L. E. J. Brouwer - Topologist, Intuitionist, Philosopher : How Mathematics Is Rooted in Life

877 Pages·2013·7.602 MB·English

by  Dirk van Dalen

L.E.J. Brouwer – Topologist, Intuitionist,
Philosopher
Dirk van Dalen
L.E.J. Brouwer – Topologist,
Intuitionist, Philosopher
How Mathematics Is Rooted in Life
DirkvanDalen
DepartmentofPhilosophy
UtrechtUniversity
Utrecht,Netherlands
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Mystic,Geometer,andIntuitionist:TheLifeofL.E.J.BrouwerbyDirkvanDalen
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In lovingmemoryofourdaughterTineke
8 July1958–14 October2009
Preface
Faireabstractiondumonded’objets(cequiestnecessairepourtravaillerdanslesmathéma-
tiquesintuitionistes)n’estpossiblequ’enéprouvantlaviecommeunrêve.
Brouwer
Among the leading scientists of all times, Brouwer occupies a somewhat un-
orthodoxpositionthatmeritsacloserlook.Ageniusisusuallysupposedtobecon-
tinuallyinvolvedinbrilliantandillustriousactivities.Mozart,forexample,wassaid
to the embodiment of music, his mind at all times and places emerged in creation
andreflection.Inmathematics,Eulerwouldbetheperfectexample—alwaysinves-
tigating, creating, publishing, until a ripe old age. Brouwer belonged to different
class of genius; gifted with a deep intuition, he had an unparalleled access to the
secretsandintricaciesofmathematicsandothersubjects,butthemanifestationsof
hisgeniuswererathertheeruptionsofaproudandisolatedvolcanothanasmooth
runningriverofclevertheorems.Indeed,Brouwerrefusedtojointheclassofspe-
cialisedacademics,whoswearallegiancetoaparticulartopic.Hefeltfreetoinvest
histimeandenergyinawiderangeofactivities,runningfrommysticism,psycho-
linguistics,art,politicstolongwalks,swimming,solitarycontemplation,tofighting
injustice.
The scientific highlights, of course, are Brouwer’s topological innovations and
thecreationofhisrevolutionaryintuitionisticmathematics.Botharemanifestations
ofhisunparalleledpowerofreflection.Hisintuitionismclearlybenefittedfrom,and
wasbasedonhismysticviews.
In the following pages the life of this unusual scientist is sketched. The scien-
tifichighlightsarehisbreakthroughintheyoungsubjectoftopologythattriggered
thetransitionfromthetraditionofCantortomoderntopology,andtheintroduction
andconsolidationofconstructivemethodsandphilosophyunderthenameintuition-
ism.Asaconfirmedinternationalisthegotentangledintheinterbellumstrugglefor
the ending of the boycott of the German and Austrian scientists. And roughly at
thesametimehewasdrawnintotheFormalism–Intuitionismconflict,knowasthe
Grundlagenstreit,whichfoundanuntimelyendintheso-calledWarofthefrogsand
themice.
vii
viii Preface
Oneshouldnotgettheimpressionthathislifewasonelongstringofconflicts,
butiscertainlytruethathisuncompromisingoppositiontoinjusticegothimmore
thanhisshareofproblems.
The present biography is a revision of the earlier two volume biography pub-
lishedbytheOxfordUniversityPress.Afterthesehad,sotospeak,passedtheirnat-
urallifespan,theOUPgracefullyagreedtoallowmetopublishthepresentversion
with Springer. The contents have here and there been updated, and some sections
havebeenpruned.
I have in the Oxford Press edition expressed my gratitude to a large number of
friends and colleagues and institutions, and I want on this place to say again how
muchthebiographyowestothem.IamindebtedtoGarthDaleswhovolunteeredto
proofreadthefirstsevenchapters.
Without the efficient and friendly support of Joerg Sixt and his staff my task
wouldhavebeenaheavyburden,theymorethandeservemythanks.
In the mean time the Selected Correspondence of Brouwer (Brouwer 2011) (in
anEnglishtranslation)hasappeared,sothereaderwillhaveaccesstoarichsource
ofbackgroundinformationnotavailableearlier.
Utrecht,theNetherlands DirkvanDalen
November2012
Contents
1 ChildandStudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 SchoolYears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 StudentinAmsterdam . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 TheReligiousCredo . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Friendship:AdamavanScheltema . . . . . . . . . . . . . . . . . 20
2 MathematicsandMysticism . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 TeachersandStudy . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 FirstResearch,Four-DimensionalGeometry . . . . . . . . . . . . 44
2.3 Marriage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Bolland’sPhilosophyCourse . . . . . . . . . . . . . . . . . . . . 54
2.5 AmongtheArtistsandVegetarians . . . . . . . . . . . . . . . . . 57
2.6 TheDelftLectures . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7 FamilyLifeinBlaricum . . . . . . . . . . . . . . . . . . . . . . . 74
3 TheDissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1 PreparationsandHesitations . . . . . . . . . . . . . . . . . . . . . 77
3.2 UnderKorteweg’sSupervision . . . . . . . . . . . . . . . . . . . 83
3.3 OntheRoleofLogic. . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4 MathematicsandtheWorld . . . . . . . . . . . . . . . . . . . . . 99
3.5 ObservationsonSetTheoryandFormalism . . . . . . . . . . . . . 101
3.6 ThePublicDefence . . . . . . . . . . . . . . . . . . . . . . . . . 115
4 Cantor–SchoenfliesTopology . . . . . . . . . . . . . . . . . . . . . . 119
4.1 TheGeometryofContinuousChange . . . . . . . . . . . . . . . . 119
4.2 LieGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3 PublishingintheMathematischeAnnalen. . . . . . . . . . . . . . 127
4.4 FixedPointsonSpheresandtheTranslationTheorem . . . . . . . 130
4.5 VectorFieldsonSurfaces . . . . . . . . . . . . . . . . . . . . . . 133
4.6 AnalysisSitusandSchoenflies . . . . . . . . . . . . . . . . . . . 137
5 TheNewTopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1 Invarianceofdimension . . . . . . . . . . . . . . . . . . . . . . . 149
ix
x Contents
5.2 TheFixedPointTheoremandOtherSurprises . . . . . . . . . . . 169
5.3 TheKarlsruheMeetingandtheContinuityMethod . . . . . . . . . 175
6 MakingaCareer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.1 FinancialWorries . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.2 FirstInternationalContacts . . . . . . . . . . . . . . . . . . . . . 199
6.3 ClimbingtheLadder . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.4 TheShortcomingsofSchoenflies’Bericht . . . . . . . . . . . . . 204
6.5 PrivaatDocent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.6 Korteweg’sCampaignforBrouwer . . . . . . . . . . . . . . . . . 214
6.7 SchoenfliesAgain . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7 TheWarYears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.1 SetsandSequences—LaworChoice?. . . . . . . . . . . . . . . . 232
7.2 TheInternationalAcademyforPhilosophy . . . . . . . . . . . . . 241
7.3 FamilyLife. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.4 AnOfferfromLeiden . . . . . . . . . . . . . . . . . . . . . . . . 250
7.5 VanEedenandtheInternationalAcademy . . . . . . . . . . . . . 253
7.6 FacultyPolitics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.7 TheFlemishCause. . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.8 AirPhotographyandNationalDefence . . . . . . . . . . . . . . . 271
8 MathematicsAftertheWar . . . . . . . . . . . . . . . . . . . . . . . 279
8.1 HowtoAppointProfessors . . . . . . . . . . . . . . . . . . . . . 283
8.2 TheReturntoTopology . . . . . . . . . . . . . . . . . . . . . . . 285
8.3 TheOffersfromGöttingenandBerlin. . . . . . . . . . . . . . . . 290
8.4 TheAcademy—HowDenjoyWasElected . . . . . . . . . . . . . 294
8.5 NegotiationswithHermannWeyl . . . . . . . . . . . . . . . . . . 297
8.6 IntuitionismandtheBegründungs-Papers . . . . . . . . . . . . . . 302
8.7 AndBrouwer—ThatIstheRevolution . . . . . . . . . . . . . . . 308
8.8 Intuitionism,theNauheimConference . . . . . . . . . . . . . . . 316
8.9 TheFailureoftheInstituteforPhilosophy . . . . . . . . . . . . . 320
9 PoliticsandMathematics . . . . . . . . . . . . . . . . . . . . . . . . 327
9.1 TheConseilandtheBoycottofGermany . . . . . . . . . . . . . . 327
9.2 TheNauheimConferenceandIntuitionism . . . . . . . . . . . . . 333
9.3 TheDenjoyConflict . . . . . . . . . . . . . . . . . . . . . . . . . 336
9.4 Weitzenböck’sAppointmentinAmsterdam . . . . . . . . . . . . . 349
9.5 KohnstammandthePhilosophyofScienceCurriculum . . . . . . 351
9.6 TheNewChronicle . . . . . . . . . . . . . . . . . . . . . . . . . 354
10 TheBreakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
10.1 TheSignificCircle . . . . . . . . . . . . . . . . . . . . . . . . . . 357
10.2 Intuitionism—PrinciplesforChoiceSequences . . . . . . . . . . . 365
10.3 IntuitionismintheMathematischeAnnalen . . . . . . . . . . . . . 376
10.4 BeyondBrouwerianCounterexamples . . . . . . . . . . . . . . . 382
10.5 Fraenkel’sRoleinIntuitionism . . . . . . . . . . . . . . . . . . . 385
10.6 Heyting’sFirstContributions . . . . . . . . . . . . . . . . . . . . 391
Contents xi
11 TheFathersofDimension . . . . . . . . . . . . . . . . . . . . . . . . 395
11.1 TheTwoRussians . . . . . . . . . . . . . . . . . . . . . . . . . . 395
11.2 TheDefinitionofDimension . . . . . . . . . . . . . . . . . . . . 398
11.3 TheVienneseConnection . . . . . . . . . . . . . . . . . . . . . . 421
11.4 TheScientificLegacyofUrysohn . . . . . . . . . . . . . . . . . . 424
12 Progress,Recognition,andFrictions . . . . . . . . . . . . . . . . . . 435
12.1 TheFirstSkirmishesintheFoundationalConflict . . . . . . . . . 435
12.2 ConsolidationandEntrenchment . . . . . . . . . . . . . . . . . . 447
12.3 TheRiemannVolume . . . . . . . . . . . . . . . . . . . . . . . . 458
12.4 InternationalRelations . . . . . . . . . . . . . . . . . . . . . . . . 463
12.5 TheDutchTopologicalSchool . . . . . . . . . . . . . . . . . . . 468
13 FromBerlintoVienna . . . . . . . . . . . . . . . . . . . . . . . . . . 491
13.1 MoreIntuitionism . . . . . . . . . . . . . . . . . . . . . . . . . . 491
13.2 FeelingsofCrisisandGermanScience . . . . . . . . . . . . . . . 493
13.3 TheBerlinLectures . . . . . . . . . . . . . . . . . . . . . . . . . 497
13.4 TheViennaLectures . . . . . . . . . . . . . . . . . . . . . . . . . 514
13.5 OtherActivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
14 TheThreeBattles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
14.1 TheGrundlagenstreit . . . . . . . . . . . . . . . . . . . . . . . . 527
14.2 TheBolognaConference . . . . . . . . . . . . . . . . . . . . . . 541
14.3 TheWaroftheFrogsandtheMice . . . . . . . . . . . . . . . . . 552
14.4 TheEndingsoftheGrundlagenstreit . . . . . . . . . . . . . . . . 588
14.5 TheMengerConflict . . . . . . . . . . . . . . . . . . . . . . . . . 595
15 TheThirties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
15.1 FreudenthalArrives . . . . . . . . . . . . . . . . . . . . . . . . . 604
15.2 IntuitionisticLogic . . . . . . . . . . . . . . . . . . . . . . . . . . 607
15.3 TheSodalitasAffair . . . . . . . . . . . . . . . . . . . . . . . . . 608
15.4 GöttingenUndertheNazi’s . . . . . . . . . . . . . . . . . . . . . 619
15.5 Bieberbach’sConversion . . . . . . . . . . . . . . . . . . . . . . 622
15.6 CompositioMathematica . . . . . . . . . . . . . . . . . . . . . . 630
15.7 GöttingenReconsidered? . . . . . . . . . . . . . . . . . . . . . . 636
15.8 DutchAffairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
16 WarandOccupation . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
16.1 OccupiedHolland . . . . . . . . . . . . . . . . . . . . . . . . . . 663
16.2 Weitzenböck’sChoice . . . . . . . . . . . . . . . . . . . . . . . . 666
16.3 FreudenthalDismissed . . . . . . . . . . . . . . . . . . . . . . . . 667
16.4 University—ResistanceorSurvival . . . . . . . . . . . . . . . . . 671
16.5 Freudenthal’sFortunes . . . . . . . . . . . . . . . . . . . . . . . . 675
16.6 TheDeclarationofLoyalty . . . . . . . . . . . . . . . . . . . . . 683
16.7 TheBrouwerFamilyinWartime . . . . . . . . . . . . . . . . . . 694
16.8 WeitzenböckinUniform . . . . . . . . . . . . . . . . . . . . . . . 698