Logic's lost genius - The life of Gerhard Gentzen

465 Pages·2007·9.251 MB·English

by  Menzler-Trott, Eckart (Author), Craig Smoryński(Translator), Edward Griffor(Translator)

HISTORY OF MATHEMATICS • VOLUME 33
Logic’s
Lost Genius
The Life of
Gerhard Gentzen
Eckart Menzler-Trott
American Mathematical Society • London Mathematical Society
Logic’s
Lost Genius
The Life of
Gerhard Gentzen

Editorial Board
American Mathematical Society London Mathematical Society
Joseph W. Dauben Alex D. D. Craik
Peter Duren Jeremy J. Gray
Karen Parshall,Chair Robin Wilson, Chair
Michael I. Rosen
This work was originallypublished in German by Birkha¨user Verlag under the title
Gentzens Problem (cid:2)c 2001. The present translationwas created under license for the
American Mathematical Society and is published by permission.
Photo of Gentzen in hat courtesy of the Archives of the Mathematisches
Forschungsinstitut Oberwolfach. Reprinted withpermission.
2010 Mathematics Subject Classification. Primary 01A60.
For additional informationand updates on this book, visit
www.ams.org/bookpages/hmath-33
Library of Congress Cataloging-in-Publication Data
Menzler-Trott,Eckart.
[GentzensProblem. English]
Logic’slostgenius: thelifeofGerhardGentzen/EckartMenzler-Trott;translatedbyCraig
Smoryn´skiandEdwardGriffor.
p.cm. (Historyofmathematics;v.33)
ISBN978-0-8218-3550-0(alk. paper)
1. Gentzen, Gerhard. 2. Mathematicians—Germany—Biography. 3. Logic, Symbolic and
mathematical. I.Title.
QA29.G467M46 2007
510.92—dc22 2007060550
AMS softcover ISBN: 978-1-4704-2812-9
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2)
service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink.
Sendreq[e-mail].
Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases,
requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).
Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe
firstpageofeacharticlewithinproceedingsvolumes.
(cid:2)c 2007bytheAmericanMathematicalSociety. Allrightsreserved.
ReprintedbytheAmericanMathematicalSociety,2016.
PrintedintheUnitedStatesofAmerica.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
TheLondonMathematicalSocietyisincorporatedunderRoyalCharter
andisregisteredwiththeCharityCommissioners.
VisittheAMShomepageathttp://www.ams.org/
10987654321 212019181716
Asa24thprobleminmyParistalkIwantedtoposetheproblem: criteria
forthesimplicityofproof,or,toshowthatcertainproofsaresimplerthan
anyothers. Ingeneral,todevelopatheoryofproofmethodsinmathemat-
ics. Under a given set of conditions there can be but one simplest proof.
Quitegenerally,iftherearetwoproofsforatheorem,youmustkeepgoing
until you have derived each from the other, or until it becomes quite evi-
dent what variant conditions (and aids) have been used in the two proofs.
Giventworoutes,itisnotrighttotakeeitherofthesetwoortolookfora
third;itisnecessarytoinvestigatethearealyingbetweenthetworoutes...
—David Hilbert (Hilbert’s notebook, Cod. Ms. D. Hil=600.3)
Itisreallyastonishinghowmanyhighlygiftedlogiciansdiedyoung(Nicod,
Herbrand, Gentzen, Spector, Ramsey, etc.).
—Kurt Go¨del to Paul Bernays
Next I would like to call your attention to a frequently neglected point,
namely the fact that Hilbert’s scheme for the foundations of mathematics
remains highly interesting and important in spite of my negative results.
What has been proved is only the specific epistemological objective which
Hilbert had in mind cannot be obtained. This objective was to prove
the consistency of the axioms of classical mathematics on the basis of ev-
idence just as concrete and immediately convincing as elementary arith-
metic. However, viewing the situation from a purely mathematical point
of view, consistency proofs on the basis of suitably chosen stronger meta-
mathematicalpresuppositions(ashavebeengivenbyGentzenandothers)
are just as interesting, and they lead to highly important insights into the
proof theoretic structure of mathematics.
—Kurt Go¨del to Constance Reid

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 Go¨ttingen 22
3. Is G¨ottingen 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 Go¨ttingen: Saunders MacLane and Gentzen as the “Type of
a Scientifically Oriented Man” (Richard Courant) 27
7. The Decision: Gentzen’s First Publication, “U¨ber die Existenz
unabha¨ngiger Axiomensysteme zu unendlichen Satzsystemen” and
His Programme for 1932 30
vii
viii 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 Go¨ttingen and What Did He Read? 37
12. Gentzen’s Life in the Early Nazi Period. The Withdrawn Manuscript
“U¨ber das Verha¨ltnis zwischen intuitionistischer und klassischer
Arithmetik” of 15 March 1933 38
13. Gerhard Gentzen’s Dissertation “Untersuchungen u¨ber das logische
Schließen” of 12 July 1933 41
14. The Penetration of the Nazis into Mathematical Research at the
University in Go¨ttingen 1933 and 1934—Or, Vahlen and Bieberbach
vs. Weber and Wegner 46
15. TheStateExaminationwith“Elektronenbahneninaxialsymmetrischen
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 gegenw¨artige Lage in der mathematischen
Grundlagenforschung”. The Extension of the Tenure of the
Unscheduled Assistantship on 1 October 1937 for a Year 77
32. Jean Cavaill`es 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 gegenw¨artige Lage in der mathematischen Grundlagenforschung” 86
35. “Neue Fassung des Widerspruchsfreiheitsbeweises der reinen
Zahlentheorie” 1938 88
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 Anfangsfa¨llen der transfiniten 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: Dingler, Steck, and
May 169
15. The Attacked: Heinrich Scholz (1884-1956) 176
16. Bieberbach, Max Steck and Jænsch 186
17. Bieberbach and Erich Jænsch 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 Dingler 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. Smoryn´ski 283