Joseph Mazur - Euclid in the Rainforest

A PLUME BOOK

EUCLID IN THE RAINFOREST

JOSEPH MAZUR is professor of mathematics at Marlboro College, where he has taught a wide range of classes in all areas of mathematics, its history, and its philosophy. He lives with his wife in Vermont.

Joseph Mazur brilliantly explores the symbiotic relationship between the physical and the mathematical worlds. He asks the questions: How do we know that the world is what we experience it to be? Can logic guide us through the rainforest of science and math and provide us with a chance to discover the underlying foundations for their truths? In his highly original search, Mazur is a brilliant forester whose graceful pursuit leads him to understand the logical bases of human reason. Mazur has given us a stylish and seductive book that convinces the mind even as it delights the soul.

PEN American Center (finalist, 2005 Martha Albrand Award for First Nonfiction)

This charming book radiates love of mathematics ... and of life. Mazur weaves elementary explanations of a wide range of essential mathematical ideas into narratives of his far-ranging travels.... This book is a treasure of human experience and intellectual excitement.

Choice (2005 Outstanding Academic Title)

A rare example of a universally appealing math book.

Mathematics Teacher

A delightfully entertaining adventure that weaves mathematical ideas with an amazing collection of real-life stories.

Science and Theology News

An unusual book ... Like a fascinating conversation that stays within certain bounds but nevertheless moves in unexpected directions.

Nature

Provocative and philosophical ... Mazur does readers the great service of setting the arcane ideas and procedures of mathematics back in the world where they belong.

Mathematical Intelligencer

Mazur successfully explores how mathematical logic and proof are essential building blocks to understanding knowledge and universal truths.... His text is devoid of complex proofs and dense mathematical language; instead, the author has drawn upon his experiences as a formative teacher to create a book rich in content that connects with real-world experiences.

Library Journal

Euclid in the

Rainforest

Discovering Universal Truth in Logic and Math

J OSEPH M AZUR

PLUME

Published by Penguin Group

Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A.

Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.)

Penguin Books Ltd., 80 Strand, London WC2R 0RL, England

Penguin Ireland, 25 St. Stephens Green, Dublin 2, Ireland (a division of Penguin Books Ltd.)

Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.)

Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi 110 017, India

Penguin Books (NZ), cnr Airborne and Rosedale Roads, Albany, Auckland 1310, New Zealand (a division of Pearson New Zealand Ltd.)

Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa

Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England

Published by Plume, a member of Penguin Group (USA) Inc. Previously published in a Pi Press edition.

First Plume Printing, August 2006

Copyright Joseph Mazur, 2005

All rights reserved

REGISTERED TRADEMARK MARCA REGISTRADA

CIP data is available

ISBN: 978-1-101-66487-2

Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.

PUBLISHERS NOTE

The scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage electronic piracy of copyrighted materials. Your support of the authors rights is appreciated.

BOOKS ARE AVAILABLE AT QUANTITY DISCOUNTS WHEN USED TO PROMOTE PRODUCTS OR SERVICES. FOR INFORMATION PLEASE WRITE TO PREMIUM MARKETING DIVISION, PENGUIN GROUP (USA) INC., 375 HUDSON STREET, NEW YORK, NEW YORK 10014.

Version_1

I N MEMORY OF D AVID F OWLER , WHO ,

WITH HUMOR AND SCHOLARLY

INSIGHT , INTRODUCED THE WORLD TO

REFRESHING NEW WAYS OF

APPRECIATING E UCLID AND THE

MATHEMATICS OF P LATOS A CADEMY

I came to understand mathematics by way of a Russian novel. On the morning of my seventeenth birthday, my older brother presented me with two books that I, as he put it, might enjoy reading. One was a 532-page paperback of Dostoyevskys Crime and Punishment; the other was a 472-page algebra textbook. My brother, having the gift of quick mathematical insight, hadnt noticed that one does not read a textbook of modern algebra in the same way that one reads a Russian novel. The next morning, I began reading the novel without ever getting out of bed and finished sometime late that night, having skipped lunch and supper. Raskolnikov, after swinging the axe with both arms and bringing the blunt side down on the old womans head, made me feverish and deliriously spellbound. On reaching the end of the novel, trust in my brothers choice of mesmeric literature was so strong that I turned to my other book, expecting it to be as riveting as the first. The next morning, I pondered a sentence on the second page for hours: Clearly, any result which can be proved or deduced from the postulates defining an integral domain will be true in any particular integral domain....

Stuck at clearly, I dressed and once again spent the day attempting to understand my new book, but this time I did not get past page five. That summer, I struggled to get past the first few chapters. What was modern algebra about? By the fourth chapter, I was well into abstractions, having completed as many exercises as I could. But I could not understand the relevance of all these mathematical abstractions to life itself.

The joys of feeling confidence in solving problems and the emotion of witnessing beauty at the end of a proof are intense. Mathematics gradually became for me as a teenager a range of mountains to climb. Challenges of ascending through the thin air of abstraction only made vistas from the summits more magnificent. With a firm proven foothold on one peak, I could see others beckoning me higher from above misty clouds covering valley paths through rainforests of flowering ideas.

In thirty years of teaching mathematics, I have collected stories of extraordinary students and fellow mathematicians trying to reach peaks from the steepest faces. Their stories are about the climb, the view from even the smallest peak, the excitement of discovery, the investigation of unknown intellectual encounters with beauty, and the confidence of feeling certain about mathematical proof. They are human stories, ultimately not so removed from the excitement of a Russian novel. I begin to see my brothers inadvertent point.