As we know, the background of bourbaki's birth is the vigorous development of new branches of mathematics and the decline of French mathematics. This decline or stagnation is not only reflected in the backwardness of emerging branches, but also in the lack of mathematical talents caused by war factors. During his career in bourbaki, J. Dieudoone commented on French mathematics and French mathematics education at that time:
..... At that time, World War I had just ended, which must have been very painful for French mathematicians ... In the war of1914-1918, the German government and the French government disagreed on scientific issues. The German government sent their scholars to engage in scientific work. The French government believes that everyone should go to the front to fight at least one or two years after the war begins, so young French scientists go to the front to do their duties like other French people. We can only show respect for this spirit of democracy and patriotism, but the consequences are a terrible massacre for the younger generation of French scientists. When we open the student register of wartime normal university, we will find a huge fault, which shows that two-thirds of the students were destroyed by the war. This situation has brought disastrous consequences to French mathematics. At that time, we were too young to go to war directly, but we entered the university a few years after the war. We should have been guided by those young mathematicians, many of whom will certainly have great prospects. They are young people brutally destroyed by the war, and their influence has been completely obliterated.
..... Of course, the last generation left is the great scholars whom we respect and admire. Masters like Picard, Mundell, E. Bole, Jordan, LeBerger, etc. All are still alive and very active, but these mathematicians are all close to 50 years old, and some are even older. There is a generation between them and us. I'm not saying that they didn't give us the best math. We have all listened to the first-year courses of these mathematicians. However, it is indisputable that a 50-year-old mathematician only knows the mathematics he learned at the age of 20 or 30, and only has some rather vague concepts about the mathematics he learned at that time (that is, at the age of 50). In fact, we can only accept this situation and there is nothing we can do.
..... I still remember the day when the book B.L. Vander Walden (1903- 1996) was first published. At that time, I knew nothing about algebra, and now I can't go to college. I hurried to these books and was shocked to see the new world open in front of me. At that time, my knowledge of algebra did not exceed that of preparatory mathematics, determinant, solvability of one-point equation and one-way curve. I had graduated from normal college, but I didn't know what an ideal was, I only knew what a group was! This will let you know what a young French mathematician knew in 1930. ...
It is precisely because of the revival of the French mathematical tradition and the reversal of the bad trends in all aspects of French mathematics that a group of young French mathematicians have gathered.
In the decade after World War II, bourbaki's popularity reached its peak. Principles of Mathematics has become a new classic and is often cited as a document. The topic of the bourbaki seminar was undoubtedly the latest achievement of mathematics at that time. In the international mathematics field, Weil, Katan, dieudonne, Xue Huali, Searle, Grothendieck and others all have important influences. It is also at this time that three tasks that determine the image of bourbaki are in full swing:
In addition to some small papers, the two most important papers published in the name of bourbaki, Architecture of Mathematics and Mathematical Basis of Mathematical Researchers, were published in 1948 and 1949 respectively. In fact, they are the program and declaration of the Bourbaki school, and they are the original documents of the Bourbaki school. At the same time, the main members of bourbaki also expressed their views on mathematics, its history and development. These two articles are very good, I recommend reading them! )
Further publication of bourbaki's main work "Principles of Mathematics". During World War II, bourbaki's Principles of Mathematics only published four volumes. Starting from 1947, the release is accelerated. 10 years, published 18 volumes, to 1959, published 25 volumes, basically separating the "basic structure of analysis". During this period, many volumes were reprinted many times. At the same time, bourbaki's thought and writing style became the object of imitation by young people, and soon "bourbaki's" became a special adjective.
The establishment of Buerbaki seminar. The report of the seminar reflects the great progress of current mathematics, which is not only a simple introduction, but also digested, absorbed and even recreated by the speaker, which is very important for mastering the current mathematical trends. It can be said that this form of seminar originated in Germany has taken root and sprouted all over France. (The bourbaki seminar seems to be still going on now. )
In addition, through the efforts of two generations of bourbaki members, algebraic topology, homology algebra, differential topology, differential geometry, theory of multiple complex variables, algebraic geometry, algebraic number theory, Lie group and algebraic group theory, functional analysis and other mathematical fields have finally merged together, forming the mainstream of modern mathematics, and the leading position of French mathematicians in the international mathematical field has also been recognized by everyone. This can be seen from the fact that they have won international mathematics prizes in succession.
1970 or so, bourbaki generally moves towards its opposite and tends to decline. At this time, the founder and the second generation of bourbaki withdrew one after another, and the influence of the younger generation could not be compared with that of the older generation. Mathematics itself has also undergone tremendous changes. Bourbaki neglected analytical mathematics, probability theory, applied mathematics and computational mathematics, especially theoretical physics and dynamical system theory, but the emphases in 1950s and 1960s-algebraic topology, differential topology and function theory of multiple complex variables were relatively stable, and mathematicians were more interested in classical and specific problems than in the construction of big theoretical system. Mathematical research tends to be more specialized and technical. From the 1970s to the mid-1980s, mathematics presented a diversified situation. Obviously, there are very few emerging disciplines in recent years, which cannot be compared with bourbaki's founding period. Although, by the mid-1980s, a new trend of great unification of mathematics was taking shape, it was already a more advanced unification based on the unification of bourbaki. On the other hand, many mathematicians with classical views simply deny this unity, and quite a few people are only keen on specific, extremely professional and even trivial problems, so it is difficult to integrate into mainstream mathematics. In fact, the third and fourth generations of bourbaki are mostly experts in a certain field. Since 1970s, the report of Buerbaki seminar has also reflected this trend of specialization and technicalization. In this case, since 1970s, fewer and fewer people have cited bourbaki's mathematical principles in their papers.
Bourbaki's failure in education is also one of the reasons for its decline. Influenced by bourbaki, the so-called "New Mathematics" movement appeared in 1950s and 1960s, which introduced abstract mathematics, especially abstract algebra, into textbooks of middle schools and even primary schools. This sudden change not only makes students unable to accept the new textbook, but also makes teachers unable to understand it, which leads to the confusion of the whole mathematics education. This is a great failure of bourbaki in education. In higher mathematics education, even the textbooks compiled by the founder of bourbaki later broke the formal system of bourbaki and adopted a more natural, concrete and gradual system. In a sense, this is the negation of negation and the return to the old tradition.
So from now on, as a school that has trained so many top mathematicians and practiced such an ambitious unified planning of mathematics, it has made great contributions to mathematics itself and mathematics education.
Any school is made up of people. As a legendary "mathematician", bourbaki will eventually grow old. But the ideas put forward by the Bourbaki School, their mathematical principles and their efforts for the unity of mathematics will still affect everyone who loves mathematics. Perhaps, when mathematics is unified again, people will find that the wonderful starting point of this career is bourbaki's career.
work
Bourbaki tried to base the whole mathematics on set theory, although many people opposed it from the beginning. Algebraic geometry has been formed for decades and hundreds of years. Whether its numerous achievements, large and small, can form a rigorous mathematical building on the basis of abstract algebra and topology has become the touchstone of bourbaki's viewpoint. At the end of 1935, bourbaki members unanimously agreed that mathematical structure should be the basic principle of classified mathematics theory. The concept of "mathematical structure" is an important invention of Bourbaki school. The source of this idea is the axiomatic method adopted by bourbaki, who opposes the classical division of mathematics into analysis, geometry, algebra and number theory, but classifies the basic disciplines in mathematics with the concept of isomorphism. They believe that all mathematics is based on three matrix structures: algebraic structure, ordered structure and topological structure. The so-called structure is "the similarity of various concepts is only because they can be applied to the collection of various elements." The nature of these elements is not specified. Defining a structure is to give one or several relationships between these elements. Establishing the axiomatic theory of a given structure from the conditions satisfied by a given relationship (they are axioms of the structure) is equivalent to deducing the logical inference of these axioms only from the axioms of the structure. "Therefore, a mathematics discipline may be composed of several structures, and each structure has different levels. For example, a real number set has three structures: an algebraic structure defined by arithmetic operations; Sequence structure; The last one is the topological structure according to the concept of limit. The three structures are organically combined. For example, Lie group is a special topological group, which is a combination of topological structure and group structure. Therefore, the classification of mathematics is no longer divided into algebra, number theory, geometry, analysis and other departments as in the past, but based on whether the structure is the same or not. For example, linear algebra and elementary geometry study the same structure, which means that they are "isomorphic" and can be treated together. In this way, they disrupted the order of the classical mathematics world from the beginning and unified the whole mathematics with a brand-new point of view. The main work of Bourbaki School is Principles of Mathematics. Its first goal of completely axiomatizing the whole mathematics is to study the so-called "basic structure of analysis". This belongs to the first part of mathematical principles.
The first part is divided into:
Set Theory Volume I Volume IV Functions of One Variable in Real Variables
Volume II Algebra Volume V Topological Vector Space
Volume III General Topology Volume VI Integral Theory
As the Bourbaki School said, "From now on, mathematics has powerful tools provided by several kinds of structural theories, which dominate the vast field with a single point of view. They used to be in a state of complete chaos, but now they have been unified by axiomatic methods. " "From this new point of view, mathematical structure constitutes the only object of mathematics, and mathematics is the warehouse of mathematical structure."
affect
Before World War II, bourbaki had only completed the first part of "Principles of Mathematics"-"Results". This booklet with less than 50 pages was first published in 1939, then the first and second chapters of General Topology were published in 1940, and the third, fourth and first chapters of Algebra were published in 1942. These four books reflect the spirit of bourbaki, and they are the basis of Principles of Mathematics. The volumes of Principles of Mathematics are arranged in strict logical order. The concept or result used in a place must have appeared in previous volumes and volumes. This strict and precise style also has its advantages: all the main results are clearly and accurately expressed and become a perfect system. Therefore, Polba's Principles of Mathematics has become a standard reference book because of its rigor and accuracy, and it is one of the most cited books in postwar mathematical literature. The thought and writing style of Bourbaki School became the object of imitation by young people, and soon "bourbaki's" became a special name, which became popular in European and American mathematics circles. For example, we all know that before a science matures, the use of nouns is very confusing. Everyone uses a set for himself, and everyone has a group of followers to follow his usage, so it is difficult to understand each other. With the prestige of bourbaki, many mathematical terms, especially the neologisms of topology and functional, are based on bourbaki. It was bourbaki's Principles of Mathematics that unified mathematical terms after the Second World War. With the unification of nouns, mathematical symbols are also unified. According to bourbaki's usage, natural number set, integer set, rational number set, real number set and complex number set commonly used in mathematical literature are all represented by n, z, q, r and c. What makes bourbaki more famous is that many of his members' work before and after the war began to be known to everyone, especially their achievements in algebraic number theory, algebraic geometry, Lie groups and functional analysis. This makes bourbaki's activities more noticeable. It can be said that bourbaki's popularity reached its peak in the mid-1960s. The topic of the bourbaki seminar was undoubtedly the latest achievement of mathematics at that time. In the international mathematics field, several members of bourbaki have important influence, and even their general reports and works have attracted many people's attention.
In the development of mathematics in the 20th century, Bourbaki School played a connecting role. They put the mathematical knowledge accumulated by human beings for a long time into a well-organized and profound system according to the mathematical structure. Their Principles of Mathematics has become a new classic, and it is also the starting point and reference guide for many research work. This system, together with their contribution to mathematics, has indisputably become an important part of contemporary mathematics and the mainstream of the booming mathematical science.