Noson S. Yanofsky - The outer limits of reason: what science, mathematics, and logic cannot tell us

The Outer Limits of Reason

The Outer Limits of Reason

What Science, Mathematics, and Logic Cannot Tell Us

Noson S. Yanofsky

The MIT Press

Cambridge, Massachusetts

London, England

2013 Massachusetts Institute of Technology

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Library of Congress Cataloging-in-Publication Data

Yanofsky, Noson S., 1967

The outer limits of reason : what science, mathematics, and logic cannot tell us / Noson S. Yanofsky.

pages cm

Includes bibliographical references and index.

ISBN 978-0-262-01935-4 (hardcover : alk. paper)

1. Knowledge, Theory of. 2. SciencePhilosophy. 3. MathematicsPhilosophy. I. Title.

Q175.32.K45Y36 2013

001.01dc23

2012050531

10 9 8 7 6 5 4 3 2 1

Please email all comments and criticisms to .

For

Shayna Leah,

Hadassah, and Rivka

Table of Contents

Preface

The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance. Karl Popper A mans got to know his limitations. Harry Callahan, Magnum Force (1973) Everything should be made as simple as possible, but not simpler. Attributed to Albert Einstein

With understanding comes ambivalence. Once we know something, we often find it boring and trite. On the other hand, the mysterious and unknown fascinates us and holds our attention. That which we do not know or understand is what interests us, and what we cannot know intrigues us even more. This book explores topics that reason tells us we cannot know because they are beyond reason.

Many books convey the amazing facts that science, mathematics, and reason have revealed to us. There are also books that cover topics that science, mathematics, and reason have not yet fully explained. This book is a little different. Here we study what science, mathematics, and reason tell us cannot be revealed. What cannot be predicted or known? What will never be understood? What are the limitations of computers, physics, logic, and our thought processes? What is beyond the bounds of reason? This book aims to answer some of these questions and is full of ideas that challenge our deep-seated beliefs about the universe, our rationality, and ourselves.

Along the way we will study simple computer problems that would take trillions of centuries to solve; consider perfectly formed English sentences that have no meaning; learn about different levels of infinity; leap into the bizarre and wonderful world of the quantum; discuss specific problems that computers can never solve; befriend butterflies that bring about blizzards; ponder particles that simultaneously dance at different parties; hear about paradoxes and self-referential paradoxes; see what relativity theory tells us about our naive notions of space, time, and causality; understand Gdels famous theorems about the limitations of logic; discover certain problems in mathematics and physics that are impossible to solve; explore the very nature of science, mathematics, and reason; wonder why the universe seems perfect for human beings; and examine the complex relationship between our mind, reason, and the physical universe. We will also attempt to peek beyond the borders of reason and see what, if anything, is out there. These and many other fascinating topics will be presented in a way that is clear and comprehensible.

While exploring these various limitations in diverse areas, we will see that many of the limitations have a similar pattern. These patterns will be investigated in order to better understand the structure of reason and its limits.

This book is not a compendium of all the diverse examples in which limitations of reason are found. Rather, our goal is to understand why these boundaries arise and why reason cannot extend beyond them. Several representative limitations in each area are selected and discussed in depth.

Rather than just listing the limitations, I aim to explain them or at least provide the intuition of why a particular area is beyond reason. It is important to realize that this book is not meant to be speculative or to have a New Age orientation. Nor is it a history book in which I gloss over the meaning of ideas in order to focus on their chronological development. This is a popular science book that will gradually and clearly explain the ideas presented.

Since I accept Stephen Hawkings dictum that every equation halves the number of readers, very few equations are found in this book. However, I do believe in the power of diagrams, charts, and graphs to simplify complex ideas. My goal is clarity.

Each chapter deals with a different area: science, mathematics, language, philosophy, and so on. These chapters are arranged from concrete to abstract. I start with simple problems of everyday language and move on to straightforward philosophical questions, ending with the abstract world of mathematics. For the most part, the chapters are independent of each other and can be read in any order. Readers are encouraged to begin with topics that most interest them. (The unifying theme of self-referential paradoxes is found in chapters .)

Acknowledgments

In a sense, this book is a collaborative effort with my friends and colleagues in the Computer and Information Science Department at Brooklyn College. They have read the chapters, corrected my mistakes, chided me when I was being silly, and encouraged me when I was stuck. They have given me the warm intellectual environment that made this book possible. I thank them!

Many Brooklyn College faculty members were kind enough to read and comment on several of these chapters: Jonathan Adler, David Arnow, George Brinton, Samir Chopra, Jill Cirasella, Dayton Clark, Eva Cogan, Jim Cox, Scott Dexter, Keith Harrow, Danny Kopec, Yedidyah Langsam, Matthew Moore, Rohit Parikh, Simon Parsons, Michael Sobel, Aaron Tenenbaum, and Paula Whitlock. Their comments have made this a far better book.

Some chapters were used in a course given at Brooklyn College. Great benefit was derived from student participation and the give-and-take of a classroom dialog. I thank the students for listening, arguing, helping, and correcting. Many students at Brooklyn College and the Graduate Center of the City University of New York read and commented on earlier drafts: Firat Atagn, Can Bakent, Hubert Bennett, Greg Benson, Be Birchall, Raila C. Brejt, Fatemeh Choopani, Simon Dexter, Aline Elmann, Madelene Feingold, Terri Grosso, Miriam Gutherc, Jay Jankelewicz, Matthew P. Johnson, Joel Kammet, Tatiana Kedel, Wai Khoo, Karen Kletter, Erdal Kse, Michael Lampis, Shalva Landy, Holly Lo Voi, Jon Lo Voi, Matthew Meyer, Valia Mitsou, Jordi Navarrette, Shoshana Neuburger, Hadassah Norowitz, Nicole Reilly, Artur Sahakyan, Connor Savage, Angela Shatashwili, Alex Sverdlov, Stanislav Turzhavskiy, Fredda Weinberg, and Karol Wysocki. I am indebted to them all.

Numerous other people looked over parts of the book and have been helpful: Ros Abramsky, Samson Abramsky, Marcia Barr, Michael Barr, Rebecca Barr, Adam Brandenburger, Richard Churchill, Melvin Fitting, Leopold Flatto, Robert J. Fogelin, Chaim Goodman-Strauss, Ariel Halpert, Eliyahu Hershfeld, Ellen Hershfeld, Faigy Hershfeld, Pinchas Hershfeld, Shai Hershfeld, Yitzchok Hershfeld, Michael Hicks, Joshua Honigwachs, Roman Kossak, Klaas Landsman, Andr Lebel, Raphael Magarik, Camille Martin, Jolly Mathen, Rochel Moskowitz, Larry Moss, Naftoli Neuburger, Janos Pach, Carol Parikh, Suri Raber, N. Raja, Barbara Rifkind, Andrei Rodin, Ariel Ropek, Evan J. Siegel, Michael Vitz, Sharon Yanofsky, and Mark Zelcer. Their criticism, comments, and helpful ideas are deeply appreciated.

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