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Authors: Noson S. Yanofsky

Outer Limits of Reason

BOOK: Outer Limits of Reason

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

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

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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. Science—Philosophy. 3. Mathematics—Philosophy. I. Title.

Q175.32.K45Y36  2013



10  9  8  7  6  5  4  3  2  1

Please email all comments and criticisms to
[email protected]




Shayna Leah,

Hadassah, and Rivka

Table of Contents



1  Introduction

2  Language Paradoxes

2.1  Liar! Liar!

2.2  Self-Referential Paradoxes

2.3  Naming Numbers

3  Philosophical Conundrums

3.1  Ships, People, and Other Objects

3.2  Hangin' with Zeno and Gödel

3.3  Bald Men, Heaps, and Vagueness

3.4  Knowing about Knowing

4  Infinity Puzzles

4.1  Sets and Sizes

4.2  Infinite Sets

4.3  Anything Larger?

4.4  Knowable and Unknowable

5  Computing Complexities

5.1  Some Easy Problems

5.2  Some Hard Problems

5.3  They're All Connected

5.4  Almost Solving Hard Problems

5.5  Even Harder Problems

6  Computing Impossibilities

6.1  Algorithms, Computers, Machines, and Programs

6.2  To Halt or Not to Halt?

6.3  They're All Connected

6.4  A Hierarchy of the Unknown

6.5  Minds, Brains, and Computers

7  Scientific Limitations

7.1  Chaos and Cosmos

7.2  Quantum Mechanics

7.3  Relativity Theory

8  Metascientific Perplexities

8.1  Philosophical Limitations of Science

8.2  Science and Mathematics

8.3  The Origin of Reason

9  Mathematical Obstructions

9.1  Classical Limits

9.2  Galois Theory

9.3  Harder Than Halting

9.4  Logic

9.5  Axioms and Independence

10  Beyond Reason

10.1  Summing Up

10.2  Defining Reason

10.3  Peering Beyond





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 man's got to know his limitations.

—Harry Callahan,
Magnum Force

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
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
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 Gödel's 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 Hawking's 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
, and


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 Atagün, Can Başkent, 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 Köse, 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, André 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.

I thank Robbert Dijkgraaf for permission to use his artwork in
figure 8.6
. I also thank C. Goodman-Strauss for permission to use many of his diagrams in
section 9.3
. My beautiful and talented daughter Hadassah was very helpful with some of the diagrams.

James DeWolf, Marc Lowenthal, Marcy Ross, and the whole MIT Press team have been very helpful in getting this book into shape. Thank you.

Karen Kletter remains the world's greatest editor and proofreader. Thank you, Karen!

However, at the end of the day, I am the sole cause of any errors that may remain.

Several other, more general debts should be acknowledged. I am grateful to my friend and research partner Ralph Wojtowicz of Baker Mountain Research Corporation for supporting my other research while I was working on this book.

In the spring of 1987 I had a chance encounter with Dr. Avi Rabinowitz on a street corner in Jerusalem. Avi is a brilliant physicist bristling with creativity and enthusiasm. We eventually became traveling companions and good friends. There are few topics that Avi cannot discuss in depth. Conversations with him usually proceed at the speed of light. We've had many intense conversations while climbing mountains in Greece and watching ridiculous science fiction movies. He is a true mentor and friend. Every page in this book contains ideas that I have discussed with him over the years. (He would probably disagree with most of what I wrote.) His influence is immense and I am forever appreciative.


Over the past few years, three people who enriched my life have passed away. They added much to my education and hence much to this book.

During my senior year at Brooklyn College, Professor Chaya Gurwitz supervised me in a research project, thereby introducing me to the rigors of higher mathematics and computer science. She taught me how to read an academic paper, to put my ideas into action, and to analyze the results. This experience piqued my interest in attending graduate school. She graciously invited me to her home for many meals, where I also became friends with her husband and eight wonderful children. She continued to guide me as a graduate student, as a teacher, as a colleague, and as a person until her untimely passing in 2008. I am truly indebted to her.


My dissertation advisor,
mon maître
, Alex Heller, was a distinguished professor in the Department of Mathematics at the Graduate Center of the City University of New York. He was a kind, gentle man. Although I graduated in 1996, we continued to meet once or twice a week until a few days before his sad passing in 2008. (In a sense, he gave me twelve years of postdoctoral research.) Our conversations meandered from mathematics, to politics, to morality, to philosophy, to history, and so on. He was an amazing genius and his range of knowledge was astonishing; in mathematics, however, it was particularly striking. While speaking with him, one got the impression that he had a magnificently clear vision of the entire structure of mathematics before him. It was a privilege to study under him and to be befriended by him.

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