I've always had trouble with the idea that mathematicians discover things, as opposed to inventing them. You see, if you discover something, the implication is that that something is, in some sense, out there. But where would mathematical entities reside, if not inside human brains and thought processes? I must say, reading this book has if not changed my mind at least made me seriously question my positions - which is really what you want from any good book. Brown's treatment is relatively accessible, but of course you will be in for a good amount of philosophy, and some not so easily digestible math. Still, the attentive reader can get the gist of the arguments without having to follow every proof presented by the author. I am a little less convinced, though still intrigued, by Brown's claim that pictures can - in some circumstances - do the work of formal proofs. Then again, that notion does appeal to my generally pluralistic attitude about methods of inquiry, and it does fit very well with the author's overall contention that mathematics is - surprisingly - a lot more like the natural sciences than one might think at first. Of course, all of this leaves completely unanswered the underlying question of the ontological status of mathematical objects. Oh well, can't get everything out of a single book.
Author(s): James Robert Brown
Series: Philosophical Issues in Science
Edition: 1
Publisher: Routledge
Year: 1999
Language: English
Pages: 181
Book Cover......Page 1
Half-Title......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 7
Preface and Acknowledgements......Page 11
CHAPTER 1 Introduction: The Mathematical Image......Page 13
The Original Platonist......Page 19
Some Recent Views......Page 20
What is Platonism?......Page 21
The Problem of Access......Page 24
The Problem of Certainty......Page 26
Platonism and its Rivals......Page 31
Bolzano’s ‘Purely Analytic Proof’......Page 33
What Did Bolzano Do?......Page 35
Different Theorems, Different Concepts?......Page 36
Inductive Mathematics......Page 37
Instructive Examples......Page 39
Representation......Page 43
Three Analogies......Page 44
So Why Worry?......Page 46
Appendix......Page 47
Representations......Page 49
Artifacts......Page 51
Bogus Applications......Page 52
Does Science Need Mathematics?......Page 54
Representation vs. Description......Page 55
Structuralism......Page 57
Early Formalism......Page 61
Hilbert’s Formalism......Page 62
Hilbert’s Programme......Page 65
Gödel’s Theorem......Page 67
Truth......Page 68
The Boolos proof......Page 69
Gödel’s Second Theorem......Page 71
The Upshot for Hilbert’s Programme......Page 72
The Aftermath......Page 73
CHAPTER 6 Knots and Notation......Page 74
Knots......Page 76
The Dowker Notation......Page 77
The Conway Notation......Page 78
Polynomials......Page 80
Creation or Revelation?......Page 81
Sense, Reference and Something Else......Page 84
The Official View......Page 86
The Frege-Hilbert Debate......Page 87
Contextual Definition......Page 88
Defining Old Terms......Page 89
Consistency and Existence......Page 90
Independence Proofs......Page 91
Graph Theory......Page 92
Lakatos......Page 96
Concluding Remarks......Page 99
CHAPTER 8 Constructive Approaches......Page 101
Brouwer’s Intuitionism......Page 102
Dummett’s Anti-realism......Page 104
Logic......Page 106
Problems......Page 107
The finite but very large......Page 108
Applied mathematics......Page 109
Negation......Page 110
Exhibiting an instance......Page 111
The loss of many classical results......Page 112
A Picture and a Problem......Page 113
Following a Rule......Page 115
Platonism......Page 117
Algorithms......Page 118
Brouwer’s Beetle......Page 119
Radical Conventionalism......Page 120
Bizarre Examples......Page 121
Naturalism......Page 122
The Sceptical Solution......Page 123
What is a Rule?......Page 124
Grasping a Sense......Page 125
Platonism versus Realism......Page 127
Surveyability......Page 128
The Sense of a Picture......Page 129
The Four Colour Theorem......Page 131
Fallibility......Page 132
Surveyability......Page 133
Inductive Mathematics......Page 134
Perfect Numbers......Page 135
Computation......Page 136
Is π Normal?......Page 138
Fermat’s Last Theorem......Page 139
Clusters of Conjectures......Page 140
Polya and Putnam......Page 141
Conjectures and Axioms......Page 142
CHAPTER 11 Calling the Bluff......Page 144
Calling the Bluff......Page 149
Math Wars: A Report from the Front......Page 151
Once More: The Mathematical Image......Page 158
Notes......Page 161
Bibliography......Page 166
Index......Page 173
