Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures | ||

Brown (James Robert) | ||

This Page provides (where held) the Abstract of the above Book and those of all the Papers contained in it. | ||

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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Follow Link for James Brown’s web-page, and for comments on the book (which has just been replaced by a second edition), follow Link.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 Wikipedia: James Robert Brown for his *Wikipedia* entry, which notes that his main claim to fame is in the area of *Thought Experiments ^{1}*, so I may need to follow him up in this area in due course.

- 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.

- Preface and Acknowledgements - xi
- Chapter 1: Introduction: The Mathematical Image - 1
- Chapter 2: Platonism - 8
- Chapter 3: Picture-proofs and Platonism - 25
- Chapter 4: What is Applied Mathematics? - 46
- Chapter 5: Hilbert and Godel - 62
- Chapter 6: Knots and Notation - 79
- Chapter 7: What is a Definition? - 94
- Chapter 8: Constructive Approaches - 113
- Chapter 9: Proofs, Pictures and Procedures in Wittgenstein - 130
- Chapter 10: Computation, Proof and Conjecture - 154
- Chapter 11: Calling the Bluff - 172
- Notes - 193
- Bibliography - 199
- Index - 208

"

- Preface and Acknowledgements - xi
- Introduction: The Mathematical Image - 1
- 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

- 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

- What is Applied Mathematics? - 46
- Representations - 47
- Artifacts - 49
- Bogus Applications - 51
- Does Science Need Mathematics? - 52
- Representation vs. Description - 55
- Structuralism - 57

- 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

- 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

- What is a Definition? - 94
- The Official View - 94
- The Frege-Hilbert Debate - 95
- Reductionism - 102
- Graph Theory - 103
- Lakatos - 107
- Concluding Remarks - 112

- 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

- 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

- 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

- Calling the Bluff - 172
- Calling the Bluff - 179
- Math Wars: A Report from the Front - 181
- Once More: The Mathematical Image - 191

- Notes - 193
- Bibliography - 199
- Index - 208

- Blue: Text by me; © Theo Todman, 2019
- Mauve: Text by correspondent(s) or other author(s); © the author(s)

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