New Waves in Philosophy of Mathematics (New Waves in Philosophy)

In this book, thirteen promising young researchers write on what they take to be the right philosophical account of mathematics and discuss where the philosophy of mathematics ought to be going. New trends are revealed, such as an increasing attention to mathematical practice, a reassessment of the canon, and inspiration from philosophical logic. Cover......Page 1 Title Page......Page 5 Contents......Page 7 Series Editors’ Foreword......Page 9 Acknowledgements......Page 10 List of Contributors......Page 11 The beginning of modern philosophy of mathematics......Page 13 The recent past......Page 14 The contemporary debate......Page 15 Part I. Reassessing the orthodoxy in the philosophy of mathematics......Page 16 Part II. The question of realism in mathematics......Page 17 Part III. Mathematical practice and the methodology of mathematics......Page 18 Part IV. Mathematical language and the psychology of mathematics......Page 19 Part V. From philosophical logic to the philosophy of mathematics......Page 20 Note......Page 21 Part I. Reassessing the Orthodoxy in the Philosophy of Mathematics......Page 23 1 Motivations philosophical and personal......Page 25 2 Fregean philosophy of mathematics......Page 27 3 The Neo-Fregean picture......Page 33 Notes......Page 42 References......Page 45 1 Reductionism......Page 47 1.3 Partial-denotation reductionism......Page 48 2 Reductionism’s response......Page 49 2.1 Context-independent objectual reductionism......Page 50 2.3 Structural reductionism......Page 51 3.1 Semantic objections......Page 52 3.2 Slippery slope objection......Page 56 3.3 Epistemological objection......Page 57 4 Meaning analysis versus explication......Page 58 Notes......Page 63 References......Page 66 Part II. The Question of Realism in Mathematics......Page 69 1 Introduction: Platonism and nominalism......Page 71 3.1 Fictionalism and nominalism......Page 75 3.2.1 The crucial idea......Page 76 3.2.2 Meeting the desiderata......Page 78 3.3.1 The crucial point......Page 82 3.3.2 On the existence of mathematical objects......Page 83 3.3.3 Meeting the desiderata......Page 85 Notes......Page 88 References......Page 90 4 Truth in Mathematics: The Question of Pluralism......Page 92 1 The emergence of pluralism......Page 93 1.2 Kant......Page 94 1.3 Reichenbach......Page 95 1.4 Carnap......Page 96 2.1.1 Some key terminology......Page 97 2.1.2 The analytic/synthetic distinction......Page 98 2.1.3 Criticism #1: The argument from free parameters......Page 100 2.1.4 Criticism #2: The argument from assessment sensitivity......Page 101 2.2 Radical pluralism......Page 102 2.3 Philosophy as logical syntax......Page 104 2.3.1 Conclusion......Page 106 3 A new orientation......Page 108 4 The initial stretch: First- and second-order arithmetic......Page 110 4.2 The problem of selection for second-order arithmetic......Page 111 5.1 An initial pass......Page 116 5.2 A more promising approach......Page 117 Notes......Page 122 References......Page 127 5 “Algebraic” Approaches to Mathematics......Page 129 1 “Assertory” views of mathematics and Benacerraf’s problems......Page 130 2 “Algebraic” views and Benacerraf’s problems......Page 132 3.1 Modal structuralism......Page 133 3.2 Ante rem structuralism......Page 134 3.3 Full-blooded Platonism......Page 135 3.4 Fictionalism......Page 136 4.1 The problem of modality......Page 137 4.2 The problem of mixed claims......Page 139 Notes......Page 143 References......Page 145 Part III. Mathematical Practice and the Methodology of Mathematics......Page 147 1 Introduction......Page 149 1.2 Explanation in mathematics (intertheoretic)......Page 150 1.4 Explaining the role of mathematics in science......Page 151 2 Mathematical coincidences......Page 152 3 From mathematical coincidences to accidental generalizations......Page 154 3.2 Natural kinds......Page 155 3.3 Explanation......Page 156 4.1 The Goldbach conjecture......Page 157 4.2 The Four-Color Theorem......Page 159 5 Mathematical accidents defined......Page 160 6.1 The end of explanation......Page 162 6.2 Axiom choice......Page 164 7 Intertheoretic mathematical accidents......Page 165 8 Conclusions......Page 167 8.2 Explanatory basis......Page 168 Notes......Page 169 Bibliography......Page 170 1 Introduction......Page 172 2 Indispensability of inconsistent mathematical objects......Page 173 3 A philosophical account of applied mathematics......Page 175 Notes......Page 180 References......Page 182 I......Page 185 II......Page 186 III......Page 189 IV......Page 196 V......Page 200 A1. From the Navier-Stokes equations to the Euler equations......Page 202 A2. From the Navier-Stokes equations to the boundary layer equations......Page 203 Notes......Page 204 References......Page 205 Part IV. Mathematical Language and the Psychology of Mathematics......Page 207 1 How can we do better?......Page 209 2 Formal tools......Page 210 3 Two projects in the philosophy of mathematics......Page 212 4 Mathematical activity......Page 213 5 Mathematical language......Page 214 5.1 Syntax......Page 215 5.2 Number words......Page 216 5.3 Quantifiers......Page 219 6 Inferential relations......Page 220 7 The status of axioms......Page 222 8 An epistemic role?......Page 224 9 Pluralism......Page 225 10 Conclusion......Page 229 Notes......Page 230 References......Page 231 1 Introduction......Page 232 2 Individuation and criteria of identity......Page 233 3 The individuation of the natural numbers......Page 235 4 Against the cardinal conception......Page 237 4.1 The objection from special numbers......Page 238 4.2 The objection from the philosophy of language......Page 239 4.4 Alleged advantages of the cardinal conception......Page 240 5.1 Refining the criterion of identity......Page 241 5.2 Justifying the axioms of Dedekind-Peano Arithmetic......Page 243 6 The metaphysical status of natural numbers......Page 245 Notes......Page 247 References......Page 249 1 Against conventional wisdom......Page 251 2.1 Intelligibility......Page 260 2.2 Intelligibility and identity......Page 261 2.3 Knowledge......Page 266 References......Page 271 Part V. From Philosophical Logic to the Philosophy of Mathematics......Page 273 12 On Formal and Informal Provability......Page 275 1 Preliminary remarks: Provable, proof, and proving......Page 277 2 Formal versus informal provability: A conceptual distinction......Page 279 3 A Gödelian perspective on informal provability......Page 285 4 The logic of informal provability......Page 295 5 Are there true but informally unprovable statements?......Page 299 Notes......Page 304 References......Page 308 2 The problem of absolute generality......Page 312 3 Indefinite extensibility without a domain......Page 317 3.1 Two candidate restrictions of plural comprehension......Page 319 3.2 Quantification without a domain......Page 323 4 The costs of the restriction......Page 325 4.1 Second-order ZFC(U)......Page 326 4.2 Proper classes......Page 327 4.3 Reflection......Page 328 5 Conclusion......Page 331 Notes......Page 332 References......Page 334 Index......Page 336 13 Promising Researchers Write On What They Take To Be The Right Philosophical Account Of Mathematics And Discuss Where The Philosophy Of Mathematics Ought To Be Going. New Trends Are Revealed, Such As An Increasing Attention To Mathematical Practice, A Reassessment Of The Canon, And Inspiration From Philosophical Logic. Philosophy Of Mathematics : Old And New / Otávio Bueno And Øystein Linnebo -- New Waves On An Old Beach : Fregean Philosophy Of Mathematics Today / Roy T. Cook -- Reducing Arithmetic To Set Theory / Alexander Paseau -- Mathematical Fictionalism / Otávio Bueno -- Truth In Mathematics : The Question Of Pluralism / Peter Koellner -- Algebraic Approaches To Mathematics / Mary Leng -- Mathematical Accidents And The End Of Explanation / Alan Baker -- Applying Inconsistent Mathematics / Mark Colyvan -- Towards A Philosophy Of Applied Mathematics / Christopher Pincock -- Formal Tools And The Philosophy Of Mathematics / Thomas Hofweber -- The Individuation Of The Natural Numbers / Øystein Linnebo -- Toward A Trivialist Account Of Mathematics / Agustín Rayo -- On Formal And Informal Provability / Hannes Leitgeb -- Quantification Without A Domain / Gabriel Uzquiano. Edited By Otávio Bueno, Øystein Linnebo. Includes Bibliographical References And Index. Thirteen promising young researchers write on what they take to be the right philosophical account of mathematics and discuss where the philosophy of mathematics ought to be going. New trends are revealed, such as an increasing attention to mathematical practice, a reassessment of the canon, and inspiration from philosophical logic. Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic

thirteen Up-and-coming Researchers In The Philosophy Of Mathematics Have Been Invited To Write On What They Take To Be The Right Philosophical Account Of Mathematics, Examining Along The Way Where They Think The Philosophy Of Mathematics Is And Ought To Be Going. As One Might Expect, A Rich And Diverse Picture Emerges. Some Broader Tendencies Can Nevertheless Be Detected: There Is Increasing Attention To The Practice, Language, And Psychology Of Mathematics, A Move To Reassess The Orthodoxy, As Well As Inspiration From Philosophical Logic.