Dutch Mathematician Luitzen Egbertus Jan Brouwer (1881-1966)
was a rebel. His doctoral thesis... was the manifesto of an angry young man taking on the mathematical establishment on all fronts. In a short time he established a world-wide reputation for himself; his genius and originality were acknowledged by the great mathematicians of his time... The Intuitionist-Formalist debate became a personal feud between the mathematical giants Brouwer and Hilbert, and ended in 1928 with the expulsion of Brouwer from the editorial board of the Mathematische Annalen by dictat of Hilbert. Forsaken, humiliated and disillusioned Brouwer abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme appeared to be fundamentally flawed and major opposition collapsed...
This book attempts to follow the `genetic' development of Brouwer's ideas, linking the man Brouwer, his Weltanschauung, his philosophy of mathematics and his reconstruction of mathematics. Brouwer's own writings, his publications as well as his unpublished papers, are its immediate and main source of reference.
It is the second volume in the new series Studies in the History and Philosophy of Mathematics, and is written for the specialist as well as for the general reader interested in mathematics and the interpretation of its status and function.
Author(s): W.P. van Stigt
Series: Studies in the History and Philosophy of Mathematics 2
Publisher: North Holland
Year: 1990
Language: English
Commentary: Scanned, DjVu'ed, OCR'ed, TOC by Envoy
Pages: 557
Tags: Intuitionistic logic continuum principle of excluded middle
Cover ......Page 1
Preface ......Page 7
Acknowledgements ......Page 17
References ......Page 19
Contents ......Page 21
Chapter I. The Brouwer Bibliography ......Page 27
1.1 Publications ......Page 28
1.2 Unpublished Papers ......Page 42
2.1 Childhood and Student-years ......Page 47
2.2 Post-graduate Study ......Page 54
2.3 Life, Art and Mysticism [1905] ......Page 57
2.4 The Foundations of Mathematics ......Page 61
2.4.1 Chapter One - The Construction of Mathematics ......Page 63
2.4.2 Chapter Two - Mathematics and Experience ......Page 65
2.4.3 Korteweg’s Rejection ......Page 66
2.4.4 Chapter Three - Mathematics and Logic ......Page 69
2.5 Towards an Academic Career ......Page 71
2.5.1 The Unreliability of the Logical Principles [1908C] ......Page 72
2.5.2 The Rome Conference 1908 ......Page 73
2.5.4 The Nature of Geometry [1909A] ......Page 75
2.5.5 Zur Analysis Situs ......Page 77
2.5.6 The New Methods and the Invariance of Dimension ......Page 79
2.5.7 International Recognition ......Page 81
2.5.8 Professor L.E.J. Brouwer, 14 October 1912 ......Page 83
2.6 Intuitionism and Formalism ......Page 87
2.6.1 Intuitionist Set Theory ......Page 88
2.7 The Signific Interlude ......Page 91
2.8.1 The Foundations of Set Theory. Part I ......Page 97
2.8.3 The New Foundation Crisis ......Page 101
2.9.1 Signific and Political Campaigns ......Page 103
2.9.2 Göttingen and Berlin Calls ......Page 106
2.9.3 The Intuitionist ‘Putsch’ ......Page 108
2.10.1 The Principle of the Excluded Middle - 1923 ......Page 111
2.10.2 The Calculus of Absurdities ......Page 113
2.10.3 La Logique Brouwerienne ......Page 114
2.11 The Reconstruction of Analysis ......Page 116
2.11.1 The Continuity Arguments ......Page 117
2.12.1 Reelle Funktionen ......Page 119
2.12.2 Die Berliner Gastvorlesungen ......Page 120
2.13 The Brouwer-Hilbert Controversy and the Annalen Affair ......Page 125
2.14 The Silent Years ......Page 129
2.15 Post-war Lectures and Final Years ......Page 133
3.1 Introduction ......Page 137
3.2 Character and Early Background ......Page 138
3.3 Philosophy, Metaphysics and Ideology ......Page 141
3.4 Mysticism ......Page 145
3.5 God and the Moral Principle ......Page 148
3.6 Kant and the French Intuitionist Tradition ......Page 153
3.7 French Mathematical Intuitionism ......Page 158
3.8 The Self, Consciousness and Mind ......Page 161
3.9 Causal Attention, Will and Cunning Activity ......Page 167
3.10 The Phase of Social Acting and Language ......Page 170
4.1.1 Introduction ......Page 173
4.1.2 Genesis of the Concept and Definitions ......Page 174
4.1.3 Brouwer’s Primordial Intuition of Mathematics ......Page 176
4.1.4 The Discrete and the Continuous in the Primordial Intuition ......Page 179
4.2 Intuitive Thinking and Mathematics ......Page 183
4.3 The Nature of Mathematics ......Page 185
4.4 Reflection on the Nature of Mathematics ......Page 188
4.5.1 Brouwer Construction ......Page 190
4.5.2 Classification of Constructions ......Page 191
4.5.3 Distinctive Features of Mathematical Constructive Thinking ......Page 194
4.6.1 Brouwer’s Idealism ......Page 197
4.6.2 The Idealized Subject ......Page 198
4.6.3 Mathematics, the Life of the Idealized Free Subject ......Page 200
4.6.4 Objectivity and Inter-subjectivity ......Page 206
4.7 The Application of mathematics ......Page 208
4.7.1 Brouwer’s Critique of Scientific Practice ......Page 210
4.7.2 The Ideal Nature of Science ......Page 211
4.7.3 The Process of Application and its Mathematical Truth ......Page 214
5.1 Society and Communication ......Page 219
5.2 The Purpose and Origin of Language ......Page 222
5.3 Language and Pure Thought ......Page 225
5.4 The Programme of Reform ......Page 227
5.5.1 The Privacy of Language ......Page 229
5.5.2 The Instability of Language ......Page 230
5.5.3 Words, Sentences and Truth ......Page 237
5.6 Brouwer’s Analysis of Mathematical and Linguistic Activity ......Page 239
5.6.1 The Language of Mathematics ......Page 241
5.6.2 The Language of Logical Reasoning ......Page 246
5.7.1 Logic, the Science of Reasoning ......Page 250
5.7.2 Pure Logic as Science and Abstraction ......Page 251
5.7.3 The Application of Logical Principles ......Page 256
5.8 Brouwer Negation ......Page 264
5.8.1 Brouwer’s Constructive Notion of Negation ......Page 265
5.8.2 Absurdity as Impossibility of ‘Fitting-in’ ......Page 266
5.8.3 Constructed Impossibility ......Page 268
5.8.4 Mathematical Absurdity and Other Notions of Negation ......Page 270
5.9 The Refutation of the PEM ......Page 273
5.9.1 The Unreliability of the PEM, 1908 ......Page 274
5.9.2 Mathematics without the PEM, 1917 ......Page 276
5.9.3 Brouwer’s Counterexamples ......Page 278
5.9.4 The Contradictority of the PEM ......Page 281
5.10.1 The Calculus of Absurdities ......Page 283
5.10.2 Logical Necessity and Truth ......Page 286
5.10.3 Logical Absurdity ......Page 288
5.10.4 Non-contradictority and Absurdity-of-absurdity ......Page 291
5.11 The Crisis in Brouwer’s Intuitionism ......Page 296
5.11.1 The Use of Language and Logic in Brouwer’s Reconstruction of Mathematics ......Page 297
5.11.2 The Pragmatism of Post-Brouwer Intuitionism ......Page 300
5.12.1 The ‘Formalist’ Schools and Formal Axiomatic Theory ......Page 305
5.12.2 The Formalization of Intuitionist Mathematics ......Page 311
5.12.3 Heyting’s Formalization of Intuitionist Logic ......Page 314
Selected Contributions to the Formalization of Intuitionist Logic and Mathematics ......Page 319
6.1 Introduction ......Page 321
6.2 Brouwerian Separable Mathematics ......Page 326
6.2.1 The Primordial Intuition as Set-construction ......Page 327
6.2.2 The Fundamental Sequence ......Page 331
6.2.2.1 The Order-type n ......Page 332
6.2.2.3 Brouwer’s Generalization of the Concept Sequence ......Page 335
6.2.3 Cardinality ......Page 338
6.2.4.1 The Operations on Ordinal Numbers ......Page 340
6.2.4.2 The Integers ......Page 341
6.2.4.3 Rationals and the Order-type q ......Page 342
6.3.1 Introduction ......Page 344
6.3.2 The Genesis and Nature of the Intuitive Continuum ......Page 349
6.3.2.1 The Construction of Intervals ......Page 351
6.3.2.2 Relations and Ordering of Intervals ......Page 352
6.3.3.1 The ‘Made-measurable Continuum’ ......Page 354
6.3.3.2 Point on the Continuum ......Page 356
6.3.3.3 Sets of Constructed Points ......Page 358
6.3.3.4 Denumerably-unfinished Systems and Definable Points ......Page 359
6.3.4 The New Set-theory 1917 ......Page 360
6.3.5 Mathematical Species ......Page 361
6.3.6 The Genesis of Species ......Page 363
6.3.7 The Hierarchy of Species ......Page 366
6.3.8 The Theory of Species ......Page 371
6.3.8.1 Union and Intersection of Species ......Page 372
6.3.8.3 Relations between Species ......Page 373
6.3.8.4 Difference, Deviation and Apartness ......Page 374
6.3.8.5 Congruence and Agreement ......Page 377
6.3.9 The Complementarity of Species ......Page 378
6.3.10 Spreads and Choice-sequences ......Page 383
6.3.11 Real Elements of the Continuum ......Page 387
6.3.11.2 Real Point or Real Number ......Page 388
6.3.11.3 Law and Choice ......Page 389
6.3.11.4 Species of Coincident Points or Point-cores ......Page 390
6.3.11.5 Point-set or Point-spread ......Page 392
6.3.11.6 Fundamental Sequences of Intervals and Finite Point-spreads ......Page 395
6.3.12 The General Spread Concept ......Page 396
6.3.13.1 Finite Spreads ......Page 404
6.3.13.2 The Brouwer Continuity Hypothesis or the Fundamental Hypothesis for Real Functions ......Page 405
6.3.13.3 The Fundamental Theorem of Finite Spreads ......Page 407
6.3.13.4 The Uniform-Continuity Theorem ......Page 410
01. Profession of Faith of L.E.J. Brouwer [BMS 1 A] (Dutch original and English translation) ......Page 413
02. Student Notebook [BMS IB] (Dutch original and English translation) ......Page 420
03. The Rejected Parts of Brouwer’s Dissertation [BMS 3B] ......Page 431
04. Intuitive Signifies [BMS 22B] (Dutch original and English translation) ......Page 442
05. Will, Knowledge and Speech (English translation of BR[1933]) ......Page 444
06. Short Retrospective Notes on the Course “Intuitionist Mathematics” [BMS 52] ......Page 458
07. Disengagement of Mathematics from Logic [BMS 49] ......Page 467
08. Notes for a Lecture [BMS 47] (Dutch original and English translation) ......Page 473
09. Changes in the Relation between Classical Logic and Mathematics [BMS 59] ......Page 479
10. ‘Second Lecture’ [BMS 66] ......Page 485
11. Real Functions [BMS 37] ......Page 495
12. Intuitionist Mathematics or Die Berliner Gastvorlesungen [BMS 32] ......Page 507
13. The Brouwer-Korteweg Correspondence ......Page 513
Bibliography ......Page 533
Index ......Page 543
Back cover ......Page 557
