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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us
     

The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

by Noson S. Yanofsky
 

ISBN-10: 0262019353

ISBN-13: 9780262019354

Pub. Date: 08/31/2013

Publisher: MIT Press

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be

Overview

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.

Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world — our ideas about space, time, and motion, and the complex relationship between the knower and the known.

Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

Product Details

ISBN-13:
9780262019354
Publisher:
MIT Press
Publication date:
08/31/2013
Pages:
424
Sales rank:
437,703
Product dimensions:
6.30(w) x 9.00(h) x 1.40(d)
Age Range:
18 Years

Table of Contents

Preface ix

Acknowledgments xi

1 Introduction 1

2 Language Paradoxes 15

2.1 Liar! Liar! 15

2.2 Self-Referential Paradoxes 19

2.3 Naming Numbers 26

3 Philosophical Conundrums 31

3.1 Ships, People, and Other Objects 31

3.2 Hangin' with Zeno and Gödel 41

3.3 Bald Men, Heaps, and Vagueness 50

3.4 Knowing about Knowing 57

4 Infinity Puzzles 65

4.1 Sets and Sizes 66

4.2 Infinite Sets 69

4.3 Anything Larger? 76

4.4 Knowable and Unknowable 85

5 Computing Complexities 97

5.1 Some Easy Problems 98

5.2 Some Hard Problems 109

5.3 They're All Connected 121

5.4 Almost Solving Hard Problems 129

5.5 Even Harder Problems 131

6 Computing Impossibilities 135

6.1 Algorithms, Computers, Machines, and Programs 136

6.2 To Halt or Not to Halt? 139

6.3 They're All Connected 146

6.4 A Hierarchy of the Unknown 152

6.5 Minds, Brains, and Computers 157

7 Scientific Limitations 161

7.1 Chaos and Cosmos 161

7.2 Quantum Mechanics 175

7.3 Relativity Theory 214

8 Metascientific Perplexities 235

8.1 Philosophical Limitations of Science 235

8.2 Science and Mathematics 252

8.3 The Origin of Reason 272

9 Mathematical Obstructions 297

9.1 Classical Limits 298

9.2 Galois Theory 304

9.3 Harder Than Halting 309

9.4 Logic 320

9.5 Axioms and Independence 331

10 Beyond Reason 339

10.1 Summing Up 339

10.2 Defining Reason 345

10.3 Peering Beyond 349

Notes 355

Bibliography 379

Index 393

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