Mathematics and Religion: Our Languages of Sign and Symbol (Templeton Science and Religion Series)

Mathematics and Religion

Templeton Science and Religion Series In our fast-paced and high-tech era, when visual information seems so dominant, the need for short and compelling books has increased. This conciseness and convenience is the goal of the Templeton Science and Religion Series. We have commissioned scientists in a range of fields to distill their experience and knowledge into a brief tour of their specialties. They are writing for a general audience, readers with interests in the sciences or the humanities, which includes religion and theology. The relationship between science and religion has been likened to four types of doorways. The first two enter a realm of “conflict” or “separation” between these two views of life and the world. The next two doorways, however, open to a world of “interaction” or “harmony” between science and religion. We have asked our authors to enter these latter doorways to judge the possibilities. They begin with their sciences and, in aiming to address religion, return with a wide variety of critical viewpoints. We hope these short books open intellectual doors of every kind to readers of all backgrounds. Series Editors: J. Wentzel van Huyssteen & Khalil Chamcham Project Editor: Larry Witham

Mathematics and Religion Our L a nguag e s of Sig n a nd S ym bol

Javier Leach

Templeton Press

Templeton Press 300 Conshohocken State Road, Suite 550 West Conshohocken, PA 19428 www.templetonpress.org © 2010 by Javier Leach All rights reserved. No part of this book may be used or reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written permission of Templeton Press. Typeset and designed by Gopa & Ted2, Inc. Library of Congress Cataloging-in-Publication Data Leach, Javier, 1942– Mathematics and religion : our languages of sign and symbol / Javier Leach. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-59947-149-5 (pbk. : alk. paper) ISBN-10: 1-59947-149-3 (pbk. : alk. paper) 1. Mathematics. 2. Religion and science. 3. Signs and symbols. I. Title. BL265.M3L43 2010 201'.6510—dc22 2010008268 Printed in the United States of America 10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1

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Contents

Preface Chapter 1: Mathematics and Natural Sciences

vii 3

Chapter 2: Metaphysical Language

16

Chapter 3: Origins of Mathematics

35

Chapter 4: Euclid and Beyond

44

Chapter 5: Dawn of Science

55

Chapter 6: Mathematics Formalized

67

Chapter 7: Propositional Logic

93

Chapter 8: Language and Meaning

106

Chapter 9: Science, Language, and Religion

120

Appendix 1: Syntax of Propositional Logic

133

Appendix 2: Semantics of Propositional Logic

136

Appendix 3: Syntax of First-Order Logic

139

Appendix 4: Semantics of First-Order Logic

143

Appendix 5: Numerical Systems: Their Role in First-Order Logic

147

vi • contents

Appendix 6: The Paradox of Russell

154

Appendix 7: The Completeness of First-Order Logic 156 Appendix 8: Jack’s Formula

158

Appendix 9: Maria’s Formula

160

Appendix 10: Example of an L1 Formalization

161

Glossary

163

Essay on Sources

173

Index

179

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Preface

At any of the world’s great tourist sites—Paris, Tokyo, or Mexico City, for example—we typically see travelers using dictionaries to translate their native tongues into the local language, reminding us that our natural languages still divide us. At the same time, a language that seems to unite us wherever we go is the language of mathematics. Whether we are traveling in Germany, Indonesia, or Brazil, we can do business together because we agree that 2 + 2 = 4. The mathematical laws of gravity and the three dimensions of space allow us to find directions to New York’s Central Park or fly home, over a curved planet, on a jetliner. This book is about our languages, but with a focus on the privileged role that mathematics has in our ability to communicate about the world around us. In this sense, mathematics is our public language, but it is more than that also. As I hope you will see, mathematics leads us through science and brings us to questions about a greater reality called metaphysical reality, which we usually approach in the context of philosophy and religion. For many years, my friends, students, and acquaintances have asked me how I—a university professor of mathematics, logic, and computer science—reconcile my profession with my other vocation as a Jesuit priest. How does a person reconcile science and religion, both intellectually and on a personal basis? This book on mathematics and religion has also allowed me to explore that question further.

viii • pr eface

Mathematics is a difficult language for the uninitiated, especially when we reach beyond elementary arithmetic, geometry, and algebra to what I call the modern formal language of mathematics— the language we apply, for example, to computer science. In this advanced realm of math, we also probe the sufficiency of mathematical systems themselves. Do these systems have the power to explain an ultimate reality, or are they more like tools we invent to simply do the job: to measure the construction of a bridge, send a rocket to the moon, or run a computer calculation? By pursuing these advanced areas of mathematics, we finally arrive at questions about the truth and consistency of any system, whether a scientific system that talks about nature or a theological system that talks about God. I have organized this book in the following way. The first chapters focus on defining and explaining the three basic languages that concern us. Each has a particular kind of perception of reality and then a system of signs or symbols (a language) to convey that perception. The first kind of perception is found in logic and mathematics, a purely mental kind of perception. It uses the language of formal signs. This language often seems to be an array of impenetrable hieroglyphics, such as the notation Σar = (0, 1, +, ×,