Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures (Philosophical Issues in Science)

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. 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 Philosophy Of Mathematics Is An Excellent Introductory Text. This Student Friendly Book Discusses The Great Philosophers And The Importance Of Mathematics To Their Thought. It Includes The Following Topics: * The Mathematical Image * Platonism * Picture-proofs * Applied Mathematics * Hilbert And Godel * Knots And Nations * Definitions * Picture-proofs And Wittgenstein * Computation, Proof And Conjecture. The Book Is Ideal For Courses On Philosophy Of Mathematics And Logic.

philosophy Of Mathematics Is Clear And Engaging, And Student Friendly The Book Discusses The Great Philosophers And The Importance Of Mathematics To Their Thought. Among Topics Discussed In The Book Are The Mathematical Image, Platonism, Picture-proofs, Applied Mathematics, Hilbert And Godel, Knots And Notation Definitions, Picture-proofs And Wittgenstein, Computation, Proof And Conjecture.

This text discusses the great philosophers and the importance of mathematics to their thought. It includes topics such as: the mathematical image; platonism; picture-proofs; applied mathematics; Hilbert and Godel; knots and nations; definitions; and picture-proofs and Wittgenstein