The Artist and the Mathematician


By Amir D. Aczel

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Nicolas Bourbaki, whose mathematical publications began to appear in the late 1930s and continued to be published through most of the twentieth century, was a direct product as well as a major force behind an important revolution that took place in the early decades of the twentieth century that completely changed Western culture. Pure mathematics, the area of Bourbaki’s work, seems on the surface to be an abstract field of human study with no direct connection with the real world. In reality, however, it is closely intertwined with the general culture that surrounds it. Major developments in mathematics have often followed important trends in popular culture; developments in mathematics have acted as harbingers of change in the surrounding human culture. The seeds of change, the beginnings of the revolution that swept the Western world in the early decades of the twentieth century — both in mathematics and in other areas — were sown late in the previous century. This is the story both of Bourbaki and the world that created him in that time. It is the story of an elaborate intellectual joke — because Bourbaki, one of the foremost mathematicians of his day — never existed.


Le peintre et son modèle (1920) by André L'Hote demonstrates the relationship between cubism and abstract mathematical ideas of transformations. (CENTRE POMPIDOU, PARIS)

Also by Amir D. Aczel
Descartes's Secret Notebook
The Mystery of the Aleph
The Riddle of the Compass
God's Equation
Fermat's Last Theorem

For Miriam, with much love

IN AUGUST 1991, Alexandre Grothendieck, widely viewed as the most visionary mathematician of the twentieth century, a man with insight so deep and a mind so penetrating that he has often been compared with Albert Einstein, suddenly burned 25,000 pages of his original mathematical writings. Then, without telling a soul, he left his house and disappeared into the Pyrenees.
Twice during the mid-1990s Grothendieck briefly met with a couple of mathematicians who had discovered his hiding place high in these rugged and heavily wooded mountains separating France from Spain. But soon he severed even these new ties with the outside world and disappeared again into the wilderness. And for ten years now, no one has reported seeing him. His mail keeps piling up uncollected at the mathematics department of the University of Montpellier in southern France, the last academic institution with which he had been associated. The few individuals whom he had once trusted to forward him the select pieces of mail he did want to receive no longer have any way of making contact with him. His children have not heard from him in many years, and two of his relatives who live in southwest France—not far from the Pyrenees—and with whom he had had limited, sporadic contact, have not had a word from him in years. They do not even know whether he is still alive. It seems as if Alexandre Grothendieck has simply vanished off the face of the earth.
During his most active period as a mathematician, from the 1950s to around 1970, a period when he completely reworked important areas of modern mathematics, lectured extensively on his pathbreaking research, organized leading seminars, and interacted with the most important mathematicians from around the world, Alexandre Grothendieck had been closely associated with the work of Nicolas Bourbaki. And some have surmised that Grothendieck's inexplicable disappearance into the Pyrenees was somehow connected with his relationship with Bourbaki.
Nicolas Bourbaki was the greatest mathematician of the twentieth century. Since his appearance on the world stage in the 1930s, and until his declining years as the century drew to a close, Bourbaki has changed the way we think about mathematics and, through it, about the world around us. Nicolas Bourbaki is responsible for the emergence of the "New Math" that swept through American education in the middle of the century as well as the educational systems of other nations; he is credited with the introduction of rigor into mathematics; and he was the originator for the modern concept of a mathematical proof. Furthermore, the many volumes of Bourbaki's published treatise on "the elements of mathematics" form a towering foundation for much of the modern mathematics we do today. It can be said that no working mathematician in the world today is free of the influence of the seminal work of Nicolas Bourbaki.
But what was the nature of the relationship between Bourbaki and Alexandre Grothendieck, and who is Nicolas Bourbaki?
ALEXANDRE GROTHENDIECK WAS born in Berlin on March 28, 1928, to Alexander (Sacha) Shapiro and Johanna (Hanka) Grothendieck. Both parents were ardent and very active anarchists.
Sacha Shapiro was born on October 11, 1889, to a religious Jewish family belonging to the Hassidic movement, in the town of Novozybkov in the region where the three borders of Ukraine, Belarus, and Russia all meet. At times, the town has been Russian, at others, Ukrainian.1
Shapiro forsook the Hassidic lifestyle of his parents and was swept with the revolutionary movements of the time. He participated in several uprisings against the Russian tsar, the first one being the abortive 1905 revolution. As a member of an anarchist movement, he took the nom de guerre Sacha Piotr. He was first arrested when he was only sixteen years old, when the attempted coup against the tsar failed. Following this arrest, Shapiro was deported to Siberia and spent ten years in jail. Released in 1917, he found himself taking part in the Russian revolution that same year. Again he was arrested, and at some point was condemned to death. He escaped, was recaptured, and ran away again. During his last escape from the Russian police he lost an arm. He then fled from Russia, assuming the false name Alexander Taranoff, and would remain a stateless person for the rest of his life. Soon after his release from prison, before taking flight, he met and married a Jewish woman named Rachel, and with her had a son named Dodek.
But their time together did not last long and, with the aid of woman named Lia, he slipped across the Russian border into Poland. He and Lia then drifted through France, Belgium, and Germany, illegally crossing national borders with unusual agility, and finally arrived in Paris, where they lived together for two years; but Sacha had a string of other relationships with women. Through his anarchist activities he met important figures of world revolutionary movements, including the Americans Alexander Berkman (1870-1936) and Emma Goldman (1869-1940).2 He moved to Berlin and there, through his involvement with anarchist groups, met Hanka Grothendieck.
Johanna (Hanka) Grothendieck was born in Hamburg on February 23, 1900, to Albert and Anna Grothendieck. Albert had a shoeshine business in town, which at the height of his career he owned; otherwise, he worked there as an employee. Hanka was close to her father, who encouraged her to pursue her gifts of acting and writing.
Soon, the girl rebelled against her parents and her lower-middle-class upbringing and took up with artists and intellectuals. She hoped to become a writer, and early on in her life she worked as an editor and writer for small papers, and also studied to become an actress. She met and worked for German impressionist painters before the urge to travel swept her away from Hamburg and took her to the center of German intellectual life: Berlin.3 Here she came in contact with artists such as Paul Klee, from whom she bought a small drawing, continued her studies of acting, and became acquainted with members of revolutionary social movements that were active in the German capital at that time.4
Taken with social causes, Hanka accepted a position writing for the newspaper Der Pranger, which focused its articles on the plight of the disenfranchised. She wrote articles on prostitution and took up the cause of sexually exploited women and girls. Through her involvement with revolutionary causes, she had met a man named Alf Raddatz, and the two were soon married and had a daughter, Maidi. Raddatz could never hold a job, was often away, and did not live with Hanka.
It was during this period of intermittent separation from her husband that she met Sacha Shapiro. He worked as a street photographer, which proved difficult as he had only one arm, but he did it well. The two moved in together. Their son, Alexandre, was born in Berlin on March 28, 1928. Since they were not married, and the mother's married name was Raddatz, Alexandre Grothendieck's name was registered as Alexander Raddatz. Only later, after his mother's divorce from Alf Raddatz, would his name be changed to Grothendieck. 5 Years later, in France, the spelling of his first name would be changed to Alexandre.
For five years, Alexandre Grothendieck lived in Berlin with his parents and his half sister, Maidi. The parents were consumed with their mutual love affair with revolution. Both were angry with authority, which they saw as the source of all evil and misery in the world, and strove to rid society of the ills of government. They truly believed that anarchist movements could bring about redemption from tyranny and freedom for the masses. Poor themselves, they lived on Brunnenstrasse, in a section of the city that was populated by indigents, Jews, and foreigners.6 Sacha's work as a street photographer brought little income, and Hanka's infrequent jobs writing for the newspaper helped little. Sacha was often away on anarchist missions to France and other countries. Soon, even the little stability the family had enjoyed would be gone.
WITH THE RISE of Nazism in 1933, the situation for the family worsened significantly. Sacha, always on lists of wanted persons and living under a false name and with forged papers, had to escape from Germany. In the middle of the night, without saying good-bye to Alexandre and his sister, Sacha left the family home, crossed the French border, and traveled to Paris. A few months later, when Hanka decided to join him there, Maidi was placed in an institution for handicapped children in Berlin, even though she was not handicapped.
Hanka wanted to place five-year-old Alexandre in a foster home. Some years earlier, she had met an unusual man, a Lutheran pastor named Wilhelm Heydorn, and she knew that he and his wife often took in foster children for a small monthly fee. Hanka wrote a letter to Dagmar Heydorn, Wilhelm's wife. She described her desperate situation, explaining that her "husband," who had only one arm, was a stateless Russian working in Paris as a street photographer and that she was badly needed there to help him earn a living. They were both writing a very important book, she wrote, and this book could not be completed unless she joined him. She said she knew that the Heydorns accepted a small number of foster children, and that she was ready to pay one hundred marks—and added that she could not afford more.
Dagmar's response was encouraging, so in early May 1939, Hanka came to the Heydorn home, bringing the boy the Heydorns would later name "the little Russian." She immediately confessed to Dagmar that she had no money at all and could not pay the hundred marks she had promised, and that there was no great book she and her husband were writing. She merely needed to help her one-armed man earn a living.
Dagmar felt sorry for the boy, who would be facing a very uncertain future if she declined to take him in. She consulted with her husband. Wilhelm Heydorn was a unique individual: he had been a Lutheran priest, an army officer, an elementary school teacher, and a practitioner of alternative medicine. For several years, he had studied to become a doctor, before giving it up for the priesthood. He was an intellectual and the founder of an outlawed political party opposed to the Nazis. Within a few years he would go underground himself, forced to hide from the authorities and live using false papers. Wilhelm agreed immediately with his wife and before they knew it Hanka was gone and little Alexandre was in their care.
For five years young Alexandre was raised by the Heydorns, who had four children of their own and had taken in several foster children whose parents had escaped following the rise of Hitler. From Alexandre's autobiographical notes, we know that the boy was unhappy and deeply missed his parents. He had very little news from his mother during this period, and heard nothing at all from his father. None of his mother's relatives in Hamburg ever came to visit him.
The Heydorn household had strict rules, which was something Alexandre was not used to, having been raised by anarchist parents who did not impose any limitations on his freedom. This made his stay in the foster home even more difficult. Still, he was always appreciative of all that the Heydorns did for him, and always very patient and polite.
Meanwhile, in 1936, Alexandre's parents, ever politically active, went to Spain to take part in the Spanish Civil War. This war was the great hope of all revolutionaries and anarchists living in the continent of Europe. Thousands of anarchists flocked into Spain, coming from as far away as Italy and beyond. They joined the Republican forces fighting the Fascists. But General Franco's fascists were better organized, better trained, and better armed.
The Fascists won the war, and soon the many thousands of foreign soldiers and Republican troops, as well as Spanish anarchists, were crossing the Pyrenees back into France, defeated and demoralized. Sacha and Hanka came back to France disillusioned and depressed. In France, like the thousands of refugees from the Spanish Civil War, the two were viewed as dangerous foreigners and remained under surveillance by the authorities. Sacha resumed his work as a street photographer in Paris, and Hanka found a job teaching young children in the southern city of Nîmes.
In 1939, just before the beginning of the Second World War, the Heydorns came to believe that they no longer could keep a child who looked Jewish, for this was now too dangerous for them. They therefore tried to contact Sacha and Hanka in France. Since their whereabouts were unknown, this was a difficult task; but through the French consulate in Hamburg a message was finally relayed to the parents that they needed to collect their son. Eleven-year-old Alexandre was put alone on a train in Hamburg and arrived in Paris in May 1939, finally to be reunited with his father.7 Hanka came back from Nîmes and joined the two of them in Paris.
Their time together, however, was to be short. Soon Sacha Shapiro was arrested and sent to the worst French internment camp, Le Vernet, in the Pyrenees. This was a camp in which about 2,000 men were kept in hideous conditions. The inmates were refugees from the Spanish Civil War, revolutionaries, Jews, and other "undesirable foreigners." Life in this camp was extremely difficult, the sanitation appalling, and the inmates constantly hungry. In 1942, the French collaborationist authorities began to deport Jewish inmates to Auschwitz. Sacha was among the first to be deported, and he died in Auschwitz in 1942. It is chilling to read the following matter-of-fact sentence in a report by a French officer to his collaborationist French commanders: "I have the honor to make it known to you that Monsieur Taranoff ... was deported on 14 August 1942 in the direction of the concentration camp of Auschwitz."8
In 1940, Hanka and her son, Alexandre, were placed in the French internment camp of Rieucros, located in a flat agricultural region in southern France—a place that gets hot and unpleasant in summer and freezing cold in winter. Conditions in this small camp, however, were better than in other places of confinement in France, and the boy was able to attend school in the nearby town of Mende. Rieucros was a women's camp, and some of the women had children with them. According to Grothendieck's memoirs, he was the oldest of these children, and the only one to study at the lycée in Mende.9
Grothendieck at the Camp of Rieucros. (CREDIT: ALEXANDRE
Reports by survivors indicate that life at Rieucros was very difficult, and that hunger and deprivations were suffered daily by the poor women who ended up here. At this, or at another camp to which she was later to be sent, Hanka contracted tuberculosis and would eventually die from this disease. But Grothendieck's memoirs do not say much about these hardships. His writings focus on his studies at the local school and the teachers, those he liked and those he cared less about. And he writes about his fascination with words and poetry, and about the magic he found in numbers.10 Alexandre learned how to rhyme from another boy at the camp, and soon all his sentences rhymed. This was a fun game to play for many hours every day.
Another friend taught him about the existence of negative numbers and games one could play with them. Then he learned how to create and solve crossword puzzles, and this game kept him occupied for days on end in the confinement of the camp. Mostly, he would spend his time alone, and as an adult he would appreciate the gift he received in the camps—the ability to spend time in complete solitude. His lonely hours would teach him how to create thoughts and derive ideas without interacting with a soul.
But clearly life was extremely difficult for the boy and his mother. At the camp were interned "dangerous foreigners": German Jews, Spanish Anarchists, and Trotskyites.11 As speakers of German, the Grothendiecks were shunned and harassed by many of the others and certainly by French people in the surrounding villages, since—paradoxically—they were viewed as the enemy rather than as victims of the Germans.
Alexandre grew up in a harsh, confrontational environment in which he was often physically attacked. In order to survive, he developed into a strong fighter and would retain and develop his boxing skills for the rest of his life. His anger at the world he knew escalated to the point that he ran away from the camp with the intention of assassinating Hitler. Fortunately for himself and his mother—since he had no chance of ever completing such a task—he was caught and returned to the camp.
Grothendieck remembers, however, that as the oldest boy in the camp and the only one to attend a high school at a village four or five kilometers away, he had the ability to leave and reenter the camp almost at will.12 He recalls that he was a good student, but not exceptional. He was already in the habit he would follow throughout his life: of concentrating only on what caught his fancy and completely ignoring the rest. Alexandre did not care what his teachers thought about him. If a topic interested him, he would spend hours on end on it; and if it didn't, he cared about it not at all.
He still remembers, however, the first bad grade he received in mathematics—a field that would become his passion and his career. A teacher had asked him and his schoolmates to prove the "three cases of equality of triangles." Grothendieck's proof was different from the one in the textbook, so the teacher marked him down, even though his proof was "every bit as correct as the one in the book."13 The teacher, apparently, had such low confidence in his own mathematical abilities that he could not recognize the value of Grothendieck's alternative proof of the theorem. He had to "report to an authority," and that authority was the textbook, Grothendieck lamented.
Vichy France was a difficult place in which to survive for a boy of Jewish origins. And things got worse when the Rieucros camp was closed in early 1942 and its 320 inmates were moved to the concentration camp at Brens, which had stricter rules and even fewer freedoms than Hanka and her son had enjoyed before.
Hanka had heard rumors about a school, supported by a Swiss charity, in the French town of Le Chambon-sur-Lignon. Here Jewish students could study and were hidden from the Nazis. She managed to send Alexandre to this school.14 Sometime after her separation from her child, Hanka was moved yet again, this time to the camp at Gurs, where she would be kept prisoner until the end of the war.
The following is a short description of Alexandre Grothendieck written by the woman who ran one of the camps at which Alexandre was interned, the camp of La Guespy. It appears to have been written shortly after the end of World War II. M. Steckler in the description below was the camp's surveyor. He used to spend many hours in this camp "ferociously" playing chess with the boy.15
Dit Alex le Poète
Allemande, russe?
Mère au camp de Gurs
Enfant très intelligent, toujours plongé dans ses réflex-
ions, ses lectures, écrivant
Très bon joueur d'échecs—parties acharnées avec M.
Réclame le silence pour écouter la musique
Sinon enfant tapageur, nerveux, brusque
(Alexandre Grothendieck
called Alex the Poet
German, Russian?
Mother at the camp at Gurs
Very intelligent child, always plunged deep in his
thoughts, his reading, a writer
Very good chess player—chess matches set against
M. Steckler
Demands silence for listening to music
Otherwise, a loud, brusque, nervous child.)
The mountain resort town of Le Chambon-sur-Lignon, in a wooded area at an elevation of 3,000 feet in central France south of Saint Etienne, had been transformed into a strong-hold of the French resistance and became a haven for the few Jews and other persecuted people who could find their way here. Virtually the entire population of this town actively hid Jews from their Nazi persecutors. The force behind this amazing countercurrent in the generally anti-Semitic and antiforeign atmosphere of wartime France was one man: the Protestant pastor André Trocmé (1901-1971).
Trocmé was a Huguenot, and was born in northern France near the Belgian border. He studied for a time in the United States, and in New York met Magda Grilli, of Italian and Russian origins, who would become his wife. When he was sent to be the parish priest of Le Chambon, he began to preach tolerance, and his teachings were well received by the local Protestant population, which had been attuned to rebellion against authority ever since the revocation of the Edict of Nantes in 1684.16 These people had a history of siding with the persecuted, and saving those the authorities were hunting down for deportation and execution. The people of Le Chambon-sur-Lignon literally risked their own lives daily in order to save the Jews living and hiding among them.
But life continued to be dangerous, and there were frequent roundups of Jews by the Nazis. Grothendieck later described how he often had to escape into the woods every time the Nazis were approaching, and hide out for several days at a time, with little or no food or water.17
Alexandre attended the local school, the Collège Cévenol, from which he eventually earned his baccalauréat, which would entitle him to enroll at a French university. His education in Le Chambon, and earlier at Mende, was spotty at best and lacked both continuity and depth. But the boy had a strong ambition, and he had a special affinity for mathematics. In addition to the fact that the teaching at these small schools in wartime France was not good, the textbooks were inadequate.
The young student found the problems in the mathematics textbooks so repetitive and trite that he stopped using them. What bothered him the most, however, was the fact that the problems in the books appeared as if out of thin air—with no reason or motivation behind them. He felt that these problems did not illuminate the material but rather were arbitrary and senseless. He therefore made up his own problems, and then spent many hours solving a problem that interested him, ignoring everything else.
What most concerned Grothendieck was the fact that none of his mathematics texts at the Collège Cévenol gave a good definition of length, area, and volume. Thus, at a high school in a village near a concentration camp, a young boy was concerned with mathematical problems that were far above the level and the place and that were, indeed, not "pulled out of thin air" as the problems in the textbooks seemed, but had an important grounding in the real world. The boy Alexandre Grothendieck was interested in the theory of measure , even though he could not have known it by that name at this time.
Grothendieck wanted to be able to find the length of a curve, the area of a triangle with given sides, and the volume of a regular solid with a given edge.18 The problems of measure theory would continue to occupy his mind after the war, when he was a student at a university. He would re-derive on his own the theory of measure, which, unbeknownst to him, had been developed a few decades earlier.
The Pyrenees about twenty kilometers south of
Grothendieck's hideout. (AMIR D. ACZEL)
WHILE ALEXANDRE AND his mother were doing everything they could in order to survive in the camps of southern France, a number of French mathematicians were also going through the upheavals of the war, in other parts of the country. The war had completely disrupted the state of French mathematics, just when it was beginning to make progress—as mathematicians started to reorganize following the disastrous outcome of the First World War, which had decimated half the graduating classes of French universities of the years 1910 to 1916.19
Some of these mathematicians, who had begun to rebuild French mathematics just a few short years before the outbreak of the Second World War, were now refugees themselves. As professors they had been fired from their positions, or for other reasons found themselves without work and with the need to hide from the authorities. French mathematics revived for a short time during the period between the two wars, and then was fatally disrupted by the onset of the Second World War. French mathematicians were now struggling to survive as the world entered its most insane period in history.

ON THE EVE of the outbreak of World War II in the summer of 1939, as fear and chaos were sweeping Europe, French mathematician André Weil was spending a peaceful and serene time with his wife, Eveline, in the bucolic setting of a villa on the island of Lökö on the Gulf of Finland. The Weils were the guests of Lars Ahlfors, the renowned Finnish mathematician, and his wife, Erna. Years later, Weil described their stay in his memoirs as follows:1
Our visit with the Ahlfors was a time of unadulterated serenity. . . . Our small island was easily explored: besides our villa, there was only a small farm with four or five cows, some distance away. It was the season of white nights, close to midsummer night's eve. The air was invariably pure and clear, transparent beyond words.
Their days were spent walking on the island, going on long outings by boat to explore nearby islands, and gathering at ten or eleven in the evening in the dining room of the villa for tea and sandwiches by the light of the setting sun. As Weil described it, "We felt we were somewhere outside of time."2


On Sale
Apr 29, 2009
Page Count
256 pages
Basic Books

Amir D. Aczel

About the Author

Amir D. Aczel, a visiting scholar in the history of science at Harvard, earned both his B.A. in mathematics and master of sciences degree from the University of California at Berkeley, and a Ph.D. from the University of Oregon. His books have been translated into fifteen languages. He lives outside Boston.

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