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Mathematics as Solace

When looking at mathematical life, we usually focus on its public face: the institutions in which mathematicians work, their interactions within their communities, their jokes and eccentricities, their prizes and competitions, their breakthrough discoveries. In the following pages we address the more personal consequences of loving mathematics. We ask, “Is mathematics a safe hiding place from the miseries of the world?”

According to Gian-Carlo Rota, “Of all escapes from reality, mathematics is the most successful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same reality we are trying to evade. All other escapes—sex, drugs, hobbies, whatever—are ephemeral by comparison. . . . The mathematician becomes totally committed, a monster, like Nabokov’s chess player who eventually sees all life as subordinate to the game of chess.”¹

While working on this book we were surprised at how many well-known mathematicians have created mathematics while in prison. We found five prisoners of war, in three different wars, plus two political prisoners and one convicted of evading military service. Alongside these, there was one who used mathematics to escape an excruciating toothache, one who was revived to life by a mathematical problem while bedridden and almost 90 years old, a novelist who was distracted by mathematics from his decades-long writer’s block, and an idealistic youngster who was helped to endure the agony of participating in a senseless, brutal bombing war.

Absorption

In its mild form, escape is absorption. You know you’re finally really getting into your problem when you dream about it every night. (No guarantee your dream will give you the solution!) They say Newton sometimes forgot both to eat and to sleep. To many, this is absent-mindedness. They say that Norbert Wiener, when walking down a corridor at MIT with a mathematical paper in his right hand, would come to an open classroom door, walk through the doorway and around the four walls of the classroom and then out again, guiding himself with his left hand against the wall, while still reading.

Blaise Pascal (1623–1662) was one of France’s most illustrious sons, both in mathematics and in literature. He had renounced mathematics and science in favor of ascetic devotion to the Blessed Virgin, but he was still able to turn to mathematics in an emergency. Lying awake one night in 1658 tortured by a toothache, Pascal tried thinking furiously about the cycloid, hoping to take his mind off the excruciating pain. (The cycloid is the curve generated by a fixed point on a circle as the circle rolls along a horizontal track.) He was pleasantly surprised to notice that the pain had stopped. Pascal interpreted this as a sign from heaven: he was not sinning to think about the cycloid rather than about the salvation of his soul! He devoted 8 days to the geometry of the cycloid, solving many of the main problems concerning it.

John Littlewood, the famous collaborator of G. H. Hardy, got a new lease on life at the age of 89. After a bad fall in January 1975, he was taken to a nursing home in Cambridge and had very little interest in life. His colleague, Béla Bollobás, suggested the problem of “determining the best constant in Burkholder’s weak L² inequality (an extension of an inequality he had worked on).” To Bollobás’s immense relief and amazement, Littlewood became interested. Although he had never heard of martingales (which is the subject of the Burkholder inequality) he was eager to learn about them—at his age and in bad health! He was able to leave the nursing home a few weeks later. “From then on,” Bollobás wrote, “he kept up his interest in the weak inequality and worked hard to find suitable constructions to complement an improved upper bound.”²

Figure 3-1. George Polya and John Littlewood. Source: The Polya Picture Album: Encounters of a Mathematician. Ed. G. L. Alexanderson. Boston: Birkhauser, 1985. Pg. 151. Reprinted by kind permission of Springer Science and Business Media.

The American novelist Henry Roth, author of Call It Sleep, lived for many years in a remote village in Maine, in the far northern United States. He was suffering from writer’s block and attempted to help support his family by raising and slaughtering ducks and geese. To survive the Maine winters, he did calculus problems. In fact, he did all the problems in George B. Thomas’ influential calculus text, and later visited Professor Thomas at MIT to tell about this feat.

Freeman Dyson is a famous physicist (who is nowadays also a regular contributor to the New York Review of Books.) In 1943 he was working 60 hours a week as a statistician for the Royal Air Force Bomber Command in the middle of a forest in Buckinghamshire. He remembers it as a long, hard, grim winter. The bomber losses he was analyzing were steadily growing higher, and the end of the war was not in sight. Hardy knew that Dyson was interested in the Rogers-Ramanujan identities,³ so he sent him a paper by W. N. Bailey that contained a new method of deriving identities of the Rogers-Ramanujan type. Dyson wrote, “In the evenings of that winter I kept myself sane by wandering in Ramanujan’s garden, reading the letters I was receiving from Bailey, working through Bailey’s ideas and discovering new Rogers-Ramanujan identities of my own. I found a lot of identities of the sort that Ramanujan would have enjoyed. My favorite was

In the cold, dark evening, while I was scribbling these beautiful identities amid the death and destruction of 1944, I felt close to Ramanujan. He had been scribbling even more beautiful identities amid the death and destruction of 1917.”⁴

Another World War II story is about Olga Taussky-Todd and her husband, the numerical analyst John Todd. They were in London during the Blitz—the intensive bombing of the city by the German air force. The couple took advantage of these raids to get some work done while taking shelter on the ground floor of their apartment building. Todd told an interviewer that “During raids we wrote papers—about six in all—while the other twenty to thirty people chatted, slept, or read.”

Prison Stories

Quite a few well-known mathematicians have served time as prisoners of war, from the Napoleonic War to World War II. At least two have been political prisoners—in the United States and in Uruguay. An impressive amount of beautiful mathematics has in fact been created in prison, where it served to help the imprisoned mathematician survive his ordeal.

Figure 3-2. Jean-Victor Poncelet. Courtesy of Smithsonian Institution Libraries, Washington, DC

A major part of projective geometry was created in prison. In November 1812 Jean-Victor Poncelet (1788–1867), a young officer in the exhausted remnant of Napoleon’s army retreating from Moscow under Marshal Ney, was left for dead on the frozen battlefield of Krasnoi. A Russian search party found him still breathing. In March 1813, after a 5-month march across the frozen plains, he entered prison at Saratov on the banks of the Volga. When “the splendid April sun restored his vitality,” he commenced to reproduce as much as he could of the mathematics he had learned at the École Polytechnique, where he had been inspired by the new descriptive geometry of Monge and the elder Carnot. In September 1814 Poncelet returned to France, carrying with him the material of seven manuscript notebooks written at Saratov. Bell writes that this work “started a tremendous surge forward in projective geometry, modern synthetic geometry, geometry generally, and the geometric interpretation of imaginary numbers that present themselves in geometric manipulations, as ideal elements of space.”

Leopold Vietoris, the Austrian topologist who died in 2002 at the age of 111, served as a mountain guide for the Austro-Hungarian army in World War I while working on his thesis, “To Create a Geometrical Notion of Manifold with Topological Means.” (A “manifold” is a generalization to higher dimensions of two-dimensional smooth curved surfaces such as spheres or cylinders.) Just before the armistice, on November 4, 1918, Vietoris was captured by the Italians. He completed his thesis while a prisoner of war.

Two other mathematicians in the Austro-Hungarian army in World War I were taken prisoner, not by Italians but by Russians. Eduard Helly of Vienna and Tibor Radó of Budapest met in a prison camp near Tobolsk in 1918. Radó had just begun university when he enlisted as a lieutenant and was sent to the Russian front. Helly was already a research mathematician; he had proved the Hahn-Banach theorem in 1912 before either Hahn or Banach. (This theorem is an essential tool in functional analysis. It permits one to extend a linear functional from a subspace to the whole space without increasing its magnitude.) Radó had studied civil engineering at the University of Budapest. In the Russian prisoner-of-war camp, Helly became Radó’s teacher. Radó escaped from the prison camp and made his way north to the arctic regions of Russia. There indigenous arctic dwellers befriended him and gave him hospitality. He slowly trekked westward for thousands of miles, reaching Hungary in 1920. It had been 5 years since he had left the university in Budapest. Helly had shown him the fascination of research in mathematics and so he went to the University of Szeged to study with the Hungarian analyst Frigyes Riesz. He assisted Riesz in his great book on functional analysis and in 1929 migrated to the United States. He founded the graduate program in mathematics at Ohio State University and was a leading authority on the theory of surface measure.

Being a mathematician could be a serious detriment in prison. The French analyst and applied mathematician Jean Leray was a German prisoner of war for 5 years in World War II. If the Germans had known about his competence in fluid dynamics and mechanics, they might have tried to force him to work for them. So he turned his minor interest in topology into a major one and while in prison did research only in topology. In fact, he created sheaf theory, which soon became one of the principal tools in algebraic topology. (We give some details about sheaf theory in the next chapter in writing about Alexandre Grothendieck.) Nevertheless, once he was free, Leray returned to analysis, leaving topology to others.

The South African statistician J. E. Kerrich was visiting Denmark when the Nazis invaded in 1940. The Danes saved the British citizens who were then in their country by agreeing to intern them so that they would not be taken to Germany. Kerrich made use of his time in confinement by tossing a coin 10,000 times and recording the results. He then wrote a little textbook, An Experimental Introduction to the Theory of Probability, based on analyzing the data from his experiment.

The French number theorist André Weil, like his compatriots Poncelet and Leray, had a spectacularly productive time in prison. In the summer of 1939, war with Germany was imminent, and Weil was under orders for military service. “This was a fate that I thought it my duty, or rather my dharma, to avoid,” he wrote in his autobiography. He departed for Finland. By bad luck, the Russians invaded Finland a few months later. “My myopic squint and my obviously foreign clothing called attention to me. The police conducted a search of my apartment. They found several rolls of stenotypewritten paper at the bottom of a closet. . . . There was also a letter in Russian, from Pontryagin.” After 3 days in prison, he was unexpectedly released at the Swedish border.⁵ Shipped back to France by way of Sweden and Scotland, Weil spent 3 months in jail in Rouen. His friend Henri Cartan wrote to him, “We’re not all lucky enough to sit and work undisturbed like you.” On April 7, 1940, he wrote to his wife Eveline, “My mathematics work is proceeding beyond my wildest hopes, and I am even a bit worried—if it’s only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?” On April 22, he wrote her, “My mathematical fevers have abated . . . before I can go any further it is incumbent upon me to work out the details of my proofs. . . .” On May 3, 1940, he was sentenced to 5 years in prison, which was immediately commuted if he agreed to serve in combat. On June 17, 1940, “the command came to abandon our machine guns and join our regiment on the beach. We were boarded on a small steamship . . . the next morning we were in Plymouth.”⁶ Weil eventually reached the United States to continue his illustrious career.

Mathematics and Politics

Chandler Davis, the editor of The Mathematical Intelligencer, was a schoolmate of one of the authors, a math grad student at Harvard when R. H. was an undergraduate English major. During the 1950s McCarthyite red scare, Davis’s career was interrupted when he refused to answer questions asked by the U.S. House of Representatives Committee on Un-American Activities. He proudly referred to his revolutionary ancestry—in the American Revolution—and refused to cooperate in proceedings that violated the First Amendment to the U.S. Constitution guaranteeing freedom of speech. Davis was fired by the University of Michigan from his job as assistant professor of mathematics. He was convicted of contempt of Congress, and after exhausting appeals he was confined for 6 months in the federal penitentiary in Danbury, Connecticut. Then, when he was released, he was totally blacklisted by universities in the United States. The great Canadian geometer Donald Coxeter invited him to apply to the University of Toronto. At first the government refused Davis entry, but after a letter-writing campaign they relented. He moved to Canada to teach at the University of Toronto. A 1994 special issue of Linear Algebra and Its Applications celebrating his contributions to matrix theory describes his time in prison: “Throughout this ordeal, Chandler maintained his research interest in mathematics. He also maintained his sense of humor. A footnote in his paper on an extremal problem, conceived while he was in prison but published afterward, reads: ‘Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author’s and are not necessarily those of the Bureau of Prisons.’ ” Davis says that this “elegant phrasing” was actually suggested to him by Peter Lax, a long-term member of the Courant Institute, whom he thanks.

Figure 3-3. Chandler Davis, mathematician at the University of Toronto. Courtesy of Sylvia Wiegand.

Let it be understood that not every imprisoned mathematician fared as well as André Weil and Chandler Davis. The Uruguayan analyst José Luis Massera writes that he was drawn into political activity under the influence of refugee mathematicians from Fascism, Luis Santalo from Spain and Beppo Levi from Italy. Massera has left us a detailed account, both of his career before prison and of his time in prison. “That epoch, around the great movement of solidarity with the Spanish people, I began the political activity which I shared with mathematics for the rest of my life.”⁷ In 1943 he became a Communist.

After completing his degrees in Montevideo, Uruguay, Massera won a Rockefeller grant to go to Stanford where he worked with Polya and Szegö. But then he became more interested in differential equations (equations involving the rate of change of the unknown function), so he transferred to the East Coast, where he commuted between New York and Princeton, working simultaneously with Richard Courant on minimal surfaces and with Solomon Lefschetz on topological methods for ordinary differential equations. In 1966, after returning to Uruguay, he published his well-known book Linear Differential Equations and Function Spaces with his student J. J. Schaffer. He was also elected as a Communist Deputy to the Parliament of Uruguay. When the military dictatorship outlawed the party in 1973, he became its underground director.

He was imprisoned in October of 1975. On the first day, while he was standing at attention with hands and ankles tied, a soldier took him by the shoulder and shoved him. He fell and fractured his right leg. “Despite which,” he later wrote, “the drill continued until they were convinced that I couldn’t stand. They laid me on a wire cot where I remained for a month without help.” He was finally taken to a military hospital where he was X-rayed and given a cane. “My own body was required to mend the fracture.” (In fact, for the rest of his life Massera walked with one leg shorter than the other.)

He remained imprisoned for 9½ years in a prison known as the Penal de Libertad; the ironic name was due to its location on the outskirts of a town of that name. The cells of the prison housed two inmates each. They were allowed 1 hour a day of recreation.

Figure 3-4. José Luis Massera of Uruguay and Lee Lorch of Canada, mathematicians and fighters for human justice. Soruce: Archivo General de la Universidad de la República (Uruguay), Subfondo Archivos Personales, José Luis Massera, Box 3 Folder 3B.

Then we could engage in sports, and those who didn’t, like in my case, walked conversing with another prisoner. Human relationships were almost exclusively limited to the cell. . . . In it one could read books of the good library that had been formed with donations that families of prisoners had made when it was permitted. Of course, political books were excluded . . . as well as mathematics books. Who knows what mysterious messages could be conveyed by those odd and incomprehensible symbols?

My cellmates were various including Communists and Tupamaros, with whom I conversed freely on the most varied topics. [Tupamaros were a movement of urban guerrillas, a tactic rejected by the Communists.] A paper factory worker, a Communist, was with me for years and we became great friends; he was very intelligent and restless, we talked on the most diverse themes. I could give him little courses on physics, chemistry, etc., which he absorbed with passion. In other cells there were prisoners who were young mathematicians like Markarian and Accinelli, whom I saw only during the recreations and collective tasks; running some risks we produced some small mathematical works like one entitled “Is it true that two plus two is always four?” which might interest and intrigue the non-mathematician prisoners.

During all this, Martha, my wife had also been imprisoned, she was tortured and interned in a women’s jail, in what was formerly a monastery. She was there for three years until some time during the year 1979; she was able to recover our apartment, which had been occupied and sacked by the military.

We can add to Massera’s own memoir a note called “Recuerdos” (Memories), part of an article written by Elvio Accinelli and Roberto Markarian, mathematicians who shared with Massera more than 3 years of imprisonment.

Written in secret, with tiny handwriting, manuscripts were carried from cell to cell by the prison inmate who delivered bread or tools, who risked with this audacity being punished and sent to the “Isla” [isolation].⁹ Those little papers circulated in open defiance and Massera wrote about dialectic, logic and mathematics, making real our affirmation that science and culture cannot be destroyed. At that time and place, to think was entirely prohibited. To demonstrate by some means what was thought, was an act of defiance and bravery, beyond the intrinsic value that the written or demonstrated material might have.

And in those conditions we conquered a space to think and discuss. . . . It was as if, despite everything, in the interstices of repression one lived freely.

One day, a book on Hilbert’s Problems, which had been sent and dedicated to Massera by Lipman Bers (then President of the American Mathematical Society), came to be seen by some prison inmates. Who knows how much pressure on the part of many international organizations allowed Massera to receive such a magnificent present in his cell.¹⁰ And who knows how much compassion there was in the prison guard that permitted that such a book be found in the prison wing.

To be a Communist was dangerous. And in that place it was also dangerous to be a mathematician. . . . Life and mathematics kept those of us who were the protagonists of this story united. Massera was for us a teacher, beyond the strictly scientific arena. And a friend with whom we shared with pride many joyous moments as well as other kinds.

Today we scarcely write or speak of these things. Nevertheless we remember those years, that some consider “empty,” often with a smile on our faces. But we forget neither the pain nor the learning of life that we experienced. It is strange to say, but those years of imprisonment and isolation had positive aspects for us. We wouldn’t be who we are, including professionally, without the “stain” of that period.

An international campaign of protest on Massera’s behalf was carried on for years, led by Laurent Schwartz in France, by Lipman Bers and Chandler Davis in the United States, and by Lee Lorch and Israel Halperin in Canada. On March 3, 1984, Massera was set free. “For months, my house was invaded by hundreds of friends who came to greet me.”

My Thoughts Are Free

Mathematicians are not the only prisoners who found in their profession comfort and solace. In a fascinating collection entitled The Great Prisoners, Isidore Abramowitz collected the letters and testimonies of men and women starting with Socrates and ending with prisoners from the 20th century. Many of these documents present deeply felt justifications for the writers, philosophers, and politicians who have been unjustly incarcerated. Others, such as Fyodor Dostoevsky, write of their experience while in jail to members of their families or the public. The similarity between the mathematicians and the men and women included in this volume is that in all these groups, when the prisoners had access to pencil and paper, they could find ways to escape from their dire circumstances.

The English writer Oscar Wilde had courted sensationalism in his writings and behavior but paid a wrenching price for his homosexuality. When accused of seduction by the father of one of his lovers, he was brought to trial, lost the case, and spent two very difficult years in prison. In a letter to a friend he wrote:

I need not remind you that mere expression is to an artist the supreme and only mode of life. It is by utterance that we live. Of the many, many things for which I have to thank the Governor there is none for which I am more grateful than for his permission to write fully and at as great a length as I desire. For nearly two years I had within a growing burden of bitterness, of much of which I have now got rid. On the other side of the prison wall there are some poor black soot-besmirched trees that are just breaking out into buds of an almost shrill green. I know quite well what they are going through. They are finding expression.¹¹

These diverse prisoners share the ability to sustain their intense inner life even under the most horrendous circumstances. When circumstances allow them to draw upon their passion and knowledge, it can become a means for survival.

Let’s end with an uplifting story. In his youth Pál Turán, a member of Erdős’s Anonymous Group (chapter 6) was imprisoned in a Fascist labor camp, from which he maintained a mathematical correspondence with Erdős. The Hungarian labor forces into which Turán was drafted were formed to support the army’s operations. They consisted of people considered too untrustworthy to be given arms in the regular army—political opponents, Gypsies, and Jews. The young men who made up these forces were unarmed, and they served under regular army officers. They were ordered to clear railway lines and build staging areas close to combat zones. When war came, they were helpless when attacked because they had no weapons. They perished in great numbers. For Turán, it was a time of great pain, but he continued to work, using mathematical problem solving as his refuge.

Figure 3-5. Pál Turán and Vera Sós, a Hungarian couple. He was a number theorist, she is a combinatorialist. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

In September 1940 I was called for the first time to serve in a labor camp. We were taken to Transylvania to work on building railways. Our main work was carrying railroad ties. It was not very difficult work, but any spectator would have recognized that most of us did it rather awkwardly. I was no exception. Once, one of my more expert comrades said so explicitly even mentioning my name. An officer was standing nearby, watching us work. When he heard my name, he asked the comrade whether I was a mathematician. It turned out that the officer, Joseph Winkler, was an engineer. In his youth he had placed in a mathematical competition; in civilian life he was a proofreader at the print shop where the periodical of the Third Class of the Academy (Mathematical and Natural Sciences) was printed. There he had seen some of my manuscripts. All he could do for me was to assign me to a wood-yard where big logs for railroad buildings were stored and sorted by thickness. My task was to show incoming groups where to find logs of a desired size. This was not so bad. I was walking outside all day long, in the nice scenery and the unpolluted air. The [mathematical] problems I had worked on in August came back to my mind, but I could not use paper to check my ideas. Then the formal extremal problem occurred to me, and I immediately felt that this was the problem appropriate to my circumstances. I cannot properly describe my feelings during the next few days. The pleasure of dealing with a quite unusual type of problem, the beauty of it, the gradual approach of the solution, and finally the complete solution made these days really ecstatic. The feeling of intellectual freedom and of being, to a certain extent, spiritually free of oppression only added to this ecstasy.¹²

After Turán’s cry of ecstasy, any more words may seem an anticlimax. But we will conclude by pointing out the advantage imprisoned mathematicians have over other scientists. Poncelet needed only pencil, paper, and his memories of the Polytechnique to lose himself in projective geometry. Turán, without even pencil or paper, was able to re-create his world of combinatorial identities and estimates. According to Vladimir Arnold, mathematics is just “the part of physics where experiments are cheap.” No need for a lab or even for a library, just your mind and its contents!

Bibliography

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