Harris - Mathematics without apologies: portrait of a problematic vocation

mathematics without apologies

mathematics without apologies

portrait of a problematic vocation

michael harris

princeton university press

princeton and oxford

Copyright 2015 by Princeton University Press

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW

press.princeton.edu

Cover illustration by Dimitri Karetnikov

All Rights Reserved

ISBN 978-0-691-15423-7

Library of Congress Control Number: 2014953422

British Library Cataloging-in-Publication Data is available

This book has been composed in Times New Roman and Archer display

Printed on acid-free paper

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To Batrice, who didnt want to be thanked

contents

preface

Wer das Buch schriebe, htte die Vorrede Schritt fr Schritt zurckzunehmen, aber sie ist das Beste daran, das Einzige was wir knnen, wir Modernen

Ich will aus solchen Vorworten zu ungeschriebenen Bchern ein Buch machen, ein modernes Buch. Und ich schrieb ebendas Vorwort dazu.

Paul Mongr, Sant Ilario

When this book was nearly done and my colleagues started asking me what it is about, I found it simplest to answer that its about how hard it is to write a book about mathematics. Thats the short answer; the unabridged version involves a few pages of explanation. Here are those pages.

Of course people are writing books about mathematics all the timeand not only for expert audiences. The most effective of these books strip away the technical jargon to convey the magical sense that pure thought can conjure a second life, a virtual world of shapes and numbers and order and rules where not only do we know that everything is as it should be but we are also satisfied that we know why. Knowing why is the specialty of mathematical reasoning, but the virtual world of pure mathematics, not designed for any practical application, is remote from our first and authentic life; those of us dedicated to that world feel (or are made to feel) obliged to justify our indulging in an activity that is charming and engrossing but that appears to bring no benefit beyond the pleasure of knowing why.

These attempts at justification are the apologies of the title. They usually take one of three forms. Pure research in mathematics as in other fields is good because it often leads to useful practical consequences (Steven Shapin calls this the Golden Goose argument); it is true because it offers a privileged access to certain truths; it is beautiful, an art form. To claim that these virtues are present in mathematics is not wrong, but it sheds little light on what is distinctively mathematical and even less about pure mathematicians intentions. Intentions lie at the core of this book. I want to give the reader a sense of the mathematical lifewhat it feels like to be a mathematician in a society of mathematicians where first and second lives overlap. But during this guided tour of what I want to call the pathos of mathematics, we will repeatedly see our intentions misrepresented, and we will be reminded how hard it is to explain what impels us along this peculiar path.

Rather than rely on apologies, this book pieces together fragments found in libraries, in the arts, in popular culture, and in the media, to create a composite portrait of the mathematical vocation. The sequence of chapters very roughly follows the trajectory from the vocations awakening, through struggles with various kinds of temptation, to its consolidation, followed by a conclusion consisting of inconclusive reflections on what we know when we know why and what it all means. Although I have consulted actual transcripts or recordings of mathematicians talking, my sources consist mainly of writings about mathematics, especially by participantsso the portrait is largely a self-portrait, though not of the author, of course!but also by (usually, but not always, sympathetic) observers. Preconceptions and misrepresentations are fair game but are usually identified as such. I have paid special attention to writing or speaking by mathematicians whose manifest content may concern truth or utility or beauty, but which exhibit an aiming at something else, the values and emotional investmentthe pathos, in other wordsinvolved in pursuing the mathematical life.

Writing the book was a process of assembling and organizing this material in connection with selected themes and unifying perspectives. The process of assembling suggested virtues rather different from those usually invoked. Alternatives I explore in this book include the sense of contributing to a coherent and meaningful tradition, which entails both an attention to past achievements and an orientation to the future that is particularly pronounced in the areas of number theory to which my work is devoted; the participation in what has been described, in other settings, as a relaxed field, not subject to the pressures of material gain and productivity; and the pursuit of pleasure of an elusive, but nevertheless specific, kind.

The alert reader will have noted, correctly, that these alternatives are no more distinctively mathematical than good, true, and beautiful. I certainly dont think they offer definitive solutions to the riddles of mathematical pathos; but they did make it possible for me to hint at a vision of the mathematical good life that I find more reliable than the standard account. Another author, presented with the same material, would assemble it in a different way and would likely reveal a different set of habits, virtues, and goals. This is only natural; I try to make the diversity of the community of pure mathematicians visible by recording their distinctive opinions, and theres no reason to assume they come to the field sharing identical motivations. At most, I hope that the reader will see coherence in my personal assemblage.

The problematic subtitle alludes to the problems that define the intellectual landscape where the mathematical life makes its home. Its conventional to classify mathematicians as problem solvers or theory builders, depending on temperament. My experience and the sources I consulted in writing this book convince me that curiosity about problems guides the growth of theories, rather than the other way around. Alexander Grothendieck and Robert Langlands, two elusive costars of the present books narrative, count among the most ambitious of all builders of mathematical theories, but everything they built was addressed to specific problems with ancient roots. Entering the mathematical life is largely a matter of seeking an orientation among such outstanding problems. In this way, as in every other way, the mathematical life is a running dialogue with human history.

The mathematical life is problematic in other ways. Trade books about mathematics typically follow a quest narrative. They share with the currently dominant model of science writing an attachment to a simple moral economy in which the forces of light and darkness are clearly delineated. The quest may be embodied in the protagonists need to overcome external obstacles or to meet an intellectual challenge; its happy ending takes the form of a triumph over a hostile or unpromising environment or the rewarding of the protagonists unique talentsor both at once. The reality is not so simple. The most interesting

But any burdens left on our conscience when we contract bargains, Faustian or otherwise, can be separated from mathematics, at least conceptually. Its not the least of the paradoxes explored in this book that the pathos of mathematics grows darkest and most problematic at its moments of greatest success. Satisfaction in solving a problem can be intense, but it is short-lived; our pathos is driven by what we have not yet understood. Andr Weil, one of the twentieth centurys dominant mathematicians, described this as achiev[ing] knowledge and indifference at the same time. We never understand more than a finite amount of the limitlessness of what mathematics potentially offers to the understanding. If anything, the situation is even more frustrating: the more we learn, the more we realize how much more we have yet to understand. This is also a kind of Faustian bargainGoethes Faust got to keep his soul until he reported to the Devil that he was satisfied with what he had seen. The mathematical soul, embodied in a historical tradition oriented to a limitless future, can rest secure in the knowledge that its dissatisfaction is guaranteed.