Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures
Brown (James Robert)
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BOOK ABSTRACT:

Follow Link for James Brown’s web-page, and for comments on the book (which has just been replaced by a second edition), follow Link. Follow Link for his Wikipedia entry, which notes that his main claim to fame is in the area of Thought Experiments1, so I may need to follow him up in this area in due course.

Amazon Product Description

    In his long-awaited new edition of "Philosophy of Mathematics", James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value? This clear and engaging book takes a unique approach, encompassing non-standard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of "Philosophy of Mathematics" contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis.
Contents
  1. Preface and Acknowledgements - xi
  2. Chapter 1: Introduction: The Mathematical Image - 1
  3. Chapter 2: Platonism - 8
  4. Chapter 3: Picture-proofs and Platonism - 25
  5. Chapter 4: What is Applied Mathematics? - 46
  6. Chapter 5: Hilbert and Godel - 62
  7. Chapter 6: Knots and Notation - 79
  8. Chapter 7: What is a Definition? - 94
  9. Chapter 8: Constructive Approaches - 113
  10. Chapter 9: Proofs, Pictures and Procedures in Wittgenstein - 130
  11. Chapter 10: Computation, Proof and Conjecture - 154
  12. Chapter 11: Calling the Bluff - 172
  13. Notes - 193
  14. Bibliography - 199
  15. Index - 208



"Brown (James Robert) - Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures"

Source: Brown (James Robert) - Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures


Contents
    Preface and Acknowledgements - xi
  1. Introduction: The Mathematical Image - 1
  2. Platonism - 8
    • The Original Platonist - 8
    • Some Recent Views - 9
    • What is Platonism? - 11
    • The Problem of Access - 15
    • The Problem of Certainty - 18
    • Platonism and its Rivals - 23
  3. Picture-proofs and Platonism - 25
    • Bolzano's ‘Purely Analytic Proof’ - 25
    • What Did Bolzano Do? - 28
    • Different Theorems, Different Concepts? - 29
    • Inductive Mathematics - 30
    • Special and General Case - 33
    • Instructive Examples - 34
    • Representation - 37
    • A Kantian Objection - 39
    • Three Analogies - 40
    • Are Pictures Explanatory? - 42
    • So Why Worry? - 42
    • Appendix - 43
  4. What is Applied Mathematics? - 46
    • Representations - 47
    • Artifacts - 49
    • Bogus Applications - 51
    • Does Science Need Mathematics? - 52
    • Representation vs. Description - 55
    • Structuralism - 57
  5. Hilbert and Godel - 62
    • The Nominalistic Instinct - 62
    • Early Formalism - 63
    • Hilbert's Formalism - 64
    • Hilbert's Programme - 68
    • Small Problems - 70
    • Godel's Theorem - 71
    • Godel's Second - Theorem 75
    • The Upshot for Hilbert's Programme - 77
    • The Aftermath - 77
  6. Knots and Notation - 79
    • Knots - 81
    • The Dowker Notation - 83
    • The Conway Notation - 84
    • Polynomials - 86
    • Creation or Revelation? - 88
    • Sense, Reference and Something Else – 92
  7. What is a Definition? - 94
    • The Official View - 94
    • The Frege-Hilbert Debate - 95
    • Reductionism - 102
    • Graph Theory - 103
    • Lakatos - 107
    • Concluding Remarks - 112
  8. Constructive Approaches - 113
    • From Kant to Brouwer - 114
    • Brouwer's Intuitionism - 115
    • Bishop's Constructivism - 117
    • Dummett's Anti-realism - 118
    • Logic - 120
    • Problems - 122
  9. Proofs, Pictures and Procedures in Wittgenstein - 130
    • A Picture and a Problem - 130
    • Following a Rule - 132
    • Platonism - 136
    • Algorithms - 138
    • Dispositions - 138
    • Knowing Our Own Intentions - 139
    • Brouwer's Beetle - 139
    • Radical Conventionalism - 140
    • Bizarre Examples - 141
    • Naturalism - 142
    • The Sceptical Solution - 144
    • Modus Ponens or Modus Tollens? - 145
    • What is a Rule? - 146
    • Grasping a Sense - 147
    • Platonism vs. Realism - 149
    • Surveyability - 151
    • The Sense of a Picture - 152
  10. Computation, Proof and Conjecture - 154
    • The Four Colour Theorem - 154
    • Fallibility - 155
    • Surveyability - 157
    • Inductive Mathematics - 158
    • Perfect Numbers - 160
    • Computation - 162
    • Is it Normal? - 164
    • Fermat's Last Theorem - 165
    • The Riemann Hypothesis - 166
    • Clusters of Conjectures - 167
    • Polya and Putnam - 168
    • Conjectures and Axioms - 170
  11. Calling the Bluff - 172
    • Calling the Bluff - 179
    • Math Wars: A Report from the Front - 181
    • Once More: The Mathematical Image - 191
  12. Notes - 193
  13. Bibliography - 199
  14. Index - 208



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  2. Mauve: Text by correspondent(s) or other author(s); © the author(s)



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