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Mathematical Culture

Mathematicians constitute a community with a long, rich history. They recognize each other as fellow members of that community. What is the culture of mathematics and of the mathematical community?

This question seems to have been seldom asked and little studied. The American topologist Raymond Wilder was one outstanding contributor to this subject. In this chapter, we attempt to continue his work and highlight some salient features of mathematical culture.

When Paul Halmos was asked, “What is mathematics?” He answered, “It is security. Certainty. Truth. Beauty. Insight. Structure. Architecture.”¹ Most mathematicians agree with Halmos and treat the aesthetic components of discovery and proof as a source of joy. Others speak of the study of shapes and patterns. The French mathematician Laurent Schwartz wrote: “I simply found mathematics beautiful—extraordinarily beautiful—and geometry in particular incomparably elegant.”² Rozsa Péter, the Hungarian author of Playing with Infinity, argued that one could do nothing better and more beautiful than to work on mathematics.³

In discovering the field in their youth, many mathematicians are delighted by the internal logic of the arguments that lead from axioms to proofs. For many, a general interest in science provides a first entry into rigorous thinking, which later leads them to mathematics after they discover that they lack the dexterity needed for laboratory sciences. (Ralph Boas wrote of his “record bill in broken glassware” when taking chemistry.⁴) Others report that their desire for certainty was not met by physics or biology. One’s personal intellectual preferences—or what psychologists call learning styles—must be consonant with the prevalent modes of thought, of the culture, and of one’s chosen field.

We start with a definition of “culture” from the American Heritage Dictionary (2006):

The totality of socially transmitted behavior patterns, arts, beliefs, institutions, and all other products of human work and thought.

These patterns, traits, and products are considered as the expression of a particular period, class, community, or population.

Culture also encompasses structured domains of human activity with characteristic symbol systems, such as musical or mathematical concepts and notations.⁵

In this chapter, we focus on four aspects of culture. The first is cognitive-emotional. We ask: What habits of mind, behavior patterns and motivation permit young mathematicians to become members of their mathematical community?

The second aspect is aesthetic. Everyone agrees that mathematics can, should, or even must, be beautiful. But it is not so easy to explain just what you mean by beautiful mathematics. What patterns, proofs, and discoveries do mathematicians call beautiful?

The third aspect is social. What shared values, stories, traditions, and institutions support the discipline? Who are the insiders and who are the outsiders?

The fourth aspect of mathematical culture we look at is how it deals with internal tensions: How does the discipline identify and address conflicts? How is a discoverer rewarded? How are prizes shared? How does competition coexist with collaboration?

Mathematical Cognition and Feelings

The acquisition of abstract modes of thought is basic to mathematical activities. It starts early in children’s lives. For instance, at birthday parties, preschoolers hold up four fingers, thereby showing that they understand the equivalence relation of whole numbers. They acquire more abstract notions when they learn number words and symbols.

Davis and Hersh describe two aspects of abstraction. One is “idealization,” that is, stripping away irrelevant details, such as the thickness of the pencil line with which a triangle is drawn. The second is “extraction,” that is, pulling out the essential features of a problem, for instance, representing a complicated maze as a graph by letting selected positions in the maze become nodes on the graph.

Mathematics progresses, both historically and psychologically, by successive expansions: counting is followed by geometric and arithmetic operations, and then by the study of more general patterns. The connection between a pattern in nature and a mathematical representation has powerful emotional appeal. It is part of the emotional and aesthetic appeal of mathematics. This connection was illustrated by Steven Strogatz’s experience with the parabola described in the previous chapter.

Participants in mathematical culture create symbols and notations, which constitute the shared tools or “language” of their community. By the use of symbols and notations, concepts are clarified and made precise. After a symbol is invented by an individual, it is appropriated, evaluated, disseminated, and applied by many mathematicians. The dynamic connection between individual cognition and communal activity takes place through conversations, academic apprenticeships, publications, conferences, and textbooks.

In ancient times, new symbols were created to meet the needs of commerce and astronomy. Scribes used Greek letters for numbers to replace the earlier reliance on tallies and clay tokens. Today, new concepts and symbols still arise from the challenges of the physical sciences as well as from the evolving internal logic of mathematics.

The evolution of the symbolic tools of mathematics has been uneven. Long periods of stasis (for instance, during the Middle Ages in Europe) were followed by periods of rapid progress. Societies with active commerce, broad geographic contacts, and emerging technologies provide a generative environment for mathematical growth. These contrast sharply with the stability of static isolated communities. An example of an extremely isolated community is the Pirahã, a small tribe of hunters and gatherers who live in the Amazon basin. The linguist Daniel Everett spent years studying the Pirahã. He found that they have no arithmetic and that their language has no number words. They communicate almost as much by singing, whistling, and humming as by using consonants and vowels. Everett suggests that their lack of number words and counting, and also a lack of cultural stories, are associated with a certain pervasive value in their conversations. The Pirahã avoid talking about any knowledge that goes beyond personal, immediate experience.⁶

The lack of number words is not limited to the Pirahã. There are hundreds of indigenous languages in Papua, New Guinea, many of which have words only for “one,” “two,” and “many.” Cross-cultural and historical analyses suggest that what we consider universal in our experience with numbers may actually be specific to societies that are economically developing and culturally open.

The habits of mind within a culture are also diverse. Even within the contemporary Western mathematical community, logic and abstraction are not the only ways that mathematicians think. They are supplemented by intuition, analogy, and visualization.

Benoit Mandelbrot is famous for his work on fractals. When he was a young child, his uncle taught him to play chess, to study maps, and to read very fast.⁷ As a student he would rely on shapes to represent mathematical functions. When he reached university age, he managed to pass the very challenging entrance examinations to France’s elite higher educational institutions, the École Normale Superieure and the École Polytechnique. He writes, “Faced with some complicated integral, I instantly related it to a familiar shape; it was usually the exact shape that motivated this integral. I knew an army of shapes I’d encountered once in some book or in some problem, and remembered forever, with their properties and peculiarities.”⁸

Mandelbrot’s strong use of visualization went against the prevailing academic trends of the postwar decades. The visualization described by Mandelbrot is one form of mathematical intuition, by which we mean a perception that is “plausible or convincing in the absence of proof.”⁹ Such intuitive perceptions are essential in discovery; they contrast with the more rigorous, deductive methods needed for justification.

For some students who find mathematics frustrating in school, the monotonous repetition of drill in algorithms contributes to a sense of alienation. In contrast, when a teacher takes a broader approach, including problem solving, guessing, and intuition, learners can engage in a freer, gamelike experimentation with ideas. They think and communicate inferentially through examples and visualization, as well as by logical rules.

Thought is hard to observe. One way to learn about it is by introspection. In Jacques Hadamard’s famous essay, “The Psychology of Invention in the Mathematical Field,” he asked his colleagues, “What internal or mental images, what kind of ‘internal words,’ do mathematicians make use of; whether they are motor, auditory, visual, or mixed, depending on the subject which they are studying?”¹⁰ Einstein’s answer has been frequently quoted. He wrote: “The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined.”¹¹

Hadamard’s own mode of thought was similar to Einstein’s. He relied on mental pictures to make discoveries and had a challenging time translating them into words. This tendency made him particularly interested in visual, intuitive, and unconscious approaches to discovery in mathematics, which would be followed by the verifying and “précising” stage.

Figure 2-1. André Weil. Source: More Mathematical People: Contemporary Conversations. Eds. Donald J. Albers, Gerald L. Alexanderson, and Constance Reid.

One description of rapid thought processes that emerge from an unknown source, what some call the “Aha” experience, is provided by André Weil, one of the founders of Bourbaki. (This group of French mathematicians reformulated the basis of their discipline in the 1930s and wrote collaboratively under their chosen name.) He writes, “Every mathematician worthy of the name experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets this word) seems to play a role. . . . Once you have experienced it, you are eager to repeat it but are unable to do so at will, unless perhaps by dogged work which it seems to reward with its appearance.”¹²

The creativity researcher Mihaly Csikszentmihalyi describes the state of deep immersion as “flow,” a condition when challenge and skill are well matched. In this state, people concentrate so deeply on a task that “they stop being aware of themselves as separate from the actions they are performing.”¹³

Mentors and advisers model both the cognitive and motivational aspects of mathematical life. As models, they teach persistence. Researchers persist in lengthy and at times frustrating labor, which may be followed by relief and joy. In 1927 Oscar Zariski had recently emigrated from Italy to America. He wrote letters to his wife in which he shared the roller coaster of emotions that frequently accompany intense mathematical work: “Now I am going through a feverish period. . . . If I will succeed in bringing this research to a good outcome . . . it will be an affirmation of my worth to the professors who have been vitally interested in this problem.”¹⁴ Weeks later, he was still nailed to his desk, still unable to solve the problem: “Ah, well, I have had a very bad period. I have gotten stuck on a hard point that I have not been able to overcome for weeks. . . . I have been close to despair, even though Prof. Coble consoled me, saying that one must always take these issues philosophically. And that mathematics is ‘a slow game.’ ”¹⁵ It took him another month to complete the research by using topological methods in algebraic geometry. This work led him to meet the famous topologist Solomon Lefschetz, with whom he shared his Russian Jewish heritage.

Creativity requires continuity of concern and intense awareness of one’s active inner life. Routine activities can be useful as a backdrop to the researcher’s deep immersion in his or her problem. The British mathematician (of Lebanese and Scottish origins) Michael Atiyah described some of his problem-solving ways:

Figure 2-2. Michael Atiyah (center), English geometer and mathematical physicist, with friends. Courtesy of Dirk Ferus.

I think that if you are actively working in mathematical research, then the mathematics is always with you. When you are thinking about problems, they are always there. When I get up in the morning and shave, I am thinking about mathematics. When I have my breakfast, I am still thinking about my problems. When I am driving my car, I am still thinking about my problems. All in various degrees of concentration. . . . There are occasions when you sit down in the morning and start to concentrate very hard on something. That kind of acute concentration is very difficult for a long period of time and not always very successful. Sometimes you will get past your problem with careful thought. But the really interesting ideas occur at times when you have a flash of inspiration. Those are more haphazard by their nature; they may occur just in casual conversation. You will be talking with somebody and he mentions something and you think, “Good God, yes, that is just what I need . . . it explains what I was thinking about last week.” And you put two things together, you fuse them and something comes out of it.¹⁶

At its most rewarding, mathematics provides aesthetic enjoyment, the satisfaction of clear and deep thinking, and the emotional roller coaster of discovery.

A prevalent myth about mathematical thinking is that somehow it is all the same kind of thing. “Either you are mathematically inclined or you’re not.” Bur there are more than one kind of mathematical talent and mathematical thinking. For those who want straightforward predictions of academic or professional success, tests like the IQ test are appealing. But the development and application of talent is quite complex. Some individuals reveal interests and skills at an early age. As a child, John von Neumann was intrigued by his grandfather’s ability to rapidly perform complex mathematical calculations (a skill for which von Neumann later became famous). He loved arithmetic, was good in classical and modern languages, and had an extraordinary memory.¹⁷

In contrast, Roger Penrose, today a preeminent leader in geometry and relativity theory, as a child loved geometry but was slow in arithmetic. In fact, he was moved back a class in grade school. “You had to add up, add and multiply, and do various things, and I just got lost—I always got lost in the middle. I was very slow, I could do these things, but only at great length, thinking about them. I suppose people tended to do these things automatically, whereas I had to figure it out each time. It was probably a help in the long run, but at the time it was a disadvantage.”¹⁸ These two mathematicians present a contrast akin to that between Mozart and Beethoven. The prodigious ease, fluency, and productivity of the young Mozart are akin to that of von Neumann. Penrose is more like the hard-working Beethoven, who constructed his music laboriously over a long period of time.

In chapter 1 we describe the varied ways that future mathematicians first discover their fascination with numbers and patterns. Some, like von Neumann, are highly focused prodigies. Another example is the recent Fields Medal winner Terence Tao, who enjoyed numbers at a very young age. He learned math and science very fast, but he did not like to write essays. “I never really got the hang of that.” And he said: “These very vague, undefined questions. I always liked situations where there are very clear rules of what to do.” Assigned to write a story about what was going on at home, Terry went from room to room and made detailed lists of the contents.¹⁹

But other future mathematicians have had broad interests, including music and poetry. They decided on a career in a more deliberate way, making a carefully considered choice as young adults.

Diversity in style is also present in mature mathematicians. Davis and Hersh describe how they differ in their reliance on “analytical” versus “analog” processes. In the former, symbolic and verbal processes play a leading role, while in the latter, geometric and visual approaches are more important. Davis recalls spending considerable time on a problem on the “theory of functions of a complex variable.”²⁰ The problem could be studied from a geometric or from an analytical point of view or by combining the two. Davis found that as he worked on the problem, textbook illustrations of spheres, maps, and surfaces came to mind, along with a certain melody:

I worked out, more or less, a body of material which I set down in an abbreviated form. Something then came up in my calendar which prevented me from pursuing this material for several years. I hardly looked at it in the interval. At the end of this period, time again became available, and I decided to go back to the material and see whether I could work it into a book. At the beginning I was completely cold. It required several weeks of work and review to warm up the material. After that time I found to my surprise that what appeared to be the original mathematical imagery and melody returned, and I pursued the task to a successful completion.²¹

This quote not only shows the diverse modalities of thought that accompany mathematical problem solving, it also reveals a form of cognitive shorthand. Davis’s observation of his own thinking process adds an interesting dimension to our understanding of inner thought processes.²²

Figure 2-3. Irving Kaplansky, Don Albers and Shiing-Shen Chern. Source: More Mathematical People: Contemporary Conversations. Eds. Donald J. Albers, Gerald L. Alexanderson, and Constance Reid.

Thus, mathematicians are not limited to a single mode of thought. They rely on intuition, logic, visual and verbal processes, inferences, and guesses. They are assisted by the many artifacts and symbol systems of their profession.

Within pure mathematics, a distinction has been made between theorizers and problem solvers. The great names from the history of mathematics—Newton, Euler, Gauss, Riemann, Cantor—are revered to this day for creating theories that are still the mainstay and generating source of mathematics. On the other hand, a mathematician is most likely to become instantly famous in his own time if he or she solves a famous problem. The problem should have been outstanding for a long time, and it should have withstood attack from many famous mathematicians. But the greatest mathematicians must be counted among both groups, they were both theorizers and problem solvers.

Italian-born mathematician and philosopher Gian-Carlo Rota (1932–1999) has written incisively and wittily:

To the problem solver, the supreme achievement in mathematics is the solution to a problem that had been given up as hopeless. It matters little that the solution may be clumsy; all that counts is that it should be the first and that the proof be correct. Once the problem solver finds the solution, he will permanently lose interest in it, and will listen to new and simplified proofs with an air of condescension suffused with boredom.

The problem solver is a conservative at heart. For him mathematics consists of a sequence of challenges to be met, an obstacle course of problems. The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable.

. . . The problem solver is the role model for budding young mathematicians. When we describe to the public the conquests of mathematics, our shining heroes are the problem solvers.

To the theorizer, the supreme achievement of mathematics is a theory that sheds sudden light on some incomprehensible phenomenon. Success in mathematics does not lie in solving problems but in their trivialization. The moment of glory comes with the discovery of a new theory that does not solve any of the old problems but renders them irrelevant.

The theorizer is a revolutionary at heart. Mathematical concepts received from the past are regarded as imperfect instances of more general ones yet to be discovered. . . . Theorizers often have trouble being recognized by the community of mathematicians. Their consolation is the certainty, which may or may not be borne out by history, that their theories will survive long after the problems of the day have been forgotten.”²³

So we can describe mathematicians as theorizers versus problem solvers or as those who prefer algebraic methods to geometric methods. Some are more rigorous, some more intuitive. But none of these categories is fixed. There are researchers in between who sometimes work one way or shift to another. By considering these varied styles, we are led to recognize and appreciate the great diversity of thinking modes that contribute to the experience of mathematicians.

In challenging the myth of homogeneity, we open the field to groups and individuals with varied talents and practices. Such recognition has both an emotional and an educational benefit. It supports the varied cognitive styles of potential members of the profession. It also validates innovative practices in reform classrooms, where young learners are encouraged to experiment with various problem-solving strategies.

Mathematical Beauty

Most mathematicians agree that the aesthetic components of discovery and proof are a source of joy for researchers and practitioners in the field. Addressing the British Association for the Advancement of Science in 1885, Arthur Cayley said, “It is difficult to give an idea of the vast extent of modern mathematics. I mean extent crowded with beautiful detail—not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream rock, wood, and flower.”²⁴ David Ruelle (the French contributor to chaos theory) writes, “The beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes.”²⁵ He has given a strikingly perceptive explanation: “The infinite labyrinth of mathematics has thus the dual character of human construction and logical necessity. And this endows the labyrinth with a strange beauty. It reflects the internal structure of mathematics and is, in fact, the only thing we know about this internal structure. But only through a long search of the labyrinth do we come to appreciate its beauty; only through long study do we come to taste fully the subtle and powerful aesthetic appeal of mathematical theories.”²⁶

In a famous essay, the British mathematician G. H. Hardy (1877–1947) gave three criteria for beautiful mathematics. He said it must be serious, deep, and surprising. Serious, he said, in contrast with chess problems, for example, which are a kind of mathematics but not serious mathematics. Deep—for example, he said the existence of infinitely many primes, as proved by Euclid, is not deep; in contrast, the prime number theorem, which gives the proportion of primes among all whole numbers, is deep. And finally—surprising, which even more than deep or serious is clearly subjective. Whether the reader is surprised depends on what the reader already knows.

To explain to the lay person what he meant by beauty in mathematics, Hardy gave two examples from Euclid that he thought were exemplary: “No fraction when squared can be equal to 2” and “Given any finite collection of prime numbers, it is always possible to find a larger prime not included in the collection.” Or, in brief, “The square root of 2 is irrational” and “There are infinitely many primes.”²⁷

Gian-Carlo Rota also attempted to explicate mathematical beauty. Unlike Hardy, he didn’t think “surprising” was necessary for “beautiful.” After listing many specific examples, he concluded that beauty is “enlightenment.” We agree that a sense of enlightenment can make one exclaim, “Beautiful!” But Rota didn’t make much of an attempt to explicate what “enlightenment” is. Surprise, depth, simplicity, clarity, and directness all contribute to the aesthetic experience of mathematics.

An early member of Harvard’s mathematics faculty, George David Birkhoff (1884–1944) published a mathematical theory of aesthetics. Although few people take it seriously as a contribution to philosophy, Birkhoff did attempt to think carefully about what is the basis of aesthetic pleasure. He proposed a formula E = M/C. Here E is aesthetic pleasure and is increased by maximizing M, the richness or inclusiveness of the artistic object, and minimizing C, the complexity of means. Although this algebraic formula may not be of much use, the two factors M and 1/C in it have been identified by other authors, as “integration” versus “simplicity.” Another way to think about this relationship is to recognize that integration is a form of simplicity.

It is useful to connect aesthetics in mathematics to aesthetics in other artistic realms. Of course, the geometric or visual parts of mathematics have well-known connections to visual art, first of all to perspective drawing and painting, and then as sources of images for artists to use in any way they wish. Drama and theater provide useful viewpoints from which to judge mathematical presentations. Certainly an effective speaker at a colloquium or seminar can use theatrical or dramatic effects and devices to enhance his or her presentation. The analogy between pure mathematics and music is even more natural. They both communicate patterns that speak for themselves, requiring no verbal justification or amplification.

Could the criteria for judging a story or a musical composition be relevant for judging a mathematical paper? Coherence; unity; arousal of interest at the beginning; a satisfying sense of completion at the end; use of recurrent themes or leitmotifs to connect the beginning, middle, and end; logical or comprehensible connection from one chapter or one movement to the next—the presence or absence of these qualities can help one to make an aesthetic evaluation. The same qualities can be sought in a mathematical talk, article, or book. This would lead to an aesthetic evaluation of the presentation of some mathematics. On the other hand, it would be a different matter to evaluate the mathematical content. And it might not be easy to separate presentation from content. Given two different proofs of the same result, one mathematician might call them mathematically equivalent, differing only in presentation; another might say that the differences between them are mathematically significant.

The question remains, how to aesthetically judge mathematical content itself. We think we can point to three important elements of beauty in mathematical content: simplicity, concrete specificity, and unexpected or surprising integration or connection of disparate elements. The Bourbaki style was ultimately unsatisfying because it bought simplicity at the expense of concreteness and multiconnectedness.

In a much quoted sentence, Hardy wrote, “The mathematician’s patterns, like the painters or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics.”²⁸ This is easily refuted. Hardy was wrong! It is not hard to find permanent parts of mathematics that no one thinks are beautiful. Rota’s article contains examples. At an elementary level, we may mention the “quadratic formula,”

which solves the general quadratic equation

This is one of the most memorized formulas in math. Not beautiful!

The truth is that a mathematician striving to solve a problem does not worry much about making it come out beautiful. The aesthetic judgment comes in at the beginning, when she chooses a problem to solve or an area to work in. Yes, at that stage, beauty attracts and ugliness repels. But once the battle is joined, anything that works is welcome. Later, after the proof is found and the result is known, one looks back and tries to refine and beautify the proof. Maybe other mathematicians also will pick up the problem and look for a different, more attractive approach. But there is no assurance that an important mathematical result will yield to a beautiful attack.

A notorious example of unbeautiful proof is computer proof. Starting with the four-color theorem, and since then followed by quite a few other examples, mathematicians have found that what they cannot do by hand they can sometimes do by machine. In 1976, when Kenneth Appel, Wolfgang Haken, and John Koch published their computer proof of the four-color theorem, Tom Tymoczko and others thought that this raised a question about the nature of mathematical proof. But it seems today that mathematicians generally accept a well-verified, stable, robust computer proof as a real proof. That is to say, they accept the truth of a theorem like the four-color theorem on the basis of the computer’s testimony. (The four-color theorem says that four colors are sufficient to color any map with no two adjacent countries having the same color.) But there is still a serious objection: “Yes, you proved it by computer, but I don’t like it!” Computer proofs seem to be an unwelcome last resort, an ugly thing that does the job. What we really want is something beautiful, not something ugly. On the other hand, some mathematicians find the work of discovering and improving computer proofs to be intellectually attractive. They can look at a neat piece of computer code and think, “That’s beautiful!”

Johann Lambert’s near-discovery of non-Euclidean geometry is a remarkable example of the aesthetic sense in conflict with the seeming demands of logic. Lambert (1728–1777) was a forerunner of Gauss, Nikolai Lobachevsky (1792–1856), and János Bolyai (1802–1860), struggling with the “problem of parallels” that gave birth to non-Euclidean geometry. He was a younger contemporary of Euler and became the leading German mathematician of his time. Although he ended up mistakenly rejecting non-Euclidean geometry as contradictory, he was very strongly tempted by the beautiful properties such a geometry would have. “There is something exquisite about this consequence, something that makes one wish that the third hypothesis were true! But all these are arguments dictated by love and hate, which must have no place, either in geometry or in science as a whole.”

This confession is a fascinating manifestation of the emotional side of mathematical life. We disagree with Lambert’s view that “love and hate have no place in geometry.” The presence of love and hate in mathematics is unavoidable—more love than hate. Such emotions are especially evident in mathematical discovery and mathematical aesthetics. Yet, of course, Lambert was right to accept consistency as the final arbiter of a mathematical concept.

Social Aspects of Mathematical Culture

From the perspective of an outsider, mathematicians appear to be solitary thinkers. The amount of collaboration that takes places between them is rarely understood or publicized. But at closer acquaintance these abstract thinkers share a lot in their work and also in their common mathematical histories.

Young mathematicians refine their abilities to work on problems and theorems through apprenticeships. An essential part of graduate education includes closely watching a more mature mathematician engaged in mathematical practice. The Hungarian-born Paul Halmos (1916–2006) wrote of his advisor Joseph Doob, with whom he studied at the University of Illinois: “He has the one absolutely necessary quality of a teacher: he can see what it is that the learner is not seeing.”²⁹

Conversations with one’s peers can also greatly contribute to mathematical habits of the mind. Halmos describes his rapport with his friend Allen Shields:

The best seminar I ever belonged to consisted of Allen Shields and me. We met one afternoon a week for about two hours. We did not prepare for the meetings and we certainly did not lecture at each other. We were interested in similar things, we got along well, and each of us liked to explain his thoughts and found the other a sympathetic and intelligent listener. We would exchange the elementary puzzles we heard during the week, the crazy questions we were asked in class, the half-baked problems that popped into our heads, the vague ideas for solving last week’s problems that occurred to us, the illuminating problems we heard at other seminars—we would shout excitedly, or stare together at the blackboard in bewildered silence—and, whatever we did we both learned a lot from each other during the year the seminar lasted, and we both enjoyed it.³⁰

One of the most carefully documented mathematical discoveries is the work of Andrew Wiles. To prove Fermat’s Last Theorem, he devoted seven long years working in the attic of his house. He relied upon the work of many of his predecessors but sustained his own momentum. As he approached the culmination, he felt the need for communication. He recruited his colleague Nick Katz as an interlocutor. In June of 1993 Wiles presented his proof at a conference in Cambridge, England (his alma mater). The announcement of the proof galvanized the international mathematical community. During the extensive review process, a gap was found that forced Wiles to go back to work. Wiles decided to go into complete seclusion. “As months went by, Dr. Wiles finally decided that he needed help. He was frightened that he would mislead himself. Spending years, or decades going in circles trying frantically to fix the gap with methods that never would work. . . . ‘You get caught in problems, trapped in them. I knew the dangers psychologically . . . I needed someone to talk to all the time . . . I wanted someone I was sure of.’ He chose a former student of his, Richard Taylor of Cambridge University.”³¹ Fortunately, Taylor was ready for a sabbatical leave and was able to join Wiles in Princeton. It is interesting that at a crucial stage a collaborator was sought, even by a man committed to solving a problem by himself, a problem that had been of interest to him throughout his entire life, starting at the age of 10. Then, with the added perspective and energy of a scientific partner, he successfully resolved one of mathematics oldest challenges.

During Wiles’ 8-year ordeal he brought together virtually all the breakthroughs in 20th century number theory and incorporated them all in one mighty proof. He created completely new mathematical techniques and combined them with traditional ones in ways that had never been considered possible. In doing so he opened up new lines of attack on a whole host of other problems.³²

Figure 2-4. Andrew Wiles, who proved Fermat’s last theorem. Courtesy of C. J. Mazzochi.

The Love of Stories

Another social feature of mathematics is telling the personal stories of accomplishments and eccentricities of mathematicians, which they discuss with great gusto when colleagues meet. Often these stories tell of sustained immersion in intense research mathematics.

Isaac Newton (1643–1727) described how he made his discoveries: “I keep the subject constantly before me and wait until the first dawnings open little by little into the full light.”³³ It was hard for Newton to switch his attention away from mathematics, even when he was hosting a party. His friend William Stukeley recalled that when he had friends in his rooms, if he went into his study for a bottle of wine and a thought came into his head, he would sit down to write and forget his friends.³⁴

Norbert Wiener (1894–1964) of MIT was well known as an extreme example of someone who could get lost in thought. Once while walking on campus, Wiener met an acquaintance, and after a while he asked his companion: “Which way was I walking when we met?” The man pointed, and Wiener said, “Good. Then I’ve had my lunch.”³⁵

Another example of absent-mindedness is John Horton Conway of Princeton University, inventor of surreal numbers and the game of “Life.” Conway’s office is incredibly chaotic:

There were papers and books scattered everywhere. Puzzles, games, charts, novelties, and other paraphernalia were piled and heaped and stacked on all horizontal surfaces. Conway realized he had a problem when he could no longer lay his hands on his latest theorem, or list of problems or new conjectures. He set about to design a physical device that would address his quandary, and imbue some order into the chaos. After some time and effort, he had produced a set of plans. He was about to go off and find a craftsman to implement his idea in wood and metal when he noticed that such an item was already standing, empty and unused, in the corner of the office. It was the filing cabinet!³⁶

It is hard to choose from the hundreds of stories illustrating the depths of engagement of mathematicians. Here is one of our favorites. It is about the topologist R H Bing (1914–1986), a famous student of Robert Lee Moore (1882–1974). (The R and H are not initials, Bing’s complete name consisted of the two letters R and H and his family name Bing.)

One day R H Bing was driving a group of mathematicians to a conference. As usual, Bing launched into a detailed discussion of a particular problem in topology that he wanted to solve. The passengers were nearly as interested in the math as Bing, but they were rather nervous because the weather was bad, visibility poor, and there was lots of traffic. Bing did not seem to be giving the matter his full attention. To make matters worse, the windshield was completely fogged over and it was nearly impossible to see. A sigh of relief was quietly breathed when Bing began to reach toward the windshield. Everyone supposed that he was going to wipe the windshield clear and pay attention to what was happening on the dark and stormy road. Instead Bing started drawing diagrams with his finger in the windshield mist.³⁷

Figure 2-5. John Conway, British-American algebraist and number theorist, with a friend in 1987. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

Figure 2-6. R H Bing and Mary Bing. Courtesy of Dolph Briscoe Center for American History, The University of Texas at Austin. Identifier: di_05557. Title: Bings. Date: 1975/08. Source: Halmos (Paul) Photograph Collection.

These narrative exchanges provide the social glue to hold a diverse and dispersed community together. In cherishing their colleagues’ foibles, mathematicians come to terms with the emotional complexity of living in an abstract world.

Through most of history, the world of mathematics has been separate from the broader cultural milieu. But recently the lives of mathematicians have become of increasing popular interest. There was A Beautiful Mind, a successful book and movie about John Nash, and then the Broadway hit play called Proof; and articles about mathematicians in magazines such as The New Yorker. These biographical and fictional representations focus on the intensity of mathematical life and the consequences of such dedication. Other themes of these stories are competition in mathematical discovery and the vagaries of success. The main characters in Proof are a retired University of Chicago mathematics professor and his daughter. In the last years of his life, the professor kept notebooks of his mathematical explorations, and no one knew what was in them. The struggle for possession of these notebooks, and the actual identity of their author, are a substantial part of the play, delving into issues of ownership, gender, and fame.

Tensions and Their Resolution

In general, mathematicians are supportive of each other as they focus on solving problems. The rules serve to decide when a problem has been solved, and there are not as many battles for credit in mathematics as in archeology or linguistics. The celebration of Wiles’ success in proving Fermat’s Last Theorem was widespread and joyous; it is an instance of the usual mutual support among mathematicians.

Yet there are exceptions to this camaraderie. Prizes, academic politics, and the value of fame and visibility in the broader culture inevitably impact the discipline. We will talk about three dramatic examples. First, there is Grigori Perelman’s sensational proof of the Poincaré conjecture (which says that under a certain simple condition, any three-dimensional manifold is continuously deformable to a sphere) and the involvement therein of the great Chinese geometer Shing-Tung Yau and two of his students. Then there is the tangled tale of the recognition of chaos as a major, nearly ubiquitous feature of nature. And finally, there is the controversy in the mathematics department at the University of California at Berkeley about the denial and then the granting of tenure to the American mathematician Jenny Harrison.

Poincaré first stated his conjecture in 1904. The conjecture was proved in dimensions 5 and greater by Steven Smale and in dimension 4 by Michael Freedman. Surprisingly, a solution in three dimensions proved to be much more difficult than in the higher dimensions.

As with many important mathematical problems, progress was cumulative. William Thurston, now at Cornell, formulated what is known as the “geometrization conjecture,” which implies the three-dimensional Poincaré conjecture. Richard Hamilton of Columbia University made a major contribution by introducing the method of the “Ricci flow” to the geometry of manifolds. In the early 1990s, after making a presentation at Berkeley, Hamilton was approached by a shy Russian, Grigory Perelman, who was spending a couple of years in the United States. At their first meeting, Perelman was just asking questions. Later he made some contributions to Hamilton’s work, the importance of which was not immediately obvious to Hamilton. While Perelman was still in the United States, he made some additional progress on a particularly difficult aspect of the problem. As his reputation grew, he received several job offers in the United States. Although he would have enjoyed collaborating with Hamilton, his overtures were ignored. He decided to return to St. Petersburg. Using the Internet, he could work alone while continuing to tap a common pool of knowledge.³⁸

Hamilton was also an important influence on the famous geometer Shing-Tung Yau, whose career accomplishments include a Fields Medal, appointments at Harvard and the Institute for Advanced Study, and an honorary professorship in mainland China. Yau and Perelman are vastly different in their ways of dealing with recognition and public acclaim. “Grisha” Perelman lives very simply with his mother and connects to mathematics via the Internet. While his work is highly respected, both in his home country and abroad, he has become a recluse during the last few years. In contrast, Yau is a very public figure whose energy, tenacity, and ambition are widely known. He grew up in Hong Kong, received his Ph.D. at Berkeley, became famous by proving the Calabi conjecture, and during his tenure at Harvard trained dozens of young mathematicians. He has also played a prominent part in China’s recent efforts to strengthen scientific research and education.

Figure 2-7. Grisha Perelman, who proved Poincaré’s conjecture. Courtesy of ICM2006 Madrid.

Perelman sent Yau an e-mail in November 2002 describing some of his results, but he did not receive a response. Subsequently Perelman started to publish aspects of his proof for the Poincaré conjecture on the web site “arXiv.org.” He provided procedures that overcame the blocks in Hamilton’s approach. Many mathematicians were impressed and hoped that the conjecture had been solved. Perelman made presentations in the United States about his new results. While traveling on the East Coast, he was eager to engage Hamilton in discussing his new work, but he was unsuccessful. In an article in Science, Yau pointed out that parts of Perelman’s proofs were sketchy. But others, including the team of Gang Tian and John Morgan, checked Perelman’s work and found it acceptable. Once the work was verified, Perelman was offered the Fields Medal.

Yau encouraged two of his younger colleagues, Huai-Dong Cau and Xi-Ping Zhu, to write up their own extended presentation of the proof for the Asian Journal of Mathematics, of which Yau is the principal editor. The publication was unusually hasty and seemed timed to reflect some degree of glory or credit on China. In a rare gesture, Perelman declined the offer of the Fields Medal. He said, “Everybody understood that if the proof is correct then no other recognition is needed.”³⁹ It is difficult to know how much the controversy over credit contributed to Perelman’s decision, and whether his withdrawal from mathematical activity is permanent. In the meantime, Yau has threatened to sue The New Yorker. They published an extensive interview with Perelman accompanied by an offensive cartoon showing Yau pulling the Fields Medal from Perelman’s neck. Several well-known mathematicians filed letters with Yau’s lawyers testifying to Yau’s devotion to and accomplishments in mathematics.

Figure 2-8. Two geometers: Robin Hartshorne of Berkeley and Shing-Tung Yau of China. Courtesy of Gerd Fischer.

This controversy illustrates an interesting distinction between “intrinsic” and “extrinsic” motivation that is made in the creativity literature by Teresa Amabile. “Intrinsic motivation” refers to the enjoyment and satisfaction that a person derives from engaging in the creative activity.⁴⁰ For a mathematician, that includes the enjoyment and satisfaction from his or her colleagues’ appreciation that the work is correct, interesting, and important. Perelman’s comment above is a beautiful example of intrinsic scientific motivation; his reward is simply his success in solving the problem.

On the other hand, there can be motivation for “extrinsic” recognition—promotions, pay raises, prizes, power, and prestige in the wider community. In the case of the Poincaré conjecture, there is actually a million-dollar prize, offered by the Clay Foundation. Are these extrinsic motivations supportive, or do they interfere with creativity? Some researchers argue that such rewards may contribute to task-focused motivation, increasing creative individuals’ level of concentration.⁴¹ But Amabile’s original work suggested the contrary—that extrinsic rewards may divide one’s attention between the creative task itself and the extrinsic goals. More recently, she has proposed the possibility of an interactive effect of the two types of motivation. In Yau’s case, this latter model seems applicable. His long, deep involvement with challenging mathematical problems is driven by intrinsic motivation. At the same time, in contrast with Perelman, Yau also thrives on public success and influence and refers with great pride to his various awards.

Which of these two men represents mathematical culture? In the view of the public, where mathematicians are seen as introverted and unconventional, Perelman fits the image. But with the discipline becoming increasingly a part of social life, with broad-ranging discussions of mathematics’ usefulness and with policy debates and cinematic depictions, Yau fits a second view of the culture—one of political relevance within academia and the larger society. This story raises the issue of commonality and diversity within the culture. To be a successful mathematician requires focused energy, problem-solving skills, persistence, creativity, and the effective use of logic. Enthusiasm for the beauty and clarity of mathematical problems unites mathematicians, who enjoy discussing great discoveries, and the varied aspects of their joint history. At the same time, differences in temperament and in the desire for public acclaim can lead to fractures in a community that is inherently interdependent. There is a continuum, of degree of concern with extrinsic versus intrinsic scientific rewards, in mathematics; Yau and Perelman occupy two widely separated positions on that continuum.

As mathematics is not an empirical science, agreement on the procedures governing the acceptance of proofs is crucial to the profession. A new finding is normally submitted to a journal. Its editors then assign “referees”—specialists to determine whether it stands up under expert scrutiny. In some cases of unexpected or very complex results, the process of verification may become contentious. This was illustrated in the Perelman-Yau controversy recounted above.

Occasionally, a mathematician may decide to forego publication of a dramatic new discovery. This was the choice made by Carl Friedrich Gauss (1777–1855) when he was working on Euclid’s parallel postulate. Gauss discovered a new geometry, completely different from that of Euclid. It was later independently discovered by the Russian Nikolai Lobachevsky and the Hungarian János Bolyai. But Gauss refrained from publishing his findings (of what later became known as non-Euclidean geometry) because he disliked controversy and feared that he would be attacked on philosophical grounds by some of the followers of Immanuel Kant. This new geometry, as developed by Lobachevsky and Bolyai, was ignored for a long time by mathematicians, only to later find its usefulness proven mathematically in Poincaré’s function theory, and physically in Einstein’s general relativity.

Our second example shows the difficulties that can arise when a new finding violates prevailing mathematical intuition. Against all expectations, it turns out that, in higher than two dimensions, “almost all” dynamical systems are unpredictable in the long run—they are chaotic. The first inkling of this fundamental feature of the physical universe came in Poincaré’s work on the stability of the solar system. (This is the most famous example of an n-body problem.)

In honor of the 60th birthday of King Oscar II of Sweden and Norway, Acta Mathematica offered a prize for the best essay on this celebrated problem. Poincaré’s contribution, including the notion of “characteristic exponents,” had a lasting impact on the study of dynamics. His qualitative, topological approach to this problem is a wellspring of topology, which became one of the main branches of mathematics in the 20th century. In Poincaré’s analysis of the n-body problem he discovered what he called the “homoclinic tangle.” In three and higher dimensions of phase space, a trajectory can loop around itself in an infinitely complex entanglement which he found it hopeless to sort out. This was the first mathematical manifestation of the phenomenon now called chaos.⁴²

Poincaré recognized the importance of this discovery but did not try to explore it in fuller detail. The full story was told by June Barrow-Green, a math historian, who wrote: “. . . [Poincaré] had found this result so shocking that it is not perhaps surprising that he did not consider the possibility of even more complex solutions.”⁴³ For the next 30 years, this insight of Poincaré was largely ignored, almost forgotten. When Hadamard and Cartan developed some of Poincaré’s ideas, they ignored chaos. It contradicted the dominant belief that mathematics and science enable us to control nature.

The 2-body problem—the motion of the earth around the sun, ignoring the perturbing effects of the other planets, or the motion of the moon around the earth, ignoring the effects of the sun—had already been solved by Newton. Hypnotized by Newton’s epochal discoveries in dynamics, mathematicians expected to go ahead and solve the 3-body, and in general, the n-body problem. Poincaré’s discovery of the homoclinic tangle was the first intimation of the profound truth, understood only in our times, of the chaotic character of almost all dynamical systems in dimensions higher than 2. They are unpredictable in the long run because unobservably small deviations in the data today eventually result in arbitrarily large differences in the outcome.

For a long time Poincaré’s successors in the mathematical theory of dynamical systems turned away from this problem of the homoclinic tangle. Such a “strange attractor” cannot arise in two dimensions. In fact, dynamical systems in two dimensions of phase space are well understood, although of course many open questions remain, and it was generally expected that similar regularity could ultimately be proved in higher dimensions as well. Mathematicians tried in vain to prove that “most” dynamical systems are “well-behaved,” that is, predictable in the distant future.

This was the project that Steve Smale of Berkeley undertook until a postcard from Norman Levinson of MIT called his attention to a paper of John Littlewood (1885–1977) and Mary Cartwright (1900–1998). Working on the Van der Pol equation of electrical engineering, they had found a kind of chaotic behavior similar to what Poincaré had discovered in the n-body problem. Smale reversed course and ultimately was able to show that in higher dimensions, unlike the plane, chaos rather than predictability is the norm.

The term “chaos” has been used to describe a wide variety of phenomena. It was introduced by Yorke and Li in relation to a different phenomenon.

Meanwhile, quite apart from these investigations in pure mathematics, the Hayashi group in Japan, particularly Yoshisuke Ueda, and Christian Mira and his group in Toulouse, France, were finding chaos by carrying out long calculations on analog computers.

These researchers were challenged by colleagues who ascribed their findings to “noise.” They preferred to ignore the implications and questioned the focus of researchers who wandered so far from the results of traditional mathematical reasoning.

In a volume called The Chaos Avant-Garde, some of these researchers recall their frustrating experiences. Christian Mira wrote of the research on chaotic manifolds in Toulouse as follows: “. . . a point is worth to be mentioned about the local background of these researchers. The complex dynamics studies of the Toulouse group have never occurred in a favorable environment. So my projects of developing this topic and applications were systematically refused during the ‘prehistoric’ times of ‘chaos,’ with the argument ‘nobody is interested by such a matter,’ which I frequently heard. . . . This in spite of the highly significant Smale’s contribution of prime interest, leading in particular to the ‘horseshoe’ map properties (1963), and other rare interesting publications.”⁴⁴

Figure 2-9. Steve Smale, leading researcher in topology and dynamical systems. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

Part of the problem was that many of the contributors to chaos theory came from applied fields, including engineering and physics. Another reason for the lack of acceptance of their findings was that some of the basic research was published in Russian journals which were not read by many western mathematicians.

A similar problem faced the Japanese researcher Yoshisuke Ueda. He was working in a laboratory where Professor Hayashi, the head of the team investigating electric oscillations, considered the unexpected chaotic results as inappropriate. He recommended that Ueda repeat the experiments “until the transient state settled to a more acceptable result.”⁴⁵ In view of this opposition, Ueda waited until Hayashi was on sabbatical leave to continue his path-breaking research. But when he tried to publish some of his findings, they were rejected. He was caught in a difficult contradiction. While his work did not meet the standard of rigorous mathematical proof, it did correspond to phenomena that were observed in the real world. “While I wanted them to hear me out a little more sympathetically I also idolized mathematics . . . I had intentionally sent off the paper during Professor Hayashi’s absence so I had a good excuse for not having it reviewed by him and it probably lacked certain fine editing because of it. But there was no way I could show the paper to Professor Hayashi since I knew he would make drastic changes and cut out what was essential to me. I couldn’t compromise, however . . . I learned the hard way that changing the already established order in this world was a truly difficult task.”⁴⁶

Although it was hard to have this research accepted, eventually the work became known as “the Japanese attractor.” Ueda refers to the early years of his research as “the dark hours before chaos was universally recognized.”⁴⁷

As more and more results emerged, confirming the prevalence of chaotic phenomena, Poincaré’s discovery of the homoclinic tangle was remembered. In summarizing the long and complex 70-year history of the discovery of chaos, the American mathematician Ralph Abraham writes, “Triggered by mathematical discovery, The Chaos Revolution is a bifurcation event in the history of the sciences, comprised of sequential paradigm shifts in the various sciences. Perhaps it is also a major transformation in world cultural history: time will tell. Meanwhile, we are struck with the personal observation of the similarity in the sociological and psychological experiences of the various pioneers who have suffered from the novelty of their ideas, and the bravery of their convictions. We are deeply in their debt.”⁴⁸

Tenure Battle at Berkeley

Finally, here is a last example of tension in mathematical life. Divisions among mathematicians are not limited to their philosophical outlook or their attitude toward public awards and recognition. They also include issues of judgment about who should be part of highly prestigious mathematics departments. Decisions concerning membership involve the quality of an individual’s research, collegiality, area of specialization, and possible hidden preferences concerning ethnicity, age, and gender. One of the best known cases is that of Jenny Harrison at the University of California, Berkeley. The central issue in the case was gender discrimination alleged by Harrison, who was denied tenure in 1986. After a long period of departmental and universitywide reviews, the university settled her case and granted her tenure and the title of full professor. Much has been written about this lengthy process, and some authors have said that the case opened a Pandora’s box of issues about tenure, discrimination, and the law.⁴⁹ In addition to printed information, a recent personal interview with Harrison by John-Steiner provides a more complete picture of how she relied on inner resources during this conflict.

Harrison’s earliest interest was nature. “I went to public school and I was basically self-taught because I wasn’t getting anything out of school. And from the age of 5 until 15 I spent all of my time that was available to me outside of school in the woods.” This attraction to the natural world has influenced Harrison throughout her life and can be seen in the way she explores mathematical landscapes.

In the interview she told John-Steiner about her visual view of the world and her enjoyment in exploring paths in the woods. An important influence in her life was her older brother, whose encouragement and sustained belief in Harrison’s talent and determination contributed to her strength and self-confidence. “He was a great teacher, and got me interested in basic problems in physics when we were teenagers.”

As a teenager she scored well in math contests, but she was primarily devoted to music. She thought that she was going to become a professional musician, but then she found she was uncomfortable performing in public. It has often been remarked that musical talent and mathematical talent seem to be associated. Of the people I have known personally, my graduate school office mate Leonard Sarason had earned a Master’s degree at Yale under the composer Paul Hindemith. Richard Courant’s eagerness to perform at the piano was notorious in his family. Courant’s student Hans Lewy became a mathematics professor at Berkeley but could have been a concert pianist. The mathematical analyst Leonard Gilman of the University of Texas in Austin performed piano classics to entertain national meetings of the American Mathematical Society.

Once Harrison switched to mathematics at the University of Alabama, her work was quickly recognized, with an offer to become a Marshall scholar at the University of Warwick in England. She received excellent training at Warwick, but she encountered sexual harassment there. The issue of sex discrimination and male/female relationships in a discipline long dominated by men is intricately involved in the Harrison case. It affected her both at Warwick and at Berkeley.

Her dissertation research is highly regarded. She completed her thesis, titled “Unsmoothable Diffeomorphisms,” in 1975. She worked as a postdoctoral fellow at the Institute for Advanced Study and as an instructor at Princeton University. She received a Miller Fellowship to Berkeley in 1977 and a subsequent offer for a tenure track assistant professorship at Berkeley. A year later, she accepted an offer from Oxford University for a position at Somerville College, where she spent 3 years while still retaining her Berkeley position. On returning to Berkeley in 1982, she announced that she had found a new and stronger counterexample to the Seifert conjecture. Harrison’s work drew on new and delicate techniques from several fields and proved very difficult to write up for publication.⁵⁰ Once the manuscript was submitted, opposition by Michel Herman, a leading expert in the field, resulted in a more than usually cautious stance by the referees. Harrison further clarified what happened.

Figure 2-10. Jenny Harrison, Berkeley professor of mathematics. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

Michel Herman was a highly regarded French mathematician who believed my methods were not only hopelessly wrong, but “dangerous” and did not hesitate to say these things to others. He feared my novel methods would be taken seriously by others and there would be no end of problems to correct. Besides that, he was working hard to find a counterexample to the Seifert Conjecture himself. Herman’s opposition was taken seriously by referees who took years to accept the papers. They were convinced they would find a fatal flaw somewhere in the long and technical work, but finally relented and the papers were accepted in 1986. Herman had been wrong. In recent years, it has become apparent how important these papers were. They led the way to my broad extension of calculus, which unites the discrete to the smooth continuum, solving an old and important problem of mathematics that dates back to Newton and Leibniz.⁵¹

In 1986, after a many-stage process going up from a vote by the math department to the dean, two committees, and the chancellor, Harrison was denied tenure. She filed a formal complaint with Berkeley’s Academic Senate Committee on Privilege and Tenure, which ruled against her. In her web page Harrison writes, “From the very beginning I only asked for a new and fair review, free from any question of gender bias.”⁵² Once denied, her next step was to file a lawsuit against the university in 1989 charging sex discrimination. The lawsuit was eventually settled by the university, based on the recommendation of seven distinguished members of the mathematics community chosen by the university. Their identities were kept strictly confidential, and they included two members of the math department who had not taken sides in the case. “If a majority of the review committee voted against me, I agreed to walk away, with no appeal. If a majority voted for me, I would be reinstated.”⁵³ The committee reviewed Harrison’s research subsequent to the original tenure decision and unanimously agreed that it was as good as the work of 10 other mathematicians who had received tenure in her department in the meantime. They recommended offering her a full professorship.

This decision at the university level was controversial in the math department. From the very beginning of this case there were deep divisions among the faculty members. One issue was, what are the criteria for tenure in an elite department such as Berkeley? Whatever these may be, the department’s record in hiring female faculty members was considered by some faculty members to be very poor. As mentioned later in this book, Julia Robinson was appointed only after she became a member of the National Academy of Sciences. After Robinson’s death, there was only one female member among 55 men in the department until Harrison’s appointment.

Many faculty members believe that part of the university’s willingness to appoint an independent committee and to accept their decision was their concern that this record of female absence would reflect poorly on Berkeley. Others argue that it was better for the dispute to be decided by knowledgeable academic experts rather than in a courtroom by a jury of 12 randomly selected citizens of Alameda County. Some members of the math department strongly supported Harrison. Her case was publicized by a Harrison support committee (chaired by Charity Hirsch, the wife of a well-known member of the department), including faculty from across the campus. It was financially supported by the legal funds of the American Association for University Professors and the American Association of University Women, who accepted Harrison’s contention that the appeal mechanism within the university was inadequate for addressing her grievances.⁵⁴

Once the decision was made by the university, the division within the department became even more pronounced. Some members objected to the department being overruled by the administration. Others were critical of Harrison’s accomplishments. In a comprehensive article on this case written by Allyn Jackson, Dorothy Wallace said that the tenure process was quite subjective: “We are expected to be able to say that the candidate is at least as good as any of the people to whom we are comparing her or him. We are never asked to justify this judgment. In fact we are never asked to define what constitutes ‘good’ or ‘as good as’. . . . I am driving at the fact that the process itself is wide open to any sort of individual or group prejudice that possibly could exist.”⁵⁵ The opposition to Harrison involves only a few members of the department, but it has been vociferous and frequently hurtful to her.

During the years of fighting for her case, Jenny Harrison suffered from tonsil cancer but continued her fight without giving in to despair. In fact, her research continued to develop, and its originality and importance have been broadly recognized during the 12 years that have followed her reinstatement. Calvin Moore has written,

During the period of appeals and litigation, Harrison had continued her research program, developing new lines of research in geometric measure theory, and her work was supported by federal grants. After her reinstatement as professor, she has continued to develop her research program in geometric measure theory aimed at understanding multivariable calculus on domains with very irregular boundaries. In this she was building on and expanding ideas of Hassler Whitney. More recently, she has published work on a theory of domains based upon an extension of the Cartan exterior calculus to a normed space of domains that includes soap films, manifolds, rough domains, and discrete atomic structures. The analysis involved in her earlier work on the Seifert conjecture counterexamples with three minus epsilon derivatives had pointed the way to these more recent results. She has supervised the doctoral work of two students.⁵⁶

Her case raises important issues about women’s determination to make a place in a traditionally masculine culture. Harrison strongly believes in maintaining one’s balance when confronted with difficult problems. She has tried to establish a balance between teaching, research, music, and love of nature. In this book, where we emphasize the emotional aspects of mathematics, this story from Berkeley reminds us how, even in a discipline that prizes objectivity and rationality, professional practice is not immune from long-held societywide assumptions about women’s contributions to outstanding intellectual work.

The lived experience of mathematicians is influenced by the long history and the sense of awe in which their discipline is held. Members of the profession still address century-old problems, while constantly renewing their techniques and modes of thought. Recently, as mathematics has captured public interest and attention, the more hidden emotional aspects have come to the fore. Mathematicians rely deeply upon each other to evaluate their proofs and findings even as they compete for prizes and fame. This tension between community and competition has recently been made more complex by the greater heterogeneity of membership in what has traditionally been a rather exclusive group.

How have we answered our question, What is the culture of mathematics? We have tried to contribute to it, by examining the cognitive, emotional, and aesthetic aspects of the field and the social and competitive tensions. All these issues are, to some degree, found in other aspects of academic and scientific life, yet all have a special and definite flavor in the life of mathematicians.

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