This page is a sub-page of the page on our Learning Object Repository.

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Related KMR-pages:

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Other relevant sources of information:

• The Artist and the Mathematician: The story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed, by Amir D. Aczel, ISBN 978.
• Récoltes et Semailles, Part 1, by Alexandre Grothendieck (in English translation).
• The Grothendieck circle.

In Swedish:

• Matterevolutionen som kom av sig: Mängdläran skakade om 1970-talets skola – men blev ett monumentalt fiasko. Här är historien om lärans uppgång och fall. Mats Karlson, Forskning och Framsteg, 2012-03-31.

/////// Quoting (Aczel, 2007 front cover):

Nicolas Bourbaki, whose mathematical publications began to appear in the late 1930s 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 to be an abstract field of human study with no direct connection with the real world. In reality, however, it is closely linked 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 also the story of an elaborate intellectual joke – because Bourbaki, one of the foremost mathematicians of his day – never existed.

Quoting Aczel (2007, back cover):

‘Nicolas Bourbaki did not exist. He was dreamt up by a playful clique of French math professors in the mid-1930s who used the Bourbaki pseudonym to tear mathematics down to its foundations. The collective tried to root out the imprecision that festered underneath the proofs of the day and replace it with more rigorous underpinnings. In so doing, the non-existent mathematician produced more important and more original work than most real-life scholars’

– The Washington Post

Quoting Aczel (2007, p. 1):

In August 1991, Alexander 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 known if 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, 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. Alexander 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 from the influence of the seminal work of Nicolas Bourbaki. But what was the nature of the relationship between Bourbaki and Alexander Grothendieck, and who is Nicolas Bourbaki?

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Quoting Aczel (2007, p. 9):

In 1940, Hanka and her son, Alexander, 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.

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 there. 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 and 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. [source] 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 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. [source] 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.

Alexander 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. [source] 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. Alexander did not care what his teachers thought of 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 every bit as correct as the one in the book.” [source] 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.

Quoting Aczel (2007, p. 13):

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 stronghold 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. [source] 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. [source]

Alexander attended the local school, the Collège Cévenol, from which he eventually earned his baccalauréat, which would entitle him to enrol 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 book 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 Alexander 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. [source] 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.

Quoting Aczel (2007, p. 49):

When the war in Europe was finally over in May 1945, seventeen-year-old Alexander Grothendieck was reunited with his mother. He was among the fortunate children separated from their parents during the war – those who both survived the war and had a parent who did so as well.

Hanka and Alexander moved to the village of Maisargues, in a wine-growing region in southern France near the city of Montpellier. Alexander enrolled at the University of Montpellier, which at that time was one of the poorest universities in France, to study mathematics. He spent much of his time making up the material he should have learned in high school, but also deriving on his own the elements of measure theory – a mathematical discipline which, unbeknownst to him, had been established around the turn of the century by the French mathematician Henri Lebesgue.

A professor who taught Grothendieck calculus, Monsieur Soula, whom he had asked about a discovery in mathematics, told him that “the last open problems in mathematics had all been resolved twenty or thirty years ago by a person named Lebesgue.” [source] Grothendieck noted the irony of this statement: ignorant of this work, he had by himself derived Lebesgue’s theory. Of course, Lebesgue’s theory was not at all the “last word” in mathematics. And the immense progress made in mathematics during the following decades of the twentieth century – in which Grothendieck would play a major role – is witness to the absurdity of the teacher’s statement. Fortunately, the student was undeterred.

While Grothendieck was a student at Montpellier, he and his mother survived on his student scholarship. They both also worked seasonally as day laborers in harvesting grapes at local vineyards, and when the harvest season was over, they worked in the making of the wine. From 1945 to 1948, mother and son lived in the small hamlet of Mairargues, virtually hidden among the vineyards, a dozen kilometres from Montpellier. They had a marvellous small garden: they never had to work at gardening and yet the earth here was so fertile, and the rains so abundant, that the garden produced a plentiful harvest of figs, spinach, and tomatoes. Their garden was at the edge of a field of splendid poppies. Grothendieck remembers his time there with his mother as “la belle vie.” [source] Often, however, his mother was sick. Her health was damaged irreparably from the privations she had suffered in the French concentration camps of the war. As a survivor, however, she was entitled to free medical care. Grothendieck and his mother never had to pay for a doctor’s services.

Grothendieck found the classes at the university boring and uninspiring. He understood mathematics at a much higher level than did his professors. At the beginning of each semester, Alexander would purchase his textbooks and immediately read them from cover to cover, assimilating all the information. For the rest of the semester, he hardly ever showed up for class. He had no need to do so.

Grothendieck graduated from the university in 1948. He had applied for a scholarship to go to Paris, which required a twenty-minute interview with a government official with knowledge of the discipline to be studied. His short interview turned out to be over two hours long, since the official, who knew mathematics well, was amazed at the depth of understanding, analysis, and creativity shown by the student. He wholeheartedly recommended that the State provide him with a scholarship. One of Alexander’s professors at Montpellier was so impressed with his abilities that he gave him a recommendation to one of the greatest French mathematicians at the time, Henri Cartan, in Paris.

At nineteen, Alexander Grothendieck descended in the mathematical community of Paris like a storm. His formal preparation in mathematics was lacking to say the least. His school classes in mathematics were inadequate and without continuity, and his university studies from Montpellier were insufficient for understanding the material at the graduate courses and seminars at the École Normale Superieure in Paris. In particular, Henri Cartan’s seminar at this elite school was exceptionally advanced and required a great deal of mathematical knowledge, which Grothendick did not have. But the ambitious and immensely talented student worked so hard that he was able to assimilate the advanced material despite lacking the prerequisites. At the seminar, he distinguished himself by carrying on conversations with the renowned professor from the back of the room, speaking with him “as if they were equals, rather than a student speaking with a professor.” [source]

In his memoirs, Grothendieck wrote:
When I finally made contact with the mathematical world in Paris, one or two years later, I finally came to understand, among other things, that the work I had done in my own corner [at Montpellier] with the means at my disposal, was that which was well known by “all the world” under the name of “Lebesgue’s theory of measure and integration.” In the eyes of two or three older students to whom I had spoken about this work (and shown my manuscript) it seemed as if I had simply wasted my time, redoing an “already known.” But I do not remember being at all disappointed. At that time, the idea of receiving credit or gaining approval in the eyes of someone for work I had done would have been strange for me. [source]

But Grothendieck also realized that in the academic backwaters where he had studied before coming to Paris he had not wasted his time. He learned on his own things that would prove important in his life: he learned that he was a mathematician. Most importantly, Grothendieck reflected, “I learned, in those crucial years to be alone” [his emphasis]. [source]

In Paris, Grothendieck fell in with a clique of mathematicians who knew each other extremely well. All of them were very talented, and they worked together creating modern mathematics. But Grothendieck was a lone wolf – he did not work well in a group. He wanted to verify for himself the mathematical facts he came across, rather than accept ideas that people took to be true “by consensus.” He felt very welcome, however, within the group in Paris. These were gifted thinkers, and for them mathematics seemed to come effortlessly, while Grothendieck felt like a mole slowly digging its way through a mountain. [source] In hindsight, he realized that “the most brilliant individuals in this group have become competent and well-known mathematicians. However, after the passing of thirty or thirty-five years, I see that they have not left on the mathematics of our time a truly profound imprint.” [source: Récoltes et Semailles]

Grothendieck’s work, however, certainly changed modern mathematics. His view was that these mathematicians had not achieved the deep results they might have been able to reach had they not lost the capacity to be alone, to work alone, and to think alone without accepting the word of any authority other than that of their own intelligence.

The group of mathematicians in which Grothendieck found himself and about which he wrote, included Henri Cartan, Claude Chevalley, André Weil, Jean-Pierre Serre, and Laurent Schwartz. All of them accepted him with friendliness, “without worry or secret disapproval – except, perhaps, André Weil.” [source]

This is a telling statement by Grothendieck, for Weil and Grothendieck were certainly opposites. One was privileged, spoiled, selfish, and somewhat lazy – Weil wanted to do only mathematics that was easy for him, and would not set his sights on overly difficult problems. He wanted to enjoy himself, to travel, to socialize, and to be with friends. Grothendieck had a deprived childhood, and everything that ever came his way did so through hard work. He was immensely ambitious, looking for the hard problem rather than the easy way out; and he was not very interested in group work. Another reason why the two men may not have gotten along very well was jealousy. Weil was the best mathematician in the group – until the arrival of the young Grothendieck. Weil must have sensed that Grothendieck could – and in fact, would – go much further in mathematics than he himself ever would be able to. He was suspicious and envious of the new arrival and wanted to keep him at arm’s length from his group.

The rest of them supported the young mathematician embarking on an adventure of a lifetime, seeking new knowledge in mathematics. But Weil had always been looked up to by everyone around him. This group of mathematicians considered him their leader and the most gifted of them all. But the young Grothendieck was very different from Weil in the way he approached mathematics: Grothendieck was not just a mathematician who could understand the discipline and prove important results – he was a man who could create mathematics. And he did it alone.

Within a few years, Grothendick would become a regular member of this group working in Paris. And he would work with other people, listen to what they would say, contribute to discussions, and help everyone around him. But mathematical research he would do all alone. And eventually, he would come to live all alone in this world.

Quoting Aczel (2007, p. 54):

Mathematics was not the only field seeing a spurt of growth and revival in Paris right after the end of World War II. The city teemed with cultural activity, as if the war had stopped all of it and now people could not get enough. The new freedom from tyranny and persecution brought out French intellectuals in droves. To these were added expatriate philosophers, writers, and artists from around the world, and they brought Paris back to its old glory of the pre-war years, a time during which artists, writers, scholars, and intellectuals had made it the capital of culture. Intellectual life was blooming in the cafes of the French capital and on its streets, and the undisputed leader of the Paris café intellectuals was the philosopher Jean-Paul Sartre.

Reviving French thinking from the ashes, Sartre brought philosophy to the street. Existentialism became the reigning philosophy. Sartre, together with Simone de Beauvoir, founded a journal, Les Temps Modernes, in which he and Beauvoir and others reviewed important cultural works. He also founded the newspaper Libération, which still exists and thrives in today’s France. Yet, his success made him vulnerable to attack, and during the late forties and early 1950s, Sartre had a number of ruptures with former allies, which left him and his new philosophy even more vulnerable. In 1949, Beauvoir reviewed in Temps Modernes a book by a young anthropologist, Claude Lévi-Strauss. Her review was very positive, but as history would show, the philosophy engendered by Lévi-Strauss would bring an end to existentialism, and inaugurate a new philosophy in its place: structuralism.

This new approach to life, the sciences, and the human condition would begin as a principle in scientific investigation, but within a few years it would sweep all areas of intellectual life. At its kernel, the idea of structuralism would be a mathematical one, but it would slowly take over the sciences, the humanities economics, and philosophy. Structuralism would begin from strict mathematical considerations, but would widen like a tidal wave – sweeping everything in the intellectual world before it.

The ideas of the new philosophy of structuralism would be seeded in the very seminars in the French capital that Alexander Grothendieck was attending in the late 1940s. Its creators would be virtually all the professors he was meeting in Paris: Cartan, Weil, Chevalley, and others. Café life in Paris would, therefore, be dominated not only by the philosophers and the writers and artists, but also by the mathematicians. And perhaps for the first time in modern history, mathematics would play a key role in the general culture – in a way that it did only in the very distant past of ancient Greece.

Since Sartre and his allies were distinctly non-mathematical in their approach to life, they would inevitably be left behind. Their philosophical theory of existentialism would end its reign as the strictly axiomatic, rigorous, and system-oriented theory called structuralism swept France and the rest of the Western world. The mathematicians would play a key role in the new milieu not only as proponents of a new and widely used approach to life, but also as connectors among practitioners in different fields: the exact sciences the social sciences, art, literature, psychology, economics, and philosophy. This would launch a new age for mathematics, one in which the role of the discipline in our culture could not be matched by any other. For, while the mathematicians strove to keep their discipline abstract and abstruse – appearing to be completely disconnected from the real world and oblivious to everything that was taking place around them – in fact, the ideas developed by these mathematicians would constitute nothing less than a revolution in human thought – one whose effects would be felt far and wide. When this revolution reached its culmination in the late 1960s, not a single area of human interest would be left untouched.

During this period, in Paris, Grothendieck was completely oblivious to the changes about to take place around him. And yet, his work and his ideas would, within a few years, form the vanguard of the new philosophy. But Grothendieck’s mind was so supple and so fecund that he would forever remain far ahead of his own time. Unbeknownst to him during this early period, he would within a few years find the defects in the new theory that would sweep Western thought. He would singlehandedly try to correct the deficiencies of the mathematical theory forming the backbone of the new philosophical and scientific approach, structuralism.

And yet, the mathematicians now philosophizing in Paris cafés alongside writers and intellectuals would find it impossible to follow Grothendieck’s lead. Frustrated with the lack of direction, he would abandon the group and set out on his own. But he would be unable to help the humanities, philosophy, social sciences, and economics recover from the conflict. Structuralism would, therefore, decline just as existentialism did, and both would give way to the new trend in Western culture: postmodernism. But this was still decades in the future. In the meantime, there was very exiting work to do – work that would affect every discipline in our world. The end of the war truly spelled a new beginning for Western culture.

Despite his genius, Grothendieck sorely needed a better background in mathematics. Henri Cartan, who recognized this fact, suggested that Alexander leave Paris and the young genius move to Nancy to work toward his doctorate at the University of Nancy under the extremely gifted and renowned professor there, Laurent Schwartz.

Grothendieck followed Cartan’s suggestion and moved to Nancy. There, he distinguished himself in his work as a graduate student, greatly impressing Schwartz and the other professors at this school. Grothendieck produced six seminal papers of mathematical scholarship, each of which would have been considered a masterpiece and would easily have qualified as a doctoral dissertation. Although his professors found it difficult to choose which of these should be used as his dissertation, he was awarded his doctorate in mathematics from the University of Nancy in 1953.

At Nancy, he was a frequent guest of Laurent and Marie-Hélène Schwartz and felt at home in their house. Schwartz and Dieudonné gave Grothendieck a number of problems on topological vector spaces, whose theory was being developed at the time, and they were astonished to find out within a few months that the young student had solved each of these exceptionally difficult problems; he thus advanced this new field significantly [source]

Grothendieck lived in Nancy with his mother, who was suffering from tuberculosis, a disease she had contracted in the concentration camps. The mother and son were renting rooms from a woman who was older than the son, and Grothendieck developed a relationship with her. Through this liason, his first son, Serge, was born.

After receiving his doctorate, Grothendieck returned to Paris and again joined the seminars at the École Normale Supérieure as well as at the Sorbonne. In addition to Henri Cartan, who ran this important seminar, Grothendieck came to know better all the mathematicians in the group associated with the work of Nicolas Bourbaki. But just who was Nicolas Bourbaki?

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Quoting Aczel (2007, p. 17):

On the eve of the utbreak 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: [source]

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 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.” [source]

After a few weeks with their hosts, André and Eveline Weil travelled farther in Finland to stay at a small hotel on the shore of lake Salla, close to the Russian border. Our days were calm and uneventful. We spent many hours on the lake in the hotel skiff, or sitting at the water’s edge. I had with me my faithful typewriter, and there I typed the outline of a report. … [source]

Weil also dictated long passages to his wife. All that time, they were taking pauses, looking at the beautiful surroundings, the hills near the Russian border. Unbeknownst to them, some people at the hotel became suspicious of a couple sitting outside, surveying the border surroundings and typing incessantly. They came to the conclusion that the two were Russian spies and reported their suspicions to the Finnish police, which opened a file on André Weil.

The pair travelled farther north and reached the Arctic Ocean where André even tried to swim for a few seconds in the icy water. Then they continued along the Finnish border with Russia, and came to Helsinki, arriving there just before the Second World War broke out. But Eveline had to return to France. She had a five-year-old son from a previous marriage whom she had left with her mother and she needed to get back to take care of him. André knew, however, that if he returned to France he would immediately be arrested. He had to stay away, and he had brought with him a large amount of cash, in U.S. dollars, to support his stay.

They were sitting in an outdoor café on the Esplanade when they heard of the declaration of war, and they felt as if they had just lost a beloved friend. [source] Eveline decided to stay a bit longer with her husband, whom she felt she might never see again. Ahlfors helped them to find a small room with the use of a kitchen not far from his own house in the city. And as the hostilities of war raged, south of them on the main part of the continent of Europe, the pair took walks in the parks along the Baltic Sea, feeding the tame squirrels that came and sat in their hands.

But by October 20, 1939, Eveline could wait no longer. Communications with France were in jeopardy of being discontinued at any moment, and she left by train to Sweden, going from there through Denmark and Holland back to France, where she was reunited with her son and her mother in a small town in the countryside.

André Weil was astute enough to understand that he was not safe in Finland. He knew that the Russians were planning to attack, and his intuition told him that Sweden was probably his safest bet. But because the war had begun, a visa would have been needed for travel to Sweden. Obtaining a visa would have required the cooperation of the French authorities – something Weil could not hope for in this case. So he stayed in Helsinki, in his little room, and continued his work.

On November 30, the Russians began their bombardment of Helsinki. Weil saw all his neighbors rush out and leave for the countryside. He followed them. Around noon, the air raid alert ended, and people began to return home. He walked toward a nearby square, to see what was happening. But his foreign clothing and his myopic squint made him stand out in the crowd. A policeman approached him, and proceeded to arrest him.

Weil was taken to the local police station. When the Finnish police headquarters was contacted about the arrest, it was discovered that there was already a file on André Weil for suspicion of spying. He was immediately transferred to the central police station in Helsinki, where he was interrogated for several hours. Then the police conducted a thorough search of his room, where they found what they believed was condemning evidence: [source]

The manuscripts they found appeared suspicious. […] There was also a letter in Russian, from Pontrjagin, I believe, in response to a letter I had written at the beginning of the summer regarding a possible visit to Leningrad; and a packet of calling cards belonging to Nicolas Bourbaki, member of the Royal Academdy of Poldevia, and even some copies of his daughter Betty Bourbaki’s wedding invitation.

There followed a “rather calm, if lengthy, interrogation at the police station.” The interrogation was conducted in German, since the police officer in charge knew that language and so did Weil. Weil was aware that the policeman was trying hard to catch him contradicting himself in his statements. At some point the policeman exclaimed: “Sie haben gelügt!” (“You have lied!” – but with a grammatical error). Weil’s immediate reaction was “Nicht gelügt, man sagt gelogen” (“it’s not gelügt, one says gelogen”). The policeman did not seem offended by this response, and stood corrected.

Unbeknownst to Weil, the Finnish authorities had concluded definitely that he was a Russian spy, and they condemned him to death. On December 3 or 4, a Finnish official named Rolf Nevanlinna was present at a state dinner also attended by the chief of police. When coffee was served, the latter approached Nevanlinna and said, “Tomorrow we are executing a spy who claims to know you. Ordinarily I wouldn’t have troubled you with such trivia, but since we are both here anyway, I am glad to have the opportunity to consult you.” Nevanlinna asked him, “What is his name?” The chief of police answered, “André Weil.” [source]

Quoting Aczel (2007, p. 21):

André Weil was born in Paris in 1906. His paternal grandfather was Abraham Weill, an important member of the Jewish community in Strassbourg. As such, he was often called upon to settle disputes between members of the community. When his wife died, he married her sister, as dictated by Jewish tradition. With her he had two sons, André’s father Bernard Weil, and his uncle, Oscar. At some point in his life, Bernard shortened his family name, losing the second l.

By law, the people of Alsace could choose either German or French citizenship. Weil’s parents and their respective families chose both chose France as their homeland and moved to Paris. Bernard Weil studied medicine, and in Paris he became a well-respected general practitioner known for the reliability of his diagnoses and the excellent treatment he provided his patients. [source]

André Weil’s mother, Selma Reinherz, was born in 1879 in the Russian city of Rostov-on-the-Don to Austrian Jews who had immigrated there. Her father was a prosperous grain merchant. Three years after she was born, following the pogroms of 1882, the family moved to Paris.

Selma an Bernard were married in 1905. Theirs was an arranged marriage, but in time they grew to love each other. Her family’s wealth helped Bernard set up his clinic in Paris and enabled him to embark on a very successful medical career. They had two children, André and Simone (who would become a famous philosopher). The family lived in an appartment on Boulevard Saint Michel in the Latin Quarter in Paris, by the Luxembourg Gardens. Every day, Selma would take Simone and André for a stroll in these beautiful gardens and the children would run and play and enjoy the best of a Parisian child’s life.

André Weil was growing up in a world of priviledge and his enviable circumstances stand in sharp contrast with those of the formative years in the life of Alexander Grothendieck. Both men would reach great heights in mathematics, but Grothendieck’s contributions to this field would ultimately be valued as greater than those of Weil. Yet, while Weil would be spared no advantage in his education and exposure to great ideas, Grothendieck would enjoy nothing of the sort and would have to develop his immensely important ideas in an intellectual vacuum while barely surviving physically.

Early on in his school days, André Weil was recognized for his amazing mathematical abilities. He learned to read between the ages of four and five, and soon he began to read books that were far above his level in school. [source]. He also started to read books on mathematics, which he enjoyed very much.

When the First World War erupted in 1914, Dr. Bernard Weil was called to serve his country in the medical corps and was assigned to various places. After a few months he became ill himself and in 1915 was sent to recuperate in a hospital in Menton, in the south of France in the Mediterranean cost. The family followed him from Paris and stayed for some time in this region. It was during this period, when he was nine years old, that André Weil became aware of his great mathematical abilities. His parents bought him a subscription of a mathematical journal for teachers, the Journal de Mathématiques Elémentaires. In this journal were published problems to be solved by the teachers, and those who solved them got to have their names in print. After he tried his hand at these problems, André Weil’s name appeared on the pages of almost every issue of the journal. The boy also developed a taste for grammar and literature, and especially enjoyed a French grammar class in which the teacher created an algebraic method of notation. This early experience with mathematical concepts applied to language inspired him and would exert an impact on his development of abstract mathematical concepts in his future mathematical career.

In 1917, after he attended various schools at the locations to which his father had been assigned – Paris, Chartres, and Laval – André Weil taught himself enough Greek and mathematics to gain acceptance to the classical section of a higher grade at school. He read the Iliad, and then devoured a book on Indo-European linguistics, which whetted his appetite to study Sanskrit. He was well on his way to becoming a Renaissance man.

But Weil’s great breakthrough in understanding and loving mathematics came a year later, after the family had moved back to Paris and he was accepted at the prestigious Lycée Saint-Louis, one of the best high schools in France. Here, his mathematics teacher was a man named M. Collin, a teacher with unusual abilities and teaching habits. Once, M. Collin had been assigned to a very difficult class, a class known to be comprised of troublemakers, at the preparatory school of Saint-Cyr. One day, he found out that the students in his class had planned to put him to the test. He entered the classroom, sat down, and spent the entire period staring at his students. He never had trouble with these students afterward.

In mathematics, M. Collin taught the important concept of rigor. André Weil wrote in his memoirs that Collin “showed me once and for all that mathematics operated by means of rigorously defined concepts.” Once the definition of a function was given, for example, the teacher did not tolerate anyone using the word “function” for anything that did not correspond precisely to the given definition. [source] This was a very important lesson, which started Weil in a direction of mathematics that would exert a strong impact on the development of the discipline throughout the entire century and up to the present time.

Mathematics had by that time often been pursued in imprecise, even vague, ways. Newton, for example, had define the derivative in calculus using the vague notion of a “fluxion,” which had not been clearly and rigorously defined. And the same was true for later mathematicians. Weil would grow up to become one of the greatest proponents of clarity, precision, and rigor in mathematics.

Weil progressed quickly and soon took his baccalaureat examinations and enrolled in the required preparatory classes for one of France’s “Great Schools,” the École Normale Supérieure (ENS). There, he was introduced to one of the greatest French mathematicians of the time, Jacques Hadamard. Later, Hadamard would become Weil’s doctoral dissertation advisor.

Even though he was getting on in years, Hadamard retained an amazing clarity of mind and an incredible sharpness in his mathematical work. He was also unusually modest and kind. He welcomed Weil into his fold. Through Hadamard, the young Weil came in contact with virtually all of France’s great mathematicians, including Henri Lebesque and Élie Cartan. The student benefited from taking courses from these renowned professors, and from attending many lectures and symposia.

In 1921, Einstein’s theory of relativity was all the rage in Europe. This was three years after Arthur Eddington provided definitive physical proof for the general theory of relativity by observing the bending of starlight around the sun during an eclipse expedition to the island of Principe in the Atlantic Ocean in 1918 – exactly as predicted by Einstein’s theory three years earlier. [source] Fifteen year old André Weil read Eddington’s book about relativity, understood both the physics and the complicated mathematics Einstein had used, and explained it to his parents during a vacation in the Black Forest.

The following year, Albert Einstein was to give a talk about relativity at the Collège de France. The most important French mathematicians and scientists, as well as members of French high society, were going to attend, and admittance to this monumental event was by invitation only. Jacques Hadamard arranged for the young Weil to be invited to this talk.

The precocious young student then turned his attention to the study of Sanskrit. He had read Indian poetry in translation and was so taken with it that he decided he had to read it in the original. Within a short time, Weil became proficient enough in Sanskrit to be able to read the Bhagavad Gita. Throughout his life he would often quote from the Gita and derive from it both artistic pleasure and spiritual guidance in life and its travails.

At the same time, he also became enamored with ancient Greek and Latin books. The boy would save his money to buy books printed in the 1500s. His favourite was a 1560 edition of the Iliad in Greek and Latin, which would become his inseparable companion for many years. He also bought Aldus’s Plato printed in 1513. Weil found some of these old books in the boxes of booksellers along the banks of the Seine, as one can still do nowadays – although, unlike today, he was able to purchase them at reasonable prices.

German mathematics had survived World War I, and Germany had thus retained its hegemony over mathematics, which it had acquired sixty years earlier. [source] Mathematics was well-organized in Germany, with many fine mathematicians at the universities of Berlin, Göttingen, and elsewhere. The Germans were leaders in analysis, algebra, and other areas of modern mathematics. They had a tradition of science and technology, and within this tradition, mathematics was very important, a subject to be taken seriously, to be studied and developed. German mathematicians were the unchallenged rulers of the discipline worldwide.

André Weil was proficient in German because of his family roots in Alsace and his grandparents’ Austrian and German origins. Both his parents spoke German and actually used it as a secret language when they did not want André and Simone to understand what they were saying. This was strong enough incentive for both children to learn this language well. André became interested in German mathematics. He read German papers in the original and studied the advances the Germans were making in this field. As an adult, he would be influencial in France’s reassuming a leadership role in mathematics vis-à-vis Germany.
As a student at the École Normale in Paris, Weil often checked out German books on mathematics from the library and learned from them about the methods and ideas that German mathematicians had been using. This was an excellent source of knowledge not used by many of his peers.

One of the German mathematicians whose work especially interested Weil was Bernhard Riemann, a man who in the nineteenth century rewrote much of geometry and topology and made important contributions to other areas such as number theory. Equally important, Weil attended Jacques Hadamard’s seminar at the Collège de France, learning much high level mathematics – material that was well beyond the usual courses at the École. “The bibli (library) and Hadamard’s seminar, that year and the following ones, are what made a mathematician out of me,” Weil wrote in his memoirs. [source]

Riemann was one of the mathematicians Weil considered a “great mind,” and, through reading him and the Greek poets, Weil came to the conclusion that “what really counts in the history of humanity are the truly great minds, and that the only way to get to know these minds is through direct contact with their works.” [source] According to Weil, his sister, the philosopher Simone Weil, also came to a similar conclusion, perhaps through his influence.

Weil was also impressed by a phrase he read in the work of the great French mathematician of the late nineteenth century, Henri Poincaré, whom Weil quotes as saying: “The value of civilizations lies only in their sciences and arts.” With such ideas in mind, Weil wrote, he had no choice but to “dive headlong into the works of the great mathematicians of the past. [source]

Weil studied the works of Riemann, and the writings of the French mathematician Camille Jordan helped him understand Riemann’s concise writings. He also read Felix Klein’s mimeographed lecture notes, which further helped him flesh out the ideas of Riemann. For a beginning university student, Weil was moving very deep and very fast into complex contemporary mathematics.

However, living in such a specialized intellectual environment, he did not want to leave behind his interests in Sanskrit, Greek and Roman poets, and music. With the aid of the English translation of the Sacred Books of the East, Weil sat in an overgrown country house his parents had built in Chevreuse and read the Bhagavad Gita from cover to cover. “The beauty of the poem affected me instantly,” he would later write, “I felt I found in it the only form of religious thought that could satisfy my mind.” [source] Judaism never meant much to Weil (or to his sister, who converted to Catholicism). He claimed not to have known that he was Jewish until the age of ten or twelve, and, once he found out about it, not to attach to this fact any great importance. [source]

After his graduation in 1925, Weil began to travel. He had always been fascinated by Italian art and he longed to see the great masterpieces, so he set out for Rome and spent some time there as well as in other Italian places: Naples, Ravello, and Sicily. In Italy, he met some of the best mathematicians that country could boast, including Volterra, who hosted him at his house for some time. He also met foreign mathematicians who were visiting Rome: Mandelbrojt and Zariski.

Then Weil applied for and obtained a Rockefeller Foundation grant allowing him to visit Germany. Finally he had the opportunity – the rare privilege, considering how young and unknown he still was – to meet some of the best German mathematicians of the day. He spent time in Göttingen with the mathematician Courant, who would later emigrate to the United States. And in Berlin, he met the renowned master of mathematical foundations, L. E. J. Brouwer.

Then, continuing his whirlwind tour of Europe, he went up to Stockholm where he met Mittag-Lefler, the foremost Swedish mathematician of all time and the founder of an important mathematical journal, for whom Weil had agreed to do some mathematical work. He stayed at Mittag-Lefler’s own villa, admiring his books and works, and hearing from him many stories about his life in mathematics. […]

Returning from Stockholm to Göttingen by way of Copenhagen, he took a commercial flight from Copenhagen to Lübeck, and from there he returned to Göttingen through Hamburg. Weil notes in his memoirs that this form of travel was still very uncommon. The young man liked it very much and decided to combine such travel with doing mathematics. He thus became the forerunner of modern mathematical and scientific researchers, who conduct much of their work nowadays through frequent travel to professional meetings worldwide. At this time, Weil also resolved to become a “universal mathematician” in the likeness of his mentor, Hadamard – someone who knows “more than non-specialists and less than specialists about every mathematical topic.” [source]

Back in Göttingen, André Weil was taken with the workings of the tightly knit mathematical group headed by Max Dehn. Here, he realized that mathematics can be pursued most effectively – and with much enjoyment – when done together is a well-organized group of highly motivated and gifted people. This idea would stay with him and guide him in his career. […]

While in Germany, Weil became interested in modern art. He visited a number of exhibitions in his travels, including one of the works of Nolde, which he saw in Hamburg, and was influenced by the new ideas of painters such as Picasso on form and shape and structure. Once, in Zürich, Weil visited an exhibition of Picasso. “Rightly or wrongly,” he wrote, “it seemed to me that this art juxtaposed the profoundly serious with the prankish – a mixture that was not without charm for the Normalien [a person from the École Normale Supérieure] I still had in me. We mathematicians were only vaguely aware of what was called “the crisis.” [source]

Modern art broke form and deliberately destroyed all norms that had existed before it, in a way that was not dissimilar to the way Einstein, in constructing his theories of relativity, had destroyed the supremacy of classical physics. These ideas would affect Weil’s intellectual development – and his approach to mathematics. Mathematicians were already breaking away from past strictures in their field to forge ahead into new territory.

While in Germany, Weil had read the manuscript of a mathematical paper in German written by Dehn. At the end of the paper, which was to be published in the Proceedings of the Berlin Academy, Weil found the following sentence (here translated into English), which was to be deleted from the published version of the paper: “It is a bourgeois, who still does algebra.” Long live the unrestricted individuality of transcendental numbers!” [source] Dehn’s dictum represented ideas that were already in the air, so to speak, in the first few decades of the twentieth century – ideas that reflected what was happening within the general culture: that the past should be left behind and that new ideas should be pursued vigorously. This trend was already apparent in the art, literature, architecture, and, of course, physics. The time had come, these visionary mathematicicans felt, to bring about similar changes in mathematics.

///// Introduce Bourbaki here

Quoting Aczel (2007, p. 99):

According to the French historian of mathematics Denis Guedj:

Animated by a profound faith in the unity of mathematics, and wishing to be ‘universal mathematicians’ Bourbaki undertook to derive the whole of the mathematical universe from a single starting point. That starting point was the theory of sets.

Twenty-five centuries earlier, Euclid based his entire Elements on the elementary notions of point, line, circle, and other geometrical concepts, and on this foundation he constructed his entire system of mathematics. Bourbaki wanted to found his entire system of mathematics in a similar way to that of Euclid. Their stated aim was to produce a new set of Euclid’s elements – “to last for the next 2,000 years.” In that spirit, they named their own series of books they planned to publish, “Eléments de mathématique.” They purposefully spelled mathematics in the singular, in French: mathématique, rather than the ususal way, mathématiques. They chose this innovative new word to stress the unity of mathematics. They were striving to write their series of books, their elements, to be “prepared according to the axiomatic methods, always maintaining as a horizon the possibility of a total formalization, our Treatise aims at perfect rigor.” [source]

Guedj notes that the Bourbaki group was not the first in history to have entertained such an ambitious project, but that they were the only ones to have advanced so far toward the realization of this goal. To carry on toward their goal, the Bourbaki group chose two other powerful methods. One was the idea of axiomatization; and the other was the general notion of structure. Axiomatization was an idea they drew directly from Euclid, as further enhanced by the German mathematician David Hilbert and others. But the second idea, the important concept of structure as applied to mathematics, was something that Bourbaki had to invent. “It counts as one of the most beautiful jewels of twentieth-century mathematics.” [source]

According to Claude Chevalley, another innovation by Bourbaki was “the principle that every fact in mathematics must have an explanation.” This principle is separate from the idea of causality, meaning that one fact causes the occurrence of another. Bourbaki held that “anything that was purely the result of a calculation was not considered by us to be a good proof.” [source]

All of the principles mentioned above are extremely useful and important in mathematics, and we could hardly imagine modern mathematics today without an explanation for every fact, an axiomatic foundation for the discipline, and the underlying concept of structure. This last principle, however, is unique in its importance and has surpassed all others. The reason for this is that the principle of structure, which, as we know, originated in linguistics, has importance far beyond the confines of mathematics. The concept is, at its core, a mathematical one; and Bourbaki has brought it into the forefront of mathematical thinking. But the concept is so powerful and so fundamental to science and to the processes of human thinking that it found applications in virtually every area of human interest. The concept of structure, in fact, started an entire revolution in human thinking and in human philosophy. Bourbaki was the key proponent of this concept and made it into the universal core of human thought.

What are the mathematical essentials of the concept of structure? Bourbaki developed a number of “mother structures,” as the group called them. Two of them were the notion of nearness and the structure of a group. The idea of nearness, or neighbourhood, comes from the mathematical field of topology. Here, we study the notion of neighbourhood. These are precise mathematical notions that define “nearness” of two points, and Bourbaki made these notions of paramount importance in defining nearness. These ideas are abstract, and they have found applications in psychology and the theory of learning, for example. A psychologist may want to know whether two concepts, such as a curve and a straight line, are “near” each other in the human mind. Here, quasi-mathematical notions based on the topological idea of nearness may be used.

The idea of the structure of a group was one that held special importance for Bourbaki, since the area in which it resides, abstract algebra, was the favourite research area of several of its members.

The idea of a group was over a hundred years old when Bourbaki tackled it. It had originated in the late eighteenth century and was developed well by the young Évariste Galois around 1830, just before this twenty-year-old genius died tragically in a duel. If we look at the set of numbers 123, their various possible orders form a group. It is called the permutation group of n elements (in this example, n = 3). The members of this group are 123, 132, 213, 231, 312, and 321. The group has six distinct elements. But this group appears in many other guises.

The idea of an abstract group is that it does not matter what the actual situation is, the application from which the group arises. What matters is only the inherent structure of the group.

Bourbaki attached great importance to this idea of structure. The internal nature of the group alone – completely independently of where this group came from, whether it is triangles, or numbers, or solutions of equations – was the key element, the structure, that Bourbaki cared about. Structure could thus be seen as a latent code or symbolism for what was “going on” mathematically in a given situation. And the situation itself no longer mattered to Bourbaki – it was only the code, the symbolism, the latent structure that was of interest to these mathematicians.

Bourbaki did not invent the idea of structure: it existed, in the non-mathematical setting of linguistics. And soon, while Bourbaki was working on its structures, the idea would transform itself into anthropology, and from there to psychology, and eventually – again through linguistics – into literature, an area in which one could hardly imagine a mathematical idea could find fertile ground. But, in fact, within a few years, the idea of structure would dominate all thinking in Western culture. And Bourbaki would be a major force in this innovation, enabling social scientists, humanists, and writers to use and understand this immensely important concept.

Sometimes an idea that has been developed earlier waits until the right person grasps it and makes it a reality. This is what would happen with the idea of structure in 1942. The great anthropologist Claude Lévi-Strauss would attempt to apply structure, learned from the linguist Roman Jakobson, in anthropology. But he would need the mathematical underpinnings of structure – and these would be supplied to him by Bourbaki. A meeting in New York that year between Lévi-Strauss and André Weil would result in the solution of a difficult problem of kinship studied by Lévi-Strauss.

André Weil would use structure – and, in fact, the structure of a mathematical group as described above – to solve Lévi-Strauss’s problem. In doing so, Weil would enable the transfer of the mathematical idea of structure into anthropology, and from it into other fields. Bourbaki would thus be instrumental in unleashing the powerful concept of structure. Our culture would never be the same after this important development.

Quoting Aczel (2007, p. 104):

Following the end of the war, the secret mathematical society of Nicolas Bourbaki flourished and assumed a key role in the development of mathematical ideas. Bourbaki began to transform mathematics by placing it on a firm theoretical foundation. Through the work of Bourbaki, the French gained prominence in mathematics as German mathematics, dominant in the prewar years, declined. There took place the birth of the New Math, based on set theory, which dominated American education and was important in the educational systems of other nations for a period of time. This approach owes its inception to the work of Bourbaki. What were the achievements of this unique group of mathematicians?

Quoting Aczel (2007, p. 119):

The members of the Bourbaki group recognized only one French mathematician of the pre-Bourbaki era as their godfather: the geometer Élie Cartan, Henri’s father. Other French mathematicians of the generation before Bourbaki were not well liked, including Lebesgue and Poincaré.

During the 1950s, when Pierre Cartier joined Bourbaki, it was not fashionable to value the work of Henri Poincaré, who was one of the most important French mathematicians of all time. Poincaré was called “the last universalist.” He was a mathematician who understood a great deal about a very wide range of mathematical topics. Poincaré derived very important results in mathematics, and a conjecture he had made a century ago now seems to have been proved by contemporary mathematician Grigori Perelman. But Bourbaki viewed Poincaré’s way of doing mathematics as old-fashioned. Poincaré’s style of doing mathematics and the style of Bourbaki were at odds with each other. [source] Poincaré was intuitive and detail-oriented. To him, mathematics was an art. He did not care about structures or axiomatics. He cared about seeing things clearly and producing results – in whatever way he could.

In the 1960s, a new generation joined the group: the fourth generation of Bourbaki. These were mostly former students of Alexander Grothendieck. Grothendieck himself was a member of Bourbaki for about ten years, and contributed greatly to the development of modern mathematics. But he came to a conflict with the group, and left it in anger. Before he left, there were frequent clashes within the group, as well as generational conflicts.

As Armand Borel wrote: “The fifties also saw the emergence of someone who was even more of an incarnation of Bourbaki in his quest for the most powerful, most general, and most basic – namely, Alexander Grothendieck.” [source] Grothendieck “quickly made mincemeat of many problems on topological vector spaces put to him by Dieudonné and Schwartz, and proceeded to establish a far-reaching theory. Then he turned his attention to algebraic topology, analytic and algebraic geometry, and soon came up with a version of the Riemann-Roch theorem that took everybody by surprise, already by its formulation, steeped in functorial thinking, way ahead of anyone else. As major as it was, it turned out to be just the beginning of his fundamental work in algebraic geometry.” [source]

Grothendieck was unhappy with Bourbaki because he felt their work and aims were not ambitious enough. Eventually, he would write his own series of books, and leave Bourbaki. By the fourth generation, the goal of the group was not as clear as it had been earlier. Grothendieck had by then developed his own, more ambitious, program outside of Bourbaki. Thus the need for Bourbaki was less obvious, and a lack of a global understanding of mathematics was beginning to manifest itself vis-à-vis these new developments. [source] The members of the group had become more specialized in their interests, and less able to see the larger picture.

But as great success came, almost overnight as it may have seemed, to the members of Bourbaki – and with it, invitations for talks and presentations around the world, prizes, and recognition for their collective achievements – there also arose a degree of resentment. Not every mathematician was happy with the new approach to mathematics. Many around the world felt that Bourbaki had gone too far, that the group was now pursuing generality for generality’s sake, and that they cared less about explaining mathematics than they cared about presenting results that were very abstract. This negative trend would haunt the group in the years to come, as the criticism spread to the various methods employed by the group.

One major criticism has been that Bourbaki was overly formal, too abstract, and much more rigorous than necessary, thus making it unnecessary difficult to read and understand mathematics, and to use it in a meaningful way. Bourbaki’s emphasis on precise definitions has found disapproval in the international mathematical community. [source] But Bourbaki’s main aim had been to improve and deepen the understanding of mathematical concepts, not to make them obscure, thus the criticism was somewhat justified. For the question arose: How far should generality and abstraction go before they become an impediment to understanding? The proofs and the rigor were only part of the vehicle toward understanding, and this vehicle could easily be overused. There is no real answer to this question. Somewhere there must be a middle ground, but its location is not clearly seen.

Quoting Aczel (2007, p. 125):

From the 1950s to the 1970s, Bourbaki reigned supreme over mathematics. But Bourbaki’s greatest contribution was the one it made to Western civilization as a whole, not only to mathematics. That contribution has been the development and promotion of the concept of structure. The idea of structure can be mathematical, and, indeed, it is best understood and used within a mathematical framework. But the idea itself was not born in the field of mathematics.

As we shall see, the modern idea of structure originated in linguistics, and its seeds were sown at the turn of the century with the first ideas in modern linguistics proposed by Saussure. These ideas were further developed in the early decades of the century by the Russian linguists Troubetzkoy and Jakobson. André Weil was aware of the emergence of structuralism in linguistics, and his own ideas in this area were crucial to the further development of structuralism by Bourbaki.

Bourbaki took the rather vague notion of structure to a much higher, and more precise, level. Since structure is something that can be very well cast in mathematical terms, Bourbaki had a great opportunity here to make an incredible contribution to civilization: it could formalize, axiomatize, and generalize this concept, making it a very precise mathematical idea. Bourbaki did just that. It did so for its own reasons and purposes: the mathematicians of Bourbaki wanted to use the idea of structure in mathematics. But in making this concept rigorous, they also gave the rest of the world a fabulous tool: the well-defined and precise idea of what structure really means. Within a few years of mathematical development of the concept of structure by Bourbaki, everyone in science, social science, the humanities, economics, and psychology was talking about structure and using this concept in their works. The age of structuralism had truly arrived once Bourbaki made this concept well-defined.

In his definitive history of structuralism, the French historian François Dosse wrote about the role of Bourbaki in promoting the concept of structure. “Bourbaki’s mathematical structures brought pure formalism into the context of structures,” he noted. “Bourbaki’s ideology has certainly strongly contributed to the mentality and activity of structuralism.” he wrote. This brought about a new “ideology of rigor” to the concept. “Bourbakism made the edifice of mathematics appear as a splendid construction.” This was due, according to Dosse, to the importation into mathematics of the concept of structure, born in the area of linguistics.

“Structuralism and phenomenology were thus directed toward the search for mathematical ideals,” Dosse continued, “Those ideals, however, are not the result of evading the real world, nor do they reside outside the realm of experience. Rather, they are a means of capturing the properties of objects and ideas. They are rooted in the realm of symbolic entities, not relevant directly to the world of the intelligible or the perceptible, but one that occupies some place in between.” These mathematical structures were the results of the efforts of the Bourbaki group, which allowed us to construct problematic objects that are only defined through symbols. … “It allows us to obtain powerful theorems that enable us to control chains of properties of objects seemingly of differentiated nature.”

Thus Bourbaki took the concept of structure out of its limited linguistic significance, made it precise and mathematically powerful, and unleashed it on the world of ideas. It would take an anthropologist, however, to then use both the linguistic model and the vastly improved mathematical concept of structure to make structuralism a viable approach in a wide variety of situations in the world.

Quoting Aczel (2007, p. 138):

In 1943, Claude Lévi-Strauss met André Weil in New York, and the exchange of ideas between them was what eventually led to Bourbaki’s ideas being introduced into anthropology. The first important example of the use of mathematical principles in anthropology was to be the solution of the difficult marriage-rules problem in tribes of Australian aborigines studied by Lévi-Strauss. This problem was solved by André Weil using purely abstract algebraic methods. Weil was so proud of his mathematical solution of Lévi-Strauss’s problem, and the connections forged between “pure” mathematics and applied science, that he continued to tell the story about this cooperation between practitioners of different fields until his death.

Lévi-Strauss had large amounts of data, but his initial analysis had completely failed to result in the desired rule he tried to extract from the data. It seemed that everything he had tried failed, and so he came to the inescapable conclusion that only a mathematician could help him solve the mystery. Thus he went to Jaques Hadamard, the famous French mathematician and uncle of Laurent Schwartz, who was also living in New York at the time, one of a group of European intellectuals who had fled Hitler and made up the émigré community that has been called “Paris on the Hudson.” Hadamard, who was then quite old, did not help Lévi Strauss. He told him: “Mathematics has for operations, and marriage is not one of them.” [source]

So Lévi-Strauss went to see André Weil, who was in New York at the time. Weil’s reaction was quite different. Weil was a superb algebraist, and while his work had always been in pure mathematics, he quickly saw that Lévi-Strauss’s problem could be attacked using the abstract theory of groups, here applied to a real-world problem. “When in doubt, look for the group!” Weil worked on the problem for a while, and solved it. What was a very complicated situation in the real world was solved brilliantly by an application of abstract algebraic techniques.

Weil explained to Lévi-Strauss that he was able to solve the problem by ignoring the actual elements of the problem: the nature of the marriages themselves. Instead, Weil concentrated on the relationships among the marriages. This idea reflected the main thought of Bourbaki: that relationships and structures were the key elements of mathematics.
The solution of his problem by the Bourbaki cofounder made Lévi-Strauss even more interested in structures and structuralism, and this encounter, along with his interaction with Jakobson, made him concentrate all his efforts on developing structural methods in anthropology. Eventually, the work would bring structuralism to prominence.

Lévi-Strauss’s data on marriage rules indicated that among the Australian tribes he studied, there were four population groups, which for simplicity, he denoted , , , and . The rule that seemed to exist in this aboriginal population dictated that a man from group XXX could marry a woman from group YYY; similarly, a man from group could marry a woman from group ; a man from a woman from ; and a man from group could marry a woman from group .

Pictorially, the marriage rules are as shown below

\,\;\;\;\;\;\,\;\;\;\;\;\,\;\;\;\;\; A \, \rightarrow \, B \, \rightarrow \, C \, \rightarrow \, D \, \rightarrow \, A \, .

The question was what happened long-term in such a population. What did these rules say about the development of long-term kinship ties within the society? What was the hidden internal structure of the complex relationships among the members of this society? This was what mathematics was called on to solve in this context.

What Weil did was to define the possibilities of marriage within the society. There were, he understood, four possible kinds of marriages within this aboriginal population. Weil defined the types of marriages as follows:

\begin{matrix} \,\;\;\;\;\;\,\;\;\;\;\;\,\; M_1 = [ \, M_{an} \, f_{rom} \, A \, , \, W_{oman} \, f_{rom} \, B \, ] \\ \,\;\;\;\;\;\,\;\;\;\;\;\,\; M_2 = [ \, M_{an} \, f_{rom} \, B \, , \, W_{oman} \, f_{rom} \, C \, ] \\ \,\;\;\;\;\;\,\;\;\;\;\;\,\; M_3 = [ \, M_{an} \, f_{rom} \, C \, , \, W_{oman} \, f_{rom} \, D \, ] \\ \,\;\;\;\;\;\,\;\;\;\;\;\,\; M_4 = [ \, M_{an} \, f_{rom} \, D \, , \, W_{oman} \, f_{rom} \, A \, ] \end{matrix} \,

Then Weil looked at the definitions of the offspring of these marriages, and their possible marriages, He conducted the analysis, and identified a mathematical group that governed this particular process. Mathematics allowed him to solve this problem. Then he wrote up his results as an appendix to the first part of Lévi-Strauss’s book. Elementary Structures of Kinship (Les Structures Élémentaires de la Paranté). (The Hague: Motoun, 1947).
In the appendix, Weil wrote, continuing the description of the types of marriages allowed in the Australian aboriginal society that Lévi-Strauss had studied: [source]

We add to the above that the children of a mother of class \, A, B, C, D \, are respectively defined as members of class \, B, C, D, A . This gives us the following table:

\begin{matrix} \,\;\;\;\;\;\,\;\;\;\;\;\,\; T_{ype} \, o_{f} \, m_{arriage} \, o_{f} \, p_{arents} & M_1 & M_2 & M_3 & M_4 \\ \,\;\;\;\;\;\;\; T_{ype} \, o_{f} \, m_{arriage} \, o_{f} \, s_{on} & M_3 & M_4 & M_1 & M_2 \\ \,\;\;\;\;\;\,\;\;\;\;\;\,\; T_{ype} \, o_{f} \, m_{arriage} \, o_{f} \, d_{aughter} & M_2 & M_3 & M_4 & M_1 \end{matrix} \,

Weil pointed out that what we have here are permutations of the marriage types of the parents. What permutation means, he explained, was that the orders of the kinds of marriages in the second and third rows are simply rearrangements of the symbols in the first row.

Permutations are studied by the abstract algebra theory of groups. As we have seen earlier, permutation groups are a common example of a mathematical group. Recall that the permutation group of the three elements , , consists of the members , , , , , and .

Weil then applied the powerful results of the mathematical theory of groups, in particular the study of permutation groups, to the analysis of marriage types. He pointed out that results from group theory allow us to make conclusions about such a population; in particular, the kind of group under analysis here could be reducible or irreducible. A reducible group of marriage types would indicate to the anthropologist that the population under study is really the grouping together of two distinct populations that never intermarry and perhaps simply occupy the same region. If the group inherent in the permutations above is irreducible, however, then this means that the population is unified and cannot be naturally broken into two or more subpopulations that do not intermarry, as happens in this example.

Each of the populations studied by Lévi-Strauss had its own mathematical group structure. Once the structure was analysed, it was possible to determine whether or not the population had non-interacting segments, or whether, on the other hand, it was uniform. This was a determination that could come about only through mathematical analysis. One could study the population for years and not find this important characteristic if one did not employ the structural mathematical analysis using group theory. For what Weil discovered was tantamount to a theorem: it was something that had to be true about a population because of an inherent structure and nothing else. Simply not finding any interaction between subgroups, if performing a strictly empirical analysis, would never be enough because one would never know whether a connection between the subgroups was possible (in which case, in the future it could appear) or whether it was not possible. Only mathematics could give the definitive answer.

Weil’s analysis was structural, and it was in the spirit of Bourbaki and its aims. Through the results derived from purely abstract mathematics, Weil was able to discover an inherent truth about the population under study. This truth could not have been discovered by the weaker, non-mathematical structural methods that had been used by the linguists. The power of mathematical structural methods had thus been brilliantly demonstrated. Lévi-Strauss was duly impressed – and grateful – and from then on dedicated all his efforts to promoting the use of rigorous structural methods in anthropology and other areas.

Lévi-Strauss took Weil’s ideas and learned from them. His entire book about kinship – a classic of anthropology and of structuralism – is unique in that it infused into anthropology the structural ideal of the new mathematics of Bourbaki.

In time, Lévi-Strauss’s book became the opus magnum of French structuralism. In it, he tried to explain the system of kinship and marriage – which comes in an enormous variety throughout societies around the world – by means of a single principle: the principle of exchange. Thus the marriages themselves do not matter at all. What matters, rather, are the relationships among the various kinds of marriages, as Weil had been able to demonstrate using the results of the abstract theory of groups.

This idea, in fact, is the heart of structuralism: Structuralism teaches us that the elements under study are unimportant – only the relationships among these elements have significance. Thus, in the example Weil analysed, the pertinent variable was how marriages of the parents affected the kinds of marriages their children could have.

The exchange that Lévi-Strauss studied is perceived to be a manifestation of fundamental structural constants within the human mind. And these structural constants may be found in other systems of human culture. One example of such a manifestation of hidden structure is language. It was for this reason that the influence of structuralism in anthropology originally came from the work in linguistics. The same structural ideas manifest themselves in psychology and other fields.

Quoting Aczel (2007, p. 146):

At first, Lévi-Strauss Elementary Structures of Kinship was embraced by the existentialists. Simone de Beauvoir reviewed the book very favourably in Temps modernes in 1949 (vol. 5, 1949, pp.943-949). But only in the late 1950s did the book begin to exert real impact on French intellectual life. By 1968, structuralism as championed in this book would replace existentialism at the apex of French philosophy. Lévi-Strauss and others such as Michel Foucault would then take Sartre’s place at the center of modern philosophy in France. In addition to Lévi-Strauss and anthropology, Roland Barthes used structuralism in literary and cultural criticism, Jaques Lacan used it in psychoanalysis, and Michel Foucault used it in philosophy. Other fields enjoyed the fruits of this revolution as well. In 1951, Lacan, Lévi-Strauss, and Benveniste began meeting regularly with the mathematician Georges-Théodule Guilbaud in order to work on structures and attempt to uncover links between the social sciences and mathematics. The ideas germinated, and from the late 1950s until the end of the 1960s – the period in which Bourbaki made its greatest contributions to mathematics – structuralism “happened.”

Structuralism, with its mathematical underpinnings, became a major social and cultural phenomenon. The trend became important after the publication of yet another book by Claude Lévi-Strauss, Structured Anthropology, published in 1958 (New York: Basic Books). Following the publication of this book, two conferences were held in 1959 with the purpose of exploring and explaining the meaning of structuralism. Notions of mathematical structure developed by Bourbaki figured prominently in these conferences.

Thus mathematical structures became the key idea in these congresses devoted to the social sciences and the application of structures in these areas. The first of these two meetings was held in Paris January 10-12, 1959, and it was named “Meanings and Uses of the Term Structure in the Human and Social Sciences.”

The second congress, held between July 25 and August 3 of the same year, took place in Cerisy-la-Salle, and its theme was “Genesis and Structure.” The key players in world structuralist movements were there: Lévi-Strauss in the Paris conference, and Jean Piaget in the meeting at Cerisy-la-Salle. Mathematics excerted a universal appeal in these congresses and in ones that would follow in the years to come.

Bourbaki had seeped into intellectual folklore because of his high profile in the mathematical community and his alledged role in educational reforms. He had become a synonym for rigor, axiomatics, and set theory [source]
Many authors, however, kept insisting that structuralistm could be pursued without mathematics.

From the structuralist work Jakobson had done in linguistics, Lévi-Strauss learned not to get bogged down by the multiplicity of terms, but to look for the simplest and most salient relationships uniting them. Jakobson looked for the smallest unit of spoken language: the phonemes. Lévi-Strauss, similarly, looked for elementary structures in anthropology. Both Lévi-Strauss’s structural analysis of kinship and Bourbaki’s structural view of mathematics aimed at unifying their respective disciplines by concentrating on underlying structures. Structuralism and Bourbakism peaked in the late 1960s, and then began a decline as postmodernism took over.

Quoting Aczel (2007, p. 155):

These fields, therefore, psychology, anthropology, linguistics, sociology, and economics (which studies the aggregate economic behavior of people) all involve the structural elements that are hidden inside the human brain and the human subconscious. To reach that subconscious level, one must study the behaviors of individuals and societies and seek to identify the latent structures in the brain.

The way the brain processes information according to Lévi-Strauss, is by using symbolism. The symbolism is what structural analysis is designed to uncover. Structure is thus a code, consisting of concise symbols. The symbolism inherent in the brain function follows mathematical rules that are tantamount to the ideas developed by Bourbaki: the notions of closeness, transformation, groupings, and other of the “mother structures” studied by the group.

The symbols in the brain determine how information is processed, but they also capture the workings of language, as uncovered by structural linguists. These codes determine how societies behave – and it was here that Lévi-Strauss made his seminal discovery, for, using the structuralist approach to his discipline, he was able to demonstrate something that would not have been possible otherwise, namely that the prohibition against incest has a positive role in the development of social structures. The discovery of structure is the discovery of codes, how codes develop, and how codes change from one to another.

Proceeding into other areas, Lévi-Strauss set the idea that the code precedes the message – be it in anthropology, sociology, or linguistics. This is the canonical message of structuralism. Structuralism aims at studying the code as such, rather than concerning itself with the actual message in the code and its context. This is in line with the work of Bourbaki in mathematics, in which the elements themselves have no meaning; only the relations between them, the codes that exist within mathematics, are of importance.

Thus, according to Lévi-Strauss, structuralism goes beyond the confines of any one discipline. It is an immensely powerful mathematical way of unlocking the inner workings of the brain or of any logical system, and it therefore finds applications in almost any discipline. Structuralism certainly took the social sciences out of their empirical stagnation and brought them into the scientific milieu of the “hard sciences.” Thus, Jakobson has said that structural linguistics is “like quantum mechanics.” [source]

“We want to learn from the linguists the secret of their success.” Lévi-Strauss wrote. “Couldn’t we, too, apply to a complex field of of our studies the rigorous methods whose usefulness linguistics verifies every day?”

Lévi-Strauss also made the important deduction that the symbols – the basic elements in language as well as in anthropology – are mathematical entities embedded in the human mind. Access to the subconscious, in which these elements reside, passes through language, as well as through social behavior. This realization unified, from a structural point of view, the three fields of psychology, anthropology, and linguistics. The unifying elements were the mathematical symbols in the subconscious brain. Thus, structural mathematical analysis could reveal the inner workings of the mind and language and the behavior of societies. Symbolism reigns in this realm, since the symbols are the latent structures in all three fields, and the symbol thus is more significant than what it signifies, be it in the mind, language, or social behavior. Thus the symbols that structural analysis reveals are the essential elements of the theory.

Quoting Aczel (2007, p. 161):

With the influence of Bourbaki, structuralism entered psychology. The Swiss psychologist Jean Piaget, in particular, was interested in using mathematics in psychology, and here the ideas of Bourbaki were especially useful to him.

“A critical account of structuralism,” Piaget wrote in 1968, “must begin with a consideration of mathematical structures.” [source] The structural laws that Bourbaki introduced, and which eventually found their way into the sciences, the humanities, and other areas, are:
• Composition
• Neighborhood
• Order
• Equivalence

Based on several decades of research in psychology, Piaget concluded that at every stage of the development of intelligence in children, thought processes occur in very structured ways based on the mathematical ideas above, the same ones now made precise by Bourbaki. Bourbaki’s structures were crucial for Piaget’s work in intelligence. Piaget believed that the acquisition of propositional logic was the key element in the intellectual maturity of a child. The mental structures that enable teenagers, and adults, to think logically are themselves modeled on mathematical structures. The structure of a group, for example, is an important example of such a mathematical structure in logic.

Similarly to the personal connection that took place between André Weil and Claude Lévi-Strauss in New York, an encounter took place between Jean Piaget and Jean Dieudonné, and this meeting also affected the development of science in a profound way and exerted Bourbaki’s influence on the progress of structuralism. The meeting took place in April 1952 at a conference on Mathematical Structures and Mental Structures, held outside Paris. Dieudonné gave a talk in which he described three of Bourbaki’s “mother structures”: composition, neighbourhood, and order. Then Piaget gave his talk, in which he described the structures he had found in childrens’ thinking. To the great surprise of both speakers, it was clear that the two of them spoke about exactly the same ideas. It was evident that there was a direct relationship between the three mathematical structures studied and promoted by Bourbaki and the three structures inherent in children’s operational thinking. [source]

At the Cerisy-la-Salle conference in 1959, mathematics was not emphasized at the beginning of the talks. But on the second day, Jean Piaget brought it up very forcefully. The structures pioneered by Bourbaki were the key elements stressed by Piaget, who summed it all up by saying that systems presented laws that were totally distinct from the laws governing the behaviors of single elements [source]. Piaget explained elsewhere that the roots of the spontaneous psychological development of arithmetic and geometric operations in the minds of children paralleled the concepts used by mathematicians. He referred to the “linear order” of science as extolled by the positivist thinkers of the Vienna Circle, arguing that it needed to be replaced by a mathematical circle [source].

For Piaget, the mathematical sciences were the basis for all science – including the “science of man” – and, in turn, mathematics itself was based on hidden structures of the human mind [source]. Bourbaki’s structures were the key elements Piaget was looking for in understanding the inner workings of the human mind. These structures determine both the workings of the brain, and, through the effects of the brain on social behavior, they also determine the behavior of an entire society – as shown by Lévi-Strauss’s work.

Piaget’s “genetic structuralism” appeared as a unified methodology that was equally adaptable for use in a wide variety of fields other than psychology. He saw in mathematical structuralism the answer to questions posed by all science. In mathematics, he commended Bourbaki for bringing structuralism to the forefront of all mathematical analysis, making it widely applicable in all forms of human investigation. He saw Bourbaki’s structures as deeply rooted elements of the human brain. [source]

A meeting was held in Paris from April 18 to 27, 1956, the theme of which was “Notions of Structure and Structures of Knowledge.” The idea here was to synthesize knowledge across disciplines with the concept of structure. The organizers of this meeting hoped to exhibit “an isomorphism between different sectors of knowledge,” thus borrowing from mathematics the key concept of isomorphism: a structure-preserving map. An isomorphism is a function from one set into another that also preserves the relationships among the elements of the set that is being transformed. This way, when we look at the set that is the result of the mapping by the isomorphism of the original set, this new set preserves all the relationships that are inherent between objects of the original set.

But the conference had overly ambitious goals, and ultimately these were not met. More questions were raised than answered and the conclusion was that “no solution has been found; the structure of knowledge has not been defined.” [source]
The question remained of whether Bourbaki’s definition of structure really meant anything outside of mathematics.

Quoting Aczel (2007, p. 184):

In 1957, Pierre Cartier made an amazing observation. He understood that a ringed space is locally isomorphic to a ringed space of the form Spec(A) – the set of all prime ideals of a commutative ring A – should be considered as a generalization of an algebraic variety. Cartier told Grothendieck about the idea, and the latter began to develop the foundations of algebraic geometry based on Cartier’s generalized notion of algebraic variety, now known as a scheme.

Grothendieck then began to plan a very ambitious project to rewrite all of algebraic geometry in a book titled Eléments de Géometrie Algébrique. He saw the book as comprising thirteen chapters, in which much of of algebraic geometry would be developed in the new language of schemes. Grothendieck was prescient enough to realize that his new approach could be instrumental in proving the conjectures made some time earlier by André Weil about the zeta function of algebraic varieties over finite fields.

Grothendieck planned to devote the final chapter of his book to this proof. But he was able to finish only four chapters of the ambitious project; those four numbered around 2,000 pages. This work was carried out jointly with Jean Dieudonné. A seminar on these topics, which Grothendieck gave during the 1960s and early 1970s, provided the details that would have been included in the remaining, unwritten chapters. [source]

Grothendieck’s book and seminars exerted an immense impact on the development of mathematics in the second half of the twentieth century. His ideas paved the way for one of his students, Pierre Deligne, to eventually prove the Weil conjectures. Furthermore, Grothendieck’s innovations in the field were instrumental in Gerd Faltings‘s proof of the Mordell conjecture – the difficult problem that Hadamard had wanted Weil to solve as his dissertation topic. These same ideas were of paramount importance in the eventual proof of Fermat’s Last Theorem by Andrew Wiles in the early 1990s.

One of Grothendieck’s great early achievements in his project was making commutative algebra part of algebraic geometry. Grothendieck thus established the role of commutative algebra as the study of the local structure of schemes. The theory of schemes also weds arithmetic with geometry, thus achieving a goal set a century earlier by the German mathematician Leopold Kronecker. But the theory of schemes not only added new objects of study to algebraic geometry, it also brought new insights and instrumental techniques to the quest of solving important problems of classical geometry. [source]

Grothendieck’s monumental work brought us new insights into the very nature of space and its points. Euclid saw points as the basic elements of geometry. Points produce lines and circles, and the intersections of lines, or lines with circles, give us points. But a point was something that Euclid defined axiomatially: a point was something which had no length or width or breadth. It was something that was defined by what it was not.

In the seventeenth century, Newton and Leibniz took up the re-examination of space and its points. For Leibnitz, the constituent of all things – spiritual as well as physical – was something he called a monad. The monads, reflecting Euclid’s points, were “windowless” elements, meaning that they had no internal structure whatsoever, and the only interesting property they possessed was the relationships they had with one another. [source]

On the other hand, the definition of points and sets that Bourbaki chose to use in its first published volume of the Élements de mathématique, the volume of set theory, is as follows: A set is composed of elements capable of having certain properties and having certain relations among themselves and with elements of other sets. [source]

Thus, points are seen as pre-existing, and the problem of mathematics is to organize them and give them structure. Grothendieck, inspired by the ideas of the German mathematician Bernhard Riemann (1826–1866), who discovered the notion of surface stacked over a plane, proposed the idea of topos. Grothendieck came to the idea of topos by replacing the open sets of a space (the basic elements in topology) by spaces stacked over the given space. The same idea can be viewed by considering the category of sheaves over the space. These notions, in fact, came from category theory, which was discovered and developed in the 1940s by the American mathematician Saunders MacLane and by Bourbaki member Samuel Eilenberg.

The main idea of category theory is to consider general objects and their transformations, rather than points. Category theory is a very abstract discipline, in which the nature of the objects one deals with does not matter; it is only the relationships among these objects that are of importance. […]

The concept of topos, the abstract idea derived from notions in category theory, was for Grothendieck the ultimate generalization of space. Grothendieck then claimed the right to transcribe mathematics into any topos he might choose. [source] Using these new ideas, Grothendieck was able to bring parts of modern mathematics to great heights that Euclid, Leibniz, Riemann, or even his own contemporaries could not even dream of. The new level of abstraction and the new and very general way of defining space allowed Grothendieck to prove important results in mathematics, ones that were out of reach for anyone before these amazing developments took place. What Grothendieck did was to inject into geometry the powerful methods and concepts of abstract algebra.

These concepts, now viewed in the context of space and geometry, made many new things possible in mathematics. They redefined space itself, and they allowed a mathematician to see things in an entirely new way. They also united various branches of mathematics, bringing a new understanding of mathematical ideas that would not have been possible otherwise.

It was this kind of amazing insight into problems that people have looked at before but could not solve that made Grothendieck’s achievements so astounding. And it is because of his great insight and understanding that he can be compared with Albert Einstein.

The theory of categories provided a suitable framework for describing general properties of objects studied by mathematicians – a framework that, following unsuccessful attempts, Bourbaki decided not to include within its own work. [source] Category theory was like a superstructure hanging above set theory, abstract algebra, topology, and other areas. This superstructure contains the essential elements of a mathematical theory, which can then be applied to sets, algebra, topology, and so on. It is thus a very powerful theory. But Bourbaki had made its decision back in the 1930s to base all its work on set theory. It did not want to go back and base its oeuvre on category theory.

Quoting Aczel (2007, p. 190):

Grothendieck did not much like the system of Bourbaki, in which every detail had to be discussed and argued about. His coverage of topics in mathematics was deeper, and it focused on particular areas, rather than following the very general approach of Bourbaki, who wanted to cover every area in an encyclopedic way. In 1960, the final break took place, and Grothendieck – frustrated and angry with many of the members – left the Bourbaki group for good. But he remained in contact with some of the members. These people were, after all, among France’s greatest mathematicians, and he wanted to continue to interact with them.

Alexander Grothendieck was the most gifted French mathematician of the 1960s. Grothendieck used his deep insights to discover hidden relationships between mathematical objects. These relationships then revealed to him hidden aspects of the objects he studied. Grothendieck invented – or discovered – new mathematical constructions.

One of the most important of these was the deep concept of motive. The word, in French, actually means pattern, rather than motive, and Grothendieck’s motive is a hidden pattern within a mathematical structure of the field of algebraic geometry. Grothendick described in his writings his aim of unifying two worlds: “the world of arithmetic in which live the spaces with no notion of continuity, and the world of continuous size, in which live the spaces in the proper sense of the word, accessible to the methods of the analyst.” [source]

The ideas in his work have to do with the mathematical objects called schemes, sheaves, and topos. Sheaves were conceived by Jean Leray and later developed further by Henri Cartan and Jean-Pierre Serre. The new theory, based on Grothendieck’s topos holds perhaps a great promise of replacing the traditional theory of sets, with its inherent paradoxes.

A scheme is a generalization of the concept of variety. One remarkable aspect of Grothendieck’s work was his introduction into mathematics of ideas of great generality. His generalization of a variety into a scheme allowed people to understand all the incarnations of a variety in many different settings.

In this sense, Grothendieck is the greatest structuralist the Bourbaki group ever could count among its members. He is a giant in mathematics who thinks in great generalities, has incredible vision, and can foresee the right course to take in research. Grothendieck always knew which paths of research in mathematics held the greatest promise of important results. His problem in life was to convince other people that his direction was the right one.

Quoting Aczel (2007, p. 196):

Grothendieck’s leaving mathematics was related to his falling out with Bourbaki. Already in the 1957 Bourbaki congress, Grothendieck had begun to have doubts about continuing to work with Bourbaki, and even doubts about continuing to do mathematics [source]. Even at this early stage he felt strongly that Bourbaki should redo its work on the foundations of mathematics, changing these from set theory to category theory, In addition to its great generality and power, category theory does not suffer from the inherent limitations of set theory. As Pierre Cartier puts it: “Set theory is too restrictive: an element is either a member of a set or not, there is nothing in between.” [source] And then, of course, there are the great paradoxes in set theory which make the discipline full of theoretical holes.

Some members of Bourbaki supported Grothendieck’s push toward category theory. But others – and these were the influential ones – felt that the foundation had already been laid and there was no going back. Grothendieck tried hard to convince them of the importance of this new proposed direction for the group. But he failed miserably. Grothendieck then got up, left in anger, and never came back to Bourbaki again. His own writings at the IHES had already begun to supplant the entrenched and inflexible work of Bourbaki.

In his memoirs, Grothendieck described his disagreement with Bourbaki, as well as his views about the group, its mission and how it operated, and the vision of structure in mathematics. He wrote as follows: [source]

“I am struck by the fact that I haven’t here thought of the vast synthesis of contemporary mathematics that attempted to present the (collective) treatise of N. Bourbaki. This happened, it seems to me, for two reasons. First, this synthesis dealt with “putting in order” a vast collection of ideas and results that had already been known, without bringing into it new thought. If there is in all this a new idea, it is that of “structure,” which revealed itself as the precious thread that runs through the entire treatise. But this idea seems to me similar to that of an intelligent and imaginative lexicography, rather than a new element of language, providing a new understanding of reality (here, mathematical reality). Additionally, since the 1950s, the idea of structure has become passé, superseded by the influx of new “categorical” methods in certain of the most dynamical areas of mathematics, such as topology or algebraic geometry. (Thus the notion of “topos” refuses to enter into the “Bourbaki sack” of structures, decidedly already too full!) In making this decision, in full cognizance, not to engage in this revision, Bourbaki has itself renounced its initial ambition, which had been to furnish both the foundations and the basic language for all of modern mathematics.”

Of course, Grothendieck is the inventor of topos as well as of other mathematical ideas that were not incorporated by Bourbaki. It is easy to understand his feelings about Bourbaki’s actions, and to understand why he left the group. History would prove him right. Bourbaki would decline following its refusal to accept new methods.

Quoting Aczel (2007, p. 204):

Why did Bourbaki and its ideas decline after the 1970s? What are some of the new approaches to mathematics that are now sweeping the academic world, and why do these ideas generally fare better than Bourbaki’s? Which of Bourbaki’s ideas are still dominant today? And why do these ideas survive?

The decade of the 1970s witnessed the decline of both Bourbaki on the one hand and French structuralism on the other. At the same time, Bourbaki also gave up its role as the cultural connector among the various fields of human study in which structuralism was important [source].

Bourbaki achieved its goals of axiomatizing mathematics, stressing structure, and promoting rigor in a discipline that had fallen into laxity in the decades before the emergence of the group. Bourbaki began its decline after the end of the 1960s precisely because it had achieved its goals so marvellously. Henceforth, mathematics was carried out in a much more rigorous and precise way than had been done prior to the group’s great contributions. There was, therefore, no more need for the group – there was nothing more to innovate.

Another factor that contributed to the decline of Bourbaki was that its members became well-known mathematicians under their own names. They were being awarded prizes and medals, the same medals they themselves had fought so strongly against in the early days of the group. The Bourbaki organization had become too powerful; and yet it now lacked its main purpose since all its goals had by now been achieved.

Bourbaki lost an incredibly important opportunity to remake its oeuvre in the new form of the theory of categories, something that would have better suited the study of structures than did the old theory of sets with its myriad problems and inadequacies. In addition, new theories in mathematics, such as chaos and fractals, as well as René Thom’s catastrophe theory, emerged and demonstrated that structuralism is not absolutely necessary for doing good mathematics. [source]

Bourbaki had a chance, through the work of Grothendieck and his students, to refound modern mathematics on the theory of categories, but Bourbaki missed that chance. In part, this missed opportunity led to the demise of Bourbaki. For mathematics remained based on a flawed system, set theory, rather than something that would have been much more appropriate. Bourbaki had a possibility in the late 1960s to redirect itself towards a more ambitious goal. Mathematics had evolved further and reached a place in which new foundations could be laid for the discipline. This direction would have been possible because of the work of one man: Alexander Grothendieck.

Quoting Aczel (2007, p. 213):

Another factor that has led to the decline of Bourbaki has been the great opposition that the group has encountered in its work. Bourbaki has brought rigor, abstraction, generality, preciseness, and structure into modern mathematics. Most mathematicians have welcomed these developments. However, there came a point at which many mathematicians felt that Bourbaki had gone too far.

In reviewing the works of the group, mathematicians outside it have noted an overreliance on generality and abstraction. Bourbaki seems to many to have reached the point that the generality is more important that specific cases. Besides the fact that this trend makes Bourbaki’s writings very hard to follow – and the group itself acknowledges that its books could not and should not be used as textbooks for teaching mathematics – this trend is contrary to intuition and to the way mathematics is actually done. In general, the human mind does not work in great generalities.

Rather, the progression is from the specific to the general. A mathematician would usually start working on a specific problem or theorem, and once it had been solved, move on to see if it can be generalized. Generalities mean more power, greater meaning, and far enhanced significance. But few start with a general statement.

Another problem is the abstraction and rigor. Abstraction and rigor in mathematics are necessary to ensure that results are precise and correct and that there are no missing or false steps in proofs. This is, of course, a very important aspect of mathematics and its promotion is certainly something that Bourbaki should be commended for. However, the abstraction and rigor should be the tool, not the purpose. In Bourbaki’s works, it often seems that the writers have turned abstraction into a goal, and rigor takes over and leaves absolutely no place for intuition or even general understanding. Bourbaki does not, in general, use pictures or other visual aids: thus it completely discourages any understanding of the material that is “human.” For how many people can see what is going on in a proof simply by following very technical details? Most mathematicians rely, at some point, on some kind of a mental picture of what is going on in a theorem or a proof.

It is this blind reliance on technical details, strict adherence to rigorous procedures, and an over appreciation of generalities at the expense of the specific case, the picture, the intuition, the human idea of a mathematical problem which have made Bourbaki disliked by some mathematicians. Having brought us their ideas of abstraction and generality and structure, Bourbaki lost its lead as the world of mathematics moved forward.

Quoting Aczel (2007, p. 213):

If he is alive, Grothendieck is still hiding in the Pyrenees. He is hiding very well now, since attempts to find him have failed. Apparently this is what he wants: to be alone, to write and destroy his own writing, and not to have any connection with people other than grocery store clerks or laborers who might do occasional work for him.

His disappearance and his anger with the world symbolize the demise of Bourbaki. For Grothendieck alone held the great hope for the future of Bourbaki. Grothendieck and his work were the next stage in the program of abstraction and generalization in mathematics that Bourbaki had embarked upon. Alexander Grothendieck was the human incarnation of the essence of Bourbaki, of the ideals that Bourbaki strove for in mathematics, for here was a man who actually thought in great generalities, and for whom axiomatic thinking was natural. Grothendieck’s oeuvre was, in fact, all about structure, so that the structuralist idealism of Bourbaki found in Grothendieck’s work its finest manifestation.

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Quoting from Grothendieck’s Récoltes et Semailles:

In those critical years I learned how to be alone. [But even] this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation [1945–1948], when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law.

By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like “volume” which was “obviously self-evident”, “generally known,” “unproblematic” etc… it is in this gesture of going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one – it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle – while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials of, or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective of thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

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