# Proof and Other Dilemmas: Mathematics and Philosophy

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*Bonnie Gold; Roger A. Simons*

MAA Press: An Imprint of the American Mathematical Society

During the first 75 years of the twentieth century, almost all work in the philosophy of mathematics concerned foundational questions. In the last quarter of the century, philosophers of mathematics began to return to basic questions concerning the philosophy of mathematics, such as "what is the nature of mathematical knowledge and of mathematical objects" and "how is mathematics related to science." Two new schools of philosophy of mathematics, social constructivism and structuralism, were added to the four traditional views (formalism, intuitionalism, logicism, and platonism). The advent of the computer led to proofs and the development of mathematics assisted by computer and to questions of the role of the computer in mathematics. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Approximately half the essays were written by mathematicians and consider questions that philosophers are not yet discussing. The other half, written by philsophers of mathematics, summarize the discussion in that community during the last 35 years. In each case, connections are made to issues relevant to the teaching of mathematics.

# Table of Contents

## Proof and Other Dilemmas: Mathematics and Philosophy

- Cover cover11
- Copyright page ii3
- Title page iii4
- Contents ix10
- Acknowledgements xi12
- Introduction xiii14
- 1 The Purpose of This Book xiii14
- 2 What is not Included in This Book xiv15
- 3 A Brief History of The Philosophy of Mathematics to About 1850 xiv15
- 4 The Foundational Problems and the Three Foundational Schools xvii18
- 5 More Recent Work That is Worth Reading but is Not Represented Here xxiv25
- 6 A Brief Overview of This Book xxvii28
- Bibliography xxix30

- I Proof and How it is Changing 134
- 1 Proof: Its Nature and Significance, Michael Detlefsen 336
- From the Editors 336
- 1 Introduction 437
- 2 Empirical Reasoning in Mathematics: Historical Background 639
- 3 Empirical Reasoning in Mathematics: Recent Proposals 942
- 4 Formalization and Rigor 1649
- 5 Visualization and Diagrammatic Reasoning in Mathematics 1952
- 6 Concluding Thoughts 2760
- References and Bibliography 2861

- 2 Implications of Experimental Mathematics for the Philosophy of Mathematics, Jonathan Borwein 3366
- 3 On the Roles of Proof in Mathematics, Joseph Auslander 6194

- II Social Constructivist Views of Mathematics 79112
- 4 When Is a Problem Solved ?, Philip J. Davis 81114
- From the Editors 81114
- 1 Introduction 82115
- 2 A Bit of Philosophy 82115
- 3 What Might Elizabeth Have Meant? 83116
- 4 Mathematical Argumentation as a Mixture of Materials 84117
- 5 From a Mathematician’s Perspective 85118
- 6 When is a Proof Complete? 88121
- 7 Applied Mathematics 88121
- 8 Some Historical Perspectives 89122
- 9 A Dialogue on When is a Theory Complete 90123
- 10 A Possible Example of Renewal from the Outside 92125
- 11 Implications for Mathematical Education 92125
- Bibliography 93126

- 5 Mathematical Practice as a Scientific Problem, Reuben Hersh 95128
- From the Editors 95128
- 1 Introduction 96129
- 2 Atiyah’s Pleasant Surprise 96129
- 3 For a Multi-Disciplined Study of Mathematical Practice 98131
- 4 Definition of “Mathematical Object” 99132
- 5 The Basic Problem 101134
- 6 Timely or Timeless? 103136
- 7 Educational Implications 105138
- 8 Conclusion 105138
- References 105138

- 6 Mathematical Domains: Social Constructs?, Julian Cole 109142
- From the Editors 109142
- 1 Introduction 109142
- 2 Ernest’s and Hersh’s View of Mathematics 110143
- 3 Social Construction and Dependence 113146
- 4 Logic and Ontological Structure 116149
- 5 Abstract Entities 119152
- 6 Why Accept Practice-Dependent Realism? 121154
- 7 Platonism and Epistemology 123156
- 8 Platonism vs. Practice-Dependent Realism 125158
- 9 Conclusion 127160
- References 127160

- III The Nature of Mathematical Objects and Mathematical Knowledge 129162
- 7 The Existence of Mathematical Objects, Charles Chihara 131164
- From the Editors 131164
- 1 Introduction 132165
- 2 What is Philosophy? 132165
- 3 The Platonic (Realistic) Conception of Mathematics 133166
- 4 Reasons for Accepting the Realist’s View 138171
- 5 The Hilbert-Frege Dispute 140173
- 6 Mathematics Regarded as a Theory About Structures 142175
- 7 The Structural Content of Theorems of Mathematics 144177
- 8 A Structural Account of Applications of Mathematics 146179
- 9 Fermat’s Last Theorem 149182
- 10 The Big Picture 152185
- References 154187

- 8 Mathematical Objects, Stewart Shapiro 157190
- From the Editors 157190
- 1 Battle Lines 158191
- 2 What Mathematical Objects are Like, or Would be Like if they Existed 160193
- 3 A Dilemma 160193
- 4 The Irrealist Horn 162195
- 5 The Realist Horn 164197
- 6 A Matter of Meaning 168201
- 7 Mathematics is the Science of Structure 172205
- References and Further Reading 175208

- 9 Mathematical Platonism, Mark Balaguer 179212
- 10 The Nature of Mathematical Objects, Øystein Linnebo 205238
- From the Editors 205238
- 1 Frege’s Argument for Mathematical Platonism 206239
- 2 Two Challenges to Mathematical Platonism 207240
- 3 From Objects to Semantic Values 209242
- 4 Reference to Physical Bodies 210243
- 5 Reference to Natural Numbers 212245
- 6 The “Thinness” of the Natural Numbers 214247
- 7 Back to the Two Challenges 216249
- 8 Conclusion 218251
- References 218251

- 11 When is One Thing Equal to Some Other Thing?, Barry Mazur 221254
- From the Editors 221254
- 1 The Awkwardness of Equality 222255
- 2 Defining Natural Numbers 224257
- 3 Objects versus Structure 225258
- 4 Category Theory as Balancing Act Rather Than “Foundations” 226259
- 5 Example: The Category of Sets 226259
- 6 Class as a Library With Strict Rules for Taking Out Books 227260
- 7 Category 228261
- 8 Equality versus Isomorphism 229262
- 9 An Example of Categorical Vocabulary: Initial Objects 230263
- 10 Defining the Natural Numbers as an “Initial Object” 231264
- 11 Light, Shadow, Dark 232265
- 12 Representing One Theory in Another 233266
- 13 Mapping One Functor to Another 234267
- 14 An Object “as” a Functor from the Theory-in-Which-it-Livesto Set Theory 235268
- 15 Representable Functors 237270
- 16 The Natural Numbers as Functor 238271
- 17 Equivalence of Categories 238271
- 18 Object and Problem 239272
- 19 Object and Equality 240273
- References 240273

- IV The Nature of Mathematics and its Applications 243276
- 12 Extreme Science: Mathematics as the Scienceof Relations as Such, R. S. D. Thomas 245278
- 13 What is Mathematics? A Pedagogical Answer to a Philosophical Question, Guershon Harel 265298
- From the Editors 265298
- 0 Introduction 266299
- 1 Mental Act, Way of Understanding, and Way of Thinking 268301
- 2 A Definition of Mathematics: Epistemological Considerations and Pedagogical Implications 272305
- 3 Long-Term Curricular and Research Goals 277310
- 4 DNR Based Instruction in Mathematics 283316
- 5 Summary 286319
- References 288321

- 14 What Will Count as Mathematics in 2100?, Keith Devlin 291324
- From the Editors 291324
- 1 What is Mathematics Today? 292325
- 2 The Last Revolution in Mathematics 293326
- 3 How and Why Mathematics Changes 294327
- 4 Bernoulli’s Utility Concept 295328
- 5 Bayesian Inference 297330
- 6 Black-Scholes Theory 299332
- 7 Mathematical Theories of Language 300333
- 8 Grice’s Maxims 303336
- 9 Conversational Implicature 304337
- 10 Sociolinguistics 305338
- 11 Why this Will be Viewed as Mathematics 309342
- References 311344

- 15 Mathematics Applied: The Case of Addition, Mark Steiner 313346
- 16 Probability—A Philosophical Overview, Alan Hájek 323356

- Glossary of Common Philosophical Terms 341374
- About the Editors 345378