“the geometry behind poincaré’s conventionalism” jeremy ......poincare’s famous sphere...
TRANSCRIPT
-
1
“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.
-
2
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.
-
3
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.
-
4
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.
-
5
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.
-
6
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.
-
7
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.
-
8
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
-
9
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.
-
10
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:
-
11
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
-
12
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.
-
13
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
-
14
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.
-
15
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.
-
16
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
WORKSCITED
Beltrami,E.1868."Saggiodiinterpretazionedellegeometrianon-Eucliden,"GiornalediMath.,VI:284-312.Ben-Menahem,Yemima.2006.Conventionalism.CambridgeandNewYork:CambridgeUniversityPress.Bottazzini,UmbertoandJeremyGray.2013.Hiddenharmony--geometricfantasies:theriseofcomplexfunctiontheory.NewYork,NY:Springer.Cayley,Arthur.1859.“SixthMemoironQuantics.”PhilosophicalTransactionsoftheRoyalSocietyofLondon149:61-90.Dunlop,Katherine.2016."PoincaréOnTheFoundationsOfArithmeticAndGeometry.Part1:Against“Dependence-Hierarchy”Interpretations."HOPOS:TheJournaloftheInternationalSocietyfortheHistoryofPhilosophyofScience6:274-308.Friedman,Michael.1999."Poincaré'sConventionalismandtheLogicalPositivists."InReconsideringLogicalPositivism(CUP),71-86.Fuchs,Leonard.1880."UbereineKlassevonFunktionenmehrererVariablenwelchedurchUmkehrungderlntegralevonLösungenderlinearenDifferentialgleichungenmitrationalenCoeffizientenentstehen."JahrschriftfürreineundangewandteMathematik89:150-169.Gimbel,Steven.2004."Un-conventionalwisdom:theory-specificityinReichenbach’sgeometricconventionalism."StudiesinHistoryandPhilosophyofModernPhysics35:457-481.Gray,Jeremy.1985.LineardifferentialequationsandgrouptheoryfromRiemanntoPoincare.Boston:Birkhaüser.Gray,Jeremy.1999.Non-Euclideangeometryinthetheoryofautomorphicfunctions.editedbyJeremyGrayandAbeShenitzer.IntroductionbyJeremyGray.Providence,R.I.:AmericanMathematicalSociety.Gray,Jeremy.2013.HenriPoincaré:ascientificbiography.Princeton,NJ:PrincetonUniversityPressKlein,Felix.1871."Ontheso-calledNon-EuclideanGeometry."Tr.ByJohnStilwellinStillwell1996,69-110.Poincaré,Henri.1880a."ExtractfromanuneditedmemoirofHenriPoincaré."Ed.byN.E.Nörlund.TransbyJohnStillwell.InStillwell1985,305-356.
-
40
Poincaré,Henri.1880b."PremierSupplément."InPoincare1997,31-74.Poincaré,Henri.1881a."Surlesfonctionsfuchsiennes."Comptesrendusdel'AcademedesSciences92(14Feb1881):333-335.TranslatedbyStillwellas"OnFuchsianFunctions,"inPoincaré1985,47-50.Poincaré,Henri.1881b."Surlesfonctionsfuchsiennes."Comptesrendusdel'AcademedesSciences92:(21Feb1881):395-398.TranslatedbyStillwellas"OnFuchsianFunctions,"inPoincaré1985,51-54.Poincaré,Henri.1882a."TheoriedesgroupesFuchsiens."ActaMathematicaI:1-62.TranslatedbyJohnStillwellas"TheoryofFuchsianGroups,"inStillwell1985,55-127.Poincaré,Henri.1882b."MemoiresurlesfonctionsFuchsiennes."ActaMathematicaI:193-294.TranslatedbyJohnStillwellas"OnFuchsianFunctions,"inStillwell1985,128-254.Poincaré,Henri.1887.“Surleshypothèsesfondamentalesdelagéométrie.”BulletindelaSociétémathématiquedeFrance15:203-216.Poincaré,Henri.1891/1913."Lesgéométriesnoneuclidiennes."Revuegénéraledessciencespuresetappliquées2(1891):769-774.Reprinted,withminoralterations,inPoincaré1902.TranslatedbyHalstedas“TheNon-EuclideanGeometries,”inPoincaré1913,29-39.Poincaré,Henri.1895/1913."L'espaceetlagéométrie."Revuedemétaphysiqueetdemorale3(1895):631-646.Reprinted,withminoralterations,inPoincaré1902.TranslatedbyHalstedas“SpaceandGeometry,”inPoincaré1913,40-54.Poincaré,Henri.1897."Réponseàquelquescritiques."Revuedemétaphysiqueetdemorale5:59-70.Poincaré,Henri.1898."OntheFoundationsofGeometry."Monist9(1898):1-43.ReprintedinWilliamEwald,ed.,FromKanttoHilbert,vol.2(1996,OUP),982-1011.Poincaré,Henri.1899."Desfondementsdelagéométrie;àproposd'unlivredeM.~Russell."Revuedemétaphysiqueetdemorale7:251-279.Poincaré,Henri.1900."Surlesprincipesdelagéométrie;réponseàM.~Russell."Revuedemétaphysiqueetdemorale8:73-86.Poincaré,Henri.1902.LaScienceetl'hypothèse.Flammarion.
-
41
Poincaré,Henri.1902/1913.“ExperienceandGeometry.”InPoincaré1902.TranslatedbyHalstedinPoincaré1913,55-65.Poincaré,Henri.1905.LaValeurdelascience.Flammarion.Poincaré,Henri.1909."L'inventionmathématique."Annéepsychologique15:445-459.Poincaré,Henri.1913.TheFoundationsofScience.Trans.byGeorgeBruceHalsted.NewYork:theSciencePress.Poincaré,Henri.1985.PapersonFuchsianFunctions.Trans,withanintroduction,byJohnStilwell.SpringerVerlag.Poincaré,Henri.1997.Troissupplémentssurladécouvertedesfonctionsfuchsiennes.J.GrayandS.Walter,eds,Berlin:AkademieVerlag.Russell,Bertrand.1897.EssayontheFoundationsofGeometry.CUP.Russell,Bertrand.1900."Surlesaxiomsdelagéométrie."Revuedemétaphysiqueetdemorale7:684-707.Stein,Howard.1977.“SomePhilosophicalPre-HistoryofGeneralRelativity."InFoundationsofSpace-TimeTheories,MinnesotaStudiesinPhilosophyofScience,Vol.8,J.Earman,C.Glymore,J.Stachel(eds.).UniversityofMinnesotaPress.Steiner,Jakob.1881.SystematischeEntwicklungderAbhängigkeitgeometrischerGestaltenvoneinander.InGesammelteWerke.Vol.1.,editedbyK.Weierstrass(Berlin:G.Reimer).Stillwell,John.1985."Translator'sIntroduction."InHenriPoincaré,PapersonFuchsianFunctions.Trans,withanintroduction,byJohnStilwell.SpringerVerlag.Stillwell,John.1996.SourcesofHyperbolicGeometry.AmericanMathematicalSociety.Stillwell,John.2010.MathematicsanditsHistory.Springer.Stump,David.1989,"HenriPoincaré'sPhilosophyofScience."StudiesinHistoryandPhilosophyofModernPhysics20:335–363.Tappenden,Jamie.2006."TheRiemannianBackgroundtoFrege'sPhilosophy."InTheArchitectureofModernMathematics:EssaysinHistoryandPhilosophy.Ed.byJ.FerreirósandJ.J.Gray.OUP.
-
42
Zahar,Elie.1997.“Poincaré’sPhilosophyofGeometry,orDoesGeometricConventionalismDeserveItsName?”StudiesinHistoryandPhilosophyofModernPhysics28:183–218.
-
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