“A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Lukas M. Verburgt If the true is what is grounded, then the ground is neither true nor false LUDWIG WITTGENSTEIN Whether all grow black, or all grow bright, or all remain grey, it is grey we need, to begin with, because of what it is, and of what it can do, made of bright and black, able to shed the former , or the latter, and be the latter or the former alone. But perhaps I am the prey, on the subject of grey, in the grey, to delusions SAMUEL BECKETT “A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het College voor Promoties ingestelde commissie in het openbaar te verdedigen in de Agnietenkapel op donderdag 1 oktober 2015, te 10:00 uur door Lukas Mauve Verburgt geboren te Amersfoort Promotiecommissie Promotor: Prof. dr. ir. G.H. de Vries Universiteit van Amsterdam Overige leden: Prof. dr. M. Fisch Universitat Tel Aviv Dr. C.L. Kwa Universiteit van Amsterdam Dr. F. Russo Universiteit van Amsterdam Prof. dr. M.J.B. Stokhof Universiteit van Amsterdam Prof. dr. A. Vogt Humboldt-Universität zu Berlin Faculteit der Geesteswetenschappen © 2015 Lukas M. Verburgt Graphic design Aad van Dommelen (Witvorm) Printing Lenoirschuring Binding Atelier Kloosterman Paper Biotop 3 Next 100 g/m² Cover paper Les Naturals Safrangelb 325 g/m² Typeface dtl Fleischmann isbn 978-90-824198-0-1 table of contents acknowledgments — 8 Introduction — 10 Structure of the book — 23 PART 1 British probability theory, logic and mathematics — 25 section 1 Logicist, idealist and quasi-empiricist probability — 26 CHAPTER 1 The objective and the subjective in mid-nineteenth-century British probability theory — 28 CHAPTER 2 Remarks on the idealist and empiricist interpretation of frequentism: Robert Leslie Ellis versus John Venn — 62 section 2 Robert Leslie Ellis: probability theory and idealism — 80 CHAPTER 3 Robert Leslie Ellis’s work on philosophy of science and the foundations of probability theory — 82 CHAPTER 4 Robert Leslie Ellis, William Whewell and Kant: the role of Rev. H.F.C. Logan — 135 section 3 John Venn: probability theory and induction — 142 CHAPTER 5 John Venn’s hypothetical infinite frequentism and logic — 143 CHAPTER 6 “A modified acceptance of Mr. Mill’s view”: John Venn on the nature of inductive logic and the syllogism — 184 section 4 British symbolic logic and algebra: the limits of abstraction — 215 CHAPTER 7 John Venn on the foundations of symbolic logic: a non-conceptualist Boole — 217 CHAPTER 8 Duncan Farquharson Gregory and Robert Leslie Ellis: second generation reformers of British mathematics — 267 CHAPTER 9 Duncan F. Gregory, William Walton and the development of British algebra: ‘algebraical geometry’, ‘geometrical algebra’, abstraction — 305 6 PART 2 The axiomatization of probability theory and the foundations of modern mathematics — 357 section 1 David Hilbert and Richard von Mises: the axiomatization of probability theory as a natural science — 358 CHAPTER 10 The place of probability in Hilbert’s axiomatization of physics, ca. 1900-1926 — 360 CHAPTER 11 Richard von Mises’s philosophy of probability and mathematics: a historical reconstruction — 414 section 2 Moscow mathematics: formalism, intuitionism and the search for mathematical content — 469 CHAPTER 12 On Aleksandr Iakovlevich Khinchin’s paper ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ — 471 CHAPTER 12 ‘Ideas of intuitionism and the struggle for a subject matter – APPENDIX in contemporary mathematics’ (1926) — 512 english translation, with olga hoppe-kondrikova section 3 Moscow probability theory: toward the Grundbegrife — 527 CHAPTER 13 On Aleksandr Iakovlevich Khinchin’s paper ‘Mises’ theory of probability and the principles of statistical physics’ — 529 CHAPTER 13 ‘Mises’ theory of probability and the principles – APPENDIX of statistical physics’ (1929) — 583 english translation, with olga hoppe-kondrikova Concluding remarks — 605 samenvatting — 613 summary — 618 note on funding and co-translatorship — 623 7 acknowledgments Paul Valéry once wrote that ‘the whole question comes down to this: can the human mind master what the human mind has made?’. This seems to assume that the human mind either makes things which it can master or makes things which it cannot master. I think that there are at least two other possibilities: there are things which the mind can make that destroy its ability to master and there are things which the human mind cannot make because they are destroyed when they are mastered. It is to the future exploration of these possibilities that this book is dedicated. I will always be grateful to Gerard de Vries, whose willingness to mistake my ignorance for the possibility of insight and to allow me to take the risk of thinking as someone who does not know implicitly what it means to think must make him a real master. There are many people whom I have never met, but who have contributed much to this book: the anonymous referees, the editors Tom Archibald, Niccoló Guicciardini, Tony Mann, David Miller and Volker Peckhaus, and Jeremy Gray, Adrian Rice, Reinhard Siegmund-Schultze, and, especially, Tilmann Sauer. The articles on Russian mathematics could not have been written without Jan Von Plato, who sent me the copies of Khinchin’s papers and shared with me many bibliographical details about Khinchin, Kolmogorov and Heyting. I am indebted to Berna Kiliç, David Vere-Jones and Sandy Zabell for providing me with some of their valuable articles and to Stephen Stigler and Menachem Fisch, whose appreciation of two of my articles I consider as a great honor. I am thankful to Lorraine Daston not only for hosting me in Department II at the Max Planck Institute for the History of Science (MPIWG) in Berlin – where I was a Pre-Doctoral Visiting Fellow from September until December 2014 –, but also for critically reading two of the articles. At the MPIWG, Donatella Germanese helped me with the translation of several passages from the work of Richard von Mises, Elena Aronova was so kind to bring me into contact with some of her colleagues in Moscow, Ellen Garske managed to obtain several documents from the Hilbert Nachlass in Göttingen, and Regina 8 Held assisted me with the many practicalities of my stay. I would like to express my gratitude to Annette Vogt whose encouragement, advice and deep knowl- edge of the history of German and Russian mathematics have been of invalua- ble importance to me. Jessica, Marco and Pim allowed me to live and work in Dahlem as a worldly monk. I thank Rob and my (former) fellow PhD-candidates Berend, Floortje, Guus, Martin, Pim and, of course, Willemine for introducing me into academic life, my new colleagues Federica, Franz, Jacques, Michiel and, of course, Huub for welcoming me to the department, Olga Hoppe-Kondrikova for her help in translating Khinchin’s papers and Aad van Dommelen for his design of the book. I also want to thank, for many diferent reasons, Andries, Jattie, Jan Bouwe, Judith, Ludo, Mathijs, Robby, Sander, Simone, Tess, Vera and Wout, the saviors of pop-music Bowie, Julien and Thijs, my long-lost friend Bart and my family; Ad, Jana, Jantien, Niek, Julia, Geraldine, Elizabeth, Oek, Jeanne, Imar, Danny, Teuntje and my grandmother and two grandfathers who passed away in recent years, Dies, Ton and Klaas. Then there are Floris, Guido, Maite and Mees – who have taught me that what goes for thinking also goes for friendship; that when you do it ‘you should burn yourself completely, like a good bonfire, leaving no trace of yourself’. I started writing the articles for this book while living with the love of my young life, Imke. My love goes to her and to my parents Hannet and Peter. 9 Introduction I would like to think of the articles that make up this book as contributions to a historical epistemology of abstraction. They raise the question of how the idea of a formal axiomatic system – characteristic of ‘modern’ mathematics and logic – that paved the way for the ‘modernization’ of probability by measure theory became possible. This question asks for an answer that describes not what this idea is, but what allowed it to become thinkable, and that understands the way in which this came about not from the perspective of the ‘modern’, but from that of the ‘non-modern’. The book, thus, investigates how this (‘modern’) idea could arise out of a (‘non-modern’) style of thought the limits of which made it as yet unthinkable. It can be read on four diferent levels. The detailed historical studies of the work of nineteenth-century and early twentieth-century ‘non-moderns’ and the general view on ‘non-modern’ proba- bility, logic and mathematics (ca. 1830-1930) that arises from them form the first two levels. They are its explicit content. This introduction and the concluding remarks open up a third and fourth level, namely a ‘post-Kuhnian’ history of exact thought and a philosophical theory of the foundations of the ‘modern- ist transformation’. These levels, thus, contain suggestions for an answer to the question mentioned above – suggestions that hint at a historical ‘break’ between a ‘non-modern’ and ‘modern’ historical a priori in which a revised a priori came to ground ‘modern’ exact thoughts. Histories of the modern Much has been written about the ‘modernist transformation’ of mathematics.1 Some of these accounts reflect on what it means to do history of mathematics2 and, in doing so, consider its relation to the new historiography of science asso- ciated with the name of Thomas Kuhn (1922-1996).3 In what is perhaps the 1 See Gray (2008) and Mehrtens (1990).