The Philosophy of Penelope Maddy (Outstanding Contributions to Logic, 31)

This volume features more than 20 essays that explore the work of one of the most important contemporary philosophers of mathematics. It will help readers to better appreciate this significant and prolific philosopher. Within philosophy of mathematics, Penelope Maddy initially advocated realism. She then went on to advance naturalism. Both of her positions became very influential in the field, along with her other work in the philosophy of logic. The contributors comment on and otherwise engage with Maddy’s work. They also weigh in on the state of set theory and its philosophy, the philosophy and history of logic, naturalism, skepticism, and the myriad other areas to which Maddy left her mark. Overall, coverage traces her influence on these various ideas over the years. It will also help readers to better understand how philosophers working at the forefront of these areas see these concepts today. These essays will be essential reading for the wide group of philosophers working in these different areas as well as graduate students studying philosophy of mathematics and logic and the other related issues to which Maddy has contributed. The volume will also appeal to logicians and set theorists in general, as well as to philosophers working in analytic philosophy more widely, as well as to those working in the history of philosophy. Introduction Contents Contributors 1 Intellectual Autobiography References Part I Philosophy of Set Theory and Mathematics 2 Generically Invariant Set Theory 2.1 Introduction 2.2 Some Philosophy 2.2.1 Naturalism and Holism 2.2.2 Meaning Skepticism 2.3 Maximize and Unify 2.4 Generically Invariant Set Theory 2.5 Some Replies to Maddy and Meadows 2.6 Does the Multiverse Have a Core? 2.7 Absolute Ordinal Definability 2.8 Final Remarks References 3 Reply to Steel References 4 Intrinsic and Extrinsic Evidence in Set Theory 4.1 Introduction 4.2 The Iterative Concept of Set 4.3 Zermelo's Concept of Set and His Categoricity Theorem 4.4 Intrinsic Evidence References 5 Reply to Martin References 6 Maddy on Classes 6.1 Introduction 6.2 The Need for Classes 6.3 Combinatorial Versus Logical Conceptions of Collection 6.4 Maddy's Development of the Logical Conception 6.5 Stability as a Way to Do Better References 7 Reply to Linnebo References 8 A General Framework for a Second Philosophy Analysis of Set-Theoretic Methodology 8.1 Introduction 8.2 Some Comments on Meta-Philosophy 8.2.1 Is A Procedural Framework Desirable? 8.2.2 Methodology and Practice 8.2.3 Means-Ends Relations 8.3 Two Case Studies From Maddy 8.3.1 Cantor's Introduction of Sets 8.3.2 Zermelo's Defense of his Axiomatization 8.4 A General Framework for a Second Philosophy Analysis of Set-Theoretic Methodology 8.4.1 The Core Procedure 8.4.2 The Evaluative Apparatus 8.4.3 Analysing the Second Philosopher's Methodology 8.5 Applying the General Framework References 9 Reply to Antos and Kant References 10 Maddy's Notion of `Maximize' and Strong Theories of Sets 10.1 Introductory Remarks 10.2 Axiom Selection and Maddy's Notion of `Maximize' 10.3 Contemporary Axiom Candidates: A Fork in the Road 10.4 A New Maximality Result in Axiom Selection 10.5 Conclusion References 11 Exploring the Generous Arena 11.1 Foundations and Organization 11.2 Two Scaffolds 11.2.1 The Model Theoretic Approach 11.2.2 Category Theoretic Approach 11.3 Foundations 11.3.1 Size 11.3.2 Material and Structural Set Theories as Foundations 11.4 Comparisons References 12 Reply to Baldwin References Part II Realism, Anti-Realism and Thin Realism 13 Second Philosophy and Realism 13.1 Introduction 13.2 Second Philosophy and Skepticism 13.3 Second Philosophy and Mathematics 13.3.1 Mathematics in General 13.3.2 Robust Realism 13.3.3 Thin Realism 13.3.4 Arealism 13.4 Realism and Second Philosophy 13.5 Conclusion 14 Reply to Blanchette 14.1 On Thin Realism 14.2 On Arealism 14.3 On the Epistemology of Depth 14.4 Blanchette’s Alternative References 15 Meditations on Second Philosophy: Anti-Nominalist Reflections on Maddy’s Semi-Nominalism 15.1 Introduction 15.2 Routleyism 15.3 Naturalism and Arealism 15.4 Mathematical Existence Theorems in Physics 15.5 Fictions in Social Science 15.6 Fictions in Mathematics 15.7 Convention 15.8 Phonemes and Genotypes References 16 Reply to Burgess References 17 Enhanced ‘If-Thenism’, Fictionalism, and Realist Anti-Platonism References 18 Reply to Leng References 19 Logical Realism—A Tale of Two Theories* 19.1 Basic Questions and Methodology 19.2 Grounding Logic in the World and the Standing of Logical Necessity 19.3 Relation Between Logic and Mathematics 19.4 The World References 20 Reply to Sher 21 Trumping Naturalism Revisited 21.1 Beyond the Platitude 21.2 Preliminaries 21.3 First Objection: No Clash 21.4 Second Objection: Definitional Failure 21.5 Third Objection: The Failure of Science 21.6 Fourth Objection: Good Philosophy is Scientific 21.7 Fifth Objection: The Idle Wheel 21.8 Sixth Objection: No Dogmatism 21.9 Seventh Objection: Schools 21.10 Summary References 22 Reply to Paseau Reference Part III History of Philosophy 23 What Do Philosophers Do? Maddy, Moore (and Wittgenstein) II 23.1 Coliva on Maddy’s Moore 23.2 Maddy on Coliva’s Moore References 24 Reply to Coliva 24.1 The Nature of the Skeptical Challenge 24.2 The Role of the KK Principle 24.3 The Comparative Strategy References 25 A Carnapian Solution to the Puzzle of Extrinsic Justifications 25.1 Introduction 25.2 The Puzzle of Extrinsic Justifications 25.3 Maddy's Solution 25.4 A Carnapian Solution 25.5 Conclusion References 26 Reply to Soysal References 27 Wittgenstein's Wayward Student: The Unauthorized Autobiography 27.1 First Things 27.2 No Laughing Matter 27.3 Naturalism 27.4 Turtles All The Way Down 27.5 Science 27.6 Second Philosophy 27.7 Mathematics 27.8 The Last Word References 28 Reply to Franks Part IV Mathematical Practice 29 Realizing Infinity References 30 Second Philosophy and the Depth Metaphor 30.1 Introduction 30.2 Rorty and the Literalization of Metaphor 30.3 Wittgenstein's Treatment of the Mathematical Metaphor 30.4 Maddy's Use of the Depth Metaphor References 31 Reply to Kennedy References 32 The Flight from Application Notes 33 Reply to Wilson References Part V Conversations on Set Theory 34 A Conversation Between Penelope Maddy and Menachem Magidor References 35 A Conversation with David Asperó 36 A Conversation with Hugh Woodin 37 On Multiversism 37.1 Background 37.2 A Partial Taxonomy of Multiversisms 37.2.1 Metaphysical Multiversism 37.2.2 Axiomatic Multiversism 37.2.3 Heuristic Multiversism 37.3 Multiversism and the Arealist 37.4 Digression: Two Contrasting Multiversisms 37.5 Conclusions References Name Index Subject Index