Carnap and the Ontology of Mathematics
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Carnap and the Ontology of Mathematics Benjamin Marschall Darwin College, University of Cambridge February 2020 This thesis is submitted for the degree of Doctor of Philosophy. Declaration This thesis is the result of my own work and includes nothing which is the out- come of work done in collaboration except as declared in the Preface and speci- fied in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no sub- stantial part of my thesis has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. It does not exceed the prescribed word limit for the relevant Degree Committee. Abstract Carnap and the Ontology of Mathematics Benjamin Marschall In this thesis I investigate Rudolf Carnap’s philosophy of mathematics. Most philosophers assume that the nature of mathematics raises deep philosophical questions, which call for theories about how we manage to know about and interact with abstract objects. Carnap’s position, in contrast, is deflationary: he aims to show that we can take mathematics at face value without having to answer questions about the metaphysical status of mathematical objects. If Carnap is right, there is thus no need for a philosophy of mathematics as it is usually understood at all. The main argument of my thesis is that Carnap’s position is unstable, since his own commitments force him to make at least some ontological assumptions about syntax, i.e. entities such as letters, strings, and proofs. My claim that Carnap needs to accept some ontological questions as being in good shape goes against the received view in the secondary literature. Since the late 1980s interest in Carnap’s philosophy of mathematics has been growing, mostly in the wake of important papers by Michael Friedman, Warren Goldfarb, and Thomas Ricketts. These and other scholars have forcefully defended Car- nap against objections by, among others, Kurt Gödel, W. V. Quine, and Hilary Putnam. The most powerful challenge to Carnap’s view, however, can actually be found in a less well-known paper by the logician E. W. Beth. The core of my thesis is thus a new interpretation of what I call Beth’s argument from non- standard models, which relies on Gödel’s incompleteness theorems and targets Carnap’s claim that mathematics is analytic. I show that my reconstruction of Beth’s argument is more charitable to the text than competing interpretations in the secondary literature, and argue that it is also more powerful since extant defences of Carnap cannot be applied. Acknowledgements I have been doing philosophy for nearly ten years now, and there are many people to thank. Starting with those who most directly contributed to making this thesis a reality, I thank Tim Button for being an amazing supervisor: always positive and encouraging, always full of good ideas, and as engaged as ever even after leaving Cambridge for UCL. I could not have written this thesis without him. Alex Oliver was an excellent secondary supervisor, especially during the final stretch when he read and commented on a full draft. I know that not everyone is in the lucky position to have both their supervisors read their thesis before submission, so I am very grateful for the support. I also thank Michael Potter who, in a memorable first-year review meeting together with Alex, suggested that looking at the history of analytic philosophy might be the best way to tackle the issues I was interested in. I had considered working on Carnap before, but was put off by the fact that this seemed hard work. I am glad that this time I received a push in the right direction at the right moment. The philosophy graduate community at Cambridge has made the time dur- ing which I have been working on this thesis particularly enjoyable. Special thanks go to Annie Bosse, Bertie Probyn, Lucy McDonald, Nathan Hawkins, Makan Nojoumian, Paula Keller, Senthuran Bhuvanendra, Sofía Melendez Gutier- rez, Wouter Cohen, and the honorary graduate students Alex Jackson and Thomas Schindler. Many of them have given helpful feedback and advice more than once, and I hope that all at least somewhat enjoyed my Carnap-anecdotes. The late Hugh Mellor not only was an unusually warm and welcoming pres- ence in the Faculty, but, to my amazement, even gave me a copy of the Schilpp- volume on Carnap with scribbled notes by Paul Feyerabend in it. He is greatly missed. In early 2020 I visited the University of Pittsburgh for a term in order to work with Thomas Ricketts. I thank Tom for being so generous with his time in order to talk through Beth and Carnap, for yet another book present (this time a copy of The Logical Syntax of Language), and for being extremely support- ive during a turbulent mid-March. The meetings with him were very energis- ing and led me to re-think my interpretation of Beth, which ultimately enabled me to give a much clearer account of the core argument. Consulting the un- published material in the Rudolf Carnap Papers was also stimulating, and I thank the always helpful staff at Archives of Scientific Philosophy. Other people who made my time in Pittsburgh pleasant were Christian Feldbacher-Escamilla, Clemens Wetcholowsky, Patrick Chandler, Sander Verhaegh, Sophia Arbeiter, and Stephen Mackereth. Before coming to Cambridge I studied in Berlin and St Andrews. For Berlin I first and foremost thank Tobias Rosefeldt for the continued support, and also Barbara Vetter, Emanuel Viebahn, Lukas Lewerentz, Karl-Georg Niebergall, Mar- tin Aigner, Olaf Müller, Roland Krause, and Timo Schobinger for being inspira- tional in various ways. For St Andrews greetings go to the Stirling Bus Crew, especially to Jakob Ohlhorst and Xintong Wei. Material drawn from this thesis has been presented at talks in Bratislava, Cambridge, Munich, Oxford, and Tilburg, and I thank the respective audiences for their comments and questions. Papers based on the thesis are or have been under review at Erkenntnis and the Canadian Journal of Philosophy, and many of the referee reports I received have been helpful and constructive. I also thank Eckehart Köhler for a long conversation about the relationship between Carnap and Gödel, Gary Ebbs for giving detailed answers to my questions about his in- terpretation of Carnap via email, Hannes Leitgeb for sending me a version of his and André Carus’ SEP entry on Carnap before it went online, and Jared Warren for sending me a pre-print of his intriguing book-length defence of mathematical conventionalism. The Arts and Humanities Research Council, the Cambridge Trust, the Aris- totelian Society, the Royal Institute of Philosophy, and the German-American Fulbright Commission provided the funding that made my research possible in the first place, and for this I am very grateful. It is unhealthy to think about philosophy all the time, and so thanks go to the members of the Cambridge University Rambling Club for igniting my passion for the East Anglian countryside. I thank my parents for always being supportive and enthusiastic about what- ever I do, even though I often struggle to explain the exact nature of my pursuits to them. Last but certainly not least, I thank Clara for everything. v Contents Introduction 1 Part I Carnap and Mathematics 5 Chapter 1 Carnap’s Internalism 6 1.1 Platonism and Frameworks . 6 1.1.1 Who’s a Platonist? . 6 1.1.2 Linguistic Frameworks . 11 1.1.3 Internal Platonism . 16 1.2 Analyticity and Conventionalism . 21 1.2.1 Truth and Linguistic Rules . 21 1.2.2 Is this Conventionalism? . 25 1.2.3 Carnap’s Deflationism . 28 Chapter 2 Gödel and Incompleteness 33 2.1 Linguistic Rules Revisited . 33 2.1.1 Syntactic Rules and Their Limits . 33 2.1.2 The Nature of Frameworks . 38 2.1.3 Towards Internalism . 41 2.2 Gödel’s Consistency Argument . 48 2.2.1 Is Mathematics Syntax of Language? . 48 2.2.2 Content and Knowledge . 52 2.2.3 Frameworks and Natural Language . 56 Chapter 3 A Quinean Interlude 61 3.1 The Carnap-Quine Debate . 61 3.2 Quine’s Triviality Objection . 65 Contents Contents Part II Beth versus Carnap 70 Chapter 4 Beth’s Argument from Non-Standard Models 71 4.1 Carnap and Carnap* . 71 4.1.1 Introducing Carnap* . 71 4.1.2 What’s the Problem? . 77 4.1.3 Metalanguages and How to Use Them . 81 4.2 Tolerance and Abstract Objects . 86 4.2.1 Against Infinitary Rules . 86 4.2.2 Ontology and Ontological Commitments . 90 4.2.3 The Determinacy of Syntax . 95 Chapter 5 Alternative Interpretations of Beth 100 5.1 Beth and Model-Theoretic Arguments . 100 5.1.1 Non-Standard Interpretations . 100 5.1.2 Ricketts on Beth and Skolem . 103 5.1.3 Exegetical Traps and Pitfalls . 107 5.2 Other Approaches . 111 5.2.1 Friedman for and against Beth . 111 5.2.2 Who Is Speaking? . 116 5.2.3 Explication . 121 Chapter 6 Radical Deflationism 127 6.1 Carnap and Semantic Facts . 127 6.2 Wittgenstein’s Radical Conventionalism . 133 6.3 Is there a Middle Way? . 137 Part III Beyond Logical Syntax 142 Chapter 7 Lines of Defence 143 7.1 Internalism and Pluralism .