A STRANGE WILDERNESS
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© 2011 by Amir D. Aczel
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Frontispiece: The wilderness of the Pyrenees lies just beyond
the Aralar mountain range in northern Spain.
For Debra
Les fleuves lavent l’Histoire.
—J. M. G. LE CLÉZIO
Mathematics is not a careful march down a well-
cleared highway, but a journey into a strange
wilderness, where the explorers often get lost.
—W. S. ANGLIN
CONTENTS
Preface
Introduction
PART I HELLENIC FOUNDATIONS
ONE God Is Number
TWO Plato’s Academy
THREE Alexandria
PART II THE EAST
FOUR The House of Wisdom
FIVE Medieval China
PART III RENAISSANCE MATHEMATICS
SIX Italian Shenanigans
SEVEN Heresy
PART IV TO CALCULUS AND BEYOND
EIGHT The Gentleman Soldier
NINE The Greatest Rivalry
TEN Geniuses of the Enlightenment
PART V UPHEAVAL IN FRANCE
ELEVEN Napoleon’s Mathematicians
TWELVE Duel at Dawn
PART VI TOWARD A NEW MATHEMATICS
THIRTEEN Infinity and Mental Illness
FOURTEEN Unlikely Heroes
FIFTEEN The Strangest Wilderness
Notes
Bibliography
Photo Credits
Index
PREFACE
I fell in love with the history of mathematics and the life stories of mathematicians when I took my first “pure math” course as a mathematics undergraduate at the University of California at Berkeley in the mid-1970s. My professor for a course in real analysis (the theoretical basis of calculus) was the noted French mathematician Michel Loève. A Polish Jew who by chance was born in Jaffa, in Turkish Palestine, he then moved to France, survived the dreaded concentration camp at Drancy (just outside Paris), and after the war immigrated to America. Loève was a walking encyclopedia of the rich intellectual life of mathematicians living in Paris in the period between the two world wars. He peppered his difficult lectures—which he delivered in abstract mathematical spaces, rarely deigning to “dirty our hands,” as he put it, “in the real line,” where all the applications were—with fascinating stories about the lives of famous mathematicians he had known and worked with. “We were all sitting at a café on the Boulevard Saint-Michel on the Left Bank, overlooking the beautiful Luxembourg Gardens, on a sunny day, when Paul Lévy brought up the mysterious conjecture by …” was how he would start a new topic.
So besides real analysis, Loève also taught us that mathematicians can live exciting lives, that they like to congregate in cafés—just as Sartre, de Beauvoir, and Hemingway did—and that they form an integral part of the general culture, or, rather, a fascinating subculture with its own peculiarities and idiosyncrasies. My interest was so piqued that later, also at Berkeley, I took a course dedicated to the history of mathematics, taught by the renowned logician Jack Silver. There I learned that the lives of mathematicians can at times be downright weird: they can get absurdly involved in grandiose political intrigue, become delusional, falsify documents, steal from each other, lead daring military strikes, carry on affairs, die in duels, and even perform the ultimate trick: disappearing completely off the face of the earth so that no one could ever find them. Silver himself was a bit of a strange mathematician: he dressed carelessly, was always disheveled, and when he finished what he wanted to say, he simply turned around and walked out of the classroom—never a “Good-bye,” “See you next time,” “I will be in my office from two to four,” or any indication at all that class was over. We all sat there, looking at each other, until one intrepid soul or another would conclude that class was over and lead the way out. Presumably, Silver simply walked over to his office to continue working on his major theorem in the foundations of mathematics, Silver’s theorem, which he proved that very same year.
As my mathematical career matured, I began to realize that behavior that may seem unusual elsewhere in society is often taken as “normal” in mathematical circles, where no one dares complain about it. My undergraduate adviser at Berkeley was the well-known topologist John Kelley. I loved Kelley so much as a professor that I ended up taking most of the courses he offered. But although he had a gift for making everything seem easier than it was, other things about his classes were hard to take—today, he would not be allowed to do many of the things he did in class. Kelley was never without a lit pipe in his mouth, and his syllabus read: “I smoke constantly in class, so if you worry about getting nicotine poisoning, don’t take my course.” He often brought into class his two huge dogs (apparently immune to nicotine), and they would plunk themselves down in front of him, blocking the aisle; when they scratched themselves, it was like a drumroll or a minor earthquake. Kelley’s red or pink shirt was usually covered with political buttons, and he would admonish us to vote for his candidates or get involved in the political issues he favored.
After graduating, earning a doctorate some years later, and spending a dozen years teaching and doing research in mathematics and statistics as a professor, I returned to my early passion for the history of mathematics. And over the last decade and a half I have authored numerous popular books about the history of mathematics and the lives of mathematicians, from Fermat and Descartes to Cantor, Grothendieck, and the mysterious Bourbaki group. What I tried to do with all these books was to show how mathematics is entwined with the general culture, to point to what makes it unique and hence different from other disciplines, to expose how mathematicians think, and to showcase the kinds of exciting, interesting, and adventurous lives some mathematicians lead. I was greatly encouraged in this pursuit when Richard Bernstein, then a book critic at The New York Times, described in a book review the Parisian café scene presented in my 1996 book, Fermat’s Last Theorem: “The scene … reminds us that the world has many worlds, with the priestly cult of mathematicians, so mystifying and inaccessible to most people, among the more esoterically interesting of them.”
The realization of just how interesting the lives of mathematicians can be then caught the attention of a number of writers of both nonfiction and fiction. My friend Simon Winchester, author of superb works of nonfiction and acclaimed biographies, approached me at an international conference in Vancouver in 2001 and, in front of a crowd of three hundred people, suggested that we cowrite a biography of the nineteenth-century French genius Évariste Galois, who derived a remarkable theory in algebra and then died in a senseless duel at age twenty. The audience clapped enthusiastically as we shook hands in agreement. That book hasn’t become a reality—at least not yet—but I am immensely grateful to Simon for encouraging my research into the life of Galois, which is discussed at length in one chapter in this present
book.
Famous novelists, too, were attracted to the rich content and texture of the lives of major mathematicians. When my book Descartes’s Secret Notebook appeared in Italian translation in 2006, Umberto Eco devoted his weekly column in the Italian news magazine L’Espresso to a thorough review of it and raised a number of interesting issues. After I responded in a letter, Eco invited me to visit him in Milan to discuss the life of Descartes. I will never forget the experience of standing with him in his thirty-thousand-volume library, which takes up much of the space of his apartment, browsing through original seventeenth-century manuscripts about the life of Descartes while sipping Calvados. I thank Umberto warmly for sharing with me his extensive knowledge of Descartes and his work, and for his enduring friendship. Descartes is the subject of one of the chapters in this book.
Three years after my book The Mystery of the Aleph, about the life of the tormented German mathematician Georg Cantor and his stunning discovery that there are various levels of infinity, was published in 2000, the novelist David Foster Wallace wrote his own account of the life of this great mathematician. Cantor often worked in a frenzied state and suffered frequent periods of depression. These dark moods, in fact, may not have been too different from Wallace’s own bouts of depression, which reportedly may have been the cause of his tragic suicide. But in an interview for the Boston Globe in 2003, Wallace dismissed any connection between genius and madness and distanced himself from Cantor’s mental problems, saying that he did not want to follow my approach of looking jointly at Cantor’s mathematics and psychology; his book explored other directions, he said. But when Cantor died in 1918 in an asylum in Halle, Germany, he had been working on a mathematically impossible problem called the continuum hypothesis. His psychology, in fact, could not be separated from his mathematical work: late in life, Cantor made the continuum hypothesis a matter of personal dogma, “decreed by God.” Some years ago, I visited the mental health facility in which Cantor spent years trying to recover, and where he ultimately died. A century later, the building is still a functioning hospital in an economically depressed part of Germany. I stood in the very room in which Cantor had worked on mathematics. I saw the claw-footed bathtub in which he was forced to immerse himself for hours as “treatment” for his depression, and I read the hospital’s records of his repeated admissions and discharges from the late 1800s to 1918, when he died of starvation inside this facility. The life of Georg Cantor is covered in another chapter in this book, and I am grateful to the late David Foster Wallace for highlighting the question of the relation between psychology and our attempts to tackle the immense complexity of the infinite.
Sometime after my book The Artist and the Mathematician, about the lives of the mathematicians André Weil and Alexander Grothendieck, as well as the secret mathematical group called Bourbaki, was published in 2006, Sylvie Weil, André’s daughter, published a poignant memoir about her life with her famous father and aunt. Her aunt was the philosopher Simone Weil, who often accompanied her brother to mathematical conferences of the Bourbaki group and became affectionately known as Bourbaki’s “mother.” Visiting Boston in 2011, Sylvie graciously shared with me many of the details I had not known about her father’s life, and I am indebted to her for her kindness. André Weil is discussed in the last chapter of this book.
This chapter, fittingly entitled “The Strangest Wilderness,” also deals with the life of Alexander Grothendieck—the mathematician who managed to completely disappear from our world, hiding somewhere in the forests or foothills of the high Pyrenees, which separate France from Spain. I am grateful to Pierre Cartier, a leading French mathematician and member of the Bourbaki group, for sharing with me his knowledge of the life of Grothendieck as well as fascinating stories about the founding and collective work of Bourbaki. Two mathematicians working in Paris, who insist on anonymity, also provided details about the life of Grothendieck.
Researching this book has been one of the greatest adventures of my life as an author. It took me to faraway corners of the world, from the island of Samos, where Pythagoras was born, to southern Italy, to Beijing and Delhi, and to countless locations in Europe—all in search of intricate details of the lives of our greatest mathematicians. I thank the mathematicians Marina Ville and Scott Petrack, my good friends, for their help and for discussing with me mathematics and the life stories of some of the mathematicians described in this book. I am indebted to Barry Mazur of Harvard, Akihiro Kanamori of Boston University, Goro Shimura of Princeton, Ken Ribet of Berkeley, and Saharon Shelah of Hebrew University for fascinating details of mathematics and its history.
My warm thanks go to my friend and agent, Albert Zuckerman, president of Writers House, for his encouragement, direction, and support throughout the process of writing this book. I am immensely grateful to my editors at Sterling, Michael Fragnito and Melanie Madden, who first suggested that I write a book about the lives of the great mathematicians. Melanie’s superb editing of the manuscript in all its stages turned a rough draft into a complete book. She possessed the right vision of how to organize the complex material into parts, chapters, and sidebars, making the stories come alive and their heroes shine. Melanie wisely made me see what needed further explanation or expansion and what could well be omitted, and I am deeply indebted to her for her great insight and talent. I am also grateful to Barbara Clark for her excellent copyediting, and to Joseph Rutt of Level C for his beautiful design.
Finally, I thank my wife, Debra, and my daughter, Miriam, for their enthusiasm for this project and for their many helpful suggestions and ideas. I hope the reader will enjoy the life stories of the great men and women that are the subject of this book: history’s greatest mathematicians.
The Greek philosopher Thales proposed the first known mathematical theorem in history during his visit to the Pyramids of Giza in Egypt in the seventh century BCE.
INTRODUCTION
Our story begins around five thousand years ago in the civilizations of Egypt and Mesopotamia. These two ancient centers of human habitation are called potamic—from the Greek word for “river” (potamos)—because they developed in major river valleys: the great Nile in the case of Egypt, and the Tigris and Euphrates in the case of Mesopotamia. These valleys provided fertile ground for the development of agriculture—a key technological advance that had originated in the Jordan River Valley some eleven thousand years ago. The first mathematicians—people who performed rudimentary estimation work—were the “rope pullers” of the Nile Valley, assigned to demarcate the boundaries between the fields of various owners after the waters of the Nile receded every year following the annual inundation. Early geometrical ideas were developed in response to these problems. Pulling ropes in a flat terrain such as the Nile Valley led to the very first ideas in what is now called Euclidean geometry, named after Euclid of Alexandria, who lived much later. It is the geometry of straight lines that we study in school today.
Equally, astronomical observations of stars and planets carried out in Egypt and in Mesopotamia led to developments in mathematics. The Babylonians, Assyrians, and other inhabitants of the Fertile Crescent of Mesopotamia kept voluminous records of astronomical phenomena, such as solar eclipses, movements of planets, and locations of stars. The analyses of these data led to basic advances in calculation. Trade and finance, along with astronomy, brought about the use of a sexigesimal (base-60) number system in Mesopotamia. Though much more complicated than our current decimal (base-10) number system, we still see remnants of the Mesopotamian number system today in our clocks and geometrical and trigonometric calculations (e.g., an hour is composed of 60 minutes, a minute of 60 seconds, and a circle has 360 [6 × 60] degrees). The cumbersome base-60 number system of the Mesopotamians still enabled them to understand squares, square roots, and other concepts thousands of years ago.
The Plimpton 322 cuneiform tablet, which contains fifteen Pythagorean triples, was discovered near Sankarah, Iraq. Scholars estimate it was written around 1800 BCE.
Babylonian and Assyrian clay tablets contain an abundance of early mathematical discoveries. For example, the celebrated Plimpton 322 tablet has a list of 15 “Pythagorean triples”—sets of three squared numbers where two of them added equal the third (e.g., 9 + 16 = 25, which, as we know, is 32 + 42 = 52). This tablet and others—including one displayed in the Louvre in Paris with a geometrical design that looks uncannily like a graphic depiction of what we call the Pythagorean theorem—have led many historians of mathematics to conclude that the ancient Mesopotamians may have developed the theorem we now attribute to Pythagoras, who lived at least twelve centuries later.
While the Babylonians and Mesopotamians impressed cuneiform signs into wet clay that they later baked in the sun, the Egyptians developed another technology for keeping records. By cutting reeds found along the Nile into strips, laying them side by side in two layers, and then mashing the layers into a sheet, they made papyrus, which they wrote on using a reed pen and carbon-based ink. The Ahmes Papyrus, named after the scribe who wrote it (it is also known as the Rhind Mathematical Papyrus, named after the Scottish collector who bought it from an Egyptian antiquarian in the nineteenth century), contains many examples of arithmetic, algebra, and geometry, including ways of solving equations and elementary mathematical problems arising from commerce and other areas of life.
The Ahmes Papyrus was written in Thebes during the Second Intermediate Period of Egypt (ca. 1650–1550 BCE). Its introductory paragraph says that the papyrus presents an “accurate reckoning for inquiring into things, and the knowledge of all things.”
Ancient India also had a thriving mathematical community that studied rudimentary computations relevant to everyday life and astronomy. The numerals we use today evolved from early Hindu numerals developed by Indian mathematicians. But the first mathematicians we know anything about—and the ones who made mathematics into an abstract, powerful science and art—were the ancient Greeks.
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