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NORTH-HOLLAND MATHEMATICS STUDIES 48 Notasde Matematica (76) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester Mathematicians and Their Times Histow of Mathematics and Mathematics of History LAURENCEYOUNG Distinguished Research Professor of Mathematics (Emeritus) at the University of Wisconsin-Madison, Wisconsin, U.S.A. Past Fellow of Trinity College, Cambridge, U.K. NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD @ North- Holland Publishing Company, I981 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, o r transmitted, in any form or b y any means, electronic, mechanical. photocopying, recording o r otherwise, without the prior permission vf the copyright o wner. ISBN: 0444861351 Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAMONEW YORK*OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017 Library of Congress Cataloging in Publication Data Young, Laurence Chisholm. Mathematicians and t h e i r times. (North-Holland mathematics s t u d i e s ; 48) Bibliography : p. I n c l u d e s index. i. Mathematics--History. I. T i t l e . QA2 1.Y68 510I.9 80 -274 13 ISBN 0-444-86135-1 ( U . S . ) PRINTED IN T H E NETHERLANDS PREFACE These lectures were given at the request of colleagues, in view of my long acquaintance with distinguished mathematicians. From early childhood, I have known almost legendary figures, known them better than people today who affect my daily life: the mailman who brings my letters to the door, the garage man who attends to my aged car, even the piano-tuner (bless him) who cares for my prized Ritmiiller from old Goettingen. As a student, I learnt a great deal from some mathematicians in this book, particularly Hardy and Littlewood, and before them the great Caratheodory -- as I mention in the text, I used to spend whole afternoons at his Munich apartment across from the English Garden. Mathematics became for me, not a store of past knowledge, but creative activity of the highest form, directed towards the future. Yet I learned that to study the newest developments is not enough: we have to go back to the sources, with a professional XX-th century outlook, to find what manner of men were able to plant the seed, and what difficulties and prejudices they had to overcome. I learnt to associate mathematics, whether of yesterday or today, not just with definitions and theorems, algorithms and proofs, still less with masses of formulae, but with the creative minds of real people. The great ones, the pioneers, are much more than names attached to.some discovery: I was taught to ask, not just what they achieved, but what they tried, and in the case of mathematicians, the way they thought, even the I can relay only a small part of this, and errors they made. naturally I say most about mathematicians I know best. I have been at pains to talk, not just of greatness, but of the things that make up our reality: we know them only too well as errors, trials, tribulations and sometimes tragedies. This may help readers generally, not only those mathematically inclined. Everyone, sooner of later, has to make a crucial decision, and not depend on others. Some decide like automata, by rule of To them I can only say with Cromwell: "I beseech you thumb. think it possible you may be mistaken." However, those willing to reflect might consider reading, as they may in this book, how others did decide, and what came of it. ..., The past is a laboratory of human experience, from which to learn. Each age can look at its lengthy past, judging each event by what came after. This is by no means the usual historical proce