Infinity and the mind : the science and philosophy of the infinite
著者
書誌事項
Infinity and the mind : the science and philosophy of the infinite
Princeton University Press, 2005
Expanded Princeton Science Library ed
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注記
Originally published: [Boston] : Birkhäuser , 1982
Includes bibliographical references (p. [329]-338) and index
内容説明・目次
内容説明
In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Godel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Godel's incompleteness theorems. His personal encounters with Godel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.
目次
Preface to the Paperback Edition ix Preface xi Chapter One: Infinity 1 A Short History of Infinity 1 Physical Infinities 9 Temporal Infinities 10 Spatial Infinities 15 Infinities in the Small 24 Conclusion 34 Infinities in the Mindscape 35 The Absolute Infinite 44 Connections 49 Puzzles and Paradoxes 51 Chapter Two: All the Numbers 53 From Pythagoreanism to Cantorism 53 Transfinite Numbers 64 From Omega to Epsilon-Zero 65 The Alefs 73 lnfinitesimals and Surreal Numbers 78 Higher Physical Infinities 87 Puzzles and Paradoxes 91 Chapter Three: The Unnameable 93 The Berry Paradox 93 Naming Numbers 95 Understanding Names 100 Random Reals 107 Constructing Reals 108 The Library of Babel 120 Richard's Paradox 126 Coding the World 130 What is Truth? 143 Conclusion 152 Puzzles and Paradoxes 155 Chapter Four: Robots and Souls 157 Godel's incompleteness Theorem 157 Conversations with Godel 164 Towards Robot Consciousness 171 Formal Systems and Machines 172 The Liar Paradox and the Non-Mechanizability of Mathematics 175 Artificial Intelligence via Evolutionary Processes 180 Robot Consciousness 183 Beyond Mechanism? 185 Puzzles and Paradoxes 187 Chapter Five: The One and the Many 189 The Classical One/Many Problem 189 What is a Set? 191 The Universe of Set Theory 196 Pure Sets and the Physical Universe 196 Proper Classes and Metaphysical Absolutes 202 Interface Enlightenment 206 One/Many in Logic and Set Theory 207 Mysticism and Rationality 209 Satori 214 Puzzles and Paradoxes 219 Excursion One: The Transfinite Cardinals 221 On and Alef-One 221 Cardinality 226 The Continuum 238 Large Cardinals 253 Excursion Two: Godel's Incompleteness Theorems 267 Formal Systems 267 Self-Reference 280 Godel's Proof 285 A Technical Note on Man-Machine Equivalence 292 Answers to the Puzzles and Paradoxes 295 Notes 307 Bibliography 329 Index 339
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