Mathematical Knowledge

what Is The Nature Of Mathematical Knowledge? Is It Anything Like Scientific Knowledge Or Is It Sui Generis? How Do We Acquire It? Should We Believe What Mathematicians Themselves Tell Us About It? Are Mathematical Concepts Innate Or Acquired? Eight New Essays Offer Answers To These And Many Other Questions. Written By Some Of The World's Leading Philosophers Of Mathematics, Psychologists, And Mathematicians, mathematical Knowledge Gives A Lively Sense Of The Current State Of Debate In This Fascinating Field. Preface......Page 6 List of contributors......Page 8 Contents......Page 10 1 Benacerraf’s worry......Page 12 2 Mathematical knowers and mathematical knowledge......Page 18 1 What is distinctive about the mathematical case?......Page 27 2 Implicationism all the way down?......Page 28 3 Mother theories......Page 30 4 Understanding and truth......Page 32 5 Why externalism?......Page 33 6 The route to knowledge......Page 35 7 Benacerraf’s problem......Page 36 8 Benacerraf’s problem generalized......Page 38 9 The real problem......Page 40 1 Introduction......Page 44 2 The problem of induction in mathematics......Page 45 3 The seemingly inappropriate use of modal language by mathematicians......Page 46 4 Explicating informal mathematical terminology......Page 48 5 Memory......Page 49 6 Direct memory, competence, and generated memory......Page 52 7 Two proofs and how one remembers them......Page 54 8 Mental arithmetic and the concept of width......Page 60 9 How does the notion of width apply to proofs?......Page 65 10 Concluding remarks......Page 69 1 Introduction......Page 70 2 Enumerative induction and discovery......Page 71 3 The descriptive question: Two case studies......Page 72 4 Hume’s problem of induction......Page 75 5 The normative question: Is enumerative induction in mathematics rationally justified?......Page 76 6 Re-examining the descriptive question......Page 79 7 Conclusions......Page 83 Marinella CAPPELLETTI & Valeria GIARDINO: The cognitive basis of mathematical knowledge......Page 85 1 Numerical cognition is innate and distinct from other cognitive skills......Page 86 2 Numbers and language......Page 90 3 Different uses of the same numeral......Page 92 4 Limits of cognitive science......Page 93 5 Concluding remarks......Page 94 Mary LENG: What’s there to know?......Page 95 1 Trading ontology for modality......Page 97 2 Fictionalism and nominalism......Page 101 3 A new Benacerraf problem?......Page 105 4 A new indispensability argument?......Page 114 5 Defending the consistency of ZFC and PA......Page 115 6 Conclusion......Page 118 1 Empiricism in the philosophy of mathematics......Page 120 2 An empiricist account of mathematical knowledge......Page 121 3 Unapplied mathematics as mathematical recreation......Page 123 4 Is all mathematics recreation?......Page 128 5 Empiricism revisited......Page 133 2 Preliminaries......Page 134 3 The pragmatic and indifference objections......Page 141 4 Weak and strong scientific platonism......Page 144 5 For the indifference objection......Page 147 6 The publication test......Page 149 7 General principles of scientific method......Page 151 8 Scientific grounds and the actual content of mathematics......Page 157 9 Conclusion......Page 160 Crispin WRIGHT: On quantifying into predicate position......Page 161 1 Basic idea and project outline......Page 164 2 Fixing the meanings of the quantifiers......Page 166 3 Extreme neutralism......Page 170 4 A neutralist heuristic......Page 171 5 Comprehension......Page 175 6 Incompleteness......Page 177 7 Impredicativity......Page 180 8 Appendix: Abstractionist Mathematical Theories......Page 182 Bibliography......Page 186 Index......Page 195 Preface 6 List of contributors 8 Contents 10 Mary LENG: Introduction 12 1 Benacerraf’s worry 12 2 Mathematical knowers and mathematical knowledge 18 Michael POTTER: What is the problem of mathematical knowledge? 27 1 What is distinctive about the mathematical case? 27 2 Implicationism all the way down? 28 3 Mother theories 30 4 Understanding and truth 32 5 Why externalism? 33 6 The route to knowledge 35 7 Benacerraf’s problem 36 8 Benacerraf’s problem generalized 38 9 The real problem 40 W. T. GOWERS: Mathematics, Memory, and Mental Arithmetic 44 1 Introduction 44 2 The problem of induction in mathematics 45 3 The seemingly inappropriate use of modal language by mathematicians 46 4 Explicating informal mathematical terminology 48 5 Memory 49 6 Direct memory, competence, and generated memory 52 7 Two proofs and how one remembers them 54 8 Mental arithmetic and the concept of width 60 9 How does the notion of width apply to proofs? 65 10 Concluding remarks 69 Alan BAKER: Is there a problem of induction for mathematics? 70 1 Introduction 70 2 Enumerative induction and discovery 71 3 The descriptive question: Two case studies 72 4 Hume’s problem of induction 75 5 The normative question: Is enumerative induction in mathematics rationally justified? 76 6 Re-examining the descriptive question 79 7 Conclusions 83 Marinella CAPPELLETTI & Valeria GIARDINO: The cognitive basis of mathematical knowledge 85 1 Numerical cognition is innate and distinct from other cognitive skills 86 2 Numbers and language 90 3 Different uses of the same numeral 92 4 Limits of cognitive science 93 5 Concluding remarks 94 Mary LENG: What’s there to know? 95 1 Trading ontology for modality 97 2 Fictionalism and nominalism 101 3 A new Benacerraf problem? 105 4 A new indispensability argument? 114 5 Defending the consistency of ZFC and PA 115 6 Conclusion 118 Mark COLYVAN: Mathematical recreation versus mathematical knowledge 120 1 Empiricism in the philosophy of mathematics 120 2 An empiricist account of mathematical knowledge 121 3 Unapplied mathematics as mathematical recreation 123 4 Is all mathematics recreation? 128 5 Empiricism revisited 133 Alexander PASEAU: Scientific Platonism 134 1 Introduction 134 2 Preliminaries 134 3 The pragmatic and indifference objections 141 4 Weak and strong scientific platonism 144 5 For the indifference objection 147 6 The publication test 149 7 General principles of scientific method 151 8 Scientific grounds and the actual content of mathematics 157 9 Conclusion 160 Crispin WRIGHT: On quantifying into predicate position 161 1 Basic idea and project outline 164 2 Fixing the meanings of the quantifiers 166 3 Extreme neutralism 170 4 A neutralist heuristic 171 5 Comprehension 175 6 Incompleteness 177 7 Impredicativity 180 8 Appendix: Abstractionist Mathematical Theories 182 Bibliography 186 Index 195 What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. - ;What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and "What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field."--Publisher's description