Preface and Acknowledgements ................................... xi
Chapter 1: Introduction: The Mathematical Image
Further Reading ................................................. 8
Chapter 2: Platonism ............................................ 9
The Original Platonist .......................................... 9
Some Recent Views .............................................. 10
What is Platonism? ............................................. 12
The Problem of Access .......................................... 16
The Problem of Certainty ....................................... 19
Platonism and its Rivals ....................................... 24
Further Reading ................................................ 25
Chapter 3: Picture-proofs and Platonism ........................ 26
Bolzano's 'Purely Analytic Proof ............................... 26
What Did Bolzano Do? ........................................... 29
Different Theorems, Different Concepts? ........................ 30
Inductive Mathematics .......................................... 31
Special and General Cases ...................................... 34
Instructive Examples ........................................... 35
Representation ................................................. 38
Seeing Induction? .............................................. 40
Three Analogies ................................................ 44
Are Pictures Explanatory? ...................................... 46
So Why Worry? .................................................. 47
Further Reading ................................................ 47
Appendix ....................................................... 47
Chapter 4: What is Applied Mathematics? ........................ 51
Representations ................................................ 52
Artifacts ...................................................... 54
Bogus Applications ............................................. 56
Does Science Need Mathematics? ................................. 57
Representation vs. Description ................................. 60
Structuralism .................................................. 62
Further Reading ................................................ 66
Chapter 5: Hilbert and Gödel ................................... 67
The Nominalistic Instinct ...................................... 67
Early Formalism ................................................ 68
Hilbert's Formalism ............................................ 69
Hilbert's Programme ............................................ 73
Small Problems ................................................. 75
Gödel's Theorem ................................................ 76
Gödel's Second Theorem ......................................... 80
The Upshot for Hilbert's Programme ............................. 82
The Aftermath .................................................. 82
Further Reading ................................................ 83
Chapter 6: Knots and Notation .................................. 84
Knots .......................................................... 86
The Dowker Notation ............................................ 88
The Conway Notation ............................................ 89
Polynomials .................................................... 91
Creation or Revelation? ........................................ 93
Sense, Reference and Something Else ............................ 97
Further Reading ................................................ 98
Chapter 7: What is a Definition? ............................... 99
The Official View .............................................. 99
The Frege-Hilbert Debate ...................................... 100
Reductionism .................................................. 107
Graph Theory .................................................. 108
Lakatos ....................................................... 113
Concluding Remarks ............................................ 117
Further Reading ............................................... 117
Chapter 8: Constructive Approaches ............................ 118
From Kant to Brouwer .......................................... 119
Brouwer's Intuitionism ........................................ 120
Bishop's Constructivism ....................................... 122
Dummett's Anti-realism ........................................ 123
Logic ......................................................... 125
Problems ...................................................... 127
Further Reading ............................................... 135
Chapter 9: Proofs, Pictures and Procedures in Wittgenstein .... 136
A Picture and a Problem ....................................... 136
Following a Rule .............................................. 138
Platonism ..................................................... 142
Algorithms .................................................... 144
Dispositions .................................................. 144
Knowing Our Own Intentions .................................... 145
Brouwer's Beetle .............................................. 145
Radical Conventionalism ....................................... 146
Bizarre Examples .............................................. 147
Naturalism .................................................... 148
The Sceptical Solution ........................................ 150
Modus Ponens or Modus Tollens! ................................ 151
What is a Rule? ............................................... 152
Grasping a Sense .............................................. 153
Platonism versus Realism ...................................... 155
Survey ability ................................................ 157
The Sense of a Picture ........................................ 158
Further Reading ............................................... 159
Chapter 10: Computation, Proof and Conjecture ................. 160
The Four Colour Theorem ....................................... 160
Fallibility ................................................... 161
Survey ability ................................................ 163
Inductive Mathematics ......................................... 164
Perfect Numbers ............................................... 165
Computation ................................................... 168
Is π Normal? .................................................. 170
Fermat's Last Theorem ......................................... 171
The Riemann Hypothesis ........................................ 172
Clusters of Conjectures ....................................... 173
Poly a and Putnam ............................................. 174
Conjectures and Axioms ........................................ 175
Further Reading ............................................... 177
Chapter 11: How to Refute the Continuum Hypothesis ............ 178
What is the Continuum Hypothesis? ............................. 179
How Could We Determine the Truth of CH? ....................... 183
Kreisel's Analogy ............................................. 185
Freiling's Refutation of CH ................................... 186
What Might the Continuum Be? .................................. 191
Two Objections ................................................ 192
What Did the Thought Experiment Contribute? ................... 195
Two Morals .................................................... 196
Appendix: Freiling's Version .................................. 196
Further Reading ............................................... 197
Chapter 12: Calling the Bluff ................................. 198
Calling the Bluff ............................................. 205
Math Wars: A Report from the Front ............................ 207
Once More: The Mathematical Image ............................. 217
Further Reading ............................................... 219
Notes ......................................................... 220
Bibliography .................................................. 226
Index ......................................................... 236
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