HISTORY OF MATHEMATICS « VOLUME 33
Logics Lost Genius
The Life of Gerhard Gentzen
Eckart Menzler-Trott
Translated by Craig Smoryiiski and Edward Griffor
The London \P^D ®AMS Mathematical Ambrican Mathbmaxical Society Society r\xf Contents
Preface to the English Edition xiii
Introduction xvii 1. Gentzen's Accomplishments xvii 2. Aims of My Life Story of Gerhard Gentzen xviii 3. Mathematical Logic and National Socialism: The Political Field xix
Chapter 1. Early Youth and Abitur 1 1. Gerhard Gentzen's Birth 1 2. Gentzen's Mother: Melanie Gentzen (1873-1968) 1 3. Gentzen's Paternal Grandparents and His Father: The Attorney and Court Official Dr. jur. Hans Gentzen (1870-1919) 2 4. Youth and Student Days of Hans Gentzen 3 5. Gentzen's Maternal Grandparents: The-Physician Alfons Bilharz and Adele Bilharz 4 6. The Shining Example in Gentzen's Family Tree: The Great, Successful Physician and Natural Scientist Maximilian Theodor Bilharz (1823-1862) 9 7. Gentzen's Sister: Waltraut Sophie Margaret Gentzen (*1911) 10 8. The Schoolchild Gerhard Gentzen 11
9. The Death of the Father Means a Move and a New School 12 10. The Beginning of Gentzen's Intellectual Activity 12 11. Gentzen's Success at School 15 12. The Abitur on 29 February 1928 16
Chapter 2. 1928-1938—Weimar Republic and National Socialism in Peace. From the Beginning of Studies to the Extension of the Unscheduled Assistantship for Another Year in Effect from 1 October 1938 21 1. Beginning of Studies in Greifswald 21 2. Continuation of Study in Gottingen 22 3. Is Gottingen the Centre of Mathematics for Gentzen? 23 4. Continuing Studies in Munich 23 5. A Semester in Berlin: Winter Semester 1930/31 26 6. Back in Gottingen: Saunders Mac Lane and Gentzen as the "Type of a Scientifically Oriented Man" (Richard Courant) 27 7. The Decision: Gentzen's First Publication, "Uber die Existenz unabhangiger Axiomensysteme zu unendlichen Satzsystemen" and His Programme for 1932 30 ,i CONTENTS
8. What Do Gentzen's Intellectual Interests and Attitude in 1931 and 1932 Appear to Be? 33 9. Political Language in Mathematics in 1918 34 10. Political Language by Hermann Weyl and David Hilbert 34 11. Whom Did Gentzen Know in Gottingen and What Did He Read? 37 12. Gentzen's Life in the Early Nazi Period. The Withdrawn Manuscript "Uber das Verhaltnis zwischen intuitionistischer und klassischer Arithmetik" of 15 March 1933 38 13. Gerhard Gentzen's Dissertation "Untersuchungen iiber das logische Schliefien" of 12 July 1933 41 14. The Penetration of the Nazis into Mathematical Research at the University in Gottingen 1933 and 1934—Or, Vahlen and Bieberbach vs. Weber and Wegner 46 15. The State Examination with "Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme" on 16 November 1933 51 16. Why Did Gentzen Join the SA? 52 17. Gentzen's Political Position: A Conjecture 53 18. Gentzen in Financial Difficulties 54 19. Financial Straits and Job Hunting 55 20. Consistency Proof for Number Theory in Discussion with Paul Bernays 57 21. "Widerspruchsfreiheit der reinen Zahlentheorie" Mirrored in the Correspondence of Bernays and Weyl 58 22. Difficulties with the "Widerspruchsfreiheit der reinen Zahlentheorie" of 11 August 1935 59 23. The Unscheduled Assistantship under Hilbert from 1 November 1935: A Productive Period for Foundational Research Begins 62 24. Consistency of Type Theory 63 25. Revising the Proof of the "Widerspruchsfreiheit der reinen Zahlentheorie" 64 26. "Die Widerspruchsfreiheit der reinen Zahlentheorie" 65 27. Gentzen Was an Intellectual Independent 69 28. Gentzen Expresses His Thanks to Turing 71 29. Correspondence between Bernays and Ackermann 1936 to 1940 72 30. The Correspondence between Bernays and Gentzen Merrily Continues 75 31. Invitation to the Parisian Descartes Congress in August 1937. The
Invitation to Lecture to the DMV Conference in Bad Kreuznach on 21 September 1937: "Die gegenwartige Lage in der mathematischen Grundlagenforschung". The Extension of the Tenure of the
Unscheduled Assistantship on 1 October 1937 for a Year 77 32. Jean Cavailles and Gerhard Gentzen 82 33. Gentzen Becomes an "Associate" of the Publication of Scholz's Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 84 34. "Die gegenwartige Lage in der mathematischen Grundlagenforschung" 86 35. "Neue Fassung des Widerspruchsfreiheitsbeweises der reinen
Zahlentheorie" 1938 »« CONTENTS ix
36. Bernays' View of Gentzen's Programme (Second Consistency Proof) 89 37. Correspondence with Paul Bernays 93 38. Extension of the Unscheduled Position for Another Year Taking Effect 1 October 1938 97 39. Bernays' Views of Hilbert's Programme and Gentzen's Place in It 100 40. Closing Thoughts on Paul Bernays 101 41. Longer Notes 102
Chapter 3. 1939-1942—From the Beginning of the War to Dismissal from the Wehrmacht and the Wartime Habilitation under Helmut Hasse 117 1. 1939: At the Highpoint of Reputation 117 2. The Second Volume of Grundlagen der Mathematik of Hilbert and Bernays Appears 119 3. Active Military Service at the Homefront as Radio Operator by the Flugwachkommando 125 4. 1939/40: Preparation for Habilitation. "Beweisbarkeit und Unbeweisbarkeit von Anfangsfallen der transfmiten Induktion in der reinen Zahlentheorie" 128 5. 1941: Encouragement from Hellmuth Kneser 135 6. 1942: Discharge from Military Service 139
Chapter 4. The Fight over "German Logic" from 1940 to 1945: A Battle between Amateurs 141 A Short Preface 141
1. Ludwig Bieberbach as Supporter of the Idea That the Validity of Mathematics Be Decided through World Views 142 2. An Advocate of Racial Purity Sets the Standards in German Mathematical Logic, Where Attempts Are Made to Confuse Scientific Results with Matters of Race 143
3. Gentzen as a "Witness" for a National-Racial Interpretation of the Mathematical Foundational Research through Steck and Requard 144 4. The Somewhat Sharper Point of View of Friedrich Requard 147 5. Ludwig Bieberbach and his Deutsche Mathematik 150 6. NS Ideology in Mathematics through Bieberbach Receives Negative Resonance Even within His Own Camp 152 7. Ludwig Bieberbach: Representative of "German Mathematics" 153 8. A Contemporary of Bieberbach's in Exile: Johann L. Schmidt 159 9. Bieberbach and Intuitionism 160 10. Formalism and Proof Theory 160 11. Applied Mathematics as Folkish Mathematics 162 12. Nazis Criticised the Lack of Support from Mathematics 163 13. Mathematical Foundational Research Remains Unmolested by the Nazis 167
14. Attacks on Mathematical Logic from Without: Dingier, Steck, and May 169 15. The Attacked: Heinrich Scholz (1884-1956) 176 16. Bieberbach, Max Steck and Jaensch 186 17. Bieberbach and Erich Jaensch 188 x CONTENTS
18. Steck's Attack on Hilbert Leads to Bieberbach's Commissioning a Defence of Mathematical Logic by H. Scholz and Publishing It in Deutsche Mathematik 190 19. Interlude: May and Dingier Provide Arguments for Steck 197 20. Steck and Scholz in Dispute 202 21. Max Steck as Denouncing "Expert Witness" and Publicist 208 22. The Exception: The Dedicated National Socialist, Logician and Historian of Mathematics, Oskar Becker, Remains Neutral 216
23. Resistance as a Mathematician Was Possible under National Social¬ ism 218
24. Kurt Reidemeister's Additional Contemplations on Politico-Scientific Power Play in "German Mathematics" 219 25. Longer Notes 221
Chapter 5. Recovery and Docent Position 1942 to 1944 233 1. Final Discharge from the German Army 233 2. Hans Rohrbach Commandeers Gerhard Gentzen to Prague through the Osenberg Initiative 234 3. Gentzen's Teaching Position in Prague: "Kepler's Laws of Planetary Motion" 236 4. The First Courses in November 1943 238 5. The Last Known Scientific Letter of Gerhard Gentzen 243 6. Gerhard Gentzen in 1944: Teaching Functions, Computing Office, and War 244
7. Hans Rohrbach's Report on the Conditions in the Mathematical Institute in Prague 246 8. Why Did Gentzen Banish Any Thought of Flight? 247
Chapter 6. Arrest, Imprisonment, Death and Nachlass 253 1. The Last Days of Freedom in the Private Sector 253 2. The Arrest of Gerhard Gentzen and the Awful Imprisonment 255 3. Gentzen's Physical Death 257 4. Is Gentzen's Death Understandable? 260 5. Rumours 261 6. Attempts to Rescue the Nachlass 263 7. The Deciphering of the Stenographic Notes 266
Conclusion 267 1. Misapprehensions about the Life of Gerhard Gentzen 267 2. Logic and Politics 267 3. Upshot 269
Tables of the Life of Gerhard Gentzen 273 Chronology 273 Contemporary Assessments of Gentzen 278 Publications of Gentzen 281
Appendix A. Gentzen and Geometry C. SMORYNSKI 283 CONTENTS xi
Appendix B. Hilbert's Programme C. Smorynski 291 1. Constructive Prologue 291 2. Problems in Paris 293 3. Hilbert and Geometry 296 4. First Steps 300 5. Enter Brouwer 301 6. Back to Hilbert 308 7. Weyl Stirs Things Up 310 8. Hilbert Responds 312
9. More on Brouwer 322 10. Outbreak of Hostilities 323 11. The Formula Game 324 12. On the Infinite 325 13. A Fragile Truce 327 14. "Hilbert's Programme" Is Born 329 15. Brouwer Takes Up Arms 331 16. Hilbert Finishes Off Brouwer 332 17. The Programme Expands 334 18. Godel's Theorem 335 19. Concluding Remarks 339
Appendix C. Three Lectures Gerhard Gentzen 343 1. The Concept of Infinity in Mathematics 343 2. The Concept of Infinity and the Consistency of Mathematics 350 3. The Current Situation in Research in the Foundations of Mathematics 353
Appendix D. From Hilbert's Programme to Gentzen's Programme Jan von Plato 367 1. Mathematical Proof 367 2. Hilbert's Programme 373 3. Gentzen's Programme 383 4. Later Developments in Structural Proof Theory 396 5. References 401
Bibliography 405
Index 433