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

by Noson S. Yanofsky

Overview

This exploration of the scientific limits of knowledge challenges our deep-seated beliefs about our universe, our rationality, and ourselves.
 
“A must-read for anyone studying information science.” —Publishers Weekly, starred review

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 intuitions about the world—including our ideas about space, time, and motion, and the complex relationship between the knower and the known.
 
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
• statements that are true but cannot be proven
 
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:	9780262529846
Publisher:	MIT Press
Publication date:	11/04/2016
Series:	The MIT Press
Edition description:	Reprint
Pages:	418
Sales rank:	154,798
Product dimensions:	8.90(w) x 14.70(h) x 1.00(d)
Age Range:	18 Years

About the Author

Noson S. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City University of New York. He is a coauthor of Quantum Computing for Computer Scientists.

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

What People are Saying About This

Endorsement

Yanofsky has brought together insights about quantum mechanics, logic, and mathematics under one rubric. Very few others could pull that off. This book has the potential to be a classic.

—Prakash Panangaden, School of Computer Science, McGill University

From the Publisher

Yanofsky has brought together insights about quantum mechanics, logic, and mathematics under one rubric. Very few others could pull that off. This book has the potential to be a classic.

—Prakash Panangaden, School of Computer Science, McGill University

Prakash Panangaden

Yanofsky has brought together insights about quantum mechanics, logic, and mathematics under one rubric. Very few others could pull that off. This book has the potential to be a classic.

Editorial Reviews

Yanofsky takes on this mindboggling subject with confidence and impressive clarity. He eases the reader into the subject matter, ending each chapter with further readings. His book is a fascinating resource for anyone who seeks a better understanding of the world through the strangeness of its own limitations and a must-read for anyone studying information science.”
—Publishers Weekly, starred review

“Yanofsky provides an entertaining and informative whirlwind trip through limits on reason in language, formal logic, mathematics—and in science, the culmination of humankind's attempts to reason about the world.”
—The New Scientist

“In my view, Outer Limits is an extraordinary, and extraordinarily interesting, book. It is a cornucopia of mind-bending ideas.”
—Raymond S. Nickerson, PsycCRITIQUES

“The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating.”
—Thomas Colin, Leonardo Reviews

From the Publisher

The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating.

—Thomas Colin, Leonardo Reviews

Reviews

The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating.

Leonardo Reviews - Thomas Colin

In my view, Outer Limits is an extraordinary, and extraordinarily interesting, book. It is a cornucopia of mind-bending ideas.

PsycCRITIQUES - Raymond S. Nickerson

Yanofsky provides an entertaining and informative whirlwind trip through limits on reason in language, formal logic, mathematics—and in science, the culmination of humankind's attempts to reason about the world.

The New Scientist

★ 10/07/2013
Rather than write about what he knows, Yanofsky (Quantum Computing for Computer Scientists) prefers to explore the topic of what he doesn't know—or rather what we as humans cannot know. In this refreshingly original and ambitious philosophical inquiry, he attempts to map the limitations of human reason by examining the established conundrums, paradoxes, impossibilities within science and technology. Divided by subject area (including language, philosophy, science, mathematics, computing), each chapter lays out an array of paradoxes and unsolvable problems, clearly and concisely guiding readers into and around the worlds of reason. The examples range in complexity and some may be more familiar than others, such as his explanations of the Liar's Paradox, "this sentence is false." The more complicated contradictions, such as George Cantor's proof that an infinite set of numbers between 0 and 1 is vastly larger than an infinite set of natural numbers (1, 2, 3, 4 . . .), the author unpacks succinctly within the framework of modern life. He writes, "It would be foolhardy to cross a modern suspension bridge if you knew that the engineer did not believe in Cantor's work." Yanofsky takes on this mindboggling subject with confidence and impressive clarity. He eases the reader into the subject matter, ending each chapter with further readings. His book is a fascinating resource for anyone who seeks a better understanding of the world through the strangeness of its own limitations and a must-read for anyone studying information science. Illus. (Sept.)

Publishers Weekly