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Friendships and Partnerships

Do mathematicians have friends? The popular image of a mathematician is that of a solitary man, alone at his desk or blackboard. In this chapter, we will see how far from the truth this picture is. While the sustained concentration needed for mathematical research does require quiet and a highly focused mind-set, an intense and prolonged independent search can come to a dead end. The researcher who listens to colleagues’ insights can break through his private perspective.

One morning early in my (Hersh’s) years as a thesis student of Peter Lax, I entered my mentor’s office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, Louis Nirenberg, also one of the partial differential specialists on the faculty of NYU’s Courant Institute. He had been on leave in England; now he was back home! At the time. I didn’t get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations (PDEs), and Peter, a world master at hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success.

In the history of mathematics many names occur in pairs—Hardy and Littlewood, Cayley and Sylvester, Weierstrass and Kovalevskaya, Polya and Szegö, Riesz and Nagy, Hardy and Ramanujan, Minkowski and Hilbert, and Lax and Phillips. Each of these pairs was different. Karl Weierstrass and Sonia Kovalevskaya were not merely mentor and pupil, they were deeply involved with each other emotionally. David Hilbert and Hermann Minkowski were close friends who gave each other vital support and stimulation, although they never collaborated on a publication. G. H. Hardy and J. E. Littlewood coauthored nearly 100 papers, over 35 years, yet they seldom met face to face. Littlewood said, “Hardy was unhappy except with bright conversation available. . . . Our habits were about as opposite as could be.”¹

We take a historical route to describe some of the most interesting friendships among colleagues and across generations. Then we tell about two very different mathematical marriages—between Grace Chisholm and Will Young, and between Julia Bowman and Raphael Robinson. We conclude our survey of mathematical friendships with a section on friendships among women, including Olga Taussky-Todd and Emmy Noether.

Mentors

Two mathematicians can relate as teacher and student, as collaborators, or as friends. A vital relationship in the lives of most successful mathematicians has been as an apprentice to a mentor. One of the most famous apprentice-mentor stories is about a 20-year-old married Russian beginner, Sonia Kovalevskaya, and a 55-year-old master of complex analysis, the bachelor Karl Weierstrass. Kovalevskaya arrived in Berlin in October 1870. She obtained an audience with the great Professor Weierstrass and pleaded with him to accept her as a student. Such a thing was then impossible. Nevertheless, Weierstrass gave her a few test problems. When she returned a week later, her solutions were not only correct, but they were also original. Weierstrass said that her paper showed “the gift of intuitive genius to a degree he had seldom found among even his older and more developed students.”² So Weierstrass recruited the prominent physiologist Emil DuBois-Reymond, the famous pathologist Rudolf Virchow, and the renowned physicist-physiologist Hermannn Helmholtz to join him in requesting permission for Sonia to register as a student at the University of Berlin. But still the faculty senate voted no! So Weierstrass offered instead to recapitulate his lectures for her and moreover to tell her about his own research. For 4 years she visited him every Sunday that she was in Berlin. And he came once a week to the apartment that she shared with a friend.³ Kovalevskaya said, “These studies had the deepest possible influence on my mathematical career. They determined finally and irrevocably the direction I was to follow in my later scientific work; all my work has been done precisely in the spirit of Weierstrass.”⁴

Weierstrass’s role in Kovalevskaya’s scientific and personal affairs far transcended the usual teacher-student relationship. He found her a refreshingly enthusiastic participant in all his thoughts, and ideas that he had fumbled for became clear in his conversations with her. In 1873, while he was on vacation in Italy, Weierstrass wrote to Sonia: “During my stay here I have thought about you very often and imagined how it would be if only I could spend a few weeks with you, my dearest friend, in such a magnificent natural setting. How wonderful it would be for us here—you with your imaginative mind, I—stimulated and refreshed by your enthusiasm—dreams and flights of fancy about so many puzzles that remain for us to solve about finite and infinite spaces, the stability of the solar system, and all the other great problems of the mathematical physics of the future. However, I learned long ago to resign myself if not every beautiful dream becomes true.” It seemed to him that they had been close throughout his entire life and “never have I found anyone who could bring me such understanding of the highest aims of science, such joyful accord with my intentions and basic principles as you.”⁵

Yet their relationship did not remain untroubled. Weierstrass disapproved of Sonia’s links with socialist circles, her literary work, and her advocacy of the emancipation of women. Many of his letters to her went unanswered; at one juncture she did not respond for 3 years. . . . Sonia Kovalevskaya died on February 10, 1891, in her fortieth year, of a pulmonary infection. Her untimely death, when she was still in her prime, caused Weierstrass much grief. He burned all her letters to him.⁶

Figure 5-1. Karl Weierstrass, the great German analyst.

Another famous friendship was between the great German mathematician David Hilbert and his colleague Hermann Minkowski. Minkowski, born in 1864, was the son of a Russian merchant. He and Hilbert first met in gymnasium, and then became close friends as students at the University of Königsberg.

Although the two had quite different personalities, they shared an enthusiastic love of mathematics and a deep, fundamental optimism.⁷ Hilbert openly expressed his need for Minkowski’s companionship and stimulation. Together with their young teacher, Adolf Hurwitz, they met at 5:00 every afternoon. They formed a friendship for life.⁸ Realizing the many benefits of their relationship, these young mathematicians tried hard to stay geographically close. Even when separated, they relied heavily on each other’s advice. In 1900 Hilbert was preparing his famous speech to the International Congress of Mathematicians in Paris, where he planned to outline 23 problems for mathematicians to solve in the coming century. Even though Minkowski was in Zurich, he was eager to help. The two of them, and Adolf Hurwitz, corresponded frequently on the form and content of the lecture. Hilbert paid close attention to their criticisms of his address, “Mathematical Problems,” until he reached the final draft. At the turn of the century, the University of Göttingen became the leading center of mathematical work under Hilbert’s leadership. But not until he was able to arrange a professorship for Minkowski was Hilbert truly satisfied. After Minkowski arrived in Göttingen in the fall of 1902, Hilbert was no longer lonely. He needed only a telephone call, or a pebble tossed up against the little corner window of his study, and he was ready to join his friend.⁹

Figure 5-2. David Hilbert. Source: Mathematical People: Profiles and Interviews. Eds. Donald J. Albers and G. L. Alexanderson. Boston: Birkhauser, 1985. Pg. 285. Reprinted by kind permission of Springer Science and Business Media.

The relationship between these two men dazzled and instructed those around them. Max Born recalled, “The conversation of the two friends was an intellectual fireworks display. Full of wit and humor and still of deep seriousness.”¹⁰ Minkowski provided a critical ear as Hilbert prepared his classes; they taught a joint seminar; they shared their interest in physics. The friendship came to a tragic end when Minkowski, at the age of 44, died of a ruptured appendix. We have written more about Göttingen in the chapter on mathematical communities.

Hardy, Littlewood, and Ramanujan

G. H. Hardy was the most influential English mathematician of the first quarter of the 20th century. (His lectures won Norbert Wiener from philosophy to mathematics.) Hardy’s beautifully written essay A Mathematician’s Apology is widely known. His apology is for a life devoted to pure mathematics. His defense is that, although he never did anything “useful,” he did add to the world’s knowledge, guided by beauty as well as truth. He wrote, “The mathematician’s patterns, like the painter’s 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.”¹¹ (In chapter 2, we took up the question of beauty in mathematics.)

The essay is melancholy. At 60, says Hardy, he is too old to have new ideas. That is why he is reduced to writing books instead of doing a mathematician’s proper business, discovering and creating new mathematics. In a foreword to a posthumous reprinting of Hardy’s essay, C. P. Snow painted a moving picture of Hardy’s life. (A physicist turned scientific bureaucrat, Snow wrote novels about political maneuvering among academics, and a much-quoted lecture and subsequent book, The Two Cultures, bemoaning the literati’s ignorance of physics.)

In his prime, wrote Snow, Hardy lived among some of the best intellectual company in the world. He was one of the most outstanding young men in his circle. “His life remained the life of a brilliant young man until he was old; so did his spirit: his games, his interests, kept the lightness of a young don’s. And like many men who keep a young man’s interests into their sixties, his last years were the darker for it.”¹²

Hardy found English analysis and number theory a quiet backwater and raised them to a Continental standard. He wrote: “I shall never forget the astonishment with which I read Jordan’s famous Cours d’analyse, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant. The real crises of my career came ten or twelve years later, in 1911, when I began my long collaboration with Littlewood, and in 1913, when I discovered Ramanujan. All my better work since then has been bound up with theirs, and it is obvious that my association with them was the decisive event of my life.”¹³

Hardy’s collaboration with Littlewood began in 1911 and lasted 35 years. All of Hardy’s major work was done with Littlewood or Ramanujan. The Hardy-Littlewood partnership dominated English pure mathematics for a generation. Hardy said Littlewood “was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.”¹⁴ The number theorist Edmund Landau said, “The mathematician Hardy-Littlewood was the best in the world, with Littlewood the more original genius and Hardy the better journalist.”¹⁵

Figure 5-3. Three influential mathematicians in conversation: Richard Courant, G. H. Hardy, and Oswald Veblen. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

Littlewood is now remembered best for his prodigious productivity in his later years. But in his prime he was the antithesis of the stereotypical mathematician: muscular and athletic, a proficient rock climber. He was a bachelor, like Hardy, but unlike Hardy, he appreciated female companionship. It was a well-known “secret” that a young woman he called his niece was actually his daughter, by a liaison with a colleague’s wife.

There is a fascinating contrast between the emotional tone of Hardy’s two great friendships—with his English collaborator John Littlewood and with his Indian collaborator Srinivasa Ramanujan.

How different was Hardy’s way with Littlewood, from Hilbert’s way with Minkowski! The Danish mathematician Harald Bohr (brother of the physicist Niels) reported:

When Hardy once stayed with me in Copenhagen, thick mathematical letters arrived daily from Littlewood, who was obviously very much in the mood for work, and I have seen Hardy calmly throw the letters into a corner of the room, saying, “I suppose I shall want to read them some day.” This was according to one of the “axioms” of the Hardy-Littlewood collaboration: “When one received a letter from the other, he was under no obligation whatsoever to read it, let alone to answer it—because, as they said, it might be that the recipient of the letter would prefer not to work at that particular time, or perhaps that he was just then interested in other problems.”¹⁶

Hardy’s student Mary Cartwright observed the Hardy-Littlewood relationship from another angle. When Hardy returned to Cambridge from Oxford as the Sadleirian chair, she asked him if he would be offering a seminar similar to the Friday evening sessions she had enjoyed at Oxford. He replied that he would probably come to some arrangement with Littlewood. Soon after, the lecture list announced a Hardy-Littlewood class, to meet in Littlewood’s rooms.¹⁷

The Oxford mathematician E. C. Titchmarsh (1899–1963) thought this seminar was a model of what such a thing should be. Mathematicians of all nationalities and ages were encouraged to present their own work, and the whole seminar was delightfully informal, with free discussion after each paper.¹⁸ Cartwright recalled that at the first Hardy-Littlewood class, Littlewood was speaking. Hardy came in late, helped himself liberally to tea and began to ask questions, as if he were trying to pin Littlewood on details. Littlewood told Hardy that he was not prepared to be heckled. Thenceforth, Hardy and Littlewood alternated classes. Cartwright does not recall them being present together at any subsequent class. Eventually, Littlewood stopped participating, though the class continued to meet in his rooms. The class became known as “the Hardy-Littlewood Conversation Class at which Littlewood is never present.”¹⁹ Hardy and Littlewood, while deeply connected mathematically, had little apparent affection for each other. Their only interaction was by mail around their deep, difficult mathematics!

Figure 5-4. Dame Mary Cartwright, the famous British analyst. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

On one famous occasion, they did have to meet. They had to deal with an unprecedented human problem. Hardy had received a letter from an unknown person far away in India:

Dear Sir, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only 20 pounds per annum. I am now about 23 years of age. I have had no University education, but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”²⁰

The occasion demanded an actual conversation with Littlewood, who lived nearby in other rooms at Cambridge. Together Hardy and Littlewood concluded that Ramanujan was not an impostor but a genius mathematician. (Letters from Ramanujan had been ignored by two other leading English mathematicians.)

Hardy quickly worked to bring Ramanujan to England. This was a delicate matter because Ramanujan was an observant Hindu and his religion forbade him to cross the ocean. At last, however, he accepted Hardy’s invitation. He spent 3 years in England during World War I, working intensely with Hardy. He produced a great body of mathematics that today, 60 years later, continues to inspire and challenge number theorists. Some of his formulas even turn out to be important in nuclear physics.

Hardy wrote, “I saw him and talked with him almost every day for several years, and above all I actually collaborated with him. I owe more to him than to anyone else in the world with one exception, and my association with him is the one romantic incident in my life. . . . I liked and admired him . . . a man in whose society one could take pleasure, with whom one could drink tea and discuss politics or mathematics. . . . I can still remember with satisfaction that I could recognize at once what a treasure I had found.”²¹

But Hardy also wrote, “Ramanujan was an Indian, and I suppose that it is always a little difficult for an Englishman and an Indian to understand one another properly.”

The friendship between the two, while scientifically and intellectually extremely successful and no doubt thrilling to both of them, probably didn’t include much real emotional communication. The cold climate, the strange food, the white-faced, distant English people, the wartime conditions, and the separation from his wife and his mother were too much for the young Indian mathematician. He contracted tuberculosis and had to enter a sanatorium. There he must have been even lonelier than before. In January or February of 1918 he threw himself onto the tracks of the London underground in front of an oncoming train. A guard spotted him and brought the train to a screeching stop. Ramanujan was bloodied, and his shins were scarred deeply. He was arrested and hauled to Scotland Yard, and Hardy was called to the scene. He marshaled all his charm and academic stature and convinced the police that the great Mr. Srinivasa Ramanujan, a Fellow of the Royal Society, simply could not be arrested. “ ‘We in Scotland Yard did not want to spoil his life,’ the officer in charge of the case said later.”²²

Figure 5-5. Srinivasa Ramanujan, the great Indian number theorist. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

A few weeks later, Hardy’s pretense became true. Back in the sanatorium, Ramanujan received a letter notifying him that he had been elected to the Royal Society. On March 13, 1919, he boarded a ship for home. Back in Madras, he was now an honored celebrity, but his disease continued to worsen, and he died in April 1920 at the age of 34.

It is remarkable how different the tone is in which Hardy writes about Ramanujan. Of Hardy’s two great mathematical collaborations, one lasted 35 years, with a fellow Englishman of similar status, education, and culture, and was conducted at arms’ length through letters that could be left unanswered even though his collaborator was only a short walk away. The other collaboration, which lasted only a few years, was conducted with a man whose mathematics was intensely interesting to Hardy but much of whose thinking was totally strange to him. That relationship was daily, face to face, with warm personal concern.

Hardy suffered a coronary thrombosis in 1939. In his Apology, written a few years later, he wrote:

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.” I was at my best at a little past forty, when I was a professor at Oxford. Since then I have suffered from that steady deterioration which is the common fate of elderly men and particularly of elderly mathematicians. A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas. It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value.²³

Snow called Hardy’s Apology “a passionate lament for creative powers that used to be and that will never come again.”²⁴ Hardy was like “a great athlete, for years in the pride of his youth and skill, so much younger and more joyful than the rest of us, who suddenly has to accept that the gift has gone.”²⁵

In fact, Hardy, like Ramanujan, attempted to kill himself by swallowing barbiturates, but he took too few. After that, Snow visited him every week. Two or three weeks before his death, the Royal Society informed him that he was to be given their highest honor, the Copley Medal. “He gave his Mephistophelian grin, the first time I had seen it in full splendour in all those months. ‘Now I know that I must be pretty near the end. When people hurry up to give you honorific things there is exactly one conclusion to be drawn.’ ”²⁶

Kolmogorov and Aleksandrov

Another remarkable friendship was that between the two famous Russian mathematicians Andrei Nikolaevich Kolmogorov (1903–1987) and Pavel Sergeevich Aleksandrov (1896–1982) Kolmogorov was one of the most original and influential mathematicians of his generation. Aleksandrov was a leading creator of topology and head of the graduate math program at Moscow University during its “golden years” (chapter 6). Shortly before his death, Aleksandrov wrote: “My friendship with Kolmogorov occupies in my life quite an exceptional and unique place: this friendship had lasted for fifty years in 1979, and throughout this half-century it showed no sign of strain and was never accompanied by any quarrel. During this period we had no misunderstanding on questions in any way important to our outlook on life. Even when our views on any subject differed, we treated them with complete understanding and sympathy.”²⁷

Three years later, in 1986, at the age of 83, Kolmogorov wrote: “Pavel Sergeevich Aleksandrov died six months before my eightieth birthday. . . . [F]or me these 53 years of close and indissoluble friendship were the reason why all my life was on the whole full of happiness, and the basis of that happiness was the unceasing thoughtfulness on the part of Aleksandrov.”²⁸

The two first met in 1920, but their close friendship began in 1929, at the ages of 26 and 33, when they traveled together for three weeks. In the Caucasus they stayed in an empty monastery on an island in Lake Sevan. Kolmogorov wrote, in his 1986 memoir, “On the island we both set to work. With our manuscripts, typewriter, and folding table we sought out the secluded bays. In the intervals between our studies, we bathed a lot. To study I took refuge in the shade, while Aleksandrov lay for hours in full sunlight wearing only dark glasses and a white panama. He kept his habit of working completely naked under the burning sun well into his old age.”²⁹

In 1930 and 1931 they traveled in France and Germany. Aleksandrov had been there before, in company with the brilliant young mathematician Pavel Uryson, his close friend and collaborator. Uryson had tragically drowned while swimming off the coast of Brittany. Kolmogorov and Aleksandrov went to Brittany to visit Uryson’s grave. “The deserted granite beaches, against which the huge waves thunder, form a complete contrast to the shores of the Mediterranean. Uryson’s grave is well tended because it is looked after by Mademoiselle Cornu in whose house Aleksandrov and Uryson were living at the time of his death. Both the gloomy nature of Brittany and the memory of Uryson inclined us to silent walks along the sea shore.”³⁰

In 1935 Kolmogorov and Aleksandrov bought, from the heirs of the famous actor and director Konstantin Stanislavskii, part of an old manor house in the village of Komarovka. Kolmogorov wrote, “As a rule, of the seven days of the week, four were spent in Komarovka, one of which was devoted entirely to physical recreation—skiing, rowing, long excursions on foot (our long ski tours covered on average about 30 kilometers, rising to 50). On sunny March days we went out on skis wearing nothing but shorts, for as much as four hours at a stretch. . . . Especially did we love swimming in the river just as it began to melt, even when there were still snow drifts on the banks.”³¹ In Komarovka they were often joined by their students, and they were visited by mathematicians from abroad, including Hadamard and Frechet from Paris, Banach and Kuratowski from Warsaw, and Aleksandrov’s collaborator Hopf from Switzerland.

Kolmogorov quotes a letter written on February 20, 1939, from Aleksandrov, at Princeton in the United States, to Kolmogorov in Germany: (They were 43 and 39 years old.) “You have written very little about your sporting activities, but I should like to have a continuous detailed report. . . . Did you swim in the Schwimmhalle? What gymnastics did you do and where? Also you have not written about how you feel. Are you coughing? Are you hoarse? How is your cold? And the main thing, how do you feel in general? It should be a very good idea for you to buy yourself cream as well as milk.”³²

A year after this memoir was published, Kolmogorov was struck by a heavy swinging door and suffered a severe head trauma. As long as he was able, he continued to lecture at the boarding school for talented youngsters that he had founded long before. But the Parkinson’s disease, with which he had been afflicted for some time, became much worse. During his last two years he could neither see nor speak. “He died at eighty-four, speechless, blind, and motionless, but surrounded by his students, who for the preceding couple of years had taken turns providing round-the-clock care at his house.”³³ Kolmogorov’s wife Anna Dmitrievna died only a few months later.

Friends and Colleagues

Many young mathematicians have established friendships and engaged in thoughtful conversations with their professors. Stan Ulam reports:

Beginning with the third year of studies, most of my mathematical work was really started with conversations with Mazur and Banach. . . . I recall (one such event) with Mazur and Banach at the Scottish Café which lasted 17 hours without interruption except for meals. . . . There would be brief spurts of conversation, a few lines would be written on the table, occasional laughter would come from some of the participants, followed by long periods of silence during which we just drank coffee and stared vacantly at each other. . . . These long sessions in the café were probably unique. Collaboration was on a scale and with an intensity I have never seen surpassed, equaled or approximated anywhere except perhaps in Los Alamos during the war years.³⁴

The need to communicate one’s insights and discoveries to a friend is present even in those with a less intense emotional bent. The Hungarian mathematician John von Neumann was known above all for his penetrating intellect. Some say that he approached emotional challenges by applying logic to them. But even “Johnny” sought companionship. As a young man, he found it in Eugene Wigner who recalled their intense conversations during their walks: “He loved to talk mathematics—he went on and on and I drank it in.”³⁵

Von Neumann left Hungary in 1921 for Berlin, Göttingen, and Princeton. His friendship with Wigner continued in Berlin where, being foreigners and not part of the social structure, they became especially close.³⁶ They reconnected in Princeton and remained friends until von Neumann died of cancer in 1957.

In 1936 Ulam was invited to the Institute for Advanced Study in Princeton by von Neumann. There they became friends. Their conversations were not limited to mathematics, they shared jokes and gossip. The two men had had similar upbringings in wealthy, cultured Jewish homes in Central Europe. Ulam’s father was a banker in Lwow, Poland, and von Neumann’s was a banker in Budapest, Hungary. During World War II Ulam and von Neumann worked in Los Alamos, and Ulam supported von Neumann’s vision of the limitless possibilities of computing. From their free play of ideas came some great advances in applied mathematics: the Monte Carlo method, mathematical experiments on a computer, cellular automata, and simulated growth patterns.

In view of his fantastic mathematical power and his ability to mobilize the diverse conceptual and economic resources that led to the first computer, one might think that von Neumann had enormous self-confidence. But Ulam writes of his friend’s self-doubts in his highly informative Adventures of a Mathematician. Gian-Carlo Rota also comments: “Like everyone who works with abstractions, von Neumann needed constant reassurance against deep-seated and recurring self-doubts.”³⁷

Another Hungarian mathematician, Paul Erdős, is legendary for the number and importance of his collaborations. In the 1920s he led a group of young men and women, the self-named “Anonymous Group,” who met weekly to explore questions in discrete mathematics. (We write about this group in chapter 6.) Long after the Anonymous Group broke up, Erdős continued to find collaborators everywhere. He published all together about 1500 papers, second only to the immortal Leonhard Euler. It’s claimed that once, when Erdős had to take a long train ride, he wrote a joint paper with the train conductor!

Figure 5-6. Paul Erdős (right), Hungarian mathematician, with one of his friends, Aryeh Dvoretsky. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

There is a number called the “Erdős number,” and every mathematician has one. If you collaborated directly with Erdős, your Erdős number is 1; if you never collaborated directly with him but did collaborate with one of his collaborators, your Erdős number is 2. And so on. If you’re a well-known mathematician active during Erdős’ career, your Erdős number is almost surely less than 10. And if you don’t have a finite Erdős number, your Erdős number is infinity. Erdős himself, of course, had Erdős number zero.

Erdős’ collaborator often had to fill in details and write up results for publication. Usually Erdős had created the problem. If you were lucky enough to be one of his many mathematical friends, from time to time he would show up at your home, announcing, “My brain is open!” For the next few days, or maybe a week or two, you or your spouse had the privilege of providing him with food, lodging, chauffeuring and long-distance telephone service. Then he would move on. If he stayed longer than you could tolerate, it was all right to ask him to leave.

Once at Stanford University he lived with his friend Gabor Szegő and gave no sign of imminent departure. One night Szegő’s wife met their friend Andras Vázsonyi at a party and said, “Erdős dropped in three weeks ago and he is still staying with us. I am at the end of my wits.” Vázsonyi told her, “No problem. Tell him to get out.” “I couldn’t do that,” she said. “We love him and could not insult him.” “Do what I said,” Vázsonyi insisted. “He will not be insulted at all.” An hour later Erdős came up to Vázsonyi and asked for a ride to a hotel. “What happened?” Vázsonyi asked innocently. “Oh, Mrs. Szegő asked me to move out because I stayed long enough,” he said, totally undisturbed.”³⁸

Erdős had a famous collaboration with the Polish probabilist Mark Kac. Erdős happened to be in the audience at the Institute for Advanced Study in Princeton when Kac gave a talk revealing the amazing fact that the number of prime factors of a random integer is distributed like a bell curve. Unfortunately, Kac was still lacking one difficult estimate to clinch the proof. At that time Erdős knew nothing about probability, so he had been dozing, but he woke up when Kac said “prime divisor.” Before the talk was over, he showed Kac the missing proof.

Something similar happened with the Norwegian number theorist Atle Selberg, but with a less happy ending. Selberg had a conversation with Erdős at a moment when he was in hot pursuit of the long missing “elementary” proof of the prime number theorem (the one that says how primes are distributed asymptotically and logarithmically). Erdős quickly supplied a proof to fill in a difficult gap that Selberg was confronting. Unfortunately, word quickly got around, and in a few days Selberg was told the exciting news that “Erdős and some Scandinavian mathematician” had found the missing elementary proof. Selberg was so offended that a permanent breach was created between them. This was all the more painful to Erdős since he never begrudged sharing ideas or credit.

Erdős is often written about as an odd eccentric. But he was more an object of love than of laughter. It is true that he had no permanent address or job; he carried all his needs in a half-full suitcase. For much of his life, his mother was a constant companion. When he left Hungary for Britain, in his early twenties, it seemed that he had never before had to butter his own bread. But for those who knew him, in his long and extremely productive life, his outstanding characteristic was his kindness and selflessness. He naively and innocently expected others to do much for him; but he was always ready to share whatever he had with anyone who needed or could use his help. When he won a prize, he immediately gave the money away to other mathematicians who needed it more than he.

Gödel and Einstein

Another famous friendship, and one of the most unusual, was between the mathematical logician Kurt Gödel, then in his thirties, and Albert Einstein, the world-renowned physicist, in his seventies. Gödel is famous for his “incompleteness theorem”: Any formal language and system of formal axioms, which are strong enough to generate the natural number system, necessarily will also include an “undecidable” sentence or formula, one that the axioms can neither prove nor disprove.

Gödel and Einstein were often seen walking together in Princeton, deep in German conversation. Although they were very different, they “appreciated each other enormously.”³⁹ Einstein was friendly, frequently joyous, while Gödel was fearful, hard to relate to, and had experienced several episodes of depression. In spite of these differences, the two men wrote enthusiastically about the value of their connection. Gödel wrote to his mother that “there was simply nobody else in the world with whom to talk, at least not in the way I could talk to Einstein.”⁴⁰ Many wondered about this unusual relationship, but to the novelist Rebecca Goldstein it is understandable: “Strange as it might seem for men as celebrated for their contributions, they were intellectual exiles. . . . I believe they were fellow exiles in the deepest sense in which it is possible for a thinker to be an exile.”⁴¹ Einstein was self-exiled by his persistent seeking for a unified field theory for many years, when quantum theory was the center of interest for theoretical physics, and Gödel was self-isolated from virtually all the rest of the world.

Figure 5-7. Albert Einstein and Kurt Gödel, friends at Princeton. Courtesy of The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA. Gift of Dorothy Morgenstern Thomas.

Einstein died first. Gödel’s only remaining regular connection was with his wife Adele. When Adele became ill and they had to be separated, he thought any food not prepared by Adele might be meant to poison him. The fatal consequence was described in the last chapter.

Mathematical Marriages

We will mention two examples of mutual support and love among mathematical married couples. Our first example is the remarkable mathematical partnership of Grace Emily Chisholm (1868–1944) and William Henry Young (1863–1944). Grace and Will left behind a mass of documents and letters that Ivor Grattan-Guinness used to write “A Mathematical Union.”⁴² His article, in which he quotes many of the Youngs’ letters, is the source of this account.

William Young won a mathematics scholarship to Cambridge at age 16. After graduation he spent his spare time and money on sports, particularly rowing. In 1888 he was appointed lecturer in mathematics at the women’s college, Girton. While working there he managed to save 6000 pounds for a future of leisurely travel. “But these plans suffered an abrupt halt, for early in 1896 he fell in love with Grace Emily Chisholm.”⁴³

Grace had passed the Cambridge senior examination in 1885 at the age of 17. Her family encouraged her to devote herself to charitable duties, such as visiting the poorest parts of London where the police dared to venture only in pairs. Nevertheless she studied mathematics and won a prestigious scholarship at Girton College, Cambridge. But she was soon disenchanted. She wrote that at Cambridge it was believed that “mathematics had reached the acme of perfection, with nothing left to do but fill in the details.” Of Professor Arthur Cayley, the famous reigning Cambridge mathematician, she wrote, “Cayley sat, like a figure of Buddha on its pedestal, dead weight on the mathematical school of Cambridge.” Cayley’s lectures were “a flow of words. . . . Polyhedra with vertices constantly springing from triangular faces, like crystals growing in a solution, trees with branches forking in all directions succeeded one another without intermission, twining this way and that round the professorial head, or emerging from under his flapping sleeves as he stood with his back to the listeners chalking and talking at the same time at the blackboard.”⁴⁴

Figure 5-8. Grace Chisholm Young. Source: More Mathematical People: Contemporary Conversations. Eds. Donald J. Albers, Gerald L. Alexanderson, and Constance Reid. Figure 5-9. William Young, British analyst, husband of Grace.

Grace heard about Mr. Young. As a tutor, he had a reputation for cramming his students through examinations, and making his girl students cry. So she decided definitely to go not to him but to Mr. Berry at King’s College. But in her third year, Mr. Berry took a term off and Grace was assigned William Young as her temporary tutor.

She first saw him while sitting on a window seat in a friend’s room. “This dreaded Mr. Young, this great mathematical gun, was a boy, hardly older than herself. The face, shaded by a simple sailor hat with his college colours, was a fine one, with a marble look, due to a pure line of feature and the clear delicacy of skin. . . . The fixed look in the eyes and the slight speaking movement of the mouth made evident that he was working out some mathematical problem as he passed and disappeared under the archway.”⁴⁵

With coaching from Will, Grace graduated with honors from Cambridge in 1892, and then went to Göttingen for graduate study. She wrote home about her professor Felix Klein who was sensitive to the presence of a female student: “. . . Instead of beginning with his usual ‘Gentlemen!’ [he] began ‘Listeners!’ (‘Meine Zuhörer’) with a quaint smile; he forgot once or twice and dropped into ‘Gentlemen’ again, but afterwards he corrected himself with another smile. He has the frankest, pleasantest smile, and his whole face lifts up with it.”⁴⁶

In 1895 Grace obtained her Ph.D. by applying Klein’s group theory to spherical trigonometry. This was a tremendous triumph, for it was the first doctorate awarded to a woman in Germany in any subject whatever. Sonya Kovalevskaya’s doctorate had been given unofficially, as she had not taken courses at Göttingen nor taken a viva voce exam.

Early in February Will proposed to Grace. “When Grace told him that she could neither marry him nor anyone else, he did not hear a word. . . . She soon fell deeply in love with him. She never dared disillusion him about their betrothal.”⁴⁷ But now Will began to think of research for the first time. He believed he would not be able to match the abilities of his talented wife, but at least he might be able to make some contribution of his own.

A year after the wedding, their first child, Frank, arrived, and they decided to move to Göttingen. Grace wrote, “At Cambridge the pursuit of pure learning was impossible. . . . Everything pointed to examination, everything was judged by examination standard. There was no interchange of ideas, there was no encouragement, there was no generosity.”⁴⁸ Her husband, like most of the other fellows, worked hard to support his family by taking work at Girton and Newnham and by presiding at local examinations. Such work allowed him to provide comforts and luxury for his wife, which he considered important. But Chisolm rejected such a view and saw it as undermining the values of an older English society.

At Göttingen, under Klein’s encouragement, Will wrote his first original paper, on problems in geometry. Research was now fully joined for both of them—for him, in his 35th year. After a few years, money needs forced Will to go back to Peterhouse at Cambridge while Grace and their child stayed in Göttingen. This would be their pattern of life for many years, Will traveling to and from the family home, Grace bringing up the family and following her other interests, and both of them working intensely on mathematics.

In 1900 Klein suggested that they read two articles by Arthur Schoenflies on Georg Cantor’s revolutionary theory of infinite sets. This was excellent advice. Set theory, and its application to mathematical analysis, was the field where Grace and Will would work for the next 25 years.

And as their work developed, an extraordinary reversal took place. . . . Will had a profound and original mathematical mind, and, in the field in which he concentrated, one of the finest minds in the world. . . . Grace became his secretary and assistant, perfectly capable of making original contributions of her own but basically needed to see that the flood of ideas that was poured out to her could actually be refined into rigorous theorems and results . . . the success of Will’s late start was due to the support he received from his talented wife. During their twenty-five years of mathematical research, they published between them three books and about 250 papers. When they were apart, their letters constantly discussed mathematical questions: when they were together, their conversation was so dominated.”⁴⁹

In a letter Will justified his frequent publication as sole author of their joint work. “Our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. Mine the laurels now and the knowledge. Yours the knowledge only. Everything under my name now, and later when the loaves and fishes are no more procurable in that way, everything or much under your name. . . . But we must flood the societies with papers. They need not all of them be up to the continental standard, but they must show knowledge which others have not got and they must be numerous.”⁵⁰

By 1904 Will had independently constructed his own theory of integration equivalent to Lebesgue’s, which had been published earlier. The Lebesgue integral is one of the keystones of functional analysis. Young’s approach was significant at the time and was preferred by some later writers. Just before Easter 1907, Will was elected a Fellow of the Royal Society.

In February 1900 their second child, Rosalind Cecilia Hildegard, was born. In December 1901 the third arrived, Janet Dorothea Ernestine, and in September 1903, the fourth, Helen Marian Kinnear. In July 1904 Grace gave birth to Laurence, who would grow up to be a mathematician and a long-time professor at the University of Wisconsin. Continuing the family tradition, Laurence’s daughter Sylvia Wiegand became an algebraist at the University of Nebraska and was president of the Association for Women in Mathematics.

The Young’s last child, Patrick Chisholm, was born in March 1908. That year they moved from Göttingen to Switzerland. In 1908 Will published nearly 20 papers; and in 1910, 22. But his applications for professorships were rejected by Liverpool in 1909, by Durham and Cambridge in 1910, and by Edinburgh and King’s College London in 1912.

In his joint biography Grattan-Guinness writes, “It seems unbelievable that a man who was producing so much profound research could not obtain a post in his own land . . . his career had been unforgivably unconventional: no regular appointments after the days of Cambridge coaching, living abroad during middle life, a sudden burst of original work after years of silence. Will had stepped out of line, and he was not allowed to step back into it again. . . . But Will was not the easiest of people to live with or to know. His letters show his impatience and over-sensitivity, and his desire to impose his view upon others.”⁵¹

Finally, in August 1913, Will obtained the Hardinge Professorship of Mathematics at the University of Calcutta, at 1000 pounds per year plus expenses for a few months’ attendance during the year. But Will wrote to Grace, “The sun is a fierce enemy. . . . What with malaria, smallpox, typhoid and sun stroke one ought to be very heavily bribed if one is to come at all.”⁵²

Now Grace started to write papers under her own name again. She wrote a series of papers on the foundations of the differential calculus, starting auspiciously with a paper in 1914 in the Swedish journal Acta Mathematica and reaching a peak with a long essay that won the Gamble Prize at Cambridge in 1915.

“When Will was at home he completely monopolized Grace’s life and duties. He could not help himself and realized that one of the advantages of his travels was that it would give Grace a period of quiet and undisturbed work.”⁵³

In 1914 the Great War came. Their oldest son Frankie interrupted his engineering course at Lausanne to volunteer for the Royal Flying Corps. On February 14, 1917, at 5000 feet Frank’s plane was attacked by nine German fighters: he was shot through the head and died.⁵⁴

Grace later wrote, “It was Sunday morning, February 18th, when the doorbell rang. Ah! That bell at an unexpected time, there was one thing it might always mean. Too true, there stood the postman with a telegram. I tore it open—War Office. . . . You hundreds of thousands who have gone through what we went through, you will have a vision of those awful hours of agony!”⁵⁵ Grace and Will heard from Émile Picard and Jacques Hadamard, who had each lost two sons; from Emile Borel, who had lost his adopted son; from E.W. Hobson, whose youngest son had had a mental breakdown; and from George Polya, who had lost his brother.

Will continued to work and during 1916 and 1917 published over 20 papers. During the summer he received the de Morgan Medal from the London Mathematical Society, an award made every 3 years for distinguished contributions to mathematics. In 1919 he became professor at the University of Wales at Aberystwyth. He was in his middle fifties. He was later described by Sir Graham Sutton, who came up to Aberystwyth in the autumn of 1920: “He was a tall, vigorous man with the most immense beard I have ever seen. One’s immediate impression was he was bursting with energy. He did all things quickly, was given to forthright speech, but when he wished he could be the most persuasive and charming of men . . . he made mathematics exciting. It was one of the more memorable experiences of my life to sit at the feet of one who knew so much mathematics.”⁵⁶

Will served a term as president of the London Mathematical Society. In his retirement speech in November 1922, he also announced his retirement from active mathematical research. Not only were his own powers declining, but Grace could no longer keep up her role in their partnership. He ended his address with a quote from Prospero’s farewell speech in The Tempest “This rough magic I here abjure . . . I’ll break my staff, bury it certain fathoms in the earth, and deeper than did ever plummet sound, I’ll drown my book.”⁵⁷

For Grace his retirement meant freedom to achieve something substantial of her own. She was troubled by gallstones and much taken up with housekeeping—running the gardens and the vineyards, making jam and wine—yet in 1929 she began The Crown of England, a historical novel in the style of Sir Walter Scott. It required an enormous amount of research from original sources, took 5 years to complete, and was nearly 400 pages long, together with a number of drawings which she prepared herself. Will took the book to several publishers during his visits to London, but none of them accepted it.

In 1929 Will became president of the International Mathematical Union. He had ambitious plans for reforming international mathematical organization, but despite much effort nothing came of it. “Disappointment and disillusion loomed large in his life from this time onwards.”⁵⁸

When World War II broke out, Grace was in England. To return to Will in Switzerland would have required a dangerous sea voyage to Spain, and after that a hazardous crossing through Spain and the unoccupied parts of France. Her health could not take the strain. Will became senile, was moved into a nursing home, and died on July 7, 1942, a few months before his 79th birthday. For Grace, his death was a relief. “That is the solution.”⁵⁹ In March 1944 she was proposed for the rare award of Honorary Fellow of Girton College; but before it could be approved she suffered a heart attack and died on the evening of the 19th, a fortnight after her 76th birthday.

Julia and Raphael Robinson

The marriage between Grace Chisholm and Will Young was controlled and shaped by the relation between the sexes as it was in the early 20th century, before the Great War, as they then called it, and between World Wars. The changes in the relations between men and women in recent decades are illuminated by the story of the marriage of Julia Bowman and Raphael Robinson, as told by Julia’s sister, the author Constance Reid, in a short book, Julia. Constance’s biography of her sister is written in the first person. As Constance was writing it in 1985, Julia was dying of leukemia, at the age of 65. But Julia did hear and approve of everything that Constance wrote.

Julia Robinson became famous for her key role in solving the 10th in the list of 23 problems that David Hilbert proposed to the 1900 International Congress. The 10th problem is: to give a method or algorithm, to determine whether an arbitrary polynomial equation with integer coefficients has an integer solution. (This is called a “Diophantine equation”.) Together with Martin Davis and Hilary Putnam of the United States, Julia Robinson achieved partial results which, when completed by the young Russian Yuri Matyasevich, proved that no such algorithm exists.

Julia’s life as a mathematician was closely bound up with her husband Raphael, a professor of mathematics at Berkeley. As a child, Julia contracted scarlet fever, then rheumatic fever, and was bedridden for nearly two years. Unknown to her at the time, she suffered heart damage that would handicap her for life. These experiences taught her patience (an important trait for mathematicians). In high school she took mathematics classes, where frequently she was the only girl. Then at San Diego State College she wanted to become a mathematician, but the course offerings were limited. (This was before that college became a University.) Not until she became a graduate student at Berkeley did Julia Bowman find the challenge and stimulation she needed.

We quote from Reid’s book as it is written, in the first person of Julia Robinson:

I was very happy, really blissfully happy, at Berkeley. Mine is the story of the ugly duckling. In San Diego there had been no one at all like me. . . . Suddenly at Berkeley, I found that I was really a swan. . . . Without question what had the greatest mathematical impact on me at Berkeley was the one-to-one teaching that I received from Raphael.⁶⁰ During our increasingly frequent walks, he told me about various interesting things in mathematics. He is, in my opinion, a very good teacher. I doubt that I would have become a mathematician if it hadn’t been for Raphael. He has continued to teach me, has encouraged me, and has supported me, in many ways, including financially.⁶¹ . . . I was offered a job as a night clerk in Washington, D C at $1200 a year. My mother thought that I should accept it, but Raphael had other ideas . . . a few weeks after the Japanese attacked Pearl Harbor, Raphael and I were married.”⁶²

At mid-century, nepotism rules were still in effect. A husband and wife couldn’t both be members of the Berkeley mathematics faculty. “Because of the nepotism rule I could not teach in the mathematics department, but this fact did not particularly concern me. Now that I was married, I expected and very much wanted to have a family.”⁶³ Julia did become pregnant, but had a miscarriage. Then she contracted viral pneumonia. Her doctor discovered that she had a serious heart ailment and advised her under no circumstances to become pregnant again. He told her mother that she would probably be dead by forty, since by that time her heart would have broken down completely.

Figure 5-10. Raphael Robinson and Julia Robinson. Courtesy of Dolph Briscoe Center for American History, The University of Texas at Austin. Identifier: di_05556. Title: Rapfael and Julia Robinson. Date: 1978/04. Source: Halmos (Paul) Photograph Collection

“For a long time I was deeply depressed by the fact that we could not have children. Finally Raphael reminded me that there was still mathematics.”⁶⁴ She began to work toward a Ph. D. with Alfred Tarski., the great logician who led the logic group at Berkeley. She did not receive a regular appointment at Berkeley until the announcement of her election to the National Academy of Sciences in 1976. When the university press office received the news, someone there called the mathematics department to find out who Julia Robinson was. “Why, that’s Professor Robinson’s wife.” “Well,” replied the caller, “Professor Robinson’s wife has just been elected to the National Academy of Sciences.”⁶⁵

“The University offered me a full professorship with the duty of teaching one-fourth time—which I accepted.”⁶⁶ In 1982 Julia was nominated for the presidency of the American Mathematical Society. “Raphael thought I should decline and save my energy for mathematics, but I decided that as a woman and a mathematician I had no alterative but to accept. I am planning to take his work as the subject of my Presidential Address at the AMS Meting in New Orleans this winter.”⁶⁷

Friendships Among Women Mathematicians

There are few records available about women and their connections to other women in mathematics. Until the impact of the women’s movement, women mathematicians were usually surrounded by men in their classes and at their jobs. A high percentage of these women were (and are) married to mathematicians, whom they frequently rely on for discussion of their work and for support.

As the participation of women increased in the mathematical profession, more friendships between them were established. In an autobiographical essay written late in her life, Olga Taussky-Todd recalls her childhood in Olmutz, Austria (now in the Czech Republic). As a child, she had strong relationships with young women with intellectual interests, including her sisters. But once she entered the University of Vienna, all her contacts were with male mathematicians. Her account of those years describes her teachers, her research, and her presentations, one of which led to a temporary appointment in Göttingen. Her task there was to edit Hilbert’s work on number theory. It was then that she met Emmy Noether, whom she admired greatly. Noether was one of the most important mathematicians of the 20th century, the principal creator of modern abstract algebra. As a woman, a pacifist, and a Jew, she was not considered fit for a professorship in Germany and remained an unpaid adjunct to the Göttingen faculty through her most important creative years.

Taussky-Todd writes, “I had the good fortune to gain her confidence through an act of concern for her that had seemed very natural to me, and we became good friends. I had been present when one of the top people of the department spoke rather harshly to her. I really did not like this. The next day I told him that this had upset me. . . . He went to apologize to Miss Noether.”⁶⁸

After both of them left Göttingen, they met at Bryn Mawr College and began spending more time together. The appeal of Bryn Mawr was due to the reputation of Anna Pell Wheeler, chair of the mathematics department and a strong advocate for women in the field. In inviting Emmy Noether to her college, Wheeler gave the program a visibility which made this department renowned for decades. By the time Olga Taussky came to the United States in 1934, Noether was ill. But she tried to hide this fact from her colleagues and students. She continued to teach and to travel to Princeton, and Taussky frequently went with her. These trips to Princeton were the highlights of her year at Bryn Mawr.⁶⁹ Her characterization of Noether evokes a complex woman: generous, brilliant, devoted to her friends and students, but also a person with limited interpersonal skills, who needed the kind of support Taussky provided.

Subsequently, Taussky returned to England where she obtained a fellowship at Cambridge University. There she met her future husband, Jack Todd, also a mathematician, with whom she had a deep emotional and intellectual bond. They worked together in applied areas during World War II. In her autobiographical essay, Taussky-Todd recounts an extraordinary number of stimulating and productive interactions with mathematicians throughout her career; but none had the same impact upon her as her friendship with Emmy Noether.

Their relationship was quite rare as illustrated by the findings of the psychologist Ravenna Helson who studied women mathematicians in the 1950s. She interviewed women who were identified by their peers as creative and found that they did not have as many friends and collaborators as their male counterparts. From this finding (which was surprising as women generally report more interpersonal ties than men do), two issues arose. One of these was that women in the 1950s had very few female colleagues. For instance, Vera Pless, a mathematician at the University of Illinois, wrote that she never saw a female peer the entire time she was a student.⁷⁰ While she did know about the work of Emmy Noether (which may have influenced her choice of algebra as her area of specialization), her actual contacts were with men. The second issue is that of time. In Helson’s interviews, “several mentioned being overworked and underpaid. One participant talked about the under-stimulation and lack of time to herself that she experienced at a women’s college; so she moved to a university where she loved the library and the intellectual stimulation but she was not promoted at the same rate as her male colleagues who had published less.”⁷¹

Figure 5-11. Olga Taussky-Todd, Austrian-American algebraist and number theorist. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach

The experience of African-American women mathematicians reflects the complexities of discrimination and mutual support. The theme of minority mathematicians and women runs through a number of chapters of this book. We first mentioned Vivienne Malone-Mayes in chapter 1, and we devote chapter 8 to issues of race.

Malone-Mayes was a graduate student at the University of Texas in the early 1960s. She recalled her experiences in that institution in a panel presentation: “My personal isolation at the University of Texas in Austin was absolute and complete, especially during the summer of 1961. At times I felt that I might as well have been taking a correspondence course. For those who completed degree programs, and for many who quit along the way, the lack of interchange with fellow students was a profound hindrance to academic achievement.”⁷²

Various universities treated women differently. As mentioned in chapter 1, the Courant Institute at New York University had a number of faculty members at midcentury who encouraged and supported women mathematicians.

Joan S. Birman, currently a professor at Columbia University, started graduate studies later in life. While at the Courant Institute, she found that her male and female fellow students were open to interacting with her, and that they were helpful to each other while working on their dissertations. This early experience encouraged her to continue to collaborate actively with colleagues throughout her career. She writes, “I’ve often wondered whether, if the mathematical community welcomed older women as graduate students in a serious and non-patronizing way, and if women rejected the myth that mathematics was a young man’s game, we might see real changes in those discouragingly low numbers.”⁷³

At the University of Michigan in the early 1970s, African-American students were admitted in greater numbers than elsewhere. Janice Brown Walker’s description of her graduate experience was positive. She wrote, “[In fall 1971] I was relieved and excited to see more than six other African-American graduate students there . . . [who] formed a closely knit group that still exists. We were a family. We celebrated successes and shared failures . . . [this group] also formed the core of a mathematical society that was organized as a forum for providing support and information to each other, presenting mathematical talks to each other, and interacting socially. . . . The sheer number of us attending made it easier to develop a group sense of power, courage, and self-esteem. Also, we were warmly accepted and supported by a number of [graduate students and] faculty members.⁷⁴

The power of friendships is in the creation of an environment where new ideas can be explored before they are open to the more rigorous, critical scrutiny of the larger, professional community. The psychologist Howard Gardner writes that creative breakthroughs are sustained by friends, peers, and partners in two ways: emotionally, “in which the creator is buoyed with unconditional support,” and cognitively, “where the supporter seeks to understand, and to provide useful feedback on, the nature of the breakthrough.”⁷⁵

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www.maths.abdn.ac.uk/courses/mx4531/chap_gamma/pdf/ram.pdf