Towards a Philosophy of Real Mathematics.

By: Corfield, DavidPublisher: Cambridge : Cambridge University Press, 2003Copyright date: ©2003Description: 1 online resource (300 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780511148897Subject(s): Mathematics--PhilosophyGenre/Form: Electronic books. Additional physical formats: Print version:: Towards a Philosophy of Real MathematicsDDC classification: 510.1 LOC classification: QA8.4 .C67 2003Online resources: Click to View
Contents:
Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- CHAPTER 1 Introduction: a role for history -- 1.1 REAL MATHEMATICS -- 1.2 THE CURRENT STATE OF PLAY -- 1.3 THE FOUNDATIONALIST FILTER -- 1.4 NEW DEBATES FOR THE PHILOSOPHY OF MATHEMATICS -- 1.5 TOWARDS A PHILOSOPHY OF REAL MATHEMATICS -- 1.6 RELATIONS WITH THE PHILOSOPHY OF PHYSICS -- 1.7 FACTORS GOVERNING THE WAY MATHEMATICS PROCEEDS -- 1.8 OUTLINE OF THE BOOK -- PART I Human and artificial mathematicians -- CHAPTER 2 Communicating with automated theorem provers -- 2.1 INTRODUCTION -- 2.2 AUTOMATED MATHEMATICAL REASONING -- 2.3 THE ARGONNE PARADIGM -- 2.4 THE ROBBINS PROBLEM -- 2.5 REFORMULATION -- 2.6 CONCLUSION -- CHAPTER 3 Automated conjecture formation -- 3.1 INTRODUCTION -- 3.2 COMBINATORIAL ENUMERATION -- 3.3 NUMERICAL IDENTITIES AND ENUMERATION -- 3.4 INDUCTIVE LOGIC PROGRAMMING -- 3.5 THE PROBLEM OF THE HOMOTOPY GROUPS OF THE SPHERES -- 3.6 RESULTS AND DIAGNOSIS -- Independent sources -- Language -- 3.7 CONCLUSION -- CHAPTER 4 The role of analogy in mathematics -- 4.1 INTRODUCTION -- 4.2 THOUGHTS ON ANALOGY -- 4.3 THE ANALOGY BETWEEN NUMBER AND FUNCTION -- 4.4 A WATERSHED IN THE USE OF ANALOGY -- 4.5 CONCLUSION -- PART II Plausibility, uncertainty and probability -- CHAPTER 5 Bayesianism in mathematics -- 5.1 INTRODUCTION -- 5.2 PROBABILITY THEORY AS LOGIC -- 5.3 QUANTITATIVE BAYESIANISM -- 5.4 WHAT MIGHT BE ACHIEVED BY BAYESIANISM IN MATHEMATICS -- Analogy -- Strategy -- Enumerative induction -- 5.5 CONCLUSION -- CHAPTER 6 Uncertainty in mathematics and science -- 6.1 INTRODUCTION -- 6.2 FOUR CASE STUDIES OF THE INTERRELATIONSHIP BETWEEN THE PLAUSIBILITY OF SCIENTIFIC HYPOTHESES AND EVIDENCE AND THE… -- The stability of the solar system -- The onset of turbulence -- The Lee-Yang circle theorem -- Quantum field theory.
6.3 MATHEMATICS AND SCIENTIFIC MODELS -- 6.4 THE PROBLEM OF OLD EVIDENCE -- 6.5 CONCLUSION -- PART III The growth of mathematics -- CHAPTER 7 Lakatos's philosophy of mathematics -- 7.1 INTRODUCTION -- 7.2 LAKATO'S STAGES OF DEVELOPMENT -- 7.3 THE METHOD OF PROOFS AND REFUTATIONS -- 7.4 THE RESEARCH PROGRAMME STAGE -- 7.5 LAKATOS ON RIGOUR -- 7.6 THE JAFFE-QUINN DEBATE -- 7.7 CONCLUSION -- CHAPTER 8 Beyond the methodology of mathematical research programmes -- 8.1 INTRODUCTION -- 8.2 PUTTING THE METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES TO THE TEST -- 8.3 DEDEKIND VERSUS KRONECKER -- 8.4 OTHER ATTEMPTS TO CONSTRUCT A METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES -- 8.5 KOETSIER AND THE METHODOLOGY OF MATHEMATICAL RESEARCH TRADITIONS -- 8.6 MADDY'S NATURALISM -- 8.7 CONCLUSION -- CHAPTER 9 The importance of mathematical conceptualisation -- 9.1 VALUES IN MATHEMATICS -- 9.2 WHAT IS A GROUPOID? -- 9.3 HOW GROUPOIDS COMPARE WITH GROUPS -- 9.4 THE FULL EXPLOITATION OF ONE-DIMENSIONALITY -- 9.5 GROUPOID ALGEBRAS USED TO COMPENSATE FOR BAD SPACES -- 9.6 APPLICATIONS AND OLD CONJECTURES -- 9.7 NEW PROSPECTS: HIGHER-DIMENSIONAL ALGEBRA -- 9.8 THE CONCEPTUAL AND THE NATURAL -- 9.9 CONCLUSION -- PART IV The interpretation of mathematics -- CAHPTER 10 Higher-dimensional algebra -- 10.1 INTRODUCTION -- 10.2 CATEGORIFICATION: MAKING DISTINCTIONS -- 10.3 HIGHER-DIMENSIONAL ALGEBRA AS AN ORGANISING LANGUAGE -- 10.4 DIAGRAMMATIC CALCULATION -- 10.5 APPLICATIONS -- 10.6 CONCLUSION -- Appendix -- MONOID -- GROUP -- RING -- IDEAL -- FIELD -- ALGEBRA -- COALGEBRA -- BIALGEBRA -- HOPF ALGEBRA -- CATEGORY -- FUNCTOR -- ADJOINT FUNCTORS -- MORITA EQUIVALENCE -- Bibliography -- Index.
Summary: Corfield sets out a variety of approaches to new thinking about the philosophy of mathematics.
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Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- CHAPTER 1 Introduction: a role for history -- 1.1 REAL MATHEMATICS -- 1.2 THE CURRENT STATE OF PLAY -- 1.3 THE FOUNDATIONALIST FILTER -- 1.4 NEW DEBATES FOR THE PHILOSOPHY OF MATHEMATICS -- 1.5 TOWARDS A PHILOSOPHY OF REAL MATHEMATICS -- 1.6 RELATIONS WITH THE PHILOSOPHY OF PHYSICS -- 1.7 FACTORS GOVERNING THE WAY MATHEMATICS PROCEEDS -- 1.8 OUTLINE OF THE BOOK -- PART I Human and artificial mathematicians -- CHAPTER 2 Communicating with automated theorem provers -- 2.1 INTRODUCTION -- 2.2 AUTOMATED MATHEMATICAL REASONING -- 2.3 THE ARGONNE PARADIGM -- 2.4 THE ROBBINS PROBLEM -- 2.5 REFORMULATION -- 2.6 CONCLUSION -- CHAPTER 3 Automated conjecture formation -- 3.1 INTRODUCTION -- 3.2 COMBINATORIAL ENUMERATION -- 3.3 NUMERICAL IDENTITIES AND ENUMERATION -- 3.4 INDUCTIVE LOGIC PROGRAMMING -- 3.5 THE PROBLEM OF THE HOMOTOPY GROUPS OF THE SPHERES -- 3.6 RESULTS AND DIAGNOSIS -- Independent sources -- Language -- 3.7 CONCLUSION -- CHAPTER 4 The role of analogy in mathematics -- 4.1 INTRODUCTION -- 4.2 THOUGHTS ON ANALOGY -- 4.3 THE ANALOGY BETWEEN NUMBER AND FUNCTION -- 4.4 A WATERSHED IN THE USE OF ANALOGY -- 4.5 CONCLUSION -- PART II Plausibility, uncertainty and probability -- CHAPTER 5 Bayesianism in mathematics -- 5.1 INTRODUCTION -- 5.2 PROBABILITY THEORY AS LOGIC -- 5.3 QUANTITATIVE BAYESIANISM -- 5.4 WHAT MIGHT BE ACHIEVED BY BAYESIANISM IN MATHEMATICS -- Analogy -- Strategy -- Enumerative induction -- 5.5 CONCLUSION -- CHAPTER 6 Uncertainty in mathematics and science -- 6.1 INTRODUCTION -- 6.2 FOUR CASE STUDIES OF THE INTERRELATIONSHIP BETWEEN THE PLAUSIBILITY OF SCIENTIFIC HYPOTHESES AND EVIDENCE AND THE… -- The stability of the solar system -- The onset of turbulence -- The Lee-Yang circle theorem -- Quantum field theory.

6.3 MATHEMATICS AND SCIENTIFIC MODELS -- 6.4 THE PROBLEM OF OLD EVIDENCE -- 6.5 CONCLUSION -- PART III The growth of mathematics -- CHAPTER 7 Lakatos's philosophy of mathematics -- 7.1 INTRODUCTION -- 7.2 LAKATO'S STAGES OF DEVELOPMENT -- 7.3 THE METHOD OF PROOFS AND REFUTATIONS -- 7.4 THE RESEARCH PROGRAMME STAGE -- 7.5 LAKATOS ON RIGOUR -- 7.6 THE JAFFE-QUINN DEBATE -- 7.7 CONCLUSION -- CHAPTER 8 Beyond the methodology of mathematical research programmes -- 8.1 INTRODUCTION -- 8.2 PUTTING THE METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES TO THE TEST -- 8.3 DEDEKIND VERSUS KRONECKER -- 8.4 OTHER ATTEMPTS TO CONSTRUCT A METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES -- 8.5 KOETSIER AND THE METHODOLOGY OF MATHEMATICAL RESEARCH TRADITIONS -- 8.6 MADDY'S NATURALISM -- 8.7 CONCLUSION -- CHAPTER 9 The importance of mathematical conceptualisation -- 9.1 VALUES IN MATHEMATICS -- 9.2 WHAT IS A GROUPOID? -- 9.3 HOW GROUPOIDS COMPARE WITH GROUPS -- 9.4 THE FULL EXPLOITATION OF ONE-DIMENSIONALITY -- 9.5 GROUPOID ALGEBRAS USED TO COMPENSATE FOR BAD SPACES -- 9.6 APPLICATIONS AND OLD CONJECTURES -- 9.7 NEW PROSPECTS: HIGHER-DIMENSIONAL ALGEBRA -- 9.8 THE CONCEPTUAL AND THE NATURAL -- 9.9 CONCLUSION -- PART IV The interpretation of mathematics -- CAHPTER 10 Higher-dimensional algebra -- 10.1 INTRODUCTION -- 10.2 CATEGORIFICATION: MAKING DISTINCTIONS -- 10.3 HIGHER-DIMENSIONAL ALGEBRA AS AN ORGANISING LANGUAGE -- 10.4 DIAGRAMMATIC CALCULATION -- 10.5 APPLICATIONS -- 10.6 CONCLUSION -- Appendix -- MONOID -- GROUP -- RING -- IDEAL -- FIELD -- ALGEBRA -- COALGEBRA -- BIALGEBRA -- HOPF ALGEBRA -- CATEGORY -- FUNCTOR -- ADJOINT FUNCTORS -- MORITA EQUIVALENCE -- Bibliography -- Index.

Corfield sets out a variety of approaches to new thinking about the philosophy of mathematics.

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