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

by Brown; James Robert

• ISBN13: 9780415960472
• ISBN10: 0415960479
• Edition: 2nd
• Format: Paperback
• Copyright: 2008-01-15
• Publisher: Routledge

Summary

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

Author Biography

James Robert Brown is Professor of Philosophy at the University of Toronto, Canada

Table of Contents

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