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Towards a Fictionalist Philosophy of Mathematics

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Towards a Fictionalist Philosophy of Mathematics Towards a Fictionalist Philosophy of Mathematics A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Humanities 2015 Robert Knowles School of Social Sciences 2 Contents Preliminary pages................................................................................................ 1 Introduction ....................................................................................................... 11 Part I: Against the Indispensability Argument 1. Introduction to Part I: Two Roads to Nominalism ............................................................ 18 1.1 The indispensability argument .................................................................................... 18 1.3 The hard road to nominalism ..................................................................................... 25 1.4 The (easy) road ahead ................................................................................................ 37 2. The Weasely Way Round .................................................................................................. 40 2.1 Introduction ................................................................................................................ 40 2.2 What is weaseling? ..................................................................................................... 42 2.3 Melia’s arguments: charity and simplicity ................................................................. 50 2.4 Two challenges: content and communication ............................................................. 60 2.5 Conclusions ................................................................................................................. 65 3. Heavy Duty Platonism ....................................................................................................... 67 3.1 Introduction ................................................................................................................. 67 3.2 Arguments from Lewis ................................................................................................ 70 3 3.3 Arguments from Crane ............................................................................................... 77 3.4 Arguments from Field ................................................................................................. 92 3.5 Conclusions ................................................................................................................ 94 4. The Indispensability Argument Dispensed With .............................................................. 96 4.1 Introduction: three challenges .................................................................................... 96 4.2 Optimality and robustness (challenge 1) .................................................................... 99 4.3 Mathematical realisation (challenge 2) ................................................................... 107 4.4 Expressionistic vs. heavy duty explanation (challenge 3) ........................................ 113 4.5 Conclusions .............................................................................................................. 119 Part II: Towards Hermeneutic Fictionalism 5. Introduction to Part II: The Study of Natural Language ................................................. 123 5.1 Hermeneutic fictionalism .......................................................................................... 123 5.2 The traditional view .................................................................................................. 124 5.3 Objections to the traditional view (I): communication and completeness ............... 133 5.4 Objections to the traditional view (II): indexicals .................................................... 139 5.5 In support of the traditional view ............................................................................. 149 5.6 The (less easy) road ahead ....................................................................................... 154 6. The Semantics of Number and Magnitude ...................................................................... 157 6.1 Introduction .............................................................................................................. 157 4 6.2 Adjectival numerals ..................................................................................................163 6.3 Specificational sentences ..........................................................................................170 6.4 Free relatives and definite descriptions....................................................................179 6.5 Number-of expressions .............................................................................................188 6.6 Puzzles and solutions ................................................................................................195 6.7 Pure arithmetic .........................................................................................................199 6.8 Conclusions ...............................................................................................................209 7. The Pragmatics of Mathematics (I): Figuring Yablo .......................................................213 7.1 Introduction ..............................................................................................................213 7.2 Mathematics as metaphor .........................................................................................215 7.3 Mathematics as non-literal .......................................................................................223 7.4 Mathematics as make-believe ...................................................................................231 7.5 Mathematics as presupposition ................................................................................238 7.6 The intended content of pure mathematics ...............................................................245 7.7 Conclusions ...............................................................................................................255 8. The Pragmatics of Mathematics (II): Evaluative Salience ..............................................257 8.1 Introduction ..............................................................................................................257 8.2 Relevance theory .......................................................................................................258 8.3 Evaluative salience and felt truth-value ...................................................................266 8.4 Mathematical utterances in applied contexts ...........................................................274 5 8.5 Mathematical utterances in pure contexts ................................................................ 280 8.6 Conclusion ................................................................................................................. 286 Conclusion ........................................................................................................ 291 Cited Works ..................................................................................................... 301 Word count: 79, 381 6 List of Figures Figure 1: syntax tree for ‘John likes himself’ 175 Figure 2: syntax tree for ‘What John likes is himself’ 176 7 Abstract In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content the truth of which does not require mathematical objects to exist. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more than we otherwise would be able to about, or yielding a greater understanding of, the physical world. Mathematical objects to not need to exist for mathematical language to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences and show that they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which shows that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show the hermeneutic fictionalist position that emerges is preferable to any alternative which interprets mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects. 8 Declaration and Copyright Statement No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. The author of this thesis (including any appendices and/or schedules to this thesis)
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