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Towards a Fictionalist Philosophy of Mathematics

Robert Knowles

[Thesis].The University of Manchester;.

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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 that does not imply the existence of mathematical objects. 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 about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics 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 that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.

Keyword(s)

Philosophy of mathematics platonism nominalism fictionalism indispensability argument

Bibliographic metadata

Type of resource:
Content type:
Type of thesis:
Author(s) list:
Degree type:
PhD
Total pages:
330
Table of contents:
ContentsPreliminary pages 1Introduction 11Part I: The Applications of Mathematics1. Introduction to Part I: Two Roads to Nominalism 181.1 The indispensability argument 181.3 The hard road to nominalism 261.4 The (easy) road ahead 372. The Weasely Way Round 412.1 Introduction 412.2 What is weaseling? 432.3 Melia’s arguments: charity and simplicity 512.4 Two challenges: content and communication 612.5 Conclusions 663. Heavy Duty Platonism 683.1 Introduction 683.2 Arguments from Lewis 713.3 Arguments from Crane 783.4 Arguments from Field 933.5 Conclusions 954. The Indispensability Argument Dispensed With 974.1 Introduction: three challenges 974.2 Optimality and robustness (challenge 1) 1004.3 Mathematical realisation (challenge 2) 1084.4 Expressionistic vs. heavy duty explanation (challenge 3) 1144.5 Conclusions 120Part II: The Language of Mathemtics5. Introduction to Part II: The Study of Natural Language 1245.1 Why be descriptive? 1245.2 The traditional view 1255.3 Objections to the traditional view (I): communication and completeness 1345.4 Objections to the traditional view (II): indexicals 1405.5 In support of the traditional view 1505.6 The (less easy) road ahead 1556. The Semantics of Number and Magnitude 1586.1 Introduction 1586.2 Adjectival numerals 1636.3 Specificational sentences 1716.4 Free relatives and definite descriptions 1806.5 Number-of expressions 1896.6 Puzzles and solutions 1966.7 Pure arithmetic 2006.8 Conclusions 2117. The Pragmatics of Mathematics (I): Figuring Yablo 2157.1 Introduction 2157.2 Mathematics as metaphor 2177.3 Mathematics as non-literal 2257.4 Mathematics as make-believe 2347.5 Mathematics as presupposition 2407.6 The intended content of pure mathematics 2477.7 Conclusions 2578. The Pragmatics of Mathematics (II): Evaluative Salience 2608.1 Introduction 2608.2 Relevance theory 2618.3 Evaluative salience and felt truth-value 2698.4 Mathematical utterances in applied contexts 2788.5 Mathematical utterances in pure contexts 2838.6 Conclusions 289Conclusion 295Cited Works 305
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 that does not imply the existence of mathematical objects. 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 about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics 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 that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.

Institutional metadata

University researcher(s):

Record metadata

Manchester eScholar ID:
uk-ac-man-scw:269355
Created by:
Knowles, Robert
Created:
24th July, 2015, 14:19:34
Last modified by:
Knowles, Robert
Last modified:
24th July, 2015, 14:19:34

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