Gnomes in the Fog

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet. The foundational debate is presented with all its brilliant contributions and its shortcomings, its new ideas and its misunderstandings.

Foreword

xiii

Introduction

xv

0.1 Introduction

xv

0.2 Audience, academic discipline and personal background

xiii

0.3 Methodological remarks

xix

0.4 Sources, structure and presentation

xx

0.5 Suggestions for further research

xxii

1 Kronecker, the semi-intuitionists, Poincare

1

(24)

1.1 Introduction

1

(3)

1.1.1 Mathematical prerequisites

2

(2)

1.2 Kronecker

4

(4)

1.2.1 Kronecker's conflicts

5

(1)

1.2.2 Kronecker's views

6

(2)

1.3 The French semi-intuitionists

8

(10)

1.3.1 The French semi-intuitionists' main conflict

9

(3)

1.3.2 The French semi-intuitionists' views

12

(6)

1.4 Poincare

18

(5)

1.4.1 Poincare's conflicts

18

(2)

1.4.2 Poincare's views

20

(3)

1.5 Conclusion

23

(2)

2 The genesis of Brouwer's intuitionism

25

(66)

2.1 Introduction

25

(1)

2.2 The early years

26

(8)

2.2.1 Brouwer's youth

26

(1)

2.2.2 Brouwer's profession of faith

26

(2)

2.2.3 Mannoury

28

(2)

2.2.4 Brouwer's mysticism

30

(4)

2.3 The first act of intuitionism

34

(14)

2.3.1 Brouwer's dissertation

35

(11)

2.3.2 The unreliability of the logical principles

46

(2)

2.4 Topology

48

(4)

2.5 Intuitionism and formalism

52

(6)

2.6 The second act of intuitionism

58

(17)

2.6.1 Intuitionistic set theory

60

(7)

2.6.2 Further development of intuitionistic mathematics

67

(8)

2.7 The Brouwer lectures

75

(6)

2.7.1 Berlin

75

(2)

2.7.2 Amsterdam

77

(2)

2.7.3 Vienna

79

(2)

2.8 The Mathematische Annalen and afterwards

81

(5)

2.9 Brouwer's personality

86

(3)

2.10 Conclusion

89

(2)

3 Overview of the foundational debate

91

(26)

3.1 Introduction

91

(2)

3.2 Quantitative inquiry

93

(4)

3.2.1 The Fortschritte

93

(2)

3.2.2 'All' public reactions to intuitionism

95

(2)

3.3 Qualitative inquiry

97

(18)

3.3.1 Themes

97

(2)

3.3.2 Tone

99

(5)

3.3.3 Currents and schools

104

(7)

3.3.4 People

111

(2)

3.3.5 Languages and media

113

(2)

3.4 Conclusion

115

(2)

4 Reactions: existence and constructivity

117

(100)

4.1 Introduction

117

(7)

4.1.1 Mathematical existence

117

(3)

4.1.2 A short history of constructivism

120

(4)

4.2 The beginning of the debate

124

(30)

4.2.1 Weyl's Graruilagenkrise

125

(8)

4.2.2 Hilbert's first reactions

133

(12)

4.2.3 Becker's phenomenology

145

(4)

4.2.4 Fraenkel's early commentaries

149

(3)

4.2.5 Baldus' rector's address

152

(2)

4.3 The debate widened

154

(29)

4.3.1 Existence in a central position

158

(17)

4.3.2 Existence as a minor subject

175

(8)

4.4 Later reactions

183

(29)

4.4.1 The Konigsberg conference

183

(7)

4.4.2 Wittgenstein

190

(8)

4.4.3 Others

198

(14)

4.5 Conclusion

212

(5)

5 Reactions: logic and the excluded middle

217

(84)

5.1 Introduction

217

(3)

5.1.1 A short history of classical logic

218

(2)

5.2 The beginning of the debate

220

(11)

5.2.1 Weyl's Grundlagenkrise

222

(3)

5.2.2 Hilbert's first reactions

225

(4)

5.2.3 Addresses: Wolff, Finsler and Baldus

229

(2)

5.2.4 Fraenkel's early commentaries

231

(1)

5.3 The debate widened

231

(40)

5.3.1 The excluded middle in a central position

236

(32)

5.3.2 The excluded middle as a minor subject

268

(3)

5.4 Later reactions

271

(25)

5.4.1 Glivenko, Heyting and Kolmogorov

271

(10)

5.4.2 Godel

281

(8)

5.4.3 Barzin and Errera

289

(7)

5.5 Conclusion

296

(5)

6 The foundational crisis in its context

301

(44)

6.1 Introduction

301

(1)

6.2 Metaphors

302

(9)

6.2.1 Crisis and revolution

302

(9)

6.3 Philosophy

311

(15)

6.3.1 Lebensphilosophie

312

(4)

6.3.2 Mathematical and philosophical intuitionism: a comparison

316

(4)

6.3.3 Contemporaries' remarks

320

(2)

6.3.4 Gottingen and Hilbert

322

(1)

6.3.5 Spengler

323

(2)

6.3.6 Summary

325

(1)

6.4 Physics

326

(7)

6.4.1 Theory of relativity

327

(2)

6.4.2 Quantum mechanics

329

(4)

6.5 Art

333

(2)

6.5.1 Constructivism

334

(1)

6.6 Politics

335

(5)

6.6.1 Mathematics and the rise of the Third Reich

335

(3)

6.6.2 Bieberbach's racial interpretation of the foundational debate

338

(2)

6.7 Moderne and Gegenmoderne

340

(2)

6.8 Conclusion

342

(3)

Conclusion

345

(28)

A Chronology of the debate

355

(8)

B Public reactions to Brouwer's intuitionism

363

(8)

C Logical notations

371

(2)

Glossary

373

(4)

Bibliography

377

(54)

Index

431

(14)

Dankwoord/ Acknowledgements

445

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Gnomes in the Fog

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