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Mathematicians: An Outer View of the Inner World

Copyright Date: 2009
Pages: 208
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  • Book Info
    Book Description:

    Mathematiciansis a remarkable collection of ninety-two photographic portraits, featuring some of the most amazing mathematicians of our time. Acclaimed photographer Mariana Cook captures the exuberant and colorful personalities of these brilliant thinkers and the superb images are accompanied by brief autobiographical texts written by each mathematician. Together, the photographs and words illuminate a diverse group of men and women dedicated to the absorbing pursuit of mathematics.

    The compelling black-and-white portraits introduce readers to mathematicians who are young and old, fathers and daughters, and husbands and wives. They include Fields Medal winners, those at the beginning of major careers, and those who are long-established celebrities in the discipline. Their candid personal essays reveal unique and wide-ranging thoughts, opinions, and humor, as the mathematicians discuss how they became interested in mathematics, why they love the subject, how they remain motivated in the face of mathematical challenges, and how their greatest contributions have paved new directions for future generations. Mathematicians in the book include David Blackwell, Henri Cartan, John Conway, Pierre Deligne, Timothy Gowers, Frances Kirwan, Peter Lax, William Massey, John Milnor, Cathleen Morawetz, John Nash, Karen Uhlenbeck, and many others.

    Conveying the beauty and joy of mathematics to those both within and outside the field, this photographic collection is an inspirational tribute to mathematicians everywhere.

    eISBN: 978-1-4008-3288-0
    Subjects: Mathematics, Art & Art History

Table of Contents

  1. Front Matter
    (pp. 1-4)
  2. Table of Contents
    (pp. 5-6)
    (pp. 7-7)
    (pp. 8-11)

    Mathematics is one of the greatest human accomplishments. It has long been an important human activity, from its early use in surveying and in construction in Babylonian and ancient Egyptian times, through the present phenomenal expansion of its use and study. By the period in which Greek mathematics flourished, it was not just important for its uses but it was also a major intellectual endeavor, and already it had developed the abstraction and rigor that continue to be a principal characteristic.

    One of the striking aspects of mathematics is its cumulative nature; it is perhaps the only truly cumulative human...


      (pp. 12-13)

      I had the great good fortune to be the youngest of four sons with a seven-year gap between my brothers and me, born into a warm and loving family. This was in Georgia, in the depths of the Depression, where my father organized interracial conferences. He was the sixth Methodist minister in lineal descent. While driving he would amuse himself by mentally representing the license plate numbers of cars as the sum of four squares.

      I attended first grade in fascist Italy and still have the notebook in which I wrote to dictation, “Mussolini ama i bambini.” What a boon...

      (pp. 14-15)

      I find mathematics amazing. Why should numbers capture our world much better than words? Why do such a variety of people, like astronomers, bakers, conductors, and those who make Zbosons, rely on numbers to get things working exactly right? I am not a philosopher. I simply observe with amazement how the most abstract and cryptic symbols from our notepads and computer screens correspond perfectly with the world around us: with how the rainbows form, how the planets orbit, how everything else happens. Part of the miracle is that math problems, however complicated, always have to work out somehow. Often the...

      (pp. 16-17)

      When I was nearly forty years old I had a revelation: a recurring dream that I’d had since age twelve was an allegory of my birth! In the dream, I was stuck in a secret passage in our house but eventually worked my way out and emerged into a sunlit cupola. After my revelation, I stopped having the dream.

      My mother says that I was a big baby and it was a difficult birth, although I don’t know what I weighed. The conversion from German to English pounds adds 10 percent, and I suspect that my mother added another 10...

      (pp. 18-19)

      I was born in Liverpool, England in late 1937. My father was a lab assistant in a big Liverpool school that two of the Beatles went to. My father was very knowledgeable about science, and was also very interested in poetry. At home he’d stride up and down, sometimes naked, reciting poetry as he shaved. He was a strange character, an interesting person, I think. My father was also an air-raid warden. Once in a while, the sirens would sound. The war ground itself into me when I was a kid. Sometimes kids wouldn’t come to school and we’d learn...

      (pp. 20-21)

      I was born on October 17, 1927 in Hamm, a town north of the Ruhr district in Germany. My father was the director of a secondary school and taught mathematics. Very early, my interest in numbers arose, and in school, my favorite subject was mathematics. I had good teachers, but I extended my knowledge by studying the mathematical books of my father and by talking to him.

      In 1937, I had to join the Deutsches Jungvolk. I was aware of the pogrom of 1938. My father said to his children, “Always remember that I do not agree with this.” I...

      (pp. 22-23)

      My life as a mathematician started — as for many others who grew up in East Europe — with the high school mathematics olympiads. Starting seriously in ninth grade, the best mathematics students compete with others to solve problems. First in their own towns, then against the others in their county, and at the end they can test their skills at an international competition. Everyone has four hours to work on three hard problems. Usually, only the best can solve more than one of them. I was an undistinguished student before, and it was quite a surprise when I did...

      (pp. 24-25)

      As a child I played chess seriously but gave it up because I wasn’t as good at it as I would have liked to be. In high school my favorite subject was mathematics. But high school mathematics is not very interesting, so I started to read books about mathematics on my own, such as G. H. Hardy and Edward Wright’sThe Theory of Numbers.

      One question which endlessly puzzles me is why mathematics exists at all. This is probably one of those ultimate questions — such as, why does the universe exist? Or, what is consciousness? — that may never...

      (pp. 26-27)

      In my own experience, mathematics in general and pure mathematics in particular has always seemed like secret gardens, special places where I could try to grow exotic and beautiful theories. You need a key to get in, a key that you earn by letting mathematical structures turn in your head until they are as real as the room you are sitting in. My father’s mother had the gift: she was one of the first women to beat most of the men in the Mathematics Tripos at Cambridge. My aunt also had it: she called the complex numbers a “delightful fiction”...

      (pp. 28-29)

      My grandfather was a self-made man who built up a baconcuring business. At that time, it was the custom that eldest sons went into the family business, so though my father clearly had mathematical ability, there was no question of his going to university. My father was a reluctant but successful and popular employer, but when it came to his turn, he made sure that his children should choose their own careers, and we have all been very grateful.

      As a little boy, I always enjoyed sums and my career has been in the theory of numbers. When I was...

      (pp. 30-31)

      Many scientists in the twentieth century emerged from complex backgrounds, forced to emigrate by the oppression of Nazi Germany. This enforced cosmopolitanism may have broadened their outlook and helped their subsequent careers. While I was not one of Hitler’s refugees, I did oscillate in my childhood between Europe and the Middle East. My mother was Scottish, my father Lebanese, and we lived in Khartoum. My secondary schooling until the age of sixteen was in Egypt, and my grandmother lived in Lebanon.

      In 1945 we moved to England and, after my studies in Cambridge, we spent much time in the United...

      (pp. 32-33)

      My parents met and married in Toronto, Canada, after emigrating from Poland in 1917. They moved to Detroit, Michigan, where I was born in 1924. I was a bright student, but unfocused. Summers were spent playing baseball and reading. I woke up to the world of science when my high school chemistry teacher introduced me to the elegantly ordered periodic table. Soon I was president of the science club and lecturing on relativity.

      I entered the University of Michigan in September 1941, choosing physics over English literature as a major; for me, electromagnetism was easier to understand than poetry. Because...

      (pp. 34-35)

      The imprint of the world in our minds is not photographic; all the brain knows of the outside world is a chaotic sequence of electric impulses and out of these it creates a structural entity: our perception of what we see and hear. Most of the time, an adult’s brain talks to itself and creates more and more refined structures within itself. The word “structure” means a mathematical structure, something which becomes more and more abstract and better and better logically organized in the course of this self-conversation. The mathematical ability of each person’s brain by far exceeds those of...

      (pp. 36-37)

      I became interested in science at an early age, perhaps five or six. I was born into a family with a particular religious outlook, and it became clear to me fairly early on that theirs was not a worldview that suited me well. However, it still had a considerable grip on my imagination, and I decided that I needed some kind of counterweight to the certainties with which I was being presented. I became interested in science at least partly out of this need. Science was an approach to building certainty about the fundamental nature of the universe that was...

      (pp. 38-39)

      I was born in China in the ancient capital city of Xian. It was during the period of the Chinese Revolution, so my family moved to Hong Kong and then to Taiwan when I was two. My father was an architect and my mother an accountant. I grew up in Taiwan and attended National Taiwan University.

      While I was growing up, I was fascinated with Chinese literature but also very good in mathematics. I found mathematics simple and elegant; I enjoyed the logical way of thinking. After World War II, the economic environment in Taiwan was extremely harsh, so it...

      (pp. 40-41)

      I grew up in a village in Hong Kong. It was a beautiful landscape with oxen and other animals. I could see the ocean and the mountains, and I went to school in this farmland in my early years. For my middle years, I went to school in the city. My father was a professor in Chinese philosophy and in economics. A professor didn’t earn much money in those days. I learned a lot from him but he died when I was fourteen. We had to struggle because we were a very poor family. I had eight brothers and sisters....

      (pp. 42-43)

      I was born in a small city in southern West Virginia. My father was an electrical engineer. And there came a time in my life there when I would go into his office and play around with some calculator machines which they happened to have but which were not so common in those days. I developed an early interest in arithmetic and studied some mathematics on my own so I could study higher-level math before I went to college. My parents also arranged for me to study part time in a local junior college while also continuing in the town...

      (pp. 44-45)

      It would be easy to say either too little or too much about the past. I was a lucky child, growing up in the optimistic post–World War II climate. We played in the rural hills of northern New Jersey, prepared ourselves for our chance at the great world of culture: art, music, science, and intellect. My mother, an artist, remains a major influence on my life although she has been dead a number of years. Through her I received just about the right amount of introduction to nontraditional life styles and intellectual ambition.

      We should remember that a girl...

      (pp. 46-47)

      I can’t remember a time when I wasn’t interested in math. I remember calculating all the powers of two when I was quite little. I thought it was horrifying when my father told me a car could run out of gas. I pondered why that should be. I figured all you have to do is just use half of what’s remaining. Then you could use half of that and so on. I wasn’t especially good at arithmetic. I made arithmetic mistakes, but I knew that math was for me and I tried to move as fast as I could in...

      (pp. 48-49)

      I grew up in rural North Carolina and went to largely rural schools and then Georgia Military Academy near Atlanta. It was a tradition in the South back then to go to military school. That’s where I really fell in love with mathematics. After being exposed to the subject by a superb teacher, Lottie Wilson, I was going to think about mathematics no matter what. I went to graduate school in math at Princeton and did a postdoc at Berkeley. I taught at Harvard for many years and then went to Duke University as provost. In 1991, I went to...

      (pp. 50-51)

      My mother was a mathematician. She studied Hilbert’s sixteenth problem and made an outstanding contribution. This is a problem of studying a dynamical system governed by two polynomials. When I was a kid, my mother often gave me logic problems to solve. Most of them were not that hard but were very interesting. I liked to think about them. My mother also told me about other things, such as historical stories and Chinese classical poems. When I was seven, the Cultural Revolution started in China and lasted ten years. During that period, universities were essentially closed and my parents went...

      (pp. 52-53)

      I have only an empty list of professional scholars in my family tree, except for mentioning my father and my uncle who failed to become such against a rather strong wish to be one. My father was remembered as a 100 percent professional merchant. But at my good age some old person in town told me that my father had an unusually strong wish for “studying” at his young age. When my father was thirteen years old, his father died and he was forced to succeed in the family business. In protest against his mother’s wish, he went on a...

      (pp. 54-55)

      When I was about twelve years old, I remember my father sitting by a large fireplace in an old farmhouse nestled in the French Alps. He was talking with three of his students. I sat nearby, happily reading a book, enjoying the snow falling outside and the cozy warmth within. Suddenly my father and his younger colleagues stopped talking and were deep in thought. The stillness startled me and seemed to last an eternity, yet they all seemed quite comfortable and absorbed in their world. After a time someone said something and everyone smiled with a quiet pleasure and delight....

      (pp. 56-57)

      I first realized that I wanted to be a mathematician during my freshman year at Princeton. I had dabbled in mathematics before — my father was an electrical engineer who owned assorted books on engineering-style mathematics (and also a ridiculously terse primer on complex function theory, translated from German). But in Princeton I discovered that mathematics is much easier than other subjects!

      Physics fascinated me, but the courses often seemed boring, with lab experiments that never quite worked. A music course taught me that I have no ear for music; a philosophy course was totally arid; and a creative writing...

      (pp. 58-59)

      Why did I choose mathematics? I’m not sure that “choose” is the right word; rather, mathematics chose me. As a very young child I always wanted to understand how things work: figuring out how to build a sturdy windmill that would turn without falling apart, out of rods and spools, was one example. Predicting the swirling patterns made by many rolling marbles was another. I was fascinated by such questions, enjoyed a certain kind of solitary play, and often didn’t want to leave it for meals when I was called, much as I don’t find it easy to stop working...

      (pp. 60-61)

      It is one thing to do research in mathematics, and quite another to explain it to nonmathematicians, or even to fellow mathematicians working in different parts of the subject. This is one of the more frustrating aspects of being a research mathematician, although it is at least partially outweighed by the way that mathematics transcends political and cultural boundaries: the authors of the next research paper I pick up to read are as likely to be Indian, Japanese, Russian, or Brazilian as to come from my own country. So instead of trying to describe what my research involves (studying moduli...

      (pp. 62-63)

      I was lucky to be born in 1938: I was never in danger of being drafted. I was also lucky to have excellent parents, both educated with a bit of graduate school, but comparatively poor. My father was a (quiet) conscientious objector during World War II and hence lost a few jobs. He then went back to graduate school at age forty-one in 1948, and we lived on a teaching assistant’s salary until I went to the University of Chicago in 1954. The lack of money can influence a child in various ways, but in my case it seemed as...

      (pp. 64-65)

      How did I get started? My mother givesSesame Streeta lot of credit, and I think that’s fair: I started reading early, because I wanted to know what the words I saw everywhere were saying. My father was a computer programmer in the ’60s, when only big companies could afford computers — and the computers themselves were enormous. My father and I played with flowcharts for mathematical problems, and so we were happily ready when the first personal computers appeared in the ’70s. At that time, to get a computer to do anything interesting, you really had to program...

      (pp. 66-67)

      My father had a large influence on my development into a mathematician, at least in a general way. I have an early memory of him saying with relish : “. . . and then I shall be able to get back to research” (presumably, after completing some chores which he had described to me). I had no idea what “research” might be, but from that time the word was tinged with glamour and romance.

      My father was an engineer (my two brothers followed him) and was often sceptical of overly theoretical work. “All they produce is paper” — I hear...

      (pp. 68-69)

      I learned to read at home. My family lived in Paris and at six, I was sent to the Lycée Buffon. After the first test paper, I told my parents I was very proud because I came in twenty-fourth. To me this high number was a proof of excellence. It was explained that it was better to be first, which I managed to be from then on.

      We spent the summer in a little village in Isère, Dolomieu, near the family smithy where my father, Élie Cartan, had spent his youth. His extraordinary gift for mathematics had been detected by...

      (pp. 70-71)

      My father, a physicist, used the word “mathematician” exclusively as a put-down. A “mathematician” was someone with no physical intuition who was caught up in irrelevant problems. He finally saw an upside to my choice of a career when he realized that as a byproduct of doing mathematics, I got to experience different cultures. The mathematical community is really international. A proof is a proof, no matter what language it is in. Furthermore, mathematicians do not have laboratories, so we can move around. I have lived in ten countries for at least a month, and I have mathematical friends from...

      (pp. 72-73)

      My father, Benedict, was a child prodigy in mathematics, musical, and also a brilliant writer. I had already regressed toward the mean. Still, as a child, I also displayed certain peculiar mathematical talents but was not exceptionally strong in computation. Outside of mathematics, I had no other demonstrable talents. However, I painted in an expressionist style and this, my mother, Nancy, convinced me, showed genius. I learned quite young that mathematics also was an art form and felt certain I could make a mark upon it. This certainty was based really on nothing but exuberant overconfidence, to which I can...

      (pp. 74-75)

      Born in London just after the Second World War, I grew up in Edinburgh. My father was an embryologist and geneticist, interested in how organisms develop but also deeply interested in art, philosophy, and the uses of science. My mother was an architect, who worked in the town-planning department of the Scottish Civil Service. She always had a job, which was unusual for the time, and brought me up to think that I would also have a career. Her father had read mathematics at Cambridge University and then became a distinguished lawyer; my talent in mathematics came from that side...

      (pp. 76-77)

      “To understand” was the goal I gave for my high school yearbook, and this is still what drives me. I love to reach understanding: first, to see something (big or little) that doesn’t make sense or is simply discordant, then to reflect and ponder, to search and stare in my mind’s eye until sometimes, miraculously, vision is transformed and mist and muddle develop into form, order, and connection.

      Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. I’ve loved mathematics all my life, although I often doubted that mathematics would turn out to be my life’s...

      (pp. 78-79)

      I was in high school during the Second World War, and although I was good in mathematics, my main scientific interest was in chemistry. However, I was very undisciplined with regard to homework and I did badly in nonscience subjects. Consequently I was rejected from admission to first-rank universities.

      Nevertheless, things turned out very well indeed. The theme of what I now want to say is that by some quirk of fate I seemed to be at the right place at the right time. In the postwar period, German academic refugees brought an infusion of European science to American universities....

      (pp. 80-81)

      I remember being fascinated by logarithms when I was six years old. My father must have explained them to me. He was a professor of electrical engineering at Princeton. From time to time, he liked to teach me some aspects of elementary mathematics.

      When I was eleven or so, I found one of my father’s engineering textbooks on a shelf at home and I spent a considerable amount of time delving into it. It was basically a calculus text, including the calculus of variations, motivated by lots of applications to engineering problems. There was a lot that I did not...

      (pp. 82-83)

      I grew up in Iran and had a happy childhood. There were no scientists in my family, but I learned a lot from my older brother, who was always interested in math and science. Around me, women were encouraged to be independent and pursue their interests. I remember watching programs on TV about notable, strong women like Marie Curie and Helen Keller. I admired people who were passionate about their work, and I was impressed with books likeLust for Life, about Vincent Van Gogh. However, as a child I dreamt of becoming a writer, and reading novels was my...

      (pp. 84-85)

      There are many things that got me interested in mathematics. In Ohio on July 20, 1969, I somehow got the shutter to stick open on the plastic Kodak camera my parents had brought back from the New York World’s Fair. Later that night, we watched Neil Armstrong, also from Ohio, set foot on the moon. When the film was developed, the timed exposure showed the moon as a white comet arcing across the evening sky. Shortly thereafter we moved to the small town of Charlotte, Vermont. In the moving boxes, I found a copy of Daniel McCracken’sGuide to Fortran...

      (pp. 86-87)

      A feature of abstract mathematics that must be surprising to the thoughtful layperson is that in mathematics the concept of infinity can be formally defined and that mathematicians can work with it. Actually mathematics is very special in that each of its concepts can be precisely defined and that this in fact must be so before mathematicians can get to work. Progress in mathematics often depends on the new concepts that are actually created by these new and precise definitions. The combinatorial patterns, geometric pictures, and algebraic calculation that are present in many branches of science are the nourishment for...

      (pp. 88-89)

      I was brought up in a rural area in Michigan. From the age of five to fifteen, I lived on a ten-acre farm. My father worked in the city in the automobile plants. I went to a one-room schoolhouse for eight years, walking more than a mile each way. With one-room schoolhouses, you don’t have too many expectations. While I was in high school, my father bought me a small chemical lab and I became obsessed with organic chemistry, even to the extent of foolishly offering to make rare chemicals needed in private industry. I turned to physics when I...

      (pp. 90-91)

      I was born and grew up in Moscow. I fell in love with math when I was in the fifth grade. Traditionally the level of education in the big cities in Russia was very high. The school program in math was rigorous and stimulating. I was fascinated with mathematical reasoning in algebra and geometry, which was beautiful and exciting. Math came naturally to me and I felt unmatched satisfaction solving difficult problems.

      There were no mathematicians in my family. My father was a well-known plant physiologist and my mother was a chemist. In the late ’40s my mother was fired...

      (pp. 92-93)

      My grandfather V. F. Kagan was a very famous Russian mathematician. He was an expert in geometry and wrote several books on Lobachevsky and Lobachevskian geometry. One of his papers written a hundred years ago was recently mentioned in theAmerican Mathematical Monthly. At the moment I decided that I would also become a mathematician, my grandmother told me that I should be aware that mathematicians think about mathematics twenty-four hours a day. She asked, “Do you really want to do this?” I thought again and decided I would do it. I try to keep this habit even now.


      (pp. 94-95)

      Both my youngest Uncle and I were born in Warsaw and grew to be mathematicians. But overly interesting times cursed his late teens and later mine and made us into altogether different persons. He became a focused full-time establishment insider, and I never did.

      An adolescent during World War I, roaming around during the Russian Revolution, my Uncle fell in love with classical French mathematical analysis, moved to its source, was soon handed its torch, and kept it burning through both fair weather and foul.

      An adolescent during World War II, I found shelter in the poor and isolated highlands...

      (pp. 96-97)

      I was born in Aba in Nigeria in 1941. My mother was a daughter of the Obong of Calabar (literally translated as “King of Calabar”). She trained as a nurse and took up an appointment in a hospital working with doctors and staff from the United Kingdom, the governing country of the then British Empire which included colonial Nigeria in quite “far away” Abeokuta in western Nigeria, where she met my father, a doctor. My mother and her friends suggested the names George and Elizabeth for me and my twin sister after the recently enthroned King and Queen of England....

      (pp. 98-99)

      My mother is British, from a family with a trade-union background and a central interest in class struggle; she met my father, who is Nigerian, while both were students of mathematics in London. My father was a very talented mathematician, and after my parents married, he went on to a position in the mathematics department of the University of East Anglia. While I was growing up, the elementary school I attended was extremely ethnically homogeneous. I was unable to escape from heavy issues concerning race, which my mother always explained in a political context. My parents separated after my father...

      (pp. 100-101)

      I grew up in a family of musicians: my father was a composer and my mother a piano teacher. At school I did well in most subjects. Though mathematics was always my favorite, there were several others that I liked almost as much, and it wasn’t until I was about eleven or twelve that it became clear that I would be likely to specialize in mathematics, and a few more years after that before I gave up all thoughts of becoming a musician. If I had become one, I would probably have tried to follow in my father’s footsteps and...

      (pp. 102-103)

      There are a number of stereotyped ideas about mathematics and mathematicians. I shall look at some of these from my own personal experience.

      “There is a small number of people with a very special talent for mathematics.”

      Certainly you need good intelligence to become a good mathematician. Many people are prepared to use the word “genius” for the student in the class who can solve all problems on a school test. I should like to reserve the word for those with very special insight. In my lifetime I have only met a handful of people like that; they do, however,...

      (pp. 104-105)

      I’ve always liked math. I remember when I was two or three and I was wandering around with my grandmother. She was washing the windows and just to play a game with me, she’d ask me to pick a number, like 3. I’d use the detergent and she’d spray a big 3 on the window and then clean it. I thought it was great fun. I had workbooks as a kid. They were simple, with equations like 3 +□= 7. What’s in the box? I thought it was really fun. Math was the only thing that really made sense to...

      (pp. 106-107)

      I have been privileged to live the life of an academic mathematician, and a privileged life it is. It has an invigorating annual renewal: the departure of one group of undergraduate and graduate students whom one has tried to provide with a knowledge of and intuition about some parts of mathematics, with a sense of the excitement and challenges of mathematics, with at least some responses to the questions they have raised or should have raised, and with the encouragement to continue to learn and use mathematics. Then there is the arrival of a new group of undergraduate and graduate...

      (pp. 108-109)

      When I was very small, I became fascinated with the idea of perpetual motion. At the age of five I thought I had invented such a machine. I elaborated on this idea and imagined many variants of it. My parents didn’t know anything about science, but they humored me. When I got older I realized it probably wouldn’t work, but it was comforting to maintain the illusion that I had some special talent and I might do something great.

      When I was ten, in 1941, I came to America from Belgium because of the war. Shortly thereafter I became very...

      (pp. 110-111)

      I was born in Prague, Czechoslovakia, and lived there until the age of seven. We migrated to Ecuador in 1939, three months after the Germans occupied Prague. My maternal grandfather was a well-known lawyer and he had a keen interest in mathematics and the sciences. I still remember that when I was five he explained to me how to estimate the height of trees using a marked walking stick and how to tell the time of day from the position of the sun. My father was an architect and from an early age he got me interested in perspective, geometry,...

      (pp. 112-113)

      As a little kid, I wanted to know how rockets work and borrowed a physics book from a library. I didn’t understand one word. My wise father explained to me that the book was full of math, which I hadn’t studied.

      I set about reading math textbooks, starting with fourth-grade arithmetic. After I got through calculus, my father took me to our local university, the University of Maryland, for tutoring. That was the start of my relationship with the University of Maryland. They were wonderful. It’s a big state school, but I had the feeling that their whole math department...

      (pp. 114-115)

      My mother, who was raised in Germany, attended schools featuring exceptionally strict discipline and particularly poor instruction in mathematics, and she thoroughly hated the subject. My father was an economist who earned his PhD without taking so much as a calculus course because he was advised by his teachers that economics had nothing to do with math. Having consistently excelled in math courses in high school, he had loved and respected the subject as much as any other student and had always wished to take up calculus on his own. My parents had two children and both became mathematicians. I...

      (pp. 116-117)

      Born in China in 1943, I spent my childhood in Macau and teenage years in Hong Kong. After my undergraduate education at the University of Hong Kong, I went to the University of Minnesota for a master’s degree and got my PhD from Princeton University in 1966. Since 1992, I have been the William Elwood Byerly Professor of Mathematics at Harvard University and served as chair of its mathematics department between 1996 and 1999. Before joining Harvard in 1982, I held faculty positions at Purdue University, the University of Notre Dame, Yale University, and Stanford University.

      Although my mathematics career...

      (pp. 118-119)

      My love of mathematics began when I was a child. My father tried to teach me Hebrew, but I foolishly resisted, so he got a friend to give me private lessons. But he liked mathematical puzzles, so much of each lesson was spent on them. I went to high school in Montreal during the Depression. To be a high school teacher was considered a desirable job, and I had excellent teachers, particularly the physics teacher, who had a PhD. I decided I would try to study physics.

      In college I majored in math and physics, with the intention of becoming...

      (pp. 120-121)

      Asked the question, how did three brothers all end up becoming mathematicians? as I often am, I have no ready answer. Genetic predisposition is belied by the fact that our parents had no relation to mathematics, our mother having a law degree from St. Petersburg, and my father largely self-educated, leaving school in the third grade, reading widely, and earning a degree in law, as well, by mail. Though he never practiced, the degree stood him in good stead in defending himself against the government in various prosecutions, after he had been thrown out of the Communist Party after many...

      (pp. 122-123)

      I was born in Moscow, Russia in July 1927 and was brought to the United States at the age of five. I am the son of Earl Browder, the expelled general secretary of the American Communist Party. My father didn’t graduate from primary school. His father was an unemployed schoolteacher and trained his children at home, but my father essentially taught himself. My father opposed World War I and was the leader of Socialist antiwar agitation in Kansas City, Missouri. He was jailed in 1917–1920 for his opposition to the war. Over the course of his lifetime, my father...

      (pp. 124-125)

      Many mathematicians, perhaps most, knew from an early age that mathematics was the most interesting thing in the world and can hardly imagine wanting to do something else. I’m one of the exceptions.

      In the spring of 1955, with the truce in Korea still holding, the Eisenhower administration decided to reduce the size of the military. One of the measures adopted was to offer early release from service (up to three months) to anyone attending graduate school. At this time, I was a private at Fort Dix and looking forward to civilian life: I applied to graduate schools for the...

      (pp. 126-127)

      One of my daughters said to me that the problem of being a mathematician is that you’re on stage all the time in the sense that you’re constantly trying to achieve something in the form of proving a theorem. That’s unending. You compete against yourself as well as others and it provides a special fascination in life.

      In my youth, there were not many women who wanted to try this. However it was, in a way, a natural profession for me. My father, John L. Synge, was a mathematician who went back and forth from Ireland to Canada. He was...

      (pp. 128-129)

      Like most mathematicians, I became fascinated with mathematics early, about age ten. I was fortunate that my uncle could explain matters that puzzled me. Mathematics had a deep tradition in Hungary, going back to the epoch-making invention of non-Euclidean geometry by John Bolyai, a Hungarian genius in the early ninteenth century. A journal for high school students, and contests, were instrumental in identifying talented young people early and they were nurtured intensively. I was tutored by Rózsa Péter, an outstanding logician and pedagogue. Her popular book on mathematics,Playing with Infinity, is still the best introduction to the subject for...

      (pp. 130-131)

      I believe that mathematics is one way the human mind can create concepts. In many ways mathematics plays a role that philosophy could have played in creating concepts that then can be used in the real world. It takes time for them to evolve and be used in the real world, but the real factory is mathematics. Its concepts have to do with shape and abstract things among others, and they are much more subtle and diverse than numbers. This is probably something that the general public does not realize. Mathematicians use numbers only when they are needed. One could...

      (pp. 132-133)

      I do not consider myself a prophet. I am simply a student. All my life I have been learning from great mathematicians such as Leonhard Euler and Carl Friedrich Gauss, from my older and younger colleagues, from my friends and collaborators, and most importantly from my students. This is how I continue working.

      Many people consider mathematics to be a boring and formal science. However, any really good work in mathematics always has in it beauty, simplicity, exactness, and crazy ideas. This is a strange combination. I understood early on that this combination is essential from the example of classical...

      (pp. 134-135)

      I grew up in New Zealand, one of two children in a family with no academic ties whatsoever. My father had briefly begun to study law, but the Second World War intervened and he never returned to study. I do recall that my mother was good with numbers, and from an early age I was eager to learn about arithmetic. I remember making up my own multiplication tables if I was sent to my room for misbehaving.

      My formal education was normal enough for New Zealand, where the schools at that time were of exceptional quality, and I began to...

      (pp. 136-137)

      While growing up, most Indian children of my generation thought they wanted to become doctors, engineers or elite officials of the administrative services of the Indian government. My initial desire was to become a doctor. This changed abruptly when as a ten-year-old, I visited a college fair at the local medical school. The anatomy exhibits of real human body parts were so discomforting that I then switched to engineering as a more acceptable alternative. I was always good in mathematics at school. Even though this only meant that I could add, subtract, multiply, and divide rapidly without making mistakes, it...

      (pp. 138-139)

      I was raised in Senegal. I mention it only because I received a prize years later by proving that “one cannot hear the shape of a drum”: in a mathematical sense there exist distinct drums which cannot be distinguished by their sounds. The question was raised while I was attending a conference in California in 1977 and it reminded me of nights in Africa when I listened to Senegalese people playing drums and dancing outside our home; I had tried to guess what the instruments were from their sounds.

      Happy coincidences like this one are often responsible for theorems. Ideas...

      (pp. 140-141)

      What did I do in my life as a mathematician? I could look at my list of publications and discuss some of my old results. However, the past counts for nothing. If I am not able to prove something new now, what I did before is worthless. So here I am, day after day, working for hours pursuing some infinitely distant goal.

      I am trying to “understand.” I am not trying to discover something new, but rather see the “essential reasons” why some results are true. I return to the source, in an attempt to discover “the mother of all...

      (pp. 142-143)

      Although there are at present many occupations that require a good deal of skill and training in advanced mathematics, mathematics itself is still often regarded as a curious profession demanding singular talents and a singular personality. My own character — apart from a certain tolerance of solitude, even a preference for it — has always seemed to me to be quite ordinary. As a child, I was more apt at arithmetic calculations than my classmates, but my geometric intuition was not remarkable, and I was never fascinated by puzzles or intellectual games. The preference for solitude was perhaps acquired during...

      (pp. 144-145)

      I prefer to close my eyes when I think about mathematics. The best work is done by night, in half sleep. Sometimes I go to bed thinking, “Ah, I have a nice lemma to prove — or disprove.” (Should I explain what a lemma is? A mountain climber needs holds to get from one level to the next one. Lemmas are the holds of a mathematician.) Of course one has to write things down later, if only for publication. Sometimes you then find that what you have thought was wrong, but that’s rare.

      My thesis is a typical case. There...

      (pp. 146-147)

      Unlike many other mathematicians that I know, I was not enamored of mathematics as a young child. I found it dull, confusing, and difficult. I was interested in, and good at, most subjects in school, but I had no interest at all in mathematics — despite being constantly told by my parents and teachers how important it was to acquire a good knowledge of the subject — and for years I regularly failed almost every mathematics examination that I took. I remember on one occasion when I was very young, I decided that I hated mathematics so much that I...

      (pp. 148-149)

      My math teacher pointed at me during a lesson and bellowed, “du Sautoy! I want to see you after the class.” I was twelve. I was terrified. Had I done something wrong? When the bell rang for the end of the class he took me round the back of the maths block. “Now I’m in real trouble,” I thought. But then my teacher proceeded to explain that he thought I should find out what mathematics is really about. According to him, the mathematics we were learning in the classroom wasn’t real mathematics. He pointed me in the direction of a...

      (pp. 150-151)

      During elementary and high school, mathematics was my favorite subject, in part because it was one of the few subjects that came easily to me. However, outside the usual interests of any teenager, my passion was tournament chess where I enjoyed success at the junior and senior levels in southern Africa. My father was very supportive of my (and my brothers’) involvement with chess while we were kids, but far less enthusiastic about my running off to Europe at age seventeen to try to make it as a chess professional. He insisted that I first get a university education and...

      (pp. 152-153)

      I grew up in a coal-mining town in the German rustbelt. My father was a physicist working as an executive in the chemical industry and so I got interested in physics. After some time, I came to mathematics because I found it more interesting. Everything in mathematics is very logical, and that appealed to me. I like it when something is definitely right or definitely wrong.

      I studied in Münster, which is near my hometown. I had a very good teacher who encouraged me to study Alexander Grothendieck’s work on algebraic geometry, although this had somehow gone out of fashion....

      (pp. 154-155)

      I grew up in Montepulciano in central Italy, south of Siena. It is a small town surrounded by walls and situated on top of a hill with many houses and six or seven churches. I went to school there and had many friends. I liked to take walks in the countryside, explore caves, bicycle, play soccer, and read mathematical books. My father was in the banking business but he also had been interested in mathematics, so there were a few mathematical books in the house, accessible to the nonexpert. When I began my interest in mathematics, my father did not...

      (pp. 156-157)

      I was born in Brussels. I am told that when I was little, I surprised people by understanding what negative numbers were. Why were they surprised? A centigrade thermometer gives you a good image for them. I was lucky because my brother and sister are older than I am. When my brother was at the university, I could look at some of his books and learned how to solve a third-degree equation. I was also lucky to meet Mr. Nijs, a high school teacher who, seeing I was interested, gave me very good books to read. I viewed math at...

      (pp. 158-159)

      I’ve played with numbers and music for as long as I can remember — since before age three, according to my parents’ records. Music was ubiquitous at home, thanks to my mother’s profession as a piano teacher, but she recalls that it was numbers that first piqued my real interest in music. Novices’ piano books mark every note 1, 2, 3, 4, or 5 for thumb, index, middle, ring, or little finger, and at first these correspond exactly to notes (because the student’s hand does not move), and often also to “scale degrees” (first through fifth notes of the scale)...

      (pp. 160-161)

      When I went to college in the ’60s, everyone was trying to change the world. Whatever mathematics can do, it’s not going to make the world a better place. After graduation, I traveled for several years in Africa, Asia, and Europe, reading mathematics, playing music, and trying to sort this all out. I came to the conclusion that if I wanted to do creative work, it was going to be in mathematics, and returned to the States to study number theory.

      In graduate school, I was taught by great mathematicians — John Tate, Jean-Pierre Serre, Raoul Bott, Barry Mazur —...

      (pp. 162-163)

      I moved every year for the first thirty years of my life, so no roots, no stability. In some sense I don’t come from anywhere. I grew up in America but have now lived in Europe for so long that I no longer feel very American.

      My childhood was unusual. Until age nine, I hardly talked to people and didn’t have friends, and I remember nothing of those years. It was thought I might be retarded. The school psychologist gave me a three-hour test and it saved my life. It turned out that I had above average intelligence, and the...

      (pp. 164-165)

      We humans have been taking part in a long conversation — through the millennia — about love, death, how we tell stories about our lives, how we imagine the almost unimaginable, how we have behaved towards one another, how we should behave with each other, and how we think about all this.

      That there is a sterling architecture behind how we think, an articulation that transcends mood, circumstance, and even culture, is one of the great gifts of being alive. No mode of thought comes closer to this architecture than mathematics — and this is what makes thinking about mathematics...

      (pp. 166-167)

      When I was ten years old, living in the beautiful university town of Cambridge, England, I had the good fortune one day to stumble across a problem in a book in my local library. On the cover was stated perhaps the most famous of all mathematical problems, at least to the amateur that I then was. It was known as Fermat’s Last Theorem and the problem was to show that although it is easy to find many squares of integers which can be written as a sum of two such squares, the same should not be true for cubes, or...

      (pp. 168-169)

      I’ve always loved mathematics. As a child, I loved shapes and I loved numbers. One of my earliest mathematical memories, from when I was about eight years old, is stacking oranges (which were meant for the family juicer!) in large pyramids. I wanted to know how many oranges would one need to make a triangular pyramid with$n$oranges on a side? I thought a lot about it, and eventually figured out that the answer is$n \left ( n+1 \right )\left ( n+2 \right ) / 6$oranges. That was a very fun and exciting moment for me! I loved that I could predict exactly how many oranges would be...

    • JOHN T. TATE
      (pp. 170-171)

      I grew up in Minneapolis as an only child. My father was an experimental physicist at the University of Minnesota. My mother knew the classics and taught high school English until I was born. My father had some books of logic and math puzzles by H. E. Dudeney which fascinated me. Although there were very few I could solve when I was a child, I liked to think about the puzzles.

      I would like to express my appreciation of my father. He never pushed me, but from time to time explained some simple fundamental idea, like the fact that the...

      (pp. 172-173)

      I don’t feel very comfortable writing about myself, but around twenty five years ago I was interviewed for what later became the bookDeveloping Talent in Young People, which looked at people who had done well in music and art, athletics, or mathematics and science. I will quote myself from the book and comment a bit on what I said then. Actually, the first quote was from my mother, who said, “I have strong feelings about pressuring children and tailoring them to fit parental expectations.” This makes me laugh now, because in fact my mother was dead set on my...

      (pp. 174-175)

      My father is a certified public accountant. In the days before calculators and spreadsheets, he often spent hours at the dining table adding long columns of numbers. My dad taught me addition (with carrying) when I was a small boy. Not long after, I learned from him the secrets of subtraction, fractions, and decimals. Perhaps this is the reason that I became fascinated by numbers as a child. In any case, I was delighted to be able to do arithmetic operations long before they were introduced in school. When I was a bit older, I spent hours with a mathematics...

      (pp. 176-177)

      I was born into a family of professional musicians and I studied violin for nine years. I finished high school early and entered the City College of New York when I was fourteen. Shortly after this, Dai Vernon, who was the greatest magician in the United States at the time, invited me to go on tour with him. I left home without even telling my parents and began a very interesting life. I loved doing magic and was very good at it. I enjoyed inventing new tricks and teaching others to do them. After about eight years, a friend recommended...

      (pp. 178-179)

      I was born into a family of intellectuals who were deeply involved in politics for several generations, either by writing books or by exercising political responsibilities at a national level in France. I have the highest respect for the fighting life of my parents, uncles, and grandparents; I have often seen their disillusions after fighting for carefully planned political proposals that were finally withdrawn. One of my reasons for choosing mathematics has been that as soon as truth is discovered, it enters immediately into reality.

      I finished my graduate studies in mathematics at the Sorbonne in Paris in 1946. I...

      (pp. 180-181)

      My parents, Juliette and Richard Massey Sr., were both educators; she was from Chattanooga, Tennessee, and he was from Charlotte, North Carolina. They met at Lincoln University in Jefferson City, Missouri, which became my birthplace. My initial fascination with numbers started when my mother would let me play with plastic numbers and cut-up old calendars.

      We moved to Saint Louis, Missouri, when I was four. There I came of age educationally during the post-Sputnik era. A fifth-grade gifted program exposed me to Euclidean geometry and number systems of differing bases. My interest in drawing and graphic arts helped me to...

      (pp. 182-183)

      The longer I live, the more I believe that our lives are controlled by chance events and the actions of others. My own life confirms this thesis. Here is a chronological account.

      My life in mathematics may well have been begun by my electric shop teacher, Mr. Brockway, at James A. Foshay Junior High School in South Central Los Angeles. When I was eleven years old, he taught me the miracle of logarithms and set me to solving problems of arranging switches (single- and double-pole) to control lights in complex ways. These “puzzles” were intrinsically combinatorial problems of the sort...

      (pp. 184-185)

      I grew up in Haifa, Israel, in a tiny apartment overlooking the Mediterranean. My father was an engineer who loved math. He was very interested in teaching me and my two brothers, but I was more inclined than my siblings. We spent lots of time solving problems and puzzles from old Russian books that he brought with him to Israel after the Second World War.

      In school I enjoyed learning everything, but mathematics in particular, from very early on. After my army service, I chose computer science as my major at the Technion. It may have been more natural to...

      (pp. 186-187)

      Imagine majestic night skies filled with sparkling points of starlight scattered like diamonds across the heavens. These were the evening visual experiences of my childhood in the tiny Central American town of Dangriga, Belize. I constantly asked questions about the universe, often times causing my elders to worry about my obsession: Does space continue forever? How did the universe come about? Why do we exist? Is there a God? This early exposure to the profound beauty and mystery of the cosmos has since gripped and steered my intellectual journey.

      At fourteen years, I immigrated to the United States. I attended...

      (pp. 188-189)

      I was born and grew up in the coal-mine region in Belgium. I studied theoretical physics, so I don’t have a single math degree. I started my research work on mathematical physics, which is basically mathematics focused on or motivated by physics. A few years after my PhD, I became interested in applying mathematics not only to understanding the physical world around us, but also to technology, where you construct things. The mathematical analysis may also lead you to construct things differently, rather than study the already existing world.

      Something funny struck me after I made that transition. I had...

      (pp. 190-191)

      My original feelings for mathematics were very much stimulated by my father, Lionel Penrose, a physician who specialized in the inheritance of mental disorders, later becoming professor of human genetics at University College London. He was a multitalented man of Quaker background, his father having been a professional artist. He much enjoyed puzzles, chess, painting, music, biology, astronomy, and mathematics. I recall many walks in the country with him, my mother, and my two brothers—and much later my young sister also—which gave him opportunities to explain things about nature.

      Both of my brothers were expert chess players (my...

      (pp. 192-193)

      I was born in 1948 in Pomona, California. My father was the director of a state hospital for the developmentally disabled, and I lived on the grounds of the hospital until I went to college. The patients were very nice, but the kids at school teased me because I was smart but living in a place where others were not so smart. Through reading science fiction as a child, I became interested in science, specifically astronomy. My goal was to be the first person on Mars. In the seventh grade, I read some of Martin Gardner’s articles on mathematical games...

      (pp. 194-196)

      I grew up in Illinois in a town called Centralia, whose main industries were railroading and coal mining. All Blacks in Centralia came in 1912 or 1919 as strikebreakers for the Illinois Central Railroad. My father worked for the railroad, and I grew up with an antiunion bias because that was what had gotten my father a job in the first place.

      I was always the smartest boy in my class although there were always three or four girls ahead of me. Math was the only subject in which I was the best of all the students. I especially liked...

    (pp. 197-197)

    This book came about because I had the wonderful opportunity to meet Mariana Cook and her husband Hans Kraus. We hit it off! Mariana sent me a book of photographs of famous scientists she had published and I immediately asked if she would do one of mathematicians. She loved the idea!

    Great mathematicians have usually been considered rather different from the rest of the population, but of course they are people just like everyone else. I think that to be great at mathematics, you probably do have to have some differences: an ability to concentrate in a certain way, detach...

    (pp. 198-199)
  8. Back Matter
    (pp. 200-200)