Loving and Hating Mathematics

Challenging the Myths of Mathematical Life

Reuben Hersh, Vera John-Steiner

• Publisher: Princeton University Press
• Year: 2010
• ISBN:
  9781400836116
• Topic: Matemáticas
• Subtopic: Matemáticas general

An exploration of the hidden human, emotional, and social dimensions of mathematics Mathematics is often thought of as the coldest expression of pure reason. But few subjects provoke hotter emotions—and inspire more love and hatred—than mathematics. And although math is frequently idealized as floating above the messiness of human life, its story is nothing if not human; often, it is all too human. Loving and Hating Mathematics is about the hidden human, emotional, and social forces that shape mathematics and affect the experiences of students and mathematicians. Written in a lively, accessible style, and filled with gripping stories and anecdotes, Loving and Hating Mathematics brings home the intense pleasures and pains of mathematical life.These stories challenge many myths, including the notions that mathematics is a solitary pursuit and a "young man's game, " the belief that mathematicians are emotionally different from other people, and even the idea that to be a great mathematician it helps to be a little bit crazy. Reuben Hersh and Vera John-Steiner tell stories of lives in math from their very beginnings through old age, including accounts of teaching and mentoring, friendships and rivalries, love affairs and marriages, and the experiences of women and minorities in a field that has traditionally been unfriendly to both. Included here are also stories of people for whom mathematics has been an immense solace during times of crisis, war, and even imprisonment—as well as of those rare individuals driven to insanity and even murder by an obsession with math.This is a book for anyone who wants to understand why the most rational of human endeavors is at the same time one of the most emotional.

+ 1 +

Mathematical Beginnings

A Passion for Numbers

How does a child first begin to become a mathematician? Is it a predisposition or some special gift? Does help and encouragement from parents contribute? What makes it possible, finally, to commit one’s life to this risky, forbidding pursuit?

In this chapter, we tell contrasting stories about the childhood, adolescence, and schooling, up through graduate school, of some future mathematicians, both famous and not so famous. We also report the experiences of youngsters in Olympiad competitions, psychologists’ investigations of prodigies, and what the parents of math prodigies are like.

A few famous mathematicians showed their interest and ability before school age. The Hungarian combinatorialist and number theorist Paul Erdős¹ (1913–1996) claimed that he independently invented negative numbers at age 4.

Stan Ulam (1909–1984) (sometimes referred to as “the father of the H-bomb”) wrote in 1976: “When I was four, I remember jumping around on an oriental rug looking down at its intricate patterns. I remember my father’s towering figure standing beside me and I noticed that he smiled. I felt, ‘He smiles because he thinks I am childish, but I know these are curious patterns.’ ”²

Carl Friedrich Gauss (1777–1855), preeminent after Archimedes (c. 287 BC–c. 212 BC) and Newton (1643–1727), could calculate before he could read. In old age, Gauss told about this childhood feat: In a class that was told to add the numbers from 1 to 100, little Gauss produced the answer in a few seconds. (He constructed 50 pairs, grouping 1 with 100, 2 with 99, and so on. Each pair totals 101, and there are 50 pairs, giving the final sum, 50 × 101 = 5050.)

The famous physicist Freeman Dyson wrote in 2004:

One episode I remember vividly, I don’t know how old I was; I only know that I was young enough to be put down for an afternoon nap in my crib. . . . I didn’t feel like sleeping, so I spent the time calculating. I added one plus a half plus a quarter plus an eighth plus a sixteenth and so on, and I discovered that if you go on adding like this forever you end up with two. Then I tried adding one plus a third plus a ninth and so on, and discovered that if you go on adding like this forever you end up with one and a half. Then I tried one plus a quarter and so on, and ended up with one and a third. So I had discovered infinite series. I don’t remember talking about this to anybody at the time. It was just a game I enjoyed playing.³

For some children, during times of personal upheaval, the simplicity and order of geometry and algebra are a comforting refuge. One example is the famous author and neurologist Oliver Sacks. During the bombardment of London in World War II, he was sent away from home and family. He writes:

For me, the refuge at first was in numbers. My father was a whiz at mental arithmetic, and I, too, even at the age of six, was quick with figures—and, more, in love with them. I liked numbers, because they were solid, invariant, they stood unmoved in a chaotic world. There was in numbers and their relations something absolute, certain, not to be questioned, beyond doubt. . . . I particularly loved prime numbers, the fact that they were indivisible, could not be broken down, were inalienably themselves. . . . Primes were the building blocks of all other numbers, and there must be, I felt, some meaning to them. Why did primes come when they did? Was there any pattern, any logic to their distribution? Was there any limit to them, or did they go on forever? I spent innumerable hours factoring, searching for primes, memorizing them. They afforded me many hours of absorbed solitary play, in which I needed no one.⁴

The childhood of the well-known mathematics educator Anneli Lax (1922–1999) was also disrupted by World War II. Mathematics was “the perfect sort of escape: I didn’t have to look up anything; I didn’t have to consult libraries or books. I could just sit there and figure things out.”⁵

The Hungarian-born physicist Eugene Wigner (1902–1995) was diagnosed with tuberculosis at the age of 11 and spent weeks in a sanatorium in Austria. To help get through this difficult period he worked on geometry problems. “Sitting in my deck chair, I struggled to construct a triangle given only the lengths for the three altitudes. This is a very simple problem which I can do now in my dreams. But then it took me several months of concentrated effort to solve it.”⁶ Wigner attended a famous gymnasium in Budapest where John von Neumann (1903–1957) was also a student. They became lifelong friends. “It was likely the best high school in Hungary; it may have been the finest in the world.”⁷ “My heart was with numbers, not words. After a few years in the gymnasium I noticed what mathematicians call the Rule of Fifth Powers: That the fifth power of any one-digit number ends with that same number. Thus, 2 to the fifth power is 32, 3 to the fifth power is 243, and so on. At first I had no idea that this phenomenon was called the Rule of Fifth Powers; nor could I see why it should be true. But I saw that it was true, and I was enchanted.”⁸

Steven Strogatz, an applied mathematician at Cornell, describes the fear and awe he felt when data he was plotting in a physics lab created a curve he had met in algebra class. He was recording how the length of a pendulum string affects the time for the pendulum to complete a swing. As he plotted the data on graph paper, he realized that

these dots were falling on a particular curve that I recognized because I’d seen it in my algebra class—it was a parabola, the same shape that water makes coming out of a fountain. I remember experiencing an enveloping sensation of fear, then of awe. It was as if . . . this pendulum knew algebra! What was the connection between the parabolas in algebra class and the motion of this pendulum? There it was, on my graph paper. It was a moment that struck me, and was my first sense that the phrase “law of nature” really meant something. I suddenly knew what people were talking about when they said that there was order in the universe, and that, more to the point, you couldn’t see it unless you knew math. It was an epiphany I’ve never really recovered from.⁹

We have limited evidence about childhood engagement in mathematics. However, the stories of those who have an early passion for numbers reveal their fascination with the patterns of mathematics. For others who are experiencing trauma, doing problems provides a refuge.

Mathematicians’ Early Teen Years

Most famous mathematicians first showed a strong interest in mathematics around middle school age. For example, John Todd (1911–2007), a well-known international leader in numerical mathematics, said, “My mathematical career started in the following way. I was in a class, a singing class. My singing was so bad the teacher said I was disturbing the class and had to leave! There were some extra classes, ones with national examinations. And so I had to be put in one of them—it was a second-year algebra class! That’s when I started learning mathematics.”¹⁰

Another mathematician, Jenny Harrison, currently a professor at the University of California, Berkeley, was primarily interested in nature and music during her teen years. In an interview with John-Steiner, she recalled her beginnings. Jenny Harrison was born in Atlanta, Georgia, and spent most of her time outside of school in the woods. This attraction to the natural world may have influenced Harrison throughout her life and can be seen in the way she explores mathematical landscapes. She talked about her visual view of the world and her enjoyment in exploring paths in the woods. This strong preference may well have contributed to her later interests in geometry. The influence and encouragement of her older brother contributed to her strength and self-confidence. He got Harrison interested in basic problems in physics when they were teenagers.¹¹

Although she clearly showed an aptitude for math early on (scoring the highest in her state in a take-home competition), her primary passion was for music. She studied the piano and believed for most of her adolescence that she would devote her life to it. Music is still a part of her life, but Harrison is basically a shy person and found that she was uncomfortable performing in public. “I knew I didn’t want to do music and I went over and shut the lid to the piano. I was intrigued by three problems and wanted to try to understand them: the nature of consciousness, time and light. The question was how best to do this. I eventually settled on mathematics as I felt it would give me answers I could trust.”¹²

Julia Robinson (1919–1985), who became famous for helping to prove that Hilbert’s 10th problem is unsolvable (specifically, that there is no formula or computer program that can always decide whether an arbitrary polynomial equation with whole number coefficients has a whole-number solution) wrote, “One of my earliest memories is of arranging pebbles in the shadow of a giant saguaro, squinting because the sun was so bright. I think that I have always had a basic liking for natural numbers. To me they are the one real thing.”¹³

Sophie Germain (1776–1831), who was to make an important contribution to proving Fermat’s Last Theorem, was born in Paris. As a girl, she had to fight hard for the right to read mathematics. Her interest in mathematics began at age 13, during the French Revolution. Because of the battles going on in the streets of Paris, Sophie was confined to her home, where she spent a lot of time in her father’s library. There she read about the death of Archimedes. It is said that on the day that his city, Syracuse, was being overrun by the Romans, Archimedes was engrossed in a geometric figure in the sand and ignored the questioning of a Roman soldier. As a result, he was speared to death.¹⁴ If someone could be so engrossed in a problem as to ignore a soldier and die for it, thought Sophie, the subject must be interesting.

Sophie began teaching herself mathematics, using the books in her father’s library. Her parents did all they could to discourage her. She began studying at night to escape them, but they went to such measures as taking away her clothes once she was in bed and depriving her of heat and light to make her stay in her bed at night instead of studying. Sophie’s parents’ efforts failed. She would wrap herself in quilts and use candles she had hidden in order to study at night. Finally her parents realized that Sophie’s passion for mathematics was “incurable,” and they let her learn. Thus Sophie spent the years of the Reign of Terror studying differential calculus without the aid of a tutor!¹⁵

Sofia Kovalevskaya (1850–1891) was the first woman to achieve full status as a professional mathematician. Born in Moscow in 1850, her interest in mathematics was stimulated in childhood by the wallpaper in her nursery. Kovalevskaya wrote, “When we moved to the country from Kaluga the whole house was painted and papered. The wallpaper had been ordered from St. Petersburg, but the quantity needed was not estimated quite accurately, so that paper was lacking for one room. Considering that all the other rooms were in order the nursery might well manage without special paper.”¹⁶ Some paper that was lying around in the attic was used for the purpose. By a happy chance the paper for this covering consisted of the lithographed calculus lectures of the analyst M. V. Ostrogradsky that her father had acquired as a young man. “It amused me to examine these sheets, yellowed by time, all speckled over with some kind of hieroglyphics whose meaning escaped me completely but which, I felt, must signify something very wise and interesting. And I would stand by the wall for hours, reading and rereading what was written there. I remember particularly that on the sheet of paper which happened to be in the most prominent place on the wall, there was an explanation of the concepts of infinitesimals and limits.”¹⁷

Figure 1-1. Sofia Kovalevskaya, pioneer Russian analyst. © Institut Mittag-Leffle

Kovalevskaya recalled her uncle speaking to her about the quadrature of the circle: “If the meaning of his words was unintelligible to me, they struck my imagination and inspired me with a kind of veneration for mathematics, as for a superior, mysterious science, opening to its initiates a new and marvel...

Table of contents

• Cover
• Half title
• Title
• Copyright
• Dedication
• Contents
• Acknowledgments
• Introduction
• 1 Mathematical Beginnings
• 2 Mathematical Culture
• 3 Mathematics as Solace
• 4 Mathematics as an Addiction: Following Logic to the End
• 5 Friendships and Partnerships
• 6 Mathematical Communities
• 7 Gender and Age in Mathematics
• 8 The Teaching of Mathematics: Fierce or Friendly?
• 9 Loving and Hating School Mathematics
• Conclusions
• Review of the Literature
• Biographies
• Notes
• Index of Names
• General Index