# what exactly is mathematics? Apocalypse of the Burbaki School

Nicolas Bourbaki, French mathematician, born in 1886, received a scholarship after graduating from university, first went to Paris, then to the University of Göttingen, and studied under Poincaré and Hilbert, respectively, in 1910He completed the defense of his dissertation. His academic career is colorful and his collaborators are countless. His reputation is skyrocketing.

—— The accomplished mathematician in the resume is actually the famous Bulbachian school, and the author of the resume is one of the founders of the school, the mathematician Andre Wey. Although thisThe fabricated Burbakis studied under Poincaré and Hilbert, but the true Burbaki school always believed in axiom systems and structuralist ideas.

From the original discussion and writing of a modern analysis textbook of more than 1,000 pages in a restaurant in Paris to the final publication of dozens of volumes of the "mathematics original" of the twentieth century, "How the Mathematical Original Works" was created,Has its own legend? How does its structuralist ideas affect the entire mathematical community, and what does it mean for today's mathematics? December 10, 2019 is the 85th anniversary of the first meeting of the Bourbachian School. This winter may be just usTime to review its history.

Writing | Ding Yi Professor, Department of Mathematics, University of Southern Mississippi, USA

For centuries, the French people have played a leading role in leading the tide of cultural progress and technological development in the modern world, and have contributed generations to generations of geniuses. In the seventeenth century, scientific philosophers Descartes, PascalTo Fermat, the "king of amateur mathematicians"; from the enlightenment thinkers Voltaire, Diderot, and Rousseau to the French Revolution of 1789; from the great novelist Balzac, Hugo, and reality in the 19th centuryThe sculptural master Rodin, to the mathematical genius Poincaré and the biological giant Pasteur who benefited mankind, and the outstanding French scholars in various industries they represented added glory to humanity.

In France in the 1930s, a mathematical group called "Burbachi" appeared, and its members were young French mathematicians who were not well known at that time. However, for a short time,This group, with its concept of mathematics and its writings, quickly rose to the sky of world mathematics like a dazzling pearl and reached its glory in the 1960s and 1970s. Its influence on the international mathematics community and even primary and secondary school mathematics education is not to be ignored.It was great at that time.

In the new century, although Burbaki is no longer as glorious and its works are rarely published, it still exists, it still holds regular seminars, still holds mathematical conferences, and still affects mathematics around the world. China ’s currentThe atmosphere and prospects of mathematics are similar to the environment at the time of the birth of Bourbaki, and the state of thinking in the mathematical world is comparable to the situation in France at that time. So revisit Bourbaki's influence on the international community in the last century.A history of the rise and fall of the mathematics community of decades of mathematics. An analysis of its mathematical worldview may give us a clue about "what exactly is mathematics?"

Burbaki's Birth

How did Bourbaki come out of nowhere? It depends on the state of French mathematics at the time. In World War I at the beginning of the last century, the German government of the host country cleverly managed to protect its young talents.I saved the front line from being a cannon fodder, thus preserving a group of future high-quality scientists. Probably they learned something from the famous French commander Napoleon a hundred years ago, "I will not kill the chicken that lays eggs."Forget the advice of the great historical people, let the young people gather under the patriotism, and go to the front battlefield without discrimination. Even the students of the Paris Normal University the Paris Normal University or the Normal University, an elite university founded in Napoleon in 1794, alsoNo exception. As a result, almost half of the mathematics students who entered the university from 1911 to 1914 lost their lives in the war; a quarter of the 331 students who entered the normal university from 1900 to 1918 did not get off the battlefield.Come back.

By the 1920s, a group of future elites in French mathematics were admitted to the Paris Normal University. Five of them were "midwives" born in Bourbaki ten years later: Henri Cartan 1904-2008, XieClaude Chevalley 1909-1984, Jean Delsarte 1903-1968, Jean Dieudonne 1906-1992, and Andre Weil 1906-1998. These fiveAmong the first members of Bourbaki, Delsat and Wey, Cartan, Dieudone, and Chevalle entered the school in 1922, 1923, 1924, and 1926, respectively. Therefore, the Paris Normal University was the cradle of Bourbaki..

In the years of their university knowledge, the group of mathematicians in Paris Normal University and the broader French mathematics community fell into disuse due to the World War. Although Charles Picard 1856-1941 and Jacques Hadamard 1865-1963, Felix Borel 1871-1956, Henri Lebesgue 1875-1941 and other world-famous mathematicians are still alive, but they are all old and well overYears of creativity. A new generation of mathematicians 20 or 20 years younger than them may have died or have not been plumped. The dreary class taught by grandfather-level professors and the use of old textbooks that have not changed for many years have allowedThey are deeply disappointed.

In 1982, in an interview, Cartan recalled the original idea and direct cause of the birth of Burbaki :

"In 1934, Wey and I both taught at the University of Strasbourg. I often talked to him about the calculus classes I taught. Because the textbooks used were not satisfactory, especially about multiple integrals and StoweMax's theorem, I've been thinking about how to teach this course in the best way. We and We discussed several times about my concerns. One day, the wind and the sun, he said to me, 'We need to solve it permanently.This question. We should write a good textbook on analysis. Then you won't complain anymore. '"

Wey, the main founder of Burbaki, in his autobiography The Apprenticeship of a Mathematician published in 1991, confirms the correctness of the memory of the previous founder :

"One winter day in 1934, I came up with a trick to stop my friend's non-stop question. I told him Jiadang, 'We have five or six people teaching the same class at different universities. Let's solve this together.Question, and then I will finally not hear your question. 'I did not expect that Burbaki was born at this moment. "

In addition to the five people mentioned above, Jean Coulomb 1904-1999, Charles Ehresmann 1905-1979, Szolem Mandelbrojt 1899-1983; Father of Fractal, MundUncle of Benoit Mandelbrodt 1924-2010 and Rene de Possel 1905-1974 also participated in the creation of this mathematical group. Of the nine founders, only born in PolandThe oldest Mandebroy was not a teacher. Poser was not a mathematician in nature, but a geophysicist. He had "leave the party" as early as 1937.

Wednesday, December 10, 1934, six of the nine-Cartan, Chevalle, Delsat, Dieudone, Posser and Wey-took advantage of PangaThe Julia seminar at the Lai Institute took the opportunity to have their first "working meeting" around the table while having lunch around the basement of a restaurant called A. Capoulada in the Latin Quarter of Paris. They were all under 30 years old,The youngest Chevalle is only 25 years old and has not yet officially taught. The remaining five do not teach at the universities of Paris: Cartan and Wey at the University of Strasbourg, Delsat at the University of Nancy, and Diedo.University of Rennes and Poser at Claremont-Ferrand University.

Their purpose is simple and straightforward: to write a new analytical textbook for French higher education to replace currently unsatisfactory textbooks-such as Edouard Goursat 1858-1936 Of Analytical Science. According to Wey's idea, it is necessary to "build calculus content for the next 25 years by writing an analytical course covering a wide range of materials", and the book should be "as far as possibleModernization. "To fulfill this mission, Delsart strongly supported the idea of collective writing, and hoped that the first volume would be published six months later in fact, the first volume was not published until 1939, and Cartan proposed a complete set of booksThe maximum length is between 1000 and 1200 pages in fact, the total number of pages in more than 30 volumes of the book with "unpredictable" completion date has exceeded 6,000 pages, and Wey suggestedSeveral subcommittees have been established to confirm the content of each chapter.

In July 1935, this group of young mathematicians held the first plenary meeting. July 16th can be considered the birth day of "collective pseudonym" Burbaki. On that day, they were busy meeting but discussing the problem without successDecided to take a break and ran to Pavin Lake, three miles away. An educated statement is :

"Several of them have enough courage to jump into the water and swim naked in the waves while yelling 'Barbaki' hundreds of times."

"Burbaki" just landed on the earth!

Scene of the 1938 rallies of the Burbaki School. | Wikipedia

Legend of Burbaki

"Burbaki" is just the pen name of writing teams such as "Liang Xiao" and "Ding Xuelei" that I often see when reading newspapers when I was young? No, it is a 19th century French general of Greek origin.Last name, the general's name is Charles Bourbaki 1816-1897.

General Burbaki graduated from France's "West Point Military Academy"-I'Ecole Speciale Militaire. He has a career in horse racing for decades and fought countless battles from Africa to Italy. In the Franco-Prussian War of 1870-1871He won first and then lost, which also reflects the scene of France at the time.

Why might this former soldier who may not know how many calculus is related to today's math talents? In fact, this is related to a prank from Paris Normal University. In 1923, a third-year college student in the mathematics department Yu Song Raoul Husson decided to make a prank on the freshmen of the first grade. He posted a poster saying that Professor Holmgren was going to make a lecture and all the freshmen had to participate. The result is as described by Wey in his autobiography :

"Wearing a fake beard and making strange noises, he introduced himself to classical function theory after introducing himself to the students, and then rose to a high level of mystery, and finally ended with the" Burbachi Theorem. "This story becameLegend, but even more legend is that a student claimed that he understood the entire lecture. "

The original creator Yu Song saw the name of Burbaki from French military history. The French-Prussian war just 55 years later. In addition, there are students of the normal university under the command of General Burbaki. His name is still in people's memory.So Yusong borrowed his name from Zhang Guan and Li Daidi to use it in mathematics!

In the 1930s, Mathematicians once again under the command of "Burbaki", they really did humorous acts. First they decided to use his last name as the name of the group they formed. In order to show it as a characterThe “real existence” of individuals, they decided to publish a signed mathematical article in the French Academy of Sciences, but this still needs a name. Wey ’s future wife, Eveline, created a connection with Burbaki.The name of the last tsar was "Nicholas", so Nicolas Bourbaki became the official full name of this mathematical organization. Wei Yi volunteered to compose the resume of Burbaki, and opened a father Lao Jia, Chen Jiashen's teacher Lao JiaWhen Elie Cartan, 1869-1951 back door, sent to the Academy of Sciences Pickup, because publishing in the Academy of Sciences requires the recommendation of an academician.

This Nicholas Bourbachi born out of nothing, is a Wey who loves language and literature [note his sister Simone Weil 1909-1943 is a French philosopher and social activist with a world reputation]Depicted by Sheng Huaqiao, he is a very accomplished mathematician. Born in 1886, after graduating from Kharkov University in his home country, he received a scholarship, went to Paris, and then to the University of Göttingen, where he studied under Poincaré.Henri Poincare 1854-1912 and David Hilbert 1862-1943, successfully defended his dissertation at his alma mater in 1910. Later, his academic career was colorful, countless collaborators, and his reputation skyrocketed. In order to avoid being queried, Wey intentionally stated that Burbaki's doctoral dissertation was destroyed after the German invasion in 1941, which is truly pictured and covered up.

Andre Wey, mathematician. | Wikipedia

After Burbaki became famous, people in mathematics circles in various countries have never witnessed public respect in public. He is also a hermit like a hermit, and has become a mystery in mathematics. If he is indeed a real person, he is better than before his death.Alexandre Grothendieck 1928-2014, or Grigory Perelman 1966-, who proved Poincaré's conjecture, was even more "hermit". Until many years later,There is an article published in the Monthly, and in the author's profile, he still refers to himself as such :

"Professor Nicholas Burbaki, formerly at the Royal College of Poldavia, now settled in Nancy, France, wrote a set of" Mathematics Originals ", which is a comprehensive series on modern mathematics published by Hermann since 1939, Ten volumes have been published. "

Burbaki's ambition

Burbaki's initial intention was to write a new analysis textbook, but they soon began to be "ambitious" because they wanted to provide set theory and other basic subjects for "analysis" writing."Preliminary knowledge" or even "relevant knowledge", but in this way, the coverage of the book is getting bigger and bigger like a snowball. No one expected that so far they have published more than thirty volumes. Why their career is onGoing bigger and bigger before the 1960s and 1970s?

The reason is that these young people who have left a reputation in the history of mathematics at the beginning had the ambition of estrangement. Two thousand years ago, the ancient Greek mathematician Euclid gathered the achievements of his predecessors in geometry and wrote "GeometricsOriginally The Elements, with almost impeccable axiom system, inferred hundreds of geometric and arithmetic theorems, which sounded the sound of the horn that led to the rational thinking of modern Western civilization, and became the second edition in print after the Bible."Immortal works."

By the 1930s, after the back-and-forth discussions and enthusiastic discussions among the Burbaki people, everyone agreed that the textbooks of modern mathematics could not keep up with the pace of modern mathematics, especially in France, which was hit hard by the World War., Lack of a unified view of mathematics, that is, they believe that "mathematics depends on structure" philosophy. They should use this creed as a writing principle, and treat mathematics as an organic whole instead of a combination of fragments of each branch to reconstruct the mathematicalTherefore, they set aside the original idea of writing only a modern analysis textbook based on calculus thinking, and decided to collectively write a magnificent masterpiece full of French literary style, which embodies the "structuralism" of contemporary mathematics.

Then, they imitated Euclid and took the title of "Elements of Mathematics", hoping to become Euclid in the 20th century and lead the trend of international modern mathematics education. 65 years later,At the end of the century, they achieved their goals very well. According to a book by Bourbaki: A Secret Society of Mathematicians translated and published by the American Mathematical Society in 2006, it is in the original FrenchAt the time of publication, the published books were: 1. "Set Theory", 2. "Algebra", 3. "General Topology", 4. "Single Real Variable Function", 5. "Topological Vector Space", 6.Integral Science, 7. Commutative Algebra, 8. Differential and Analytic Clusters, 9. Lie Groups and Lie Algebras, 10. Spectral Theory. There are several volumes in almost every book.

Volume 1 of The Original Mathematics, Set Theory 1970 | Wikipedia

The writing process of these books is a bit like the creation of the revolutionary modern Peking opera fifty years ago in China: excellence and refinement into steel. They are the products of "renovation", one belongs to art and one belongs to science, but they all belong to the same philosophical speculation.In the collective view of Bourbachi, Van Der Walden's "Algebra" is a model of mathematical writing, so he decided to write each chapter in a style like it. But the basis of Van Der Walden's bookIt was the notes of Emmy Noether 1882-1935 and Emil Artin 1898-1962. He didn't need to be disturbed by others, but he could complete the masterpiece by himself. See "A hundred years ago, she became the first female mathematics lecturer in German history.

Burbaki is different. They are "writing teams." Before each book is written, everyone discusses how to write, what to write, and the order of material layout, etc., everyone must express their opinions and make suggestions.Any member has the veto power as a permanent member of the United Nations, and only after unanimous agreement decides to write. At this time, a volunteer takes over the task and he will freely write a preliminary draft according to the ambiguous plan set by everyone. One or twoYears later, the first draft was read out at the Burbaki meeting. At the "review" meeting, they followed the advice of the outstanding American mathematician Herbert Federer 1920-2010 about revising their mathematical works "It seems that you are the author's most vicious"The enemy" is like a lot of criticism and relentless criticism of the first draft. As a result, the body of a manuscript disappears completely, so I write the second draft, even the third draft, the fourth draft, and even the sixth, seventh, and eighth drafts.. The last publication the reader saw was such a well-finished product.

As mentioned earlier, the Burbakis believe in "structuralism." They think that mathematics studies structures: algebraic structures, topological structures, and ordinal structures. Algebraic structures are concerned with algebraic operations, topological structures, and "continuous"Concepts are related, and they are the object of analytic research, while the order structure considers the relationship of size like real numbers. The dozens of books written by Bourbachi are all the crystallization of structuralist ideas. These elaborate works focus on theoretical concepts.The structural analysis of the book is divided into different structures. The material in the book is organized and summarized. Each one is right. The topic is in the right position. It has a strict logic and has a great impact on contemporary and subsequent mathematicians and students. From Europe to Asia, from North AmericaIn South America, the nutrition of many adult math parents is one big book after another!

Burbaki event

As a mathematical community, although it is "semi-secret", Bulbachi has its own public mathematical activities. Since the first plenary session that lasted for a week in July 1935, until 1939, such meetings will be held annually.After World War II broke out, members could not gather, but sporadic connections still existed, and all activities resumed after the war. Until the Rockefeller Foundation of the United States began to provide funding in 1948, the members of Burbaki participated in at their own expense.The activities fully demonstrate their unparalleled love for mathematics.

Unlike the formal mathematical community, Burbaki also has no red tape like "learning bylaws." Its formal membership system reflects the tenet that "math is the cause of young people.""Must quit" is an unwritten rule. Young mathematicians want to participate in events and listen to respect, but if they want to become new members, they must be able to withstand the blazing fire that Bulbachi discussed during mathematics.Psychological preparation, and the passion to add fuel to make the fire more prosperous. Those timid people sitting side by side listening to silence or those who are less active in mathematics will not be invited to participate in the event. Many yearsIn the past, full members of Burbaki generally remained around a dozen.

France and the Soviet Union have a tradition of seminars in mathematics, France has a seminar in Adama and a seminar in Gaston M. Julia 1893-1978, and the Soviet Union has Andrey Kolmogorov 1903-1987 and Israil Gelfand 1913-2009. Since 1948, Bourbachy has held seminars, held three times a year on the weekends in February, June, and November each year.The next five invited reports were attended by about 200 people. The contents of the report are the latest results in various aspects of mathematics, and the final collection was published. This year ’s last Bourbachi seminar, the report includes Wei Dong, a postdoctoral fellow at the Beijing International Center for Mathematical ResearchYi and collaborators' results.

Burbaki's influence

Through the publication of dozens of volumes of mathematical works, Burbaki became one of the most influential mathematical schools in the last century, and its reputation reached its peak in the 1960s and 1970s.

Wei locked the Parisian publishing house Hermann to work with them from the beginning. The editor Enrique Freymann published a first edited by Chevalle and Wey in honor of an outstanding logician who died early.This collection of papers later contributed to the sale of the Burbaki series. In fact, Chevalle even claimed that the publisher had long encouraged both of them to write analytical tutorials to replace Gulsa.Possibly adventurous, Freymann supported Burbaki's cause with great enthusiasm from the beginning. He gave him all his help, and was deeply moved by the first volume of Set Theory, published in 1954.Dedicated to him who died unfortunately during proofing.

Burbaki's mathematical view is consistent with the formalist school of one of the three universities of modern mathematics. They are disciples of German Hilbert, not receivers of French Poincaré. Axiom system and abstractionStructuralism is the two flags that they have been waving like a day for decades. They believe in: the unity of mathematics, axiom methods, and the study of structure. This was in their article signed by Nicholas Burbaki in 1947The structure of mathematics has further elaboration. The mathematics book they wrote shows the central hall, brick and tile structure, door and window corridor, ceiling surface, cantilever eaves and other architectural elements of modern purely mathematical buildings.The beauty of rich mathematical architecture. The delicate writing attitude makes almost every book the best, and the cooperation of publishers, during the honeymoon period between mathematicians and publishers until the 1970s, the volumes of "Mathematics Originals"After the publication's sales are booming, not only are many mathematicians buying them, but libraries around the world also need to hold one, for a long timeBecame the main source of profit for the publishing house.

Hilbert believes in the axiomatic system. He tries to incorporate all existing theories into a limited and complete set of axioms and proves that these axioms are compatible. Poincaré believes that mathematics is a reflection of the real world. | Wikipedia

However, if you think that the main achievement of Burbaki is only a dozen large books with more than 30 volumes, it is very wrong. The members of Burbaki, whether the founders or the Mesozoic, or even the young generation, manyThey are all top mathematicians, and the most famous of them is probably Wey. He is one of the greatest mathematicians in the world in the last century. His profound work in the field of algebraic number theory and algebraic geometry has a profound impact. The same is the foundation.Diedone, who is the main writer of this mathematical circle, has worked hard in many fields throughout his life, written works, etc., and later wrote a history of mathematics in functional analysis and other disciplines. His own book, "Basics of Modern Analysis"”Foundations of Modern Analysis, which was selected as a graduate textbook by the Department of Mathematics of Nanjing University in the late 1970s. At that time, two classmates of pure mathematics—Tian Gang and Wang Hongyu—listened in this course.According to Professor Su Weiyi, one of the teaching teachers, the two of them have the highest test scores in the class. It can be seen that Bourbachi and his members have also continued to young students in the east.Affect students.

Almost all French Fields winners are later members of Bourbaki, such as Laurent Schwartz 1915-2002, Jean-Pierre Serre 1926-, GrothendieckAnd the young man who died early in the year, Jean-Cristophe Yaccoz 1957-2016. Adrien Douady 1925-2006, who has made outstanding contributions in the field of fractals, is also among them. Among the early members,The only foreigner, Samuel Eilenberg 1913-1998, was from Poland and later in the United States. The category theory created by him and American mathematician Saunders MacLane 1909-2005 is now a hot topic in theoretical physics.Mathematical tools.

Burbaki has not only influenced the world with its mathematics and writing for decades, but also created some mathematical concepts and symbols, many of which have become famous overnight, such as the original French filtre and the English translation filter. MostA well-known example is the symbol of the empty set ?, which was created by Wey. The three functional terms now widely used are "injection", "surjection", and "bijection".It was also an invention of Burbaki.

Burbachi, as a group of mathematicians, has also entered the art world, and has a certain impact on structuralist artistic schools. However, its impact on elementary mathematics education has led to the vigorous "new mathematics in the 1970s"The birth of the movement. "New Mathematics" will defeat Euclid, and "Set Theory" will be brought into the classroom of primary and secondary schools. However, this effect is negative, because the practice of "New Mathematics" is almost a fieldDisaster. Japanese mathematician Kodahiko 1915-1997, a Japanese mathematician who lives in the United States and teaches at Johns Hopkins University, saw his daughter fall victim to this experiment, and in his book "The Lazy Man""New Mathematics!"

From a historical perspective, Bourbaki contributed greatly to the entire mathematics, affecting several generations of mathematicians. However, its mathematical outlook and philosophical ideas have also been criticized. The most fierce crusaders are probably non-Russian mathematiciansVladimir Arnold 1937-2010 is the best. As far as mathematical philosophy is concerned, Arnold is a follower of Poincaré, not a fan of Hilbert. Of course, this does not prevent him from solving the latter's "23 problems"One of them. The outstanding disciple of Kolmogorov has a unique personality, sharp words, and relentless criticism. I read in one of his articles that he interviewed a job-seeking French mathematician.His major is linear algebra, but he ca n’t answer the simple question, “What is the sign difference of the quadratic form xy?” This French professor may have read too much Bourbachy ’s work, and paid too much attention to “general” and ignored it."Specificity", which is what Arnold opposes most. Arnold probably admires the famous saying "art originates from life but is higher than life." He believes that mathematics is a reflection of the real world, not the first.Studies of the structure and then apply it to the real world to his famous saying is:.. "Mathematics is part of physics"

This also proves the shortcomings of Bourbaki from another aspect: too much emphasis on pure mathematics and neglect of applied mathematics, such as the statistics that are particularly popular now are not considered by them. This is another criticism criticized by many mathematicians.On the one hand, it is ironic that Mundbro, the nephew of Mundbroi, the oldest founder of Bourbachi, was a vanguard against Bourbagi. When he was less than 30 years old, he even fled the country.And went to the United States, where he pioneered fractal geometry that Burbaki probably didn't appreciate much in the 1960s.

Today, 85 years have passed since the first working meeting of Burbaki. Its history of growth and rise and fall, its immortal merits and negative effects on world mathematics, have apocalyptic effects on Chinese mathematics. ModernThe development and application of mathematics should be the unity of Newton-Poincaré and Hardy-Hilbert. It should be the complementation of concrete and abstract. It should be the organic combination of application and theory. This is probably the Chinese mathematics effort to surpass the world.A feasible way of advanced level.

written on December 10, 2019

85th Anniversary of the First Meeting of the Burbaki School

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