvariantunderlinear

fractionaltransformations

𝐹 𝑧 = 𝐹 !"!!!"!!

.26

So,insteadofbeingdoublyperiodic,Fwouldbeinvariantunderlinearfractional

transformations:

𝑧⟼ !"!!!"!!

.

Hefurthershowed(Poincaré1880a,346ff.)thatthepathofFwouldbeconfined

insideacurvilinearpolygon oαγα’untilitspathcrossesaboundary(say,oα),at

whichpointFwouldtraceoutanidenticalpathwithinthenewcurvilinearpolygon

oαγ1α’1.WhenFcrossesaboundaryofthisnewpolygon(say,oα’1),itagainrepeats

itspathwithinyetanewcurvilinearpolygonoα1γ2α’1–andsoon(seefigure12).

Eachofthesepolygonscanbegeneratedfromtheoriginalpolygonbyrepeated

applicationsoflinearfractionaltransformations,andareboundedbyarcsofcircles

25MyunderstandingofPoincaré’sworkonFuchsianfunctionsdrawsonaseriesofworksbyJohnStillwellandJeremyGray:(Stillwell1985),(Gray1986,ch.6),(Gray1999),(BottazziniandGray2013),(GrayandWalter1997),(Gray2013).26(1880a,318).ThisfactfollowsmoreorlessimmediatelyfromthefactthatFistheinverseofthequotientofsolutionstoadifferentialequation,whereeverysolutionisalinearcombinationoftwosolutions,f(z)andϕ(z).

25

thatmeetatrightanglesacircleHH’centeredontheorigin.Therewillfurthermore

beaninfinitenumberofthesepolygons,whichwillcovertheinteriorofthecircle

HH’(figure1).ItfollowsthatFonlyexistswithinthediskHH’.(1880a,352).

ButisFsingle-valued,asFuchsclaimed?Thisalldependsonwhetherthe

curvilinearpolygonsgeneratedwithinthediskasFtracesitspatheveroverlap:if

thepolygonsdooverlap,thenFwillbemulti-valuedintheoverlappingregion

(1880a,351).Toshowthatthereisnooverlap,Poincarétakesthediskcoveredin

curvilinearpolygonsandprojectsitstereographicallyandthenorthogonallyonto

theequatorialplaneofthesphereinthewaydescribedinsection1(seefigures6,7,

and8).Thecurvilinearpolygonsareprojectedontorectilinearpolygons,andsince

simpleelementarygeometricalreasoningshowsthattheserectilinearpolygonsdo

notoverlap,therectilinearpolygonsdon’teither.Fisthuswell-defined,single

valued,andmeromorphic.

Poincarécalledthesefunctions“Fuchsianfunctions.”Moreformally

(Poincaré,1881a,47-8):aFuchsianfunctionisanymeromorphicfunctionthatis

invariantundera“Fuchsiangroup,”whereaFuchsiangroupisadiscontinuous27

groupoflinearfractionaltransformationsonthecomplexplanethatleaveinvariant

acirclearoundtheorigin.Thatis,aFuchsiangroupisagroupofoperationsonthe

complexplanethatleavethetessellationofthediskHH’intocurvilinearpolygons

intact.EachFuchsiangroupcorrespondstoawayoftessellatingthediskHH’with

curvilinearpolygons,andeachelementofagroupcorrespondstoacurvilineartile.

27Agroupisdiscontinuousifitdoesnotcontainaninfinitesimaloperation.Fuchsiangroupshavetobediscontinuous,becausetheycorrespondtowaysofmovingthepointswithinthediskthatleavethetilingintact,andnotileisinfinitesimal.

26

Inhisprizeessay,PoincaréhadfoundanexampleofaFuchsianfunction–a

functionwhoseFuchsiangroupcorrespondstothetessellationofthediskwitha

certainkindofquadrilateral.ButthiswasjustoneexampleofaFuchsianfunction,

andjustonewayoftessellatingHH’.Inthecaseofellipticfunctions,the

correspondingtessellationsoftheplanearesimpletounderstand,sincetheyareall

akintoparallelogramtessellations.28ThetessellationscorrespondingtoFuchsian

groups,ontheotherhand,areinfinitelyvarious,andtheprojectofprovinggeneral

propertiesofFuchsianfunctionscouldnotproceedunlessthereweresomegeneral

waytosurveyallofthewaysthatHH’couldbetessellatedintocurvilinearpolygons

whosesidesarecirclesmeetingHH’inrightangles.29AsPoincaréputitinapaper

from1881:“ItisnecessaryfirsttoconstructallFuchsiangroups;thisIhavedone

withtheaidofnon-Euclideangeometry”(1881a,48).

Inthemonthaftersubmittinghisprizeessay,Poincarérealizedthatthe

rectilinearpolygonsontowhichhehadprojectedthecurvilinearpolygonsofHH’in

factweretheKlein-Beltramimodelofhyperbolicgeometry,andsothetessellations

28Stillwell1985,19.29TherelationbetweenFuchsiangroupsandFuchsianfunctionsisrathersubtle.Fuchsianfunctionscannotbepairedup1-1withFuchsiangroups,anditneedstobeshownthatforeveryFuchsiangroupthereexistFuchsianfunctions.Onthefirstpoint,PoincarédiscoveredthatanytwoFuchsianfunctionsthatcorrespondtothesameFuchsiangrouparerelatedalgebraically(1881b)–afactthateventuallyledhimtohisfamousuniformizationtheorem.Onthesecondpoint,PoincaréprovedthateveryFuchsianfunctioncouldbeconstructedasthequotientoftwo“theta-Fuchsian”functionsthatcorrespondtothesameFuchsiangroup,whereatheta-FuchsianfunctionΘisameromorphicfunctionsuchthat

Θ𝑎𝑧 + 𝑏𝑐𝑧 + 𝑑 = Θ(𝑧)(𝑐𝑧 + 𝑑)

!!withmaninteger.Theexistenceoftheta-Fuchsianfunctionscouldthenbeprovedbytheconvergenceofaninfiniteseries.(Thisresultisannouncedin1881a,andprovedsystematicallyin1882b).

27

ofHH’inducedbyFuchsiangroupswerealsomodelsofhyperbolicgeometry.30Ina

supplementtotheprizeessaywrittenonJune281880(Poincaré1880b),Poincaré

describedthesituationasfollows:

Thereisadirectconnectionbetweentheprecedingconsiderationsandthe

non-EuclideangeometryofLobachevskii.Whatindeedisageometry?Itis

thestudyofagroupofoperationsformedbythedisplacementsonecan

applytoafigurewithoutdeformingit.InEuclideangeometrythisgroup

reducestorotationsandtranslations.Inthepseudo-geometryof

Lobachevskiiitismorecomplicated…TostudythegroupofoperationsM

andN[viz,theoperationsthatmoveonepolygoninHH’ontoanother]is

thereforetohavetodothegeometryofLobachevskii.Thepseudogeometry,

asaconsequence,isgoingtofurnishuswithaconvenientlanguagefor

expressingwhatwewillhavetosayaboutthisgroup.31

TherelationbetweenFuchsiangroupsandnon-Euclideanisometrieswaslaidout

systematicallyinsections1and2of1882a.There,Poincarédefinedtwofigures

withinHH’ascongruentiftheycanbetransformedintooneanotherbyalinear

fractionaltransformationwherea,b,c,anddarerealnumbers.32Heusedthis

30Thisrealization–which,Poincaréclaimed,hithimoutoftheblueashewasboardingabusonaminingexpedition--wasfamouslydescribedinPoincaré1909.JeremyGrayhasshownthatthisrealizationmusthavetakenplacebetweenMay29andJune281880(1986,266-8).31QuotedandtranslatedinGray1986,258-9.Graydiscoveredthesesupplements,whichwerepreviouslyunpublished,anddescribedtheircontentsinGray1986.Theyhavesincebeenpublished(inFrench)asPoincaré1997.32Therequirementthata,b,c,anddberealnumbersforcesthelinearfractionaltransformationtokeeptherealaxisinvariant,andthusmodelsnon-Euclideangeometryinthehalfofthecomplexplanelyingabovetherealaxis.Infact,in1882a,PoincarérepresentsFuchsiangroupsusingtheupper-halfplanemodelinsteadof

28

notiontodefinestraightline,length,area,anddistance.Thepreviouslyintractable

problemofidentifyingFuchsiangroupshadthusbeenreducedtothetractable

problemofidentifyingnon-Euclideanisometries.

Withthisapplicationofnon-Euclideangeometry,Poincaréwasableto

achieveextraordinaryresultsthatgeneralizeinpowerfulwaystheresultsobtained

usingellipticfunctions.Thisisbecause,asPoincaréputitinthesupplementfrom

June281880,“theFuchsianfunctionsaretothegeometryofLobachevskiiwhatthe

doublyperiodicfunctionsaretothatofEuclid”(1880b,translatedinGray1986,

269).Morespecifically,aFuchsiangroupwherethelinearfractionaltransformation

issuchthat(a+d)2=4isagroupofdiscontinuousEuclideanisometries(Poincaré

1882a,58)–namely,thegroupoftranslationsthatkeepsthetilingoftheplaneinto

parallelogramsintact(figure11).InthiswayFuchsianfunctionscomprise“avery

extensiveclassoffunctionsofwhichtheellipticfunctionsareaspecialcase”(1881b,

54).Justasellipticfunctionshadbeenusedtointegratealgebraicdifferentials,

Poincaréshowedthatamuchwiderclassofequations,lineardifferentialequations

withalgebraiccoefficients,couldbesolvedusingFuchsianfunctions(1882a,55).I

notedabovethatellipticfunctionscanparameterizecertainkindsofalgebraic

curves,so-calledellipticcurves–aclassthatincludesallcubicsandsomeother thePoincarédiskmodel.(TheupperhalfplanemodelresultsfromthePoincarédiskbyprojectingitstereographicallybackontothesouthernhemisphere(figure8),switchingthesouthernandnorthernhemispheresonthesphereandthenprojectingthenorthernhemispherefromapointontheequatorontoaplanetangenttothesphereatthepointontheequatoroppositethepointofprojection.)Heswitchedfromthedisktothehalf-planemodelinresponsetoanobjectionfromKlein,whodoubtedthateveryFuchsiangroupcouldberepresentedasanon-Euclideantessellationofthediskmodel(Gray1986,280,285-7).Poincaréusesthehalf-planemodelto“translate”hyperbolicgeometryintoEuclideangeometryin1891/1913,33-34.

29

specialcases.In1881,Poincaréannouncedthediscoveryofhisuniformization

theorem,thatanyalgebraiccurvewhatsoevercanbeparameterizedbyFuchsian

functions.Thatis,ifA(x,y)=0istheequationofanalgebraiccurve,itcanbe

rewrittenasA(f(t),φ(t))=0,withfandφFuchsianfunctions(Gray1999,81).

SectionIII:FromFuchsianFunctionstoConventionalism

WenowunderstandwhatPoincarémeantwhenhespokeofthe“applications”of

hyperbolicgeometry,and“theaidthatKleinandIhavegottenfromthemforthe

integrationoflineardifferentialequations.”Poincarédidnotsimplyfindamodelof

non-Euclideangeometryinasurprisingplace.Rather,thisapplicationofhyperbolic

geometryincomplexanalysisprovideddeepandpowerfultheoremsinthetheory

ofalgebraiccurvesandthetheoryoflineardifferentialequations–resultsthat

couldnothavebeenobtainedotherwise.Thisworkwasthehighpointofthe

extremelyactiveandfruitfulnineteenthcenturyworkincomplexanalysisthat

beganwithGauss.Itbroughttogetheranalysis,geometry,andalgebraina

surprisingandmutuallyilluminatingway.

ItisfurtherclearthatthismathematicalworkonFuchsianfunctions

providedPoincaréthemotivationforhisconventionalism.Toseethepossibilityof

applyingnon-Euclideangeometry,Poincaréhadtomaketwonovelconceptual

moves.First,hehadtothinkofthegroupofoperationsthatleavethevaluesofa

Fuchsianfunctioninvariantaswaysofmovingaspatialobject–thegraphofthe

function–aroundinspacewithoutchangingitsshapeorsize.Second,hehadto

30

conceiveofgeometryasfundamentallythestudyofthegroupofrigidmotionsof

bodiesinspace.Asheputthissecondpointinhisfirstsupplementtohisprizeessay,

whichhewrotejustoneortwoweeksafterseeingthatFuchsiangroupsarenon-

Euclideanisometries:

Whatindeedisageometry?Itisthestudyofagroupofoperationsformedby

thedisplacementsonecanapplytoafigurewithoutdeformingit.(1880b,

translatedinGray1986,258)

Butoncegeometryisconceivedofinthisway,itisashortsteptoconcludingthat

Euclideangeometryisnomoretruethannon-Euclideangeometry.Afterall,many

differentkindsofgroupsofrigidbodycoordinatetransformationscanbedefinedon

oneandthesamecomplexplane.Andthisisapointthat,again,Poincaréasserts

explicitlyinPoincaré1887–amathematicalpaperthatappearedfouryearsbefore

hisfirstphilosophicalpaper:

Ifgeometryisnothingbutthestudyofagroup,onemaysaythatthetruthof

thegeometryofEuclidisnotincompatiblewiththetruthofthegeometryof

Lobachevsky,fortheexistenceofagroupisnotincompatiblewiththatof

anothergroup.(1887,215,quotedabove)

Afterall,thechoiceofagroupofdisplacementsinthecomplexplaneamountsto

choosingwhichwaytotessellatetheplane.Butoneandthesamecomplexplanecan

betessellatedinmanyways.Itmakesnomoresensetoask“Whichistherightway

31

totessellatetheplane,figure1orfigure11?”thanitdoestoask“Whicharethetrue

functionsinthecomplexplane,ellipticorFuchsianfunctions?”33

Thiswayofconceivingofgeometry–asthestudyofthegroupofrigid

displacementsofbodies–iscontroversial.Itisafarcryfromtheolderviewthat

geometryisthestudyofextensivemagnitude,anditisalsoopposedtothenewer

viewthatgeometryisfundamentallythestudyofspace.34Moreover,itisalso

opposedtoRussell’sviewthatgeometryisfundamentallythestudyofdistance,

wheredistanceisaprimitivenotionnotdefinableintermsofothernotions.In 33TherearetwowaysinwhichPoincaré’spresentationofhisconventionalisminhislater,philosophicalpapersrefinestheargumentpresentedintheoffhandphilosophicalcommentshemakesin1880band1887.First,in1887PoincarésaysthatEuclideangeometryis“nomoretrue”thanitsnon-Euclideanrivals.Itakethistobealessperspicuouswayofsayingwhathecametosaylater,thatthechoiceofametricforspaceisnotamatteroftruthorfalsehoodatall,butinsteadamatterofconvention(wheremattersofconventionareopposedtomattersoffact).Second,thegroupofrigiddisplacementsinspaceisacontinuousgroup,notadiscontinuousgroup.Poincaréwaswellawareofthisfactfromhisearliestdiscussionsin1881aand1882a,wherehewouldfirstdefinethehyperbolicgroupof(continuous)displacementsandthendefineaFuchsiangroupbytakingadiscontinuoussubgroupthatleavesthefundamentalcircleHH’invariant.Inhislater,philosophicalpapers,suchas1898,hewouldmakeclearthatgeometryisthestudyofthecontinuousgroupofdisplacements–whichisaLiegroup,notaFuchsiangroup.Itakethesetwochangestoberefinementsoftheargumentfirstadumbratedintheseoffhandphilosophicalcommentsinhismathematicalpapers,thoughnotrefinementsthatinanywayalterthespiritoftheargument.34ForPoincare,spaceisnottheprimitivenotionofgeometry–thenotionofagroupofdisplacementsis.Hemakesthispointexplicitlyin1895/1913.There,afterdescribing“aparticularclassofphenomenawhichwecalldisplacements,”heassertsthatthe“lawsofthesephenomenaconstitutetheobjectofgeometry”(48);“itisfromthepropertiesofthisgroupwehavederivedthenotionofgeometricspace”(52). JeremyGrayhasnicelyemphasizedthedistinctivenessofPoincaré’sconceptionofgeometry.CommentingonPoincaré1898,hewrites:“Poincaréinsistedonananalysisofdistancethatwastheopposite,headmitted,oftheoneheldbyHelmholtz,Lie,andalmosteveryoneelse.Thesemathematicianssaidthatthematterofthegroupexistedbeforeitsform,thematterbeingthree-dimensionalmanifoldofspace.Whereasforhimself,saidPoincaré,theisomorphismclassofthegroupweusetoconstructspacecomesfirst”(Gray2013,56).

32

1882a,Poincarébeginswiththeideaofalinearfractionaltransformationthat

leavesthefundamentalcircleinvariant.Hethendefinestwofiguresascongruentif

theycanbetransformedintooneanotherbysomechosenlinearfractional

transformationthatleavesthecircleinvariant.Last,thedistancefromAtoBequals

thedistancefromCtoDifthecirculararcthatconnectsAandBiscongruenttothe

circulararcthatconnectsCtoD.ForPoincaré,distanceisnotprimitive,butis

defined.

Inthisway,RussellgetssomeofPoincaré’sargumentright,andsomeofit

wrong.AsIexplainedinsectionI,hepurportedtoidentifythreeimportantfeatures

ofPoincaré’sargument:first,thedifferentgeometriesarealldefinedinthesame

underlyingcomplexplane;second,distanceisadefined,notprimitivenotion;third,

distanceisafour-placerelation,notatwoplacerelation.Russellbelievedthatthe

argumentforconventionalismreliedessentiallyonthethirdpurportedfeature,

whichherejectedasartificial.WecannowseethatRussellgotPoincarébadly

wrong.MissingfromRussell’sinterpretationistheallimportantnotionofagroupof

rigiddisplacements;instead,heassimilatesPoincaré’smathematicalworktoKlein’s

approach,whichremainedweddedtoprojectivewaysofthinkingthatPoincaré

rejected.Still,thoughRussell’sthirdclaimmissesthemark,heiscorrecttoseethat

Poincaré’sargumentdoesrelyonthefirsttwopoints:bothEuclideanandnon-

EuclideangeometryareconstructedinPoincaré’swaybytakingthesame

underlyingcomplexplaneanddefininganotionofdistanceintermsofsomemore

fundamentalmathematicalnotionthatisimportedfromanotherareaof

mathematics.

33

InsectionI,InotedthatPoincaré’searliestargumentsforconventionalism–

theargumentsthatRussellknewwhenwritingEFG–arepresentedasargumentsby

elimination:somefeatureofgeometryiseitherananalytictruth,asyntheticapriori

truth,empiricaltruth,oraconvention;butitisnotanyofthefirstfourdisjuncts;so,

itisaconvention.Theinterpretivequestion,again,iswhichdisjunctPoincaréwas

hopingtoeliminatebyinvokinghismodelsofhyperbolicspaceinEuclideanspace.It

isuncontroversial,ofcourse,thatthesemodelsshowthatthereisnocontradiction

innegatingEuclid’saxiomofparallels,andsoEuclid’saxiomscannotallbeanalytic.

Butarethesemodelsintendedtodomorethanthis?OnDuhemianreadings,these

modelsdemonstratefurthertheempiricalunderdeterminationofgeometry,andso

showthatgeometryisnotempirical.IarguedinsectionI,ontheotherhand,that

Poincaré’sdiscussionofthemodelin1895/1913makesclearthatheintendeditto

underminetheclaimthatEuclid’saxiomofparallelsisasyntheticaprioritruth.

Furtherconfirmationofthisreadingisprovidedbythepassage–quotedalreadyin

thefirstparagraphofsectionII–wherePoincarépointstothe“applications”ofnon-

Euclideangeometryinthetheoryoflineardifferentialequations.There,after

concludingthatEuclid’saxiomisnotananalytictruthhesays,“Thisisnotall.

Lobachevski'sgeometry,susceptibleofaconcreteinterpretation,ceasestobea

logicalexerciseandiscapableofapplications”(emphasisadded).The

“interpretation”ofhyperbolicgeometryisofcoursetheuseofmodelsofnon-

Euclideangeometryinthecomplexplane.Inotherwords,theexistenceofthese

modelsshowsthatnon-Euclideangeometrydoesnotcontraveneanyanalytictruth;

theirapplicationshowsthatnon-Euclideangeometryisnotjustlogicallypossible–

34

itdoesnotviolateanysyntheticaprioritrutheither.Amathematiciansuchas

BeltramiwhocouldproduceamodelofhyperbolicgeometryinEuclideangeometry

wouldbeabletoseethatitislogicallypossible.Amathematicianwhounderstood

theapplicationsofhyperbolicgeometrywouldbeabletoseethatitisreallypossible

aswell.

WecanfruitfullycomparewhatPoincarésaysaboutLobachevskii’sgeometry

withthe“geometries”ofvariablecurvatureproposedbyRiemann.Afewpageslater,

Poincaréwrites“thesegeometriesofRiemann,inmanywayssointeresting,could

neverthereforebeotherthanpurelyanalytic”(1891/1913,37)However,though

mostmathematiciansthinkofLobachevski'sgeometryalso“onlyasamerelogical

curiosity,”Poincarédisagrees,arguingthatitsimpossibilityisnotshownby

“syntheticapriorijudgments,asKantsaid”(37).Anotherwayofputtingthisclaimis

thatRiemann’sgeometries,butnotLobachevskii’s,areinconsistentwithsome

necessaryfeatureofgeometry.Andthisnecessaryfeatureofgeometryisprecisely

whatPoincaréidentifiedinJune1880,whenhefirstannounceshisdiscoverythat

hisgraphicalrepresentationsofFuchsiangroupsaremodelsofLobachevskiian

geometry:thatgeometryisthestudyofagroupofoperationsformedbythe

displacementsonecanapplytoafigurewithoutdeformingit.Riemann’sgeometries

ofvariablecurvature,sincetheydonotallowdisplacementwithoutdeformation,

arethusnotreallygeometries.Theyarejustlogic,notgeometry.

Butwhyconceiveofgeometryasfundamentallythestudyofgroupsofrigid

35

displacements?35NoargumentisgiveninthemathematicalpapersPoincaré1880b

and1887,norinhisfirstphilosophicalpaper,1891/1913.However,onevery

powerfulmotivationforthisconceptionisprovidedbytheapplicationsthat

PoincarémadeofhyperbolicgeometryinthetheoryofFuchsianfunctions.These

applicationsrequiredthinkingofFuchsiangroupsaslikegroupsofdisplacementsof

rigidbodies,andgeometryasthestudyofsuchgroups.Thinkingofgeometryinthis

wayallowsustotransfergeometricreasoningaboutcomplexellipticfunctionsinto

geometricreasoningaboutFuchsianfunctions–itallowsustomaketheall-

importantanalogybetweenellipticfunctionsandFuchsianfunctions.The

extraordinarypoweroftheresultsobtainedmakesthisreconceptualizationvery

attractive.Furthermore,thiswayofthinkingofgeometryalsoallowedforPoincaré

tounifyalgebra,geometry,andthetheoryoffunctionsinahighlyilluminatingway–

thekindofwaythatwouldstronglysuggesttoaworkingmathematicianthathehad

hitontherightwayofthinkingofgeometry.

Thispaperbeganwithaquestion,"HowdoesPoincaréargueforhis

conventionalismaboutgeometry?"Thispaperhasidentifiedoneway–infact,the

35Thehistoricalquestion–Fromwhomdidhegetthisidea?–isnoteasytoanswer.OnemightbetemptedtoconcludethathegotitfromKlein’sErlangerProgramm,ifonlytherewereanyhistoricalevidencethatPoincaréknewofKlein’sworkinJune1880.WhentheKlein-Poincarécorrespondencebeginsoneyearlaterin1881,itisclearthatPoincarédidnotknowKlein’swork,norindeedvirtuallyanyGerman-languagework.AmoreplausiblehypothesisisthatPoincaréwasinspiredtodevelopthisideafromreadingHelmholtz’sessays,whichweretranslatedintoFrenchinthe1870s(Gray2013,40).Intheseessays,Helmholtz–thoughhecertainlydoesnotdeveloptheviewthatgeometryisthestudyofthegroupofdisplacements–arguesthatwecometoknowEuclid’saxiomfrommovingarigidmeasuringrodthroughspace.

36

earliestway,whichwascontainedalreadyinhismathematicalpapersfromthe

1880s--thatPoincaréarguedforconventionalism.Thislineofargumentis

distinctiveinasmuchasitinnowayreliesonDuhemianunderdetermination,indeed

doesnotdependonfactsabouttheapplicationofgeometrytophysicalscienceatall.

Instead,thisargumentarisesnaturallyfromreflectingontheverypowerfulresults

Poincaréobtainedbyapplyingonemathematicaltheory(metricgeometry)to

another(thetheoryofFuchsianfunctions).Moreover,Ibelievethatthe

mathematicalresultsdescribedinthispaperexplainclearlywhyPoincaré,having

donethekindofmathematicalworkthathehaddone,foundconventionalismso

natural,whereasothermathematicianswhowerealsoawareofEuclideanmodelsof

hyperbolicgeometryweredrawntootherphilosophicalviews.Whatinitiallyand

powerfullydrewPoincarétoconventionalism,then,werenotjustphilosophical

reflectionsonspace,geometry,andphysics,butveryspecificfeaturesofhisworkin

puremathematics.

Ofcourse,Poincaré'sargumentsforconventionalismevolvedandwere

expandedovertime,partiallyasaresponsetocriticisms,andpartiallyasaresultof

Poincaré–whosewide-ranginggeniusventuredintonearlyallofthesciences–

expandinghisreflectionsonspaceandgeometryintophysicsandpsychology.For

example,aswe'veseen,Poincaré'sargumentfortheconventionalityofaspatial

metricwasoftenposedasaquadrilemma.Theexistenceofmodelsofnon-Euclidean

geometryshowthatthetruthofaparticularmetriccannotbeanalytic;thepowerful

applicationsofbothEuclideanandhyperbolicmetricsincomplexanalysisshowthat

itcannotbesyntheticapriori;andsince(Poincaréarguedinitially)itcouldnot

37

possiblybeempirical,itmustbeconventional.However,inPoincaré'searliest

philosophicalpapers(1891/1913,38and1895/1913,53),theargumentagainst

empiricismisweakandundeveloped.Heargues,briefly,thatgeometryisanexact

andunrevisablescience,whereasallempiricalsciencesareonlyapproximatelytrue

andaresubjecttoconstantrevision.Partiallyasaresultofreadingandreviewing

Russell1897(whicharguedthatmetricgeometryisempirical),Poincarébeganto

refineandexpandhisargumentsagainstempiricism.Theseargumentsarecollected

inCh.5ofScienceandHypothesis(Poincaré1902/1913),whichconsistslargelyof

reprintedsectionsofhisexchangewithRussell(Poincaré1899andPoincaré

1900).36Furthermore,geometryisintimatelyconnectedwithperceptual

psychology,sinceweperceiveobjectsinspaceandgeometricalnotions(suchas

point,line,anddistance)appearinperceptualpsychologyaswell.Poincarébeganin

1895/1913(andcontinuinginchapters3and4ofPoincaré1905)toputtogethera

richtheoryoftheoriginsofspatialnotionsinperception–atheorythatreinforces 36Asmentionedearlier(notes11and16)thepenultimateparagraphofPoincaré1891/1905–apassagedevotedtothepossibilityofdeterminingthecorrectgeometryempiricallybymeasuringstellarparallaxes–wasmovedtoch.5ofSH,whereitappearsontheopeningpage(page81)ofthepaper,followedinthesubsequentpagesbyotherparagraphsthathadappearedinPoincaré1899and1900.Inthispassage,Poincarépointsoutthat,ifLobachevskiangeometryweretrue,theparallaxofdistantstarswouldhaveapositivelimit,providedthatlightraystravelalwaysinstraightlines.Butshouldexperimentsshowthatindeedstellarparallaxesdohaveapositivelimit,weareleftwithtwochoices:rejectEuclideangeometryorrejecttheclaimthatlightraystravelinstraightlines.Thelatterwouldbemoreadvantageous,andso"Euclideangeometryhasnothingtofearfromnewexperiments." Poincarédoesnotconnectthislineofreasoningtothespheremodelin1895/1913,which(asremarkedabove)isusedtoshowthatwecanimagineanon-Euclideanworld.Thereasoninginthispassageisverybrief,thoughclearlysuggestive.Thereisgoodreason,then,whyPoincaréneededtofurtherexpandanddeepenhisarguments(inthepaperspublishedin1899andlater)againsttheclaimthatthemetricofspaceisempirical.

38

theargumentforconventionalismbycapturingtheobjectivityofspatialperceptions

ingroup-theoreticterms.

Theselaterdevelopmentsarepowerfulandsuggestive.Itisnotsurprising,

then,thatdebateovertheconventionalismofgeometrybegantofocusmoreon

Poincaré'sreflectionsontherelationsbetweengeometry,physics,andpsychology,

asopposedtohisoriginalargumentsbasedontherelationsbetweengeometryand

otherbranchesofpuremathematics.Inconcertwithotherhistoricaldevelopments

–hereweshouldcertainlyincludethereceptionandexpansionofPoincaré's

argumentsbylogicalempiricistssuchasSchlickandReichenbach–theconnections

betweenPoincaré'sconventionalismandhismathematicalworkinFuchsian

functionswasde-emphasized,leadingtothesortofreadingsrecountedinthe

openingpagesofthispaper.Buttheseconnectionsareworthremembering,

becausetheymotivateadistinctiveargumentforconventionalismbasedonthe

loosenessoffitbetweentwomathematicaltheories.Andtheyshowclearlythatit

wasnoaccidentalfactthatamathematicianwhoobtainedthekindofresultsusing

thekindofmethodsPoincaréemployedwasledtoconventionalism.

39

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Figures

Figure1:thePoincaréDiskmodelofhyperbolicgeometry

Figure2:Projectionontotheplaneofthedistancefunctiononasphere

Figure3:BeltramiModelofHyperbolicGeometry

Figure4:Thequadrilateralconstruction

Figure5:TheKleinmodel

Figure6:TheBeltrami-Kleinmodelontheequatorialplane

Figure7:TheBeltrami-Kleinmodelprojectedontothesouthernhemisphere

Figure8:TheBeltrami-KleinmodelprojectedontothePoincarémodel

Figure9:TheLemniscateofBernoulli

Figure10:Thesinefunction

Figure11:Tessellationofthecomplexplanebyellipticfunctions

Figure12:TessellationofthediskinthecomplexplanebyaFuchsianfunction