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The Epistemology of Abstractionism

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The Epistemology of Abstractionism The epistemology of abstractionism Alexander Christoph Reinhard Olderneier June, 2012 Submitted in accordance with the requirements for the degree of PhD. The University of Leeds Department of Philosophy The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright mate­ rial and that no quotation from the thesis may be published without proper acknowledgement. The right of Alexander C. R. Oldemeier to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patent Act 1988. © 2012 The University of Leeds and Alexander Oldemeier Acknowledgements First and foremost, I thank my patient main supervisor, Robbie Williams, for his great support. I have learned so much. I also thank my supervisor John Divers. Without his guidance, I might not have pursued this project in the first place. My research has been made possible by the kind support of the Leeds Phi­ losophy Department, the AHRC, the Jacobsen 'frust of the Royal Institute of Philosophy, and the Northern Institute of Philosophy. I am deeply grateful for that. I also thank all my fellow PhD students at Leeds, from whom I received a lot of valuable feedback and criticism. In particular: Thomas Brouwer, Michael Bench-Capon, Mirja Holst, Wouter Kalf, Stephan Kraemer, and Duncan Wat­ son. Moreover, I am deeply grateful for all the support I received from the ~orthern Institute of Philosophy (NIP). Thanks to the NIP, I am the proud owner of a Frege T-Shirt. lowe a great deal to its members. In particular: Aaron Cotnoir, Dylan Dodd, Sharon Coull, Filippo Ferrari, Andreas Fjellstad, Aidan McGlynn, Giacomo Melis, Grant Reaber, Bjorn Schotte, Paula Sweeney, and Crispin Wright. I profited immensely from discussions with and constructive criticism from the following philosophers I did not yet mention: Philip Ebert, Ole Hjortland, Joachim Horvath, Nikolaj Pedersen, and Walter Swetly. I am grateful to everyone who was involved in developing the free software that helped me complete and manage this project. It would have been much more difficult to produce all this without Mercurial, ~1E;X, MiK1E;X, LyX, JabRef, Dia, and FreeMind. And I thank everyone who made my non-philosophy time in Leeds an excellent time. In particular, my long-time housemates: Billy, Hermione, Katie, Joe, Lucy, and Oli. Ben and Alasdair. And also: Joe, Martin, and Vasiliki for their hospitality in times of writing up. Moreover, I thank all the people who made my time an excellent time in general. In particular, my good friends from Germany: Anja, Annika, Daniel, David, Georg, Isabella, and Moritz. Without your emotional support over many years, directly and indirectly, I would not have been able to complete this project. The same holds true for my parents, Andrea and Matthias Oldemeier, who enabled me to study philosophy in the first place, for my brothers Dominik and Philipp, my sister Michelle, for Tina, and for Sofia. There is (much) more to say about you, but I do not know how to do justice to you in this context. Let me put it simply: you have been crucial. Unfortunately, it is highly probable that this list is incomplete. If you would have liked to see your name on this list, but did not find your name on it, please accept my apologies. Munich, 30 Oct 2012 ii Abstract I examine the nature and the structure of basic logico-mathematical kllowledgp. What justifies the truth of the Dedekind-Peano axioms and the validit.y of Modus Ponens? And is the justification we possess reflectively available? To make progress with these questions, I ultimately embed Hale's and Wright's neo-Fregeauism iu a general internalistic epistemological framework. In Part I, I provide an introduction to the problems in the philosophy of mathemat­ ics to motivate the investigations to follow. I present desiderata for a fully satisfactory epistemology of mathematics and discuss relevant positions. All these positions turn out to be unsatisfactory, which motivates the abstractionist approach. I argue that abstractionism is in need of further explication when it comes to its central epistemo­ logical workings. I fill this gap by embedding neo-Fregeanism in an internalistic epistemological framework. In Part II, I motivate, outline, and discuss the consequences of the frame­ work. I argue: (1) we need an internalistic notion of warrant ill our epistemology and every good epistemology accounts for the possession of such warrant; (2) to avoid scepticism, we need to invoke a notion of non-evidential warrant (entitlement); (3) because entitlements cannot be upgraded, endorsing entitlements for mathematical axioms and validity claims would entail that such propositions cannot be claimed to be known. Because of (3), the framework appears to yield sceptical consequences. In Part III, I discuss (i) whether we can accept these consequences and (ii) whether we have to accept these consequences. As to (i), I argue that there is a tenable solely entitlement­ based philosophy of mathematics and logic. However, I also argue that we can over­ come limitations by vindicating the neo-Fregean proposal that implicit definitions can underwrite basic logico-mathematical knowledge. One key manoeuvre here is to ac­ knowledge that the semantic success of creative implicit definitions rests on substantial presuppositions - but to argue that relevant presuppositions are entitlements. iii Contents Introduction 1 I Old solutions to old problems 8 1 Epistemology of mathematics: issues and options 9 1.1 Puzzles about arithmetical knowledge ..... 9 1.1.1 Puzzle 1: how are the axioms justified? . 10 1.1.2 Puzzle 2: incompatible constraints 10 1.2 Desiderata for a solution .... 14 1.2.1 Arithmetical Platonism 14 1.2.2 Reconstructing arithmetical knowledge. 17 1.2.3 Additional desiderata for the reconstructive project. 22 1.3 Unsatisfying approaches · . 24 1.3.1 Godelian Platonism 25 1.3.2 Fregean logicism · . 27 1.3.3 The indispensability argument 33 1.3.4 Nominalistic positions 37 1.4 Intermediate conclusion · .. 41 2 Neo-Fregeanism (abstractionism) 42 2.1 Frege's Theorem ......... 42 2.2 Neo-Fregeanism: two (Fregean) aims 44 2.2.1 Neo-Fregean Logicism 45 2.2.2 Neo-Fregean Platonism. 47 2.3 Implicit definition. 49 2.3.1 Explicit vs. implicit definitions 49 2.3.2 Three dimensions of achievement 52 2.3.3 Proposed conditions for semantic and epistemic success 54 2.3.4 The epistemology of implicit definition 56 2.3.5 Hermeneutic reconstruction again . 62 2.4 Three objections to neo-Fregeanism .... 67 iv 2.4.1 The Caesar problem .. 67 2.4.2 The Bad Company objection 72 2.4.3 Epistemic rejectionism 79 2.5 Intermediate conclusion · . 82 II An epistemological framework 84 3 Internalism 85 3.1 Motivating the distinction · .......... 85 3.1.1 Analysing knowledge and justification 85 3.1.2 Scepticism . 87 3.2 Epistemic warrant and the internalism VB. externalism debate 89 3.2.1 The notion of warrant 89 3.2.2 Warrant pluralism · . 91 3.2.3 The question of internalism vs. externalism 92 3.3 Relevance Internalism about the external world 96 3.3.1 Accessible warrant · ........ 96 3.3.2 Arguing for Relevance Internalism 98 3.3.3 The Traditional Epistemic Project 99 3.3.4 Scepticism. 103 3.3.5 Wright's Instability Argument . 105 3.4 Wright's notion of being in a position to claim a warrant. 110 3.5 Wright Internalism about Arithmetic ........ 113 3.5.1 The argument from mathematical practice . 113 3.5.2 The Argument from Analogy ........ 115 3.5.3 Arithmetic and the Traditional Epistemic Project. 117 3.6 Intermediate conclusion ........ 117 4 Scepticism and non-evidential warrant 119 4.1 Cognitive projects, belief-forming methods, and foundationalism . 119 4.1.1 Cognitive projects and their belief-forming methods 119 4.1.2 Foundationalism .................... 123 v 4.1.3 Two basic belief-forming method:; . 123 4.2 Conservativism ............. 124 4.2.1 Conservativi:;m about Perception 124 4.2.2 Deduction............. 127 4.3 Scepticism, circularity, and non-evidential warrant 128 4.3.1 A sceptical challenge ..... 129 4.3.2 Circularity at the second-level . 130 4.3.3 Internalistic warrant by default 132 4.4 Entitlement of cognitive project . 134 4.5 Leaching and Justification Generation 140 4.5.1 The Leaching Worry ..... 140 4.5.2 Two models of Justification Generation. 142 4.6 Pragmatism and epistemic consequentialism 147 4.7 Intermediate conclusion . 153 5 Transmission-failure 155 5.1 The weak status of entitlement 156 5.1.1 Entitlement, evidence, and knowledge 156 5.1.2 The epistemic value of claimable knowledge (and evidence) 159 5.1.3 We should prefer internalistic knowledge over entitlement 161 5.2 Transmission-failure .... 162 5.2.1 Mooreau upgrading . 162 5.2.2 Wright's diagnosis 163 5.2.3 Consequences.. 167 5.3 Rule-circular arguments 169 5.3.1 Transmission-failure 169 5.3.2 The failure of Justification Generation 172 5.4 Two responses ......... 174 5.4.1 Discharged assumptions 175 5.4.2 Entitlement for rules . 176 5.5 Intermediate conclusion: unavoidable sceptical consequences? 177 vi III New solutions to old problems 180 6 Entitled mathematics 181 6.1 Entitlement and arithmetic 181 6.1.1 Axioms as entitlements 182 6.1.2 A regress argument ... 184 6.1.3 Wright on entitlement and arithmetic 186 6.1.4 Wright on avoiding a leaching worry for arithmetic 189 6.1.5 Transmission-failure ................ 190 6.1.6 Presupposition expansion and extended leaching 193 6.2 Semi-sceptical foundationalism 195 6.2.1 The idea ...
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