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Friedrich Riesz' Beiträge Zur Herausbildung Des Modernen

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Friedrich Riesz' Beiträge Zur Herausbildung Des Modernen Friedrich Riesz’ Beitr¨age zur Herausbildung des modernen mathematischen Konzepts abstrakter R¨aume Synthesen intellektueller Kulturen in Ungarn, Frankreich und Deutschland Dissertation zur Erlangung des Grades “Doktor der Naturwissenschaften” am Fachbereich 08: Physik, Mathematik und Informatik der Johannes Gutenberg-Universit¨at in Mainz Laura Regina Rodr´ıguezHern´andez geboren in Tulancingo, Hidalgo (Mexiko) Mainz, den 10. Juni 2006 Zusammenfassung Der ungarische Mathematiker Friedrich Riesz forschte zwischen 1899 und 1908 in den mathematischen Milieus von Budapest, G¨ottingen und Paris. Diese Tatsa- che motiviert die Suche nach Elementen aus den besagten mathematischen Kul- turen, die zur Herausbildung eines abstrakten Raumbegriffes in dem von Riesz 1906 ver¨offentlichten Text Die Genesis des Raumbegriffs“ beitrugen. ” Sowohl fur¨ seine Fragestellungen als auch fur¨ seinen methodischen Zugang fand Riesz in Frankreich und in G¨ottingen Anregungen: Henri Poincar´esBeitr¨age zur Raumdiskussion, Maurice Fr´echets Ans¨atze einer abstrakte Punktmengen- lehre, David Hilberts Charakterisierung der Stetigkeit des geometrischen Raum- es. Diese Impulse aufgreifend wollte Riesz ein Konzept schaffen, das die Forde- rungen von Poincar´e,Hilbert und Fr´echet in entsprechendem Maße erfulle.¨ So schlug Riesz einen allgemeineren Begriff des mathematischen Kontinuums vor, mit dem sich Fr´echets Konzept der L-Klasse, Hilberts Mannigfaltigkeitsbegriff und Poincar´eserfahrungsgem¨aßes Konzept der Stetigkeit des wirklichen‘ Raum- ’ es auf einen gemeinsamen Nenner bringen ließen. Fur¨ die Durchfuhrung¨ eines solchen Projekts wandte Riesz mengentheoretische und axiomatische Methoden an, die er der Analysis in Frankreich und der Geometrie bei Hilbert entnommen hatte. Riesz’ aufnahmebereite Haltung spielte dabei eine wichtige Rolle. Diese Haltung kann wiederum als ein Element der ungarischen mathematischen Kultur gedeutet werden, welche sich damals ihrerseits stark an den Entwicklungen in Frankreich und Deutschland orientierte. Daruber¨ hinaus enth¨alt Riesz’ Arbeit Ans¨atze einer konstruktiven Mengenleh- re, die auf Ren´eBaire zuruckzuf¨ uhren¨ sind. Aus diesen unerwarteten Ergebnissen ergibt sich die Aufgabe, den Bezug von Riesz’ und Baires Ideen zur sp¨ateren in- tuitionistischen Mengenlehre von L.E.J. Brouwer und Hermann Weyl weiter zu erforschen. Inhaltsverzeichnis Einleitung 1 Teil I 7 1 Sozial- und wissenschaftshistorischer Kontext 7 1.1 Zur Biographie von Friedrich Riesz ................. 7 1.2 Ungarn, Deutschland und Frankreich um 1900 ........... 14 1.3 Zur Raumdiskussion um 1900 .................... 18 1.4 Zu Mengenlehre und Axiomatik um 1900 .............. 19 2 Zur ungarischen mathematischen Kultur 23 2.1 Institutionen, Forschung und Lehre ................. 23 2.1.1 Ausbau des Bildungssystems und die Grundung¨ der Unga- rischen Akademie der Wissenschaften ............ 23 2.1.2 Nationale und internationale Forschungsinteressen ..... 27 2.1.3 Die F¨orderung des Nachwuches ............... 32 2.2 Mengenlehre, Axiomatik, Geometrie, Analysis ........... 34 2.2.1 Mengenlehre .......................... 34 2.2.2 Axiomatik ........................... 35 2.2.3 Geometrie ........................... 36 2.2.4 Analysis ............................ 38 2.3 Wissenschaftliches Milieu ....................... 39 2.3.1 Gyula (Julius) K˝onig ..................... 41 2.3.2 J´ozsef(Josef) Kursch´ak¨ ................... 43 2.3.3 Lip´ot(Leopold) Fej´er ..................... 44 2.3.4 Alfr´edHaar .......................... 46 2.3.5 Marcel Riesz .......................... 49 2.3.6 D´enesK˝onig .......................... 50 3 3 Zur franz¨osischen mathematischen Kultur 51 3.1 Institutionen, Forschung und Lehre ................. 51 3.1.1 Universit¨at versus Grandes Ecoles‘´ ............. 51 ’ 3.1.2 Analysis: eine franz¨osische mathematische Tradition .... 54 3.1.3 Lehre und Lehrbucher¨ .................... 56 3.2 Axiomatik, Mengenlehre und moderne Analysis .......... 57 3.2.1 Rezeption der Cantorschen Mengenlehre .......... 57 3.2.2 Axiomatik und deskriptive Definitionen‘ .......... 61 ’ 3.2.3 Borels Maßtheorie ....................... 67 3.2.4 Baires moderne Theorie der reellen Funktionen ...... 75 3.2.5 Lebesgues Integrationstheorie ................ 81 3.2.6 Fr´echets Calcul Fonctionnel‘ ................. 85 ’ 3.3 Poincar´e uber¨ die Stetigkeit des Raumes .............. 93 3.3.1 Poincar´esPhilosophie der Geometrie ............ 94 3.3.2 Das physikalische Kontinuum‘ bei Poincar´e ........ 99 ’ 4 Zur mathematischen Kultur in G¨ottingen 111 4.1 Institutionen, Forschung und Lehre ................. 111 4.1.1 G¨ottingen als wissenschaftliches Zentrum .......... 111 4.1.2 Forschungsschwerpunkte ................... 113 4.1.3 Dozenten und Studenten ................... 116 4.2 Axiomatik, Mengenlehre und Grundlagen der Geometrie ..... 119 4.2.1 Mengenlehre .......................... 119 4.2.2 Axiomatik bei Hilbert (1894-1905) ............. 124 4.2.3 Axiomatik und Mengenlehre: Ausblick ........... 135 4.3 Hilbert uber¨ die Stetigkeit des Raumes ............... 137 Teil II 145 5 Die Genesis des Raumbegriffs“ 145 ” 5.1 Probleme ............................... 147 5.2 Werkzeuge ............................... 151 5.2.1 Theorie des physikalischen Kontinuums ........... 152 5.2.2 Theorie des mathematischen Kontinuums .......... 154 5.2.3 Theorie der mehrfachen Ordnungstypen .......... 162 5.3 Strategie ................................ 163 6 Das Problem der Stetigkeit des Raumes 169 6.1 Anschluß an Hilberts topologischen Zugang zu den Grundlagen der Geometrie ............................... 169 6.2 Anschluß an Poincar´esIdeen uber¨ die Stetigkeit des Raumes ... 178 6.2.1 Anschluß an Poincar´espsychologische Ansichten zu Zeit und Raum I .......................... 180 6.2.2 Zu einem abstrakten Begriff des physikalischen Kontinuums .......................... 181 6.2.3 Anschluß an Poincar´espsychologische Ansichten zu Zeit und Raum II ......................... 183 6.2.4 Zu Poincar´esProblem der Stetigkeit des Raumes ..... 184 7 Ans¨atze einer abstrakten Punktmengenlehre 189 7.1 Elemente aus Baires Punktmengenlehre fur¨ Folgen ganzer Zahlen 189 7.1.1 Gruppen ganzer Zahlen und physikalische Punkte n-ter Ord- nung .............................. 190 7.1.2 System spezieller Umgebungen ................ 197 7.2 Elemente aus Fr´echets Theorie der L-Klassen ............ 201 8 Axiomatik bei Riesz 207 8.1 Anschluß an axiomatische Praxis in Frankreich ........... 208 8.2 Anschluß an Axiomatik Hilbertscher Pr¨agung ........... 213 9 Rezeption 219 9.1 In Europa ............................... 221 9.1.1 Maurice Fr´echet ........................ 221 9.1.2 Arthur Schoenflies ...................... 228 9.1.3 Leopold Vietoris ....................... 230 9.1.4 Heinrich Tietze ........................ 232 9.1.5 L.E.J. Brouwer und Hermann Weyl ............. 232 9.2 In den USA .............................. 237 9.2.1 Mathematiker um Eliakim Hasting Moore ......... 237 9.2.2 Ralph Eugene Root ...................... 239 9.2.3 Edward Wilson Chittenden ................. 240 9.2.4 Robert Lee Moore ...................... 241 9.2.5 Theophil Henry Hildebrandt ................. 242 9.3 Schlußbemerkungen zur Rezeption .................. 243 10 Schlußbemerkungen zur Bedeutung der Rieszschen Synthesen 247 10.1 Zusammenfassung ........................... 247 10.2 Zur Bedeutung der Rieszschen Synthesen .............. 249 10.3 Schlußwort ............................... 252 A Zitate 253 A.1 Aus Die Genesis des Raumbegriffes“ ................ 253 ” B Tabellen 257 B.1 Tabellen zu Kapitel 2 ......................... 257 B.2 Tabellen zu Kapitel 4 ......................... 260 C Archivalien 265 C.1 Briefwechsel zwischen Riesz und Hilbert .............. 267 C.2 Briefwechsel zwischen Riesz und Fr´echet .............. 270 Literaturverzeichnis 273 Verzeichnis der Archivalien 290 Abbildungsverzeichnis 7.1 Baumdiagramm fur¨ die Bairesche Struktur in Riesz’ System . 196 9.1 Brouwers ¨ortlich verschiedene‘ Punkte . 233 ’ Tabellenverzeichnis 7.1 Bairesche und Rieszsche Begriffe im Vergleich . 190 B.1 Geometrie, Mengenlehre und Analysis in Ungarn . 257 B.2 Professoren an den Universit¨aten in Budapest und Koloszv´ar. 258 B.3 Auslandsaufenthalte ungarischer Mathematiker . 259 B.4 Hilberts Assistenten . 260 B.5 Hilberts Lehrveranstaltungen . 260 B.6 Minkowskis Lehrveranstaltungen . 263 Einleitung Ihrem Titel nach verspricht diese Dissertation, Beitr¨age zur Herausbildung des modernen mathematischen Konzepts abstrakter R¨aume in den Arbeiten des un- garischen Mathematikers Friedrich Riesz aus der Zeit zwischen 1906 und 1908 aufzuzeigen. Die gew¨ahlte Perspektive stutzt¨ sich auf das wissenschaftshistori- sche Konzept lokaler mathematischer Kulturen. So besagt die zentrale These, daß Riesz’ abstrakte Raumbegriffe als Synthesen lokaler mathematischer Kultu- ren in Ungarn, Deutschland und Frankreich entstand. Zun¨achst sollen die eben benutzten Begriffe gekl¨art werden. So gilt festzuhalten: Das Konzept des mathematischen Raumes als solches gibt es nicht. Stattdessen sind dem modernen mathematischen Konzept abstrak- ter R¨aume verschiedene Raumbegriffe in der modernen Mathematik untergeord- net: Vektorr¨aume, topologische R¨aume, unendlich dimensionale R¨aume, normier- te R¨aume, metrische R¨aume, Banach- und Hilbert-R¨aume und so weiter. Als konkrete Beispiele fur¨ diese abstrakten Raumbegriffe k¨onnen unter anderem Fol- genr¨aume und Funktionenr¨aume angegeben werden. Die abstrakten Raumbegrif- fe der modernen Mathematik beschreiben jeweils eine abstrakte
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