IDEAS AND EXPLORATIONS Brouwer’s Road to Intuitionism IDEEEN¨ EN VERKENNINGEN; Brouwer’s weg naar het intu¨ıtionisme (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof.Dr. W.H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op 27 februari 2004 des middags te 16.15 uur door Johannes John Carel Kuiper geboren op 4 december 1935, te Arnhem Promotor Prof.Dr. D. van Dalen Faculty of Philosophy Utrecht University The research for this thesis was made possible by a grant of the Netherlands Organization for Scientific Research (NWO). Voor de vijf vrouwen rondom mij: Janny, Ingeborg, Simone, Jessica en Rita. Acknowledgements Life sometimes takes an unexpected and unforseen turn. After retiring at the age of 57 from 35 years of civil aviation (part of that time having filled with studies in theoretical physics at the Leiden University and in numerous hotel rooms), I started a study in philosophy at the Utrecht University, thereby spe- cializing in logic and the foundations of mathematics. The unexpected turn came with a telephone call from Dirk van Dalen, with the invitation to join the ‘L.E.J. Brouwer-group’ immediately after having completed my philosophy study. He thereby offered me the opportunity to write a doctoral thesis on the early work of Brouwer and on the beginning of his road to intuitionism. The result of four years research, reading, writing, crossing out and rewrit- ing again is this dissertation. Writing a doctoral thesis is often a lonely job, especially when the author is constantly threatened by a hostile computer. It will be obvious that I have to express my gratitude in the first place to my thesis advisor Dirk van Dalen for the numerous hours that he spent on my drafts and on the long discussions and explanations of the many topics that are discussed in this dissertation. This all at the cost of the precious time he needed so badly for writing at the same time two biographies about the life and work of Brouwer (one in Dutch and a scientific one in English). It is of course a relief for me that this dissertation is now completed, but undoubtedly it must be a relief for him too (as well as for his wife Dokie, who is almost just as busy with the work on the biographies). In the second place I want to thank Mark van Atten. He was willing to share his thorough and detailed knowledge of the philosophical foundations of mathematics with me by reading through my drafts and manuscripts, pointing to many smaller and larger imperfections; he even was willing to look at the smallest details in punctuation marks and typing errors. I also owe many thanks to Dennis Dieks for several long discussions about Brouwer’s ideas on the character of physical laws and on the nature and aim of physical practice and research. I thank Theo Verbeek and Maarten van Houte, who were so kind to read through, and comment on, what I had to say about Kant, who was of such great influence on Brouwer’s ideas about the ultimate foundations of all of mathemat- ics. Finally, my gratitude goes to the whole staff of the faculty of philosophy. Despite the rather exceptional combination of being the lower in scientific rank v vi ACKNOWLEDGEMENTS and (almost) the senior in age, I have spent wonderful years here. It should, again, be said that the whole project was also made possible by the NWO, the Netherlands Organisation for Scientific Research, in the form of a grant, which is gratefully acknowledged. Ten slotte het ‘thuisfront’: Lieve Janny, we kunnen, na alle drukte van de afgelopen jaren, natuurlijk niet zeggen dat we ons ‘mijn pensioen anders hadden voorgesteld’, want het was allemaal opgenomen in onze planning, hoewel de laatste vier jaren er natuurlijk als een verrassing extra bij kwamen. Maar jij hebt gelukkig je eigen interesse met je studie kunstgeschiedenis, een onderwerp dat mij ook boeit. Jouw interesse voor wat ik doe blijkt uit het feit dat jij met veel genoegen de Brouwer-biografie hebt gelezen en nu dus over een en ander mee kunt praten. En er was gelukkig nog tijd over voor onze kinderen en kleinkinderen en, afgezien dan misschien van het laatste half jaar, voor geregeld museum- en concertbezoek en voor de vakanties samen. Moge dit alles, naast het verdere werk aan het Brouwer project, nog lang voortgang vinden. Contents Acknowledgements v Preface xi 1 Sets and the continuum before 1907 1 1.1 Cantor (1845–1918) . 2 1.1.1 The discovery of non-equivalent infinite sets . 4 1.1.2 On the equivalence of R1 and Rn .............. 5 1.1.3 On linearly ordered point sets . 6 1.1.4 Irrational numbers, the continuum, the continuum hy- pothesis, reducible and perfect sets . 7 1.1.5 Generation principles (Erzeugungsprinzipien) and the Lim- itation principle (Hemmungsprinzip) for finite and trans- finite numbers . 10 1.1.6 The Cantor-Bernstein theorem and the definition of fur- ther new concepts . 12 1.1.7 Beitrage zur Begr¨undung der transfiniten Mengenlehre . 13 1.1.8 The linear continuum . 14 1.1.9 Order types . 15 1.1.10 Fundamental sequences and perfect sets . 17 1.1.11 Well-ordering . 17 1.2 Dedekind (1831–1916) . 19 1.3 Poincar´e(1854–1912) . 23 1.3.1 The first stage in the construction of the mathematical continuum . 24 1.3.2 The second stage . 24 1.3.3 The measurable quantity . 25 1.3.4 A continuum of the third order? . 25 1.3.5 Mathematical existence and complete induction . 26 1.4 Zermelo (1871–1953) . 27 1.5 Schoenflies (1853–1928) . 28 1.6 Bernstein (1878–1956) . 28 1.7 The paradoxes . 30 1.8 Final remarks . 31 vii viii CONTENTS 2 Brouwer’s ur-intuition of mathematics 33 2.1 Introduction . 33 2.2 The dissertation about the ur-intuition . 35 2.3 Mannoury’s opposition . 39 2.4 Barrau’s objections against the ur-intuition . 40 2.5 The ur-intuition in the Rome and Vienna lectures . 42 2.6 Interpretation of the ur-intuition . 43 2.6.1 The construction of natural numbers . 44 2.6.2 The rational numbers . 46 2.7 The ur-intuition in the inaugural lecture . 48 2.8 The ur-intuition in the notebooks . 50 2.9 On the status of spoken or written signs . 56 2.10 The notebooks on spoken or written signs . 60 2.11 Summary and conclusions of this chapter . 61 3 The continuum, intuitive and measurable 63 3.1 Introduction . 63 3.1.1 Aristotle . 64 3.1.2 Georg Cantor . 65 3.1.3 Hermann Weyl . 65 3.1.4 Otto H¨older . 67 3.1.5 Emile´ Borel . 68 3.1.6 A modern sound . 69 3.2 A few other publications by Brouwer . 69 3.3 Properties of the intuitive continuum . 75 3.4 The notebooks and the intuitive continuum . 84 3.4.1 The continuum, its possible construction from logical prin- ciples alone . 85 3.4.2 The continuum, the intuition . 86 3.4.3 The physical continuum . 88 3.4.4 The mathematical versus the intuitive continuum . 89 3.4.5 The mathematical continuum is the measurable contin- uum with rational scale . 90 3.4.6 The continuum is no point set . 91 3.4.7 The vast majority of the irrational numbers can only be approximated and not defined . 92 3.4.8 The continuum; is there a way to get more grip on it, apart from a constructed scale on it? . 95 3.4.9 The second number class does not exist as a finished totality 99 3.4.10 Attempts to introduce the ‘in practice unmeasurable num- bers’..............................101 3.4.11 Pro and contra the pure and intangible continuum . 105 3.5 Summary and conclusions . 111 CONTENTS ix 4 The possible point sets 113 4.1 Introduction . 113 4.2 Set construction . 114 4.3 The principle of the excluded middle . 121 4.4 The third construction rule . 122 4.5 The review of Schoenflies’ Bericht . 126 4.6 The lecture notes 1915/’16 on Set theory . 128 4.7 Addenda and corrigenda to the dissertation . 130 4.8 The ‘Begr¨undung’ papers, 1918/19 . 133 4.9 Intuitionistische Mengenlehre, 1919 . 135 4.10 The notebooks on set formation . 137 4.10.1 Sets, general . 137 4.10.2 Sets, constructibility as condition for their existence . 139 4.10.3 Sets and the Russell-paradox . 142 4.10.4 Sets, limitations resulting from the method of construction 142 4.10.5 The perfect set cannot be constructed . 144 4.10.6 The second number class . 145 4.10.7 A third and a fourth cardinality . 146 4.10.8 Later developments, suggested in the notebooks . 147 4.11 Concluding remarks about sets . 161 5 The ‘continuum problem’ 165 5.1 Introduction . 165 5.2 The solution in the dissertation . 166 5.3 Two other publications on this problem . 170 5.3.1 The Rome lecture . 170 5.3.2 The inaugural address . 171 5.4 The dissertation again . 173 5.5 The notebooks and the continuum problem . 174 5.6 Concluding remarks . 177 6 Mathematics and experience 179 6.1 Introduction . 179 6.2 Man’s desire for knowledge and control . 181 6.2.1 Comments on Brouwer’s views on physics . 186 6.2.2 The notebooks on causality and man’s desire to rule .