View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository PROOF, RIGOUR & INFORMALITY: A VIRTUE ACCOUNT OF MATHEMATICAL KNOWLEDGE Fenner Stanley Tanswell A Thesis Submitted for the Degree of PhD at the University of St Andrews 2017 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/10249 This item is protected by original copyright PROOF, RIGOUR & INFORMALITY: A VIRTUE ACCOUNT OF MATHEMATICAL KNOWLEDGE Fenner Stanley Tanswell This thesis is submitted in partial fulfilment for the degree of PhD at the University of St Andrews Date of Submission 31st August 2016 Declaration 1. Candidate's declarations: I, Fenner Stanley Tanswell, hereby certify that this thesis, which is approx- imately 65000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2013 and as a candi- date for the degree of Doctor of Philosophy in September 2013; the higher study for which this is a record was carried out in the University of St An- drews between 2013 and 2016. Date signature of candidate 2. Supervisor's declaration: I hereby certify that the candidate has fulfilled the conditions of the Resolu- tion and Regulations appropriate for the degree of Doctor of Philosophy in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date signature of supervisor Date signature of supervisor 3. Permission for publication: In submitting this thesis to the University of St Andrews I understand that I am giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. I also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or 1 research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. I have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the publication of this thesis: No embargo on print copy. No embargo on electronic copy. Date signature of candidate signature of supervisor signature of supervisor 2 Abstract This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in for- mal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from G¨odel'sparadox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the im- portance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathemat- ical rigour. 3 Acknowledgements I am very grateful for the extensive support, encouragement, feedback and friendship I've benefitted from in the last three years in producing this the- sis. First of all, I would like to thank the family of Caroline Elder for the generosity in providing the Caroline Elder PG scholarship in her memory. Thanks also for the further SASP scholarship I received, as well as travel funding from the Indo-European Research Training Network in Logic. Thanks to my supervisors Aaron Cotnoir and Patrick Greenough, who have both been fantastic throughout and have had a huge influence on my think- ing and writing in this thesis. Their comments and feedback have been invaluable and led to massive improvements in the work presented here. My particular interest in proofs and the philosophy of mathematical prac- tice started under the supervision of Benedikt L¨owe while doing the Master of Logic in Amsterdam. In particular, the first two chapters were strongly influenced by Benedikt and the work I did there for my master's thesis, so many thanks. I have also had direct feedback on previous drafts from Noah Friedman- Biglin, Bruno Jacinto, Josh Habgood-Coote, Asgeir´ Berg Matth´ıasson,Alex Yates, Caroline Torpe Touborg, Mark Bowker, Ryo Ito, Patrik Hummel, Brian King, Claire Field, Morgan Thomas, Chris Sangwin and Colin Rit- tberg. Many thanks to all. I would also like to thank Jody Azzouni who was a great help for chap- ters 1 and 5, and who provided excellent comments and discussion on the overgeneration problem which led to a more robust objection. Thanks also to Robert Thomas, Holger Andreas and Peter Verd´ee,who provided use- ful feedback on chapters 1 and 2 towards publication. Thanks too to three anonymous referees in writing reports on earlier versions of the first two chapters. Thanks to Stephen Read for extra guidance and discussion on the philoso- phy of mathematics, as well as for running the primary seminars I have been 4 involved with in Arch´e. Thanks also to Lynn Hynd for the organisational brilliance in keeping Arch´erunning, and to the other staff who ran the phi- losophy department over the last three years: Laura, Shaun, Emmy, Lucy, Rhona and Katie. The work in chapter 4 formed part of the content of a joint Prize Semi- nar for St Andrews undergraduates with Josh Habgood-Coote, so thanks to him for the huge amount of feedback and help on knowing-how. Thanks also to the students who attended, asked insightful questions and pushed us on many philosophical points. The work in chapter 5 was presented alongside Colin Rittberg's work on the- oretical virtues for set theory theory as a three-lecture series in Aberdeen. Thanks to Colin for extensive discussion in the run up to these and for invit- ing me to Brussels twice. Anti-thanks to ISIS who tried to stop us. Thanks to audiences at the many talks I've given in St Andrews and Stirling, as well as audiences and organisers in Ume˚a,Bristol, Munich, Cambridge, Storrs, New Delhi, Paris, Brussels and Aberdeen. Thanks also to the wider St Andrews community, members of the philosophy department and Arch´e, for being great people. I spent two months in 2014 at the University of Connecticut, with grati- tude to the Arch´eTravel Fund which paid for it. I would like to thank those who I interacted with while there and who made me incredibly welcome: Jc Beall, Dave Ripley, Morgan Thomas, Nathan Kellen, Hanna Gunn, Andrew Parisi, Damir Dzhafarov, Reed Solomon, Brown Westrick, Daniel Silvermint, Keith Simmons, Nate Sheff and Suzy Killmister. I am also really thankful to the many people who welcomed me into their community in Coventry CT, with special thanks to Holly Parker and Joram Echeles. Thanks to Aikman's and Rendez-Vous for the good living. I am surrounded by friends who have made doing a PhD possible. In addi- tion to those mentioned above, many thanks to Joe Slater, Laurence Carrick, Mat Chantry, Ally Holmes, Calum McConnell, Carley Hollis, Elspeth Gille- spie, Alex Purser, Beth Allison, Bella Bandell, Ste Broadrick, Paul Black, Alper Yavuz, Marissa Wallin, Gabriela Rino Nesin, Maja Jaakson, Aadil Kurji, Nathaniel Forde, Jessica Olsen, Hraban Luyat, Ravi Thakral, Ryan Nefdt and the many others I have failed to mention. You're all top bananas. Most of all, I am indebted to my family, in particular mum & Chris, dad and Thecla: thank you all. Finally, thanks to Lea who has given me confidence, patience, love and support over the last three years. 5 Publication A version of chapter one has appeared in Philosophia Mathematica in 2015 with the title \A Problem with the Dependence of Formal Proofs on Infor- mal Proofs". This is in the bibliography under (Tanswell 2015). A version of chapter two is forthcoming in the collection Logical Studies of Paraconsistent Reasoning in Science and Mathematics, edited by Holger Andreas & Peter Verd´ee as part of Springer's Trends in Logic series, with the title \Saving Proof from Paradox: G¨odel'sParadox and the Inconsis- tency of Informal Mathematics". This is in the bibliography under (Tanswell forthcoming). 6 Contents Introduction 9 0.1 The Twin Faces of Orthodoxy . 10 0.2 The Philosophy of Mathematical Practice . 14 0.3 Outline . 16 1 A Problem with the Dependence of Informal Proofs on For- mal Proofs 20 1.1 Introduction .