This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Tatton-Brown, Oliver M W Title: Rigour, Proof and Soundness General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. Rigour, Proof and Soundness By Oliver Tatton-Brown Department of Philosophy School of Arts University of Bristol A dissertation submitted to the University of Bristol in accordance with the requirements for award of the degree of Doctor of Philosophy in the Faculty of Arts. March 2020 Word count: 79746 Abstract The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested and illustrated by considering a case study discussed in the literature. On the view put forward here one can obtain a manner of informal, rigorous mat- hematics founded on any of a variety of proof systems. The latter part of the thesis is concerned with the question of how we should decide which of these competing proof systems we should base our mathematics on: i.e., the question of which proof system we should take as a foundation for our mathematics. A novel answer to this question is proposed, in which the key property we should require of a proof system is that for as many different kinds of structures as possible, when the proof system allows a gene- ralization about that kind of structure to be proved, the generalization actually holds of all real examples of that kind of structure which exist. This is the requirement of soundness of the proof system (for each kind of structure). It is argued that the best way to establish the soundness of a proof system may be by giving an interpretation of its axioms on which they are established as true. As preparation for this discussion, the thesis first investigates the logical and conceptual basis of various arithmetic concepts, with the results obtained used in the final discussion of soundness. Dedication and Acknowledgements To Alex and Amy I would like to extend my sincere gratitude to a number of people who have made this work possible. Leon Horsten has been a wonderful guide into academic philosophy, the source of too many fascinating conversations to count, and has been admirably patient and nurturing with my various proto-ideas. Catrin Campbell-Moore has done much sterling work identifying places where the thesis was lacking in clarity or force. In a few key conversations Philip Welch has made various remarks that have contributed greatly. As well as the above, a variety of people have provided very helpful (and often very detailed) comments on my papers, including in particular Daniel Waxman, Brendan Larvor, Marcus Giaquinto, Yacin Hamami, and various anonymous referees. I am also thankful for crucial remarks from members of the audience at talks I have given, including at the FSB research seminars in Bristol, and at conferences in Sicily, London Ontario, Cambridge, and Leuven. Finally I would like to thank Mungo for being my initial companion in my first days of interest in philosophy, and Alex and Amy for their endless help and support. This work was generously funded by the AHRC, grant AH/L503939/1, via the SWWDTP. Chapter II is an edited version of Tatton-Brown (2019b), and chapter IV is an edited version of Tatton-Brown (2019a). Author’s Declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University’s Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate’s own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. SIGNED: ........................................................ 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Contents Introduction 1 I Understanding Rigour 5 1 Questions about proof .............................. 5 2 Initial remarks on rigour ............................. 9 3 A rigorous eduction ................................ 15 4 The concept of rigour ............................... 27 5 Disagreements about validity .......................... 31 6 Naive set theory? ................................. 34 7 Formalizability .................................. 36 8 Further questions ................................. 45 9 The choice of proof systems ........................... 47 II Intuition and Permissible Actions 49 1 Alexander’s lemma ................................ 49 2 De Toffoli and Giardino’s account of proof ................... 52 3 De Toffoli and Giardino’s account of Alexander’s argument . 61 4 The “legitimate operations” ........................... 64 5 Termination of the process ............................ 74 III Rigour, Pictures and Knot Theory 79 1 Rigorous use of pictures I ............................ 80 2 Jones’s argument ................................. 85 3 Definitions ..................................... 90 4 Structure of the argument ............................ 92 5 “Short stretches”? ................................ 95 6 The “over the shoulder” manoeuvre . 102 7 Assessment of Jones’s argument . 111 8 Rigorous use of pictures II ............................112 IV Ancestrals, Primitive Recursion and Isaacson’s Thesis 117 1 The thesis .....................................120 2 The argument ...................................122 ix CONTENTS 3 The ancestral and the double ancestral . 123 4 Primitive recursion and the double ancestral . 129 5 Double ancestral arithmetic . 133 V Ancestrals and plurals 137 1 Plural double ancestral logic . 138 2 Finiteness .....................................141 3 Equinumerosity ..................................143 4 Arithmetic operations ..............................148 5 Abstraction and cardinalities . 151 VI Sound Foundations 155 1 Introduction ....................................155 2 Proof, and proof .................................160 3 Interpretations of mathematics . 164 4 Realizations of mathematical concepts . 172 5 The eliminative constraint ............................194 6 Foundational goals ................................204 7 Arguing for soundness ..............................211 8 Summary .....................................234 9 Implications ....................................235 Conclusion 239 Appendices 241 A The Smooth Case of Alexander’s Lemma 243 1 The proof .....................................243 2 Smooth and periodic functions . 287 3 Smooth knots and projections . 296 B Ontologically innocent second order logic 321 C Complete ordered field structure on a continuously ordered open interval 331 D Interpreting mathematics in terms of a complete ordered field 341 Bibliography 359 x List of Figures I.1 Very basic proof example ........................... 15 I.2 Basic proof example .............................. 17 I.3 Slightly more complex proof example ..................... 19 I.4 Levels of detail ................................. 22 II.1 A knot diagram ................................. 51 II.2 A knot diagram winding around an axis ................... 52 II.3 The over the shoulder manoeuvre ....................... 63 II.4 Avoiding K with the triangle [a, b, c] ..................... 68 II.5 The result of replacing [a, b] with [a, c] ∪ [c, b] . 69 III.1 Removing squares of different colours from a chessboard . 82 III.2 The over the shoulder manoeuvre ....................... 86 III.3 The over the shoulder manoeuvre, avoiding unfixed short sections . 94 III.4 The separating hyperplane theorem . 104 III.5 A planar isotopy ................................107 III.6 More complicated planar isotopy . 107 III.7 Zooming in on a wiggly path . 109 IV.1 The “ancestor of the same generation” relation . 124 IV.2 The double ancestral of φ and ψ . 125 IV.3 Primitive recursion via the double ancestral . 131 xi Introduction A mathematical proof of a proposition is often taken to be amongst the best kinds of evidence for