Table Of Contentmaria georgiadou
ConstantinCarathéodory
MathematicsandPolitics
inTurbulentTimes
Springer-Verlag BerUn Heidelberg GmbH
maria georgiadou
Constantin
Carathéodory
Mathematics and Politics
in Turbulent Times
Springer
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Foreword
The life and times of Constantin Carathéodory (1873–1950), a brilliant German
mathematician of Greek descent, should surely interest mathematicians, German
and Greek academics, historians of science as well as many others. The author of
thisdocumentarybiographyhasresearchedwithgreatcarealargepartofCarathéo-
dory’scorrespondence,papersandotherrelevantdocumentsandarchives,andhas
constructed a narrative that is driven by Carathéodory’s own words and thoughts.
She has provided excellent personal, historical and mathematical background in-
formation, as well as lean and precise comments and interpretation. The result is
stunninglyeffective.Carathéodoryandhistimespringtolife:hismathematicsand
science,hisloveofhistory,hispolitics,hisunshakeablebeliefinthepowerofthe
Germanintellectualtradition,hisemotionalattachmenttoGreece,hissetbacks,his
lostcauses,andhisdisappointments.Weseetheworldchangingaroundhim,some-
timestoofast.Butwealsoseehisowneffortstomakechanges,forhewasenormously
influential both in the academic world and more broadly in science and education
politics. The dramatic events of the first half of the twentieth century changed the
worldandseeingtheminthisgrippingnarrativethroughtheactionsandthoughtsofa
leadingintellectual,mathematicianandacademicprovidesauniqueandfascinating
perspective.
Carathéodorydidnotdecidetogointomathematicsuntilhewastwenty-seven,
afterhehadstudiedengineeringattheMilitarySchoolofBelgiumandworkedfor
a few years as an engineer. He was born in 1873 in Berlin, where his family was
residingwhilehisfatherwasthediplomaticrepresentativeoftheOttomanEmpire
in Germany. He studied mathematics at first in Berlin, attracted to the lectures of
H.A.Schwarzonfunctiontheory.HethenwenttoGöttingen,receivingadoctoral
degree in 1904 with a dissertation entitled “On the discontinuous solutions in the
calculus of variations”. He was close to Klein and Hilbert in Göttingen but his
dissertationwassignedbyMinkowski.HisfirstteachingpositionwasinGöttingen,
wherehestayeduntil1908.HesubsequentlymovedtoBonn,HanoverandBreslau
beforereturningtoGöttingenin1913asfullprofessor.In1918hewenttoBerlinand
stayedtilltheendof1919.HismeteoricriseintheGermanacademicworldwas,of
course,adirectrecognitionofhismanyseminalcontributionsinfunctiontheory,the
theoryofmeasure,first-orderpartialdifferentialequations,thecalculusofvariations
andthefoundationsofthermodynamics.
Hismathematicalinterestsweredeeplyrootedingeometryandmechanics,but
Carathéodory was above all a powerful analyst who understood and appreciated
VI Foreword
theimportanceofsimplicity,naturalnessandeleganceinmathematics.Heactively
sought these qualities in his work. He was also very interested in the origin and
historicalevolutionofimportantmathematicalideasandmethods,whichareoften
lostinmoreefficientandabstractformulations.Hisworkinfunctiontheoryisclosely
tiedtothegeometryofconformalmapping,elegantandunexpectedusesofSchwarz’s
lemma, the various variational forms of the Riemann mapping theorem and many
results that are part of harmonic analysis today. He published his beautiful lecture
notesonconformalmappingin1932and,verylateinhislife,hewroteanelegant
treatiseoncomplexfunctiontheorythatappearedin1950.
Functiontheorymusthavebeenhisfirstandenduringlove.Itsuitedhismathemat-
icalstyleverywellandhemusthavelikedlecturingonit.Thecalculusofvariations
must not have been very far behind, however, as his favorite area in mathematics.
He had a very specific point of view here: that the elaboration of the equivalence
betweentheHuygensprincipleandFermat’sprincipleinOpticsisinfactcentralto
themathematicalfoundationsofthecalculusofvariations.InCarathéodory’swork
thisidealeadstoafar-reachinggeneralizationandextensionoftheHamilton–Jacobi
theory, in which the level surfaces of the action, or travel time, that are solutions
of the Hamilton–Jacobi equation, are also generated by families of extremals of
theassociatedLagrangefunctional.Hisworkispresentedinhisfamous1935book
VariationsrechnungundpartielleDifferentialgleichungenersterOrdnung(Calculus
of Variations and Partial Differential Equations of the First Order), translated into
Englishin1965.Thisisstilloneofthebestbooksintheareaeventhoughthemodern
theory of control and dynamical systems is often oriented toward the inclusion of
stochasticeffects,whichenteredmuchmorerecently.
ThewaythattheHamilton–Jacobitheoryemergesinhigh-frequencyasymptotics
inwavepropagationwassurelyknowntoCarathéodorybutitisnotdiscussedinhis
book. It played a very important role in the development of quantum mechanics,
whichCarathéodoryfollowedclosely,andbeforethatinoptics.Themathematical
foundationsofhigh-frequencyasymptoticsthatconnectHamilton–Jacobitheoryto
wavepropagationdidnotemergeuntilmanyyearsafterCarathéodory’swork,inthe
sixtiesandseventies,sohecouldnotdealwithitmathematicallyevenifheunderstood
itphysically.Itisinteresting,however,thathisintenselygeometricalpointofview
isbecomingincreasinglyimportanttoday,inconnectionwithlevel-setmethodsfor
theefficientnumericalcomputationoftraveltimesforimagereconstruction.
Carathéodorybeganhisworkonmeasuretheoryaround1913andinthefollow-
ingfiveyears,untilthepublicationofhisbookVorlesungenüberreelleFunktionen
(LecturesonRealFunctions)in1918,madeseveralfundamentalcontributionsthat
solidified his standing as a leading mathematician of his generation in Germany
andaroundtheworld.Histheoryofoutermeasure,andtheCarathéodoryextension
theorem, are now part of the standard material taught in every graduate-level real
variables course. For forty years, until the sixties, his real variables book was one
ofthebestsourcesforlearningmeasureandintegrationtheory.Whatmakesitlook
abitstrangetodayistheabsenceofbasicHilbertspacetheoryandfunctionalanal-
ysis,whichdidnotemergeuntilthethirties.Anotherfieldthatdidnotexistbefore
thethirtiesisthemeasure-theoreticfoundationofprobabilitytheory,establishedby
Foreword VII
Kolmogorov in 1933, which provided a major thrust for the further development
of general measure theory. For a mathematician with Carathéodory’s command of
classicalanalysis(functiontheory,calculusofvariations)itissomewhatsurprising
thatatagefortyhisresearchintereststurnedintoamoreabstractdirection,measure
andintegrationtheory.
Hiselegantmathematicalformulationofthermodynamicsattractedtheattention
ofseveralphysicistsbutremainedsomehowoutsidethemainstream.Thephysicist
MaxBorn,alifelongfriendofCarathéodory,appreciatedandpromoteditbutmathe-
maticalfoundationsofestablishedphysicalfieldsseemtopenetrateslowlyandwith
difficultyintotheworldofphysics.Itisinterestingtoseehowhisinvolvementwith
applicationsofmathematics,thermodynamicsinparticular,aswellashisbackground
inengineeringinfluencedhisappointmentsandpromotions.Itappearsthattheydid
playarolebutnotabigone.ItisalsointerestingtoseehowtheGermanmathematics
communityviewedappliedmathematicsjustbeforetheFirstWorldWar.Itisonlya
partialview,ofcourse,butitseemstobemainlyasawaytomaketheteachingof
mathematicsmoreeffective.Carathéodorywaslessinterestedinapplicationsthanin
thewaymathematicsinteractsintimatelywithphysicsatabasiclevel,asinthegen-
eraltheoryofrelativityandinquantummechanics.Anexceptionishisverydetailed
studyofaberrationsinopticalinstrumentsinthelatethirties.
Carathéodory’slifechangeddramaticallyinthefallof1919whenheaccepteda
mandatefromE.Venizelos,theprimeministerofGreece,toorganizeanewuniver-
sityinSmyrna.HeleftBerlinattheendofthatyearandspentthenextfouryears
in Athens and Smyrna before returning to Germany as professor of mathematics
in Munich. He remained there for the rest of his life, resuming his brilliant career
almostasifnothinghadhappened.Butagreatdealhadhappened,andithadtouched
Carathéodoryinaverypersonalway,evenifitdidnotdiminishhisself-confidence
andhiscreativedrive.AcceptingVenizelos’callinghadclearlybeenanemotional
decision,notarationalone,forCarathéodorybelongedtoanextendedfamilyofdis-
tinguisheddiplomatsandknewatfirsthandthecomplexandunstablepoliticalscene
inGreeceatthetimeoftheVersaillesPeaceConference.Bythefallof1920,when
Venizeloswasvotedoutofoffice,Carathéodorywasunabletorecruitevenclosefor-
mercolleaguestothenewuniversityandwasatoddswiththeroyalistgovernmentin
AthensasaVenizelist(aliberal).Heclungtothenoblecauseofbuildingauniversity,
a civilizing mission if ever there was one. In the words of Arnold Toynbee, a war
correspondentfortheManchesterGuardianinAsiaMinorduringthisperiod,Cara-
théodory the scientist-humanist looked like fish out of water. The background and
eventsofthisperiodarepresentedwithsuchclarityandprecisioninthisbiography
that even readers familiar with the complicated and little-known historical details
will find the narrative fascinating. Focusing on Carathéodory’s doomed efforts to
buildauniversityinSmyrnaisspecial,personalandeventragic.Butitisalsoarare
glimpseattheendofanerainEurope.
Remarkably, Carathéodory was able to continue his research in mathematics
throughtheturbulentperiodinSmyrnaandAthens,andnodoubtkeephismental
balancebydoingso.AfterreturningtoGermanyasprofessorinMunichhereceived
manyhonorsandmuchinternationalrecognition.In1928hebecamethefirstVisiting
VIII Foreword
LectureroftheAmericanMathematicalSocietyandvisitedandlecturedatseveral
Americanuniversities,includingHarvard,theUniversityofTexasatAustin,Stanford
andBerkeley.HewasconsideredforapermanentappointmentatHarvardbuthedid
notreceiveanoffer.Hedid,however,receiveanofferofaprofessorshipatStanford,in
early1929,whichhedidnotaccept.TheAmericanWestwasbeautifulandattractive
butattheotherendoftheworldforacosmopolitanEuropeanlikeCarathéodory.His
academiclifeinMunichwasfulfillingandproductiveandtherewas,ofcourse,no
hintthatastormwasabouttogripGermanyandEurope.
Carathéodory’srelationswithGreecetookanotherinterestingturnwhenVenize-
losreturnedtopowerandappointedhim,inearly1930,advisertotheGovernment
forthereorganizationofGreekuniversities.HetookaleaveofabsencefromMunich
forabouttwoyearstoconductanin-depthstudyofuniversitiesinGreece.Hispres-
tigewasenormous,butevenanapoliticalacademicofhisstature,whosurelyacted
inthebestinterestsofGreekuniversities,couldnotovercomethepoliticaldivisions
thatplaguedtheGreekuniversitysystem.Heeventuallyterminatedhisformalrela-
tionswithGreecein1932,andfromthenonhisacademicpositionremainedfixed
inMunichuntilhisretirementin1938attheageofsixty-five.
ThearrivalofnationalsocialisminMarch1933tookmanyacademicsinGermany
bysurprise,especiallythespeedwithwhichracialandpoliticaldiscriminationsetin,
aswellasthelossofpersonalfreedomandtheobligationtodemonstratecompatibil-
itywithandadherencetothenewregime.CarathéodoryhadbeenborninGermany,
studiedmathematicsinGermany,becameadistinguishedprofessorinseveralGer-
manuniversities,buthewasnotaGerman.HewasanethnicGreek,aChristian,and
wasabletofunctioninNaziGermanyaslongashedidnotopenlyopposetheregime
evenifhedidsoprivately.HisbehaviorandactionsduringtheNaziperiod,meticu-
louslyresearchedandaccuratelyreportedinthisbiography,wereexactlylikethose
ofmanyGermanacademicsofhisgeneration.Heactivelyandenergeticallyhelped
hisJewishfriendsandcolleagueswhobecamepersecutedundesirablesintheirown
countryalmostovernight,andsometimes,inlateryears,intervenedontheirbehalf
throughmorepowerfulcolleagueswhowereclosertotheregime.Persecutionwas
notunknowntoCarathéodorywhocamefromafamilyofethnicGreeksthathadrisen
toprominentpositionsintheOttomanEmpire.Atanymoment,eventsoutoftheir
control could, and did many times, turn the state against them with unpredictable
intensity. He no doubt understood very well that his academic status and prestige
could hardly protect him should his loyalty to the state be questioned, especially
since he was not German. Where he crossed the line, however – and the author is
verycarefulhere–isinallowinghimselftobeusedbytheNazisintheirinternational
propaganda.Whenhe,asadistinguishedacademic,representedGermanmathemat-
icsatinternationalmeetingsthatthestateallowedhimtoattend,hewasbeingused.
DuringthewarhecontinuedtohelpcolleaguesoutsideGermanywhonowwerenot
only losing their positions and their property but also were increasingly in danger
oftheirlives.However,neitherhenorotherengagedacademicscoulddomuch,and
thereisnoindicationatwhattimeCarathéodorybecameawareoftheHolocaust,or
whathisreactionwastotheinvasionandoccupationofGreecebytheGermansin
April1941.
Foreword IX
Carathéodorytriedtore-establishhiscontactsoutsideGermanyafterthewarbut
hequicklyfoundthattheworldhadchangedcompletely.HiscolleaguesinAmerica
helpedhimgenerouslybutnotenthusiastically.Whenhetriedtodifferentiatehimself
from other Germans as an ethnic Greek living in Germany he discovered that the
Americanoccupyingforcesdidnotunderstandthedifference.Whenhetriedtomove
to Greece with his family he was rebuffed by the Greeks. The only escape he had
wasinmathematics,andthatiswhenhewrotehisbookonfunctiontheory.Hewas
diagnosedwithanenlargedprostateinthespringof1947anddiedinFebruary1950.
Carathéodory is a towering figure in the world of mathematics. His work on
first order partial differential equations and the calculus of variations is still cited
andstudiedtoday,asarehiscontributionsinrealandcomplexanalysis.Helivedin
complicatedandturbulenttimes,andhadrootsinmanyculturesandtraditions,most
ofwhicharefadingmemoriestoday.Hewasacultivatedandengagedintellectual,
andthisbiographyisafittingtributetohislifeandwork.
GeorgePapanicolaou
