Cover -- Copyright page -- Title page -- Contents -- Acknowledgements -- Introduction -- 1 The Purpose of This Book -- 2 What is not Included in This Book -- 3 A Brief History of The Philosophy of Mathematics to About 1850 -- 4 The Foundational Problems and the Three Foundational Schools -- 4.1 Other Philosophers in This Period -- 5 More Recent Work That is Worth Reading but is Not Represented Here -- 5.1 Logicians with a Philosophical Bent -- 5.2 Philosophers -- 5.3 People Working in the History and Philosophy of Mathematics -- 5.4 People Working in Mathematics Education -- 5.5 Mathematicians -- 6 A Brief Overview of This Book -- 6.1 Views on Mathematical Objects -- 6.2 Views on Proof -- 6.3 What is Mathematics? -- 6.4 Social Constructivism -- 6.5 Philosophy of Probability -- Bibliography -- I Proof and How it is Changing -- 1 Proof: Its Nature and Significance, Michael Detlefsen -- From the Editors -- 1 Introduction -- 2 Empirical Reasoning in Mathematics: Historical Background -- 3 Empirical Reasoning in Mathematics: Recent Proposals -- 4 Formalization and Rigor -- 5 Visualization and Diagrammatic Reasoning in Mathematics -- 6 Concluding Thoughts -- References and Bibliography -- 2 Implications of Experimental Mathematics for the Philosophy of Mathematics, Jonathan Borwein -- From the Editors -- 1 Mathematical Knowledge as I View It -- 2 Introduction -- 3 Philosophy of Experimental Mathematics -- 4 Our Experimental Mathodology -- 5 Finding Things versus Proving Things -- 6 Conclusions -- References -- 3 On the Roles of Proof in Mathematics, Joseph Auslander -- From the Editors -- 1 Proof as a Defining Feature of Mathematics -- 2 The Roles of Proof -- 3 Computers and Proof -- 4 Four Examples -- References -- II Social Constructivist Views of Mathematics -- 4 When Is a Problem Solved ?, Philip J. Davis -- From the Editors.

1 Introduction -- 2 A Bit of Philosophy -- 3 What Might Elizabeth Have Meant? -- 4 Mathematical Argumentation as a Mixture of Materials -- 5 From a Mathematician's Perspective -- 6 When is a Proof Complete? -- 7 Applied Mathematics -- 8 Some Historical Perspectives -- 9 A Dialogue on When is a Theory Complete -- 10 A Possible Example of Renewal from the Outside -- 11 Implications for Mathematical Education -- Bibliography -- 5 Mathematical Practice as a Scientific Problem, Reuben Hersh -- From the Editors -- 1 Introduction -- 2 Atiyah's Pleasant Surprise -- 3 For a Multi-Disciplined Study of Mathematical Practice -- 4 Definition of "Mathematical Object" -- 5 The Basic Problem -- 6 Timely or Timeless? -- 7 Educational Implications -- 8 Conclusion -- References -- 6 Mathematical Domains: Social Constructs?, Julian Cole -- From the Editors -- 1 Introduction -- 2 Ernest's and Hersh's View of Mathematics -- 3 Social Construction and Dependence -- 4 Logic and Ontological Structure -- 5 Abstract Entities -- 6 Why Accept Practice-Dependent Realism? -- 7 Platonism and Epistemology -- 8 Platonism vs. Practice-Dependent Realism -- 9 Conclusion -- References -- III The Nature of Mathematical Objects and Mathematical Knowledge -- 7 The Existence of Mathematical Objects, Charles Chihara -- From the Editors -- 1 Introduction -- 2 What is Philosophy? -- 3 The Platonic (Realistic) Conception of Mathematics -- 4 Reasons for Accepting the Realist's View -- 5 The Hilbert-Frege Dispute -- 6 Mathematics Regarded as a Theory About Structures -- 7 The Structural Content of Theorems of Mathematics -- 8 A Structural Account of Applications of Mathematics -- 9 Fermat's Last Theorem -- 10 The Big Picture -- References -- 8 Mathematical Objects, Stewart Shapiro -- From the Editors -- 1 Battle Lines.

2 What Mathematical Objects are Like, or Would be Like if they Existed -- 3 A Dilemma -- 4 The Irrealist Horn -- 5 The Realist Horn -- 6 A Matter of Meaning -- 7 Mathematics is the Science of Structure -- References and Further Reading -- 9 Mathematical Platonism, Mark Balaguer -- From the Editors -- 1 Introduction -- 2 The Fregean Argument for Mathematical Platonism -- 3 The Epistemological Argument Against Platonism -- 4 Concluding Remarks -- Bibliography -- 10 The Nature of Mathematical Objects, Øystein Linnebo -- From the Editors -- 1 Frege's Argument for Mathematical Platonism -- 2 Two Challenges to Mathematical Platonism -- 3 From Objects to Semantic Values -- 4 Reference to Physical Bodies -- 5 Reference to Natural Numbers -- 6 The "Thinness" of the Natural Numbers -- 7 Back to the Two Challenges -- 8 Conclusion -- References -- 11 When is One Thing Equal to Some Other Thing?, Barry Mazur -- From the Editors -- 1 The Awkwardness of Equality -- 2 Defining Natural Numbers -- 3 Objects versus Structure -- 4 Category Theory as Balancing Act Rather Than "Foundations" -- 5 Example: The Category of Sets -- 6 Class as a Library With Strict Rules for Taking Out Books -- 7 Category -- 8 Equality versus Isomorphism -- 9 An Example of Categorical Vocabulary: Initial Objects -- 10 Defining the Natural Numbers as an "Initial Object" -- 11 Light, Shadow, Dark -- 12 Representing One Theory in Another -- 13 Mapping One Functor to Another -- 14 An Object "as" a Functor from the Theory-in-Which-it-Livesto Set Theory -- 15 Representable Functors -- 16 The Natural Numbers as Functor -- 17 Equivalence of Categories -- 18 Object and Problem -- 19 Object and Equality -- References -- IV The Nature of Mathematics and its Applications -- 12 Extreme Science: Mathematics as the Scienceof Relations as Such, R. S. D. Thomas -- From the Editors -- 1 Introduction.

2 Historical Context -- 3 Scientific Context -- 4 Mathematics as a Science -- 5 Ontological Consequences -- 6 Epistemological Consequences -- References -- 13 What is Mathematics? A Pedagogical Answer to a Philosophical Question, Guershon Harel -- From the Editors -- 0 Introduction -- 1 Mental Act, Way of Understanding, and Way of Thinking -- 2 A Definition of Mathematics: Epistemological Considerations and Pedagogical Implications -- 3 Long-Term Curricular and Research Goals -- 4 DNR Based Instruction in Mathematics -- 5 Summary -- References -- 14 What Will Count as Mathematics in 2100?, Keith Devlin -- From the Editors -- 1 What is Mathematics Today? -- 2 The Last Revolution in Mathematics -- 3 How and Why Mathematics Changes -- 4 Bernoulli's Utility Concept -- 5 Bayesian Inference -- 6 Black-Scholes Theory -- 7 Mathematical Theories of Language -- 8 Grice's Maxims -- 9 Conversational Implicature -- 10 Sociolinguistics -- 11 Why this Will be Viewed as Mathematics -- References -- 15 Mathematics Applied: The Case of Addition, Mark Steiner -- From the Editors -- References -- 16 Probability-A Philosophical Overview, Alan Hájek -- From the Editors -- 1 Personal and Pedagogical Prologue -- 2 Introduction -- 3 The Formal Theory of Probability -- 4 Interpretations of Probability -- 5 Conclusion -- 6 Personal and Pedagogical Epilogue -- Bibliography -- Glossary of Common Philosophical Terms -- About the Editors.