Mathematics without Apologies: Portrait of a Problematic Vocation

Mathematics without Apologies: Portrait of a Problematic Vocation

by Michael Harris


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Mathematics without Apologies: Portrait of a Problematic Vocation by Michael Harris

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

Product Details

ISBN-13: 9780691175836
Publisher: Princeton University Press
Publication date: 05/30/2017
Edition description: Reprint
Pages: 464
Sales rank: 1,217,699
Product dimensions: 6.00(w) x 9.20(h) x 1.10(d)

About the Author

Michael Harris is professor of mathematics at the Université Paris Diderot and Columbia University. He is the author or coauthor of more than eighty mathematical books and articles, and has received a number of prizes, including the Clay Research Award, which he shared in 2007 with Richard Taylor.

Read an Excerpt

Mathematics Without Apologies

Portrait of a Problematic Vocation

By Michael Harris


Copyright © 2015 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-8552-7


Introduction: The Veil

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

— David Hilbert, Paris 1900

The next sentence of Hilbert's famous lecture at the Paris International Congress of Mathematicians (ICM), in which he proposed twenty-three problems to guide research in the dawning century, claims that "History teaches the continuity of the development of science." We would still be glad to lift the veil, but we no longer believe in continuity. And we may no longer be sure that it's enough to lift a veil to make our goals clear to ourselves, much less to outsiders.

The standard wisdom is now that sciences undergo periodic ruptures so thorough that the generations of scientists on either side of the break express themselves in mutually incomprehensible languages. In the most familiar version of this thesis, outlined in T. S. Kuhn's Structure of Scientific Revolutions, the languages are called paradigms. Historians of science have puzzled over the relevance of Kuhn's framework to mathematics. It's not as though mathematicians were unfamiliar with change. Kuhn had already pointed out that "Even in the mathematical sciences there are also theoretical problems of paradigm articulation." Writing in 1891, shortly before the paradoxes in Cantor's set theory provoked a Foundations Crisis that took several decades to sort out, Leopold Kronecker insisted that "with the richer development of a science the need arises to alter its underlying concepts and principles. In this respect mathematics is no different from the natural sciences: new phenomena [neue Erscheinungen] overturn the old hypotheses and put others in their place." And the new concepts often meet with resistance: the great Carl Ludwig Siegel thought he saw "a pig broken into a beautiful garden and rooting up all flowers and trees" when a subject he had done so much to create in the 1920s was reworked in the 1960s.

Nevertheless, one might suppose pure mathematics to be relatively immune to revolutionary paradigm shift because, unlike the natural sciences, mathematics is not about anything and, therefore, does not really have to adjust to accommodate new discoveries. Kronecker's neue Erscheinungen are the unforeseen implications of our hypotheses, and if we don't like them, we are free to alter either our hypotheses or our sense of the acceptable. This is one way to understand Cantor's famous dictum that "the essence of mathematics lies in its freedom."

It's a matter of personal philosophy whether one sees the result of this freedom as evolution or revolution. For historian Jeremy Gray, it's part of the professional autonomy that characterizes what he calls modernism in mathematics; the imaginations of premodern mathematicians were constrained by preconceptions about the relations between mathematics and philosophy or the physical sciences:

Without ... professional autonomy the modernist shift could not have taken place. Modernism in mathematics is the appropriate ideology, the appropriate rationalization or overview of the enterprise. ... it became the mainstream view because it articulated very well the new situation that mathematicians found themselves in.

This "new situation" involved both the incorporation of mathematics within the structure of the modern research university — the creation of an international community of professional mathematicians — and new attitudes to the subject matter and objectives of mathematics. The new form and the new content appeared at roughly the same time and have persisted with little change, in spite of the dramatic expansion of mathematics and of universities in general in the second half of the twentieth century.

Insofar as the present book is about anything, it is about how it feels to live a mathematician's double life: one life within this framework of professional autonomy, answerable only to our colleagues, and the other life in the world at large. It's so hard to explain what we do — as David Mumford, one of my former teachers, put it, "I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate" — that when, on rare occasions, we make the attempt, we wind up so frustrated at having left our interlocutor unconvinced, or at the gross misrepresentations to which we have resorted, or usually both at once, that we leave the next questions unasked: What are our goals? Why do we do it?

But sometimes we do get to the "why" question, and the reasons we usually advance are of three sorts. Two of them are obviously wrong. Mathematics is routinely justified either because of its fruitfulness for practical applications or because of its unique capacity to demonstrate truths not subject to doubt, apodictically certain (to revive a word Kant borrowed from Aristotle). Whatever the merits of these arguments, they are not credible as motivations for what's called pure mathematics — mathematics, that is, not designed to solve a specific range of practical problems — since the motivations come from outside mathematics and the justifications proposed imply that (pure) mathematicians are either failed engineers or failed philosophers. Instead, the motivation usually acknowledged is aesthetic, that mathematicians are seekers of beauty, that mathematics is in fact art as much as science, or that it is even more art than science. The classic statement of this motivation, due to G. H. Hardy, will be reviewed in the final chapter. Mathematics defended in this way is obviously open to the charge of sterility and self-indulgence, tolerated only because of those practical applications (such as radar, electronic computing, cryptography for e-commerce, and image compression, not to mention control of guided missiles, data mining, or options pricing) and because, for the time being at least, universities still need mathematicians to train authentically useful citizens.

There are new strains on this situation of tolerance. The economic crisis that began in 2008 arrived against the background of a global trend of importing methods of corporate governance into university administration and of attempting to foster an "entreprenurial mindset" among researchers in all potentially useful academic fields. The markets for apodictically certain truths or for inputs to the so-called knowledge economy may some day be saturated by products of inexpensive mechanical surrogate mathematicians; the entrepreneurial mindset may find mathematics a less secure investment than the more traditional arts. All this leaves a big question mark over the future of mathematics as a human activity. My original aim in writing this book was to suggest new and more plausible answers to the "why" question; but since it's pointless to say why one does something without saying what that something is, much of the book is devoted to the "what" question. Since the book is written for readers without specialized training, this means it is primarily an account of mathematics as a way of life. Technical material is introduced only when it serves to illustrate a point and, as far as possible, only at the level of dinner-party conversation. But the "why" will never be far off, nor will reminders of the pressures on professional autonomy that make justification of our way of life, as we understand it, increasingly urgent.

The reader is warned at the outset that my objective in this book is not to arrive at definitive conclusions but rather to elaborate on what Herbert Mehrtens calls "the usual answer to the question of what mathematics is," namely, by pointing: "This is how one does mathematics."*And before I return to the "why" question, I had better start pointing.


How I Acquired Charisma

J'ai glissé dans cette moitié du monde. pour laquelle l'autre n'est qu'un décor.

— Annie Ernaux

My mathematical socialization began during the prodigious summer of 1968. While my future colleagues chanted in the Paris streets by day and ran the printing presses by night, helping to prepare the transition from structuralism to poststructuralism; while headlines screamed of upheavals — The Tet offensive! The Prague spring! Student demonstrations in Mexico City! — too varied and too numerous for my teenage imagination to put into any meaningful order; while cities across America burst into flames in reaction to the assassination of Martin Luther King and continued to smolder, I was enrolled in the Temple University summer program in mathematics for high school students at the suggestion of Mr. Nicholas Grant, who had just guided my class through a two-year experimental course in vector geometry. It was the summer between tenth and eleventh grades and between the presidential primaries and the unforgettable Democratic National Convention in Chicago. Like many of my classmates, I was already a veteran of partisan politics. Over the course of the summer, the certainty of the religious, patriotic, and familial narratives that had accompanied the first fourteen years of my own life were shaken, in some cases to the breaking point. How convenient, then, that a new and timeless certainty was ready and waiting to take their place.

At Temple that summer, I discovered the Men of Modern Mathematics poster that I subsequently rediscovered in nearly every mathematics department I visited around the world: at Swarthmore, where the marvelous Mr. Grant drove me during my senior year to hear a lecture by Philadelphia native L. J. Mordell, an alumnus of my high school and G. H. Hardy's successor at Cambridge to the Sadleirian Chair of Pure Mathematics; near the University of Pennsylvania mathematics library, where I did research for a high school project; and through all the steps of my undergraduate and graduate education. The poster was ubiquitous and certainly seemed timeless to my adolescent mind but had, in fact, been created only two years earlier by IBM. Its title alludes to Eric Temple Bell's Men of Mathematics, the lively but unreliable collection of biographies that served as motivational reading that summer at Temple. You will have noticed at least one problem with the title, and it's not only that one of the "men" in Bell's book and (a different) one on the IBM poster are, in fact, women. Whole books can and should be devoted to this problem, but for now let's just be grateful that something (though hardly enough) has been learned since 1968 and move on to the topic of this chapter: the contours and the hierarchical structure of what I did not yet know would be my chosen profession when I first saw that poster.

It was designed, according to Wikipedia, by the "famous California designer team of Charles Eames and his wife Ray Eames," with the "mathematical items" prepared by UCLA Professor Raymond Redheffer." Each "man" is framed by a rectangle, with a portrait occupying the left-hand side, a black band running along the top with name and dates and places of scientific activity, and Redheffer's capsule scientific biography filling the rest of the space, stretched to the length of "his" lifespan. As my education progressed, I began to understand the biographies, but at the time most of the names were unfamiliar to me. With E. T. Bell's help, we learned some of the more entertaining or pathetic stories attached to these names. That's when I first heard not only about the work of Nils Henrik Abel and Evariste Galois (see figure 2.1) in connection with the impossibility of trisecting the angle and with the problem of solving polynomial equations of degree 5 — they both showed there is no formula for the roots — but also how they both died at tragically young ages, ostensibly through the neglect of Augustin-Louis Cauchy, acting as referee for the French Académie des Sciences. What surprised me was that Abel and Galois both had portraits and biographies of standard size, while Cauchy, of whose work I knew nothing at the time, belonged to the very select company of nine Men of Mathematics entitled to supersized entries. The other eight were (I recite from memory) Pierre de Fermat, Sir Isaac Newton, Leonhard Euler, Joseph-Louis Lagrange, Carl Friedrich Gauss (the "Prince of Mathematicians"), Bernhard Riemann, Henri Poincaré, and David Hilbert.

Maybe Bell's book and the IBM poster should have been entitled Giants of Mathematics, with a special category of Supergiants, including the nine just mentioned plus Archimedes and a few others from antiquity (the poster's timeline starts in AD 1000). The hierarchy admitted additional refinements, the Temple professors told us. It was generally agreed — the judgment goes back at least to Felix Klein, if not to Gauss himself — that Archimedes, Newton, and Gauss were the three greatest mathematicians of all time. And who among those three was the very greatest, we asked? One of our professors voted for Newton; the others invited us to make up our own minds.

The field of mathematics has a natural hierarchy. Mathematicians generally work on research problems. There are problems and then there are hard problems. Mathematicians look to publish their work in journals. There are good journals and there are great journals. Mathematicians look to get academic jobs. There are good jobs and great jobs. ... It is hard to do mathematics and not care about what your standing is.

In Wall Street every year bonus numbers come out, promotions are made and people are reminded of where they stand. In mathematics, it is no different. ... Even in graduate school, I found that everyone was trying to see where they stood.

That's hedge-fund manager Neil Chriss, explaining why he quit mathematics for Wall Street. But his analogy between finance and mathematics doesn't quite hold up. For mathematicians, the fundamental comparisons are with those pictures on the wall. "To enter into a practice," according to moral philosopher Alasdair MacIntyre, "is to enter into a relationship not only with its contemporary practitioners, but also with those who have preceded us. ..." Adam Smith, writing in the eighteenth century, found these relations harmonious:

Mathematicians and natural philosophers ... live in good harmony with one another, are the friends of one another's reputation, enter into no intrigue in order to secure the public applause, but are pleased when their works are approved of, without being either much vexed or very angry when they are neglected.

Two centuries later, one meets a more varied range of personality types:

In the 1950's there was a math department Christmas party at the University of Chicago. Many distinguished mathematicians were present, including André Weil. ... For amusement, the gathered company endeavored to ... list ... the ten greatest living mathematicians, present company excluded. Weil, however, insisted on being included in the consideration.

The company then turned to the ... list of the ten greatest mathematicians of all time. Weil again insisted on being included.

Weil soon moved to the Institute for Advanced Study (IAS) in Princeton, and when, in the mid-1970s, a Princeton University graduate student asked him to name the greatest twentieth-century mathematician, "the answer (without hesitation) was 'Carl Ludwig Siegel.'" Asked next to name the century's second-greatest mathematician, he "just smiled and proceeded to polish his fingernails on his lapel." Fifteen years later my colleagues in Moscow proposed a different ranking: A. N. Kolmogorov was by consensus the greatest mathematician of the twentieth century, with a plurality supporting Alexander Grothendieck for the second spot.

Hierarchy and snobbery are, naturally, not specific to mathematics. "Democracy should be used only where it is in place," wrote Max Weber in the 1920s. "Scientific training ... is the affair of an intellectual aristocracy, and we should not hide this from ourselves." In the nineteenth century, Harvard professor Benjamin Peirce, perhaps the first American mathematician to enjoy an international reputation, could "cast himself ... as the enemy of sentimental egalitarianism ... a pure meritocrat with no democracy about him." Nowadays, of course, mathematicians are no less committed to democracy than the rest of our university colleagues. But we do seem peculiarly obsessed with ordered lists. A lively exchange in 2009 on the collective blog MathOverflow aimed at filling in the gap between the last Giants of Bell's book and the winners in 1950 of the first postwar Fields Medals, awarded every four years to distinguished mathematicians under 40 and still the most prestigious of mathematical honors. The discussion generated several overlapping lists of "great mathematicians born 1850–1920" and at least one novel graphic representation, as shown in figure 2.2.


Excerpted from Mathematics Without Apologies by Michael Harris. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

Preface ix
Acknowledgments xix
Part 1
Chapter 1. Introduction: The Veil 3
Chapter 2. How I Acquired Charisma 7
Chapter α. How to Explain Number Theory at a Dinner Party 41
(First Session: Primes) 43
Chapter 3. Not Merely Good, True, and Beautiful 54
Chapter 4. Megaloprepeia 80
Chapter β. How to Explain Number Theory at a Dinner Party 109
(Second Session: Equations) 109
Bonus Chapter 5. An Automorphic Reading of Thomas Pynchon’s Against the Day (Interrupted by Elliptical Reflections on Mason & Dixon) 128
Part II 139
Chapter 6. Further Investigations of the Mind-Body Problem 141
Chapter β.5. How to Explain Number Theory at a Dinner Party 175
(Impromptu Minisession: Transcendental Numbers) 175
Chapter 7. The Habit of Clinging to an Ultimate Ground 181
Chapter 8. The Science of Tricks 222
Part III 257
Chapter γ. How to Explain Number Theory at a Dinner Party 259
(Third Session: Congruences) 259
Chapter 9. A Mathematical Dream and Its Interpretation 265
Chapter 10. No Apologies 279
Chapter δ. How to Explain Number Theory at a Dinner Party 311
(Fourth Session: Order and Randomness) 311
Afterword: The Veil of Maya 321
Notes 327
Bibliography 397
Index of Mathematicians 423
Subject Index 427

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