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Objects and Objectivity Alternatives to Mathematical Realism

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Objects and Objectivity Alternatives to Mathematical Realism Objects and Objectivity Alternatives to Mathematical Realism Ebba Gullberg Umeå Studies in Philosophy 10 Objects and Objectivity Alternatives to Mathematical Realism Ebba Gullberg Umeå 2011 © Ebba Gullberg 2011 Series editors: Sten Lindström and Pär Sundström Department of Historical, Philosophical and Religious Studies Umeå University SE-901 87 Umeå, Sweden ISBN 978-91-7459-180-4 ISSN 1650-1748 Printed in Sweden by Print & Media, Umeå University, Umeå Distributor: Department of Historical, Philosophical and Religious Studies, Umeå University, SE-901 87 Umeå, Sweden. Till Ylva Abstract This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I ex- amine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on exis- tence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathe- matics and, hence, that mathematical truth cannot be used to draw the vii viii ABSTRACT conclusion that mathematical objects exist in an external/ontological sense. Keywords: Philosophy of mathematics, mathematical realism, onto- logical realism, semantic realism, platonism, the semantic argument, the indispensability argument, the non-uniqueness problem, Benacer- raf's dilemma, the irrelevance challenge, Field, Carnap, Balaguer, Yablo, the internal/external distinction, fictionalism. Acknowledgements For every hour I have spent writing this dissertation I have probably spent at least two hours discussing it, arguing about it, and forgetting about it with my advisor, Sten Lindström. For his constant support and encouragement, as well as his many comments and suggestions regarding my text, I am deeply grateful. My assistant advisor, Tor Sandqvist, has been less involved in this project as far as quantity is concerned. However, this has been more than made up for by the quality of his advice. Without his careful reading and his clarifying comments and questions, this work would have missed out on several substantial improvements, both philosophical and linguistic. In this connection I also wish to thank Peter Melander, who acted as my assistant advisor during my first few years as a doctoral student. Large parts of this dissertation have been presented and discussed at seminars in Umeå and elsewhere. I am grateful to everyone who has taken the time to be present at these occasions. In particular, I want to thank Anders Berglund, Jonas Nilsson, Lars Samuelsson, Pär Sund- ström, Bertil Strömberg, Inge-Bert Täljedal, and Jesper Östman, who have all been regular participants in the department's Friday philosophy seminar and have contributed with many interesting discussions on the topics of my texts. Thanks to a grant from STINT (The Swedish Foundation for Inter- national Cooperation in Research and Higher Education) I was able to spend five months visiting the philosophy department at UCLA in the spring of 2007. This was a very rewarding experience and I want to thank everyone who made it possible. All work and no play makes Ebba a dull girl. My time as a doctoral student would have been considerably less agreeable without friends and activities to distract me. In this connection I especially want to thank my fellow musicians in Umeå musiksällskap, in particular the viola section. I also want to thank two of my dearest friends, Susanna Lyne and Johanna Wängberg, for keeping in touch over the years. ix x ACKNOWLEDGEMENTS Last but not least I want to thank my family: my parents Rolf and Barbro, my sister Åsa and her family, my brothers Jesper and Tobias and their families, and my husband, Robert. They have all reminded me of the fact that there is a world outside “the philosophy room”. This also applies to my daughter, Ylva, to whom I have dedicated this dissertation. There are several obvious ways in which she delayed its being finished, but there are also ways in which she made it possible for me to finish it at all. Ebba Gullberg Umeå, April 2011 Contents Abstract vii Acknowledgements ix 1 Introduction 1 1.1 Background and aim . 1 1.2 Outline of the dissertation . 3 2 Mathematical realism 9 2.1 What is mathematical realism? . 9 2.1.1 Existence and ontological realism . 10 2.1.2 Truth and semantic realism . 14 2.1.3 Platonism . 17 2.1.4 Platonistic interpretations of mathematical theories 19 2.2 Arguments for mathematical realism . 35 2.2.1 The semantic argument . 36 2.2.2 The indispensability argument . 44 2.3 Summary . 51 3 Difficulties with mathematical realism 53 3.1 The non-uniqueness problem . 55 3.2 The conflict between semantics and epistemology . 65 3.3 The irrelevance challenge . 75 3.4 Preliminary assessment . 80 4 Field's nominalism 83 4.1 Introduction . 83 4.2 Field's program . 84 4.2.1 Conservativeness . 85 4.2.2 Nominalization . 87 4.3 Some objections . 89 xi xii CONTENTS 4.3.1 Is Field's theory really nominalistic? . 89 4.3.2 Is mathematics really conservative? . 91 4.3.3 How far can Field's program be extended? . 94 4.4 Concluding remarks . 95 5 Carnapian doublespeak 97 5.1 The internal/external distinction . 98 5.2 Quine's criticism . 103 5.3 Metaontology and neo-Carnapianism . 108 6 Fictionalism 113 6.1 Balaguer's fictionalism . 116 6.1.1 Correctness and objectivity . 117 6.1.2 Applicability and indispensability . 119 6.2 Yablo's figuralism . 122 6.2.1 Mathematics as metaphorical make-believe . 122 6.2.2 Ontology and objectivity . 125 6.3 Concluding remarks . 128 7 Separating truth from ontology 131 7.1 Introduction . 131 7.2 A modified internal/external distinction . 138 7.3 What happens to the arguments for ontological realism? 141 7.4 Internal truth and existence . 144 8 Coda 149 8.1 Realism without ontology; fictionalism with truth . 151 8.2 Concluding remarks . 156 Bibliography 157 Index 167 Sammanfattning 173 Chapter 1 Introduction 1.1 Background and aim Consider the following sentences: (1) 5 + 7 = 12; (2) The sum of the numbers 5 and 7 is the number 12. They are similar in many respects, and there are certainly circumstances under which we would be prepared to say that they represent only slightly different ways of conveying the same information: that five and seven make twelve. We get a clearer idea of how the two sentences are different when we try to specify what it is that we are talking about when we say that five and seven make twelve. Judging from the formulation (1), it seems that we could be talking about almost any kind of object—apples, grains of sand, people, and so on. We can think of (1) as a shorthand for the fact that five objects of any kind together with seven other objects of any kind make twelve objects. The formulation (2), on the other hand, apparently does not lend itself so easily to interpretations that are based on the counting of concrete objects. In fact, when we read it carefully, (2) says not only that five and seven make twelve, but also that there is a number, namely the number 12, which is the sum of the numbers 5 and 7. When we go beyond simple arithmetic and deeper into mathematics, we will run into more and more statements that explicitly assert the existence of entities that do not seem to be part of our ordinary concrete world. Moreover, many of these statements will not have immediate concrete counterparts that allow us to interpret them as being about some familiar, everyday objects. So if we commit ourselves to their 1 2 CHAPTER 1. INTRODUCTION truth, it appears that we are also committed to the existence of abstract entities. There are many possible attitudes that we can assume toward the abstract objects of mathematics. At one extreme there is platonism, which involves a full acceptance of mathematical entities as eternally existing, independent of human thought and language. Exceptions and modifications can then lead us through various views until we reach the opposite extreme, where we have nominalism and a complete rejection of the existence of abstract objects.
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