Preface |
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xi | |
Credits |
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xiii | |
Acknowledgments |
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xvii | |
Overture |
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1 | (4) |
The Ideal Mathematician |
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5 | (8) |
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Manifesto |
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13 | (4) |
Self-introduction |
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17 | (4) |
Chronology |
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21 | (14) |
Mathematics Has a Front and a Back |
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35 | (6) |
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Part 1 "Mostly for the right hand" |
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41 | (132) |
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43 | (4) |
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True Facts About Imaginary Objects |
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47 | (4) |
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Mathematical Intuition (Poincare, Polya, Dewey) |
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51 | (22) |
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51 | (8) |
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59 | (2) |
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61 | (2) |
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63 | (3) |
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Mental Models Subject to Social Control |
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66 | (2) |
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68 | (4) |
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72 | (1) |
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To Establish New Mathematics, We Use Our Mental Models And Build On Established Mathematics |
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73 | (16) |
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73 | (2) |
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75 | (2) |
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Mathematicians' proof vs. axiomatic proof |
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77 | (1) |
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Mathematicians' proof is semantic, not syntactic |
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78 | (2) |
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Established mathematics is fallible |
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80 | (2) |
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Published vs. private, rigorous vs. plausible |
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82 | (1) |
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Established mathematics is not controversial |
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83 | (2) |
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85 | (4) |
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How Mathematicians Convince Each Other or "The Kingdom of Math is Within You" |
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89 | (26) |
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90 | (1) |
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What some mathematicians say they are doing |
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91 | (4) |
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The Materialist versus the Platonist: Changeux and Connes |
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95 | (6) |
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What, then, is a mathematicians' proof? |
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101 | (1) |
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Relation between formal proof and mathematicians' proof |
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102 | (2) |
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Aristotle, Kant, and Locke |
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104 | (1) |
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104 | (1) |
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105 | (3) |
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108 | (1) |
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108 | (1) |
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108 | (7) |
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On the interdisciplinary study of mathematical practice, with a real live case study |
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115 | (10) |
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125 | (6) |
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125 | (3) |
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2 Lived experience as "foundation" |
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128 | (3) |
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Inner Vision, Outer Truth |
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131 | (6) |
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Mathematical Practice as a Scientific Problem |
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137 | (10) |
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Atiyah's pleasant surprise |
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137 | (1) |
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137 | (2) |
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For a multi-disciplined study of mathematical practice |
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139 | (1) |
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140 | (2) |
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142 | (2) |
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144 | (1) |
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144 | (3) |
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Proving is Convincing and Explaining |
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147 | (10) |
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147 | (1) |
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II Proof among professional mathematicians |
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147 | (1) |
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III Three meanings of "proof" |
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148 | (1) |
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IV Variation in proof standards |
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149 | (1) |
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150 | (3) |
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VI Proof in our classrooms |
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153 | (2) |
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155 | (2) |
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Fresh Breezes in the Philosophy of Mathematics |
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157 | (6) |
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157 | (1) |
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158 | (2) |
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160 | (3) |
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Definition of mathematics |
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163 | (4) |
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Introduction to "18 Unconventional Essays on the Nature of Mathematics" |
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167 | (6) |
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Part 2 "Mostly for the left hand" |
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173 | (60) |
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175 | (2) |
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Rhetoric and Mathematics (with Philip J. Davis) |
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177 | (14) |
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Part 1 Mathematics as Rhetoric |
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178 | (4) |
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Part 2 Rhetoric in Mathematics |
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182 | (6) |
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188 | (3) |
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Math Lingo vs. Plain English: Double Entendre |
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191 | (4) |
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195 | (4) |
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199 | (6) |
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205 | (2) |
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207 | (6) |
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Ethics for Mathematicians |
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213 | (4) |
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Under-represented Then Over-represented: A Memoir of Jews in American Mathematics |
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217 | (10) |
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Paul Cohen and Forcing in 1963 |
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227 | (6) |
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Part 3 Selected book reviews |
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233 | (30) |
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235 | (2) |
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Review of Not Exactly...In Praise of Vagueness by Kees van Deemter |
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237 | (4) |
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Review of How Mathematicians Think by William Byers |
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241 | (6) |
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Review of The Mathematician's Brain by David Ruelle |
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247 | (4) |
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Review of Perfect Rigor by Masha Gessen |
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251 | (4) |
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Review of Letters to a Young Mathematician by Ian Stewart |
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255 | (2) |
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Review of Number and Numbers by Alain Badiou |
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257 | (6) |
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263 | (8) |
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An amusing elementary example |
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265 | (2) |
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Annotated research bibliography |
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267 | (2) |
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269 | (2) |
Curriculum Vitae |
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271 | (2) |
List of articles |
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273 | (6) |
Index |
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279 | |