Kim-Erik Berts The Certainty of Mathematics A Philosophical Investigation Mathematics is often said to give us knowledge that it more reliable than that of other sciences. Kim-Erik Berts What do we mean by ‘mathematical certainty’? Do we have a clear understanding of this concept? Kim-Erik CertaintyThe Berts of Mathematics | 2016 | This investigation is devoted to the certainty of The Certainty of Mathematics mathematics. The starting point is that we must attend to our practice of mathematics. Through A Philosophical Investigation this approach, mathematical knowledge is seen to involve a skill in working with the concepts and symbols of mathematics, and its results are seen to be similar to rules. This normativity indicates the sense in which mathematics is certain. 9 789517 658423 Åbo Akademi University Press | ISBN 978-951-765-842-3 Kim-Erik Berts (b.1980) MA, Philosophy (Åbo Akademi University, 2006) BA, Philosophy (ÅAU, 2004) Åbo Akademi University Press Tavastgatan 13, FI-20500 Åbo, Finland Tel. +358 (0)2 215 3478 E-mail: [email protected] Sales and distribution: Åbo Akademi University Library Domkyrkogatan 2–4, FI-20500 Åbo, Finland Tel. +358 (0)2 -215 4190 E-mail: [email protected] THE CERTAINTY OF MATHEMATICS The Certainty of Mathematics A Philosophical Investigation Kim-Erik Berts Åbo Akademis förlag | Åbo Akademi University Press Åbo, Finland, 2016 CIP Cataloguing in Publication Berts, Kim-Erik. The certainty of mathematics : a philosophical investigation / Kim-Erik Berts. - Åbo : Åbo Akademi University Press, 2016. Diss.: Åbo Akademi University. ISBN 978-951-765-842-3 ISBN 978-951-765-842-3 ISBN 978-951-765-843-0 (digital) Painosalama Oy Åbo 2016 For Agnes and Gunnar Acknowledgements I have always been fascinated by mathematics. As a child, the fact that I could handle huge numbers was thrilling. Later on, this fascination was evoked by learning new things in mathematics, but it was also sometimes mixed with the frustration that can accompany the struggle with mathematical concepts. Dur- ing my studies at Åbo Akademi University, it became clear to me that it was the philosophical questions concerning mathematics that captured my interest. This philosophical interest in mathematics was also what gave me the impulse to attend university courses in mathematics to begin with. I am very grateful that I have been given the opportunity to devote so much time and attention to philosophical thinking about mathematics. This thesis is the outcome of a ten-year process. The first years were spent working full-time on the thesis. To be able to devote oneself full-time is vital to the success of this kind of project. The last four years, however, writing was done during time off from my work as a teacher of philosophy and mathematics at Vasa gymnasium. Although this has slowed down the process, teaching philosophy and mathem- atics has given me many important insights into how we learn and understand mathematics – not to mention the need to express oneself as clearly as possible. I would like to thank my supervisors Professor Lars Hertzberg and Professor Martin Gustafsson for their generous help and support during this process. Lars has created a lively research environment at the Department of Philosophy at Åbo Akademi University, and I am very lucky that I have had the chance to be a part of it. Lars’s way of doing philosophy is deeply inspiring and it has greatly influenced the work put forth in this thesis. Martin became my supervisor when he was appointed to the Philosophy chair in 2010. I was very happy to see his sharp, meticulous, and helpful comments improve the text in new ways. I have been very fortunate to have Lars and Martin as my supervisors. The first year that I was working on this thesis I spent at the Department of Philosophy at Uppsala University. During this year, I had the opportunity to discuss the philosophy of mathematics with Professor Sören Stenlund. I would like to express my gratitude to Sören for his many constructive comments on drafts of the chapters but, more importantly, for the insights into the subject that he has given me in numerous discussions. The influence of his way of doing philosophy of mathematics can be seen throughout this thesis. During the first course in philosophy that I attended at Åbo Akademi Univer- sity, Kim Solin made an interesting comment about the relation between math- ematics and the world. This comment was enough to make me head for the De- partment of Mathematics and begin my studies there. As he has also become my vii viii Acknowledgements good friend and colleague, I have had the opportunity to learn much from him – in our work together and in our many stimulating discussions. I am deeply grateful for this. I would also like to thank the participants in the Research seminar at the De- partment of Philosophy at Åbo Akademi University for many inspiring discus- sions. In particular, I would like to mention Hugo Strandberg and Göran Tor- rkulla for their valuable comments on drafts of the chapters. The atmosphere at the department is encouraging and inspiring and I am happy to be a part of it. Thanks to teachers, colleagues, and friends at the department: Jonas Ahl- skog, Benjamin Alm, Joel Backström, Stina Bäckström, Antony Fredriksson, Ylva Gustafsson, Camilla Kronqvist, Olli Lagerspetz, Mari Lindman, Yrsa Neu- man, Hannes Nykänen, Marcus Prest, Hans Rosing, Patrick Sibelius, Åsa Slotte. Thank you also to the participants in the Seminar of the Philosophy of Lan- guage at the Department of Philosophy at Uppsala University and the Seminar for the History and Pedagogy of Mathematics at the Department of Mathematics in Uppsala, in particular Anders Öberg for his thought provoking comments. I am also grateful for the support and encouragement that my parents, Mar- gareta and Lars-Erik, constantly give me. My embarking on a several-year-long project in the philosophy of mathematics, associated with a constant insecur- ity with regard to income, has not been met with suspicion, but with interested questions and encouraging words. For my wife Johanna and our children Agnes and Gunnar, who have witnessed the writing process first hand, I reserve a spe- cial ‘Thank you!’. The discussions that we have had about writing and learning mathematics have, each in their own way, given me many insights that relate directly to this thesis. During several summer holidays and many weekends, I have been working on the text instead of taking time off with my family and I am grateful that they have given me this time. Now, I can finally answer Agnes’s and Gunnar’s repeated question ‘Pappa, är din avhandling färdig nu?’ with ‘Ja!’ *** My work on this thesis has been enabled by the generous support of Stiftelsen för Åbo Akademi, Waldemar von Frenckells stiftelse, Vera och Greta Oldbergs stiftelse, The Finnish Society of Sciences and Letters, Stiftelsens för Åbo Akademi forskningsinstitut, Finnish Academy of Science and Letters, and Seniorernas råd vid Åbo Akademis studentkår. Vasa, 28 October 2016 Kim-Erik Berts Contents 1 Introduction 1 2 Certainty 7 2.1 Hilbert on Certainty 10 2.2 The Infallibility of Mathematical Methods 12 2.3 Consistency 16 2.4 Eternal Truths 19 2.5 Deductive and Empirical Sciences 20 2.6 Uncertainty 21 2.7 Certainty and Being Certain 23 3 Knowledge 37 3.1 From the Science of Quantity to a Body of Truths 39 3.2 Benacerraf and the Contemporary Discussion 45 3.3 Two Perspectives on Mathematics 49 3.4 Studying a Mathematical Object 54 3.5 Truth and Referential Semantics 57 3.6 Structuralism 61 3.7 Concluding Remarks 69 4 Formality 73 4.1 The Formal and the Intuitive 74 4.2 Historical Background to Formal Systems 78 4.3 Meaningless Signs 82 4.4 Advantages of Symbolism 89 4.5 Where is Genuine Mathematics? 95 4.6 Formal and Informal Proofs 98 5 Proof 103 5.1 The Role of Conviction 104 5.2 The Role of Understanding 111 5.3 Proof and Concept-formation 114 5.4 Proof and Experiment 121 ix x Contents 5.5 Proof and Surveyability 126 5.6 Simple Deductions 136 5.7 Concluding Remarks 142 6 Conclusion 143 Svensk sammanfattning – Swedish summary 147 Bibliography 149 1. Introduction A traditional question in philosophy concerns what, if anything, we can know with certainty. Mathematics is the only field of knowledge concerning which there has been some rough consensus that it actually gives us certain knowledge. Even though some philosophers have preferred not to limit the range of certainty to mathematics alone, this discipline is often put forward as an ideal: ‘this par- agon of reliability and truth’, as David Hilbert proclaimed.1 Certainty is an enticing concept: we want to be certain, we want to have cer- tain knowledge. We want to be sure that we have not made any mistakes and to have a safeguard against error. This is a desire that is motivated by practical reasons: life is easier if one can avoid (at least sometimes) the consequences of error or of being wrong. It is motivated by social reasons too: being wrong can be embarrassing, and being in the know is deeply satisfying. In many situations, the search for certainty makes us turn to mathematics. Mathematical techniques are often what brings certainty. They allow us to over- view situations in such a way that we see clearly what must be the case. When we have settled something using mathematical tools, we can rely on that knowledge. From a philosophical perspective, the question arises: Why is it that mathem- atics has this status? This question is raised already by the role that mathematics plays in our ordinary lives, and the urgency of it is only strengthened by the role that mathematics plays in the sciences.