New obscurantism and Russian enlightenment article. Vladimir Arnold

“SCHOOL IS A TEST OF WHETHER PARENTS CAN PROTECT THEIR CHILD OR NOT” Imagine that you, an adult, live such a life. You get up before dawn and go to work that you don’t like at all. In this job, you spend six or seven hours doing something that you generally don’t like and that you don’t see any point in. You absolutely do not have the opportunity to devote yourself to the work that interests you, that you like. Several times a day, your bosses (and there are quite a few of them) evaluate your work, and very specifically - with points on a five-point system. I repeat: several times a day. You have a certain book in which the points received, as well as comments, are recorded. Any boss can reprimand you if he notices that you are not behaving in a way that seems right to him, the boss. Let's say you're walking too fast down a corridor. Or too slow. Or speak too loudly. Any boss, in principle, can easily insult you or even hit you in the hand with a ruler. Complaining about your boss is theoretically possible, but in practice it is a very long procedure, few people get involved in it: it’s easier to endure. Finally, you return home, but even here you have no opportunity to be distracted, because at home you are obliged to do something necessary, to do something you don’t like. The boss can call your child at any time and tell all sorts of nasty things about you so that the younger generation can influence you. And in the evening the child will scold you for walking too fast along the service corridor or getting few points. Or he might deprive you of your nightly glass of cognac - you don’t deserve it. Four times a year you are given final grades on your work. Then the exams begin. And then - the most terrible exams, so incomprehensible and difficult that you have to prepare for them for several years. Have I exaggerated school life that much? And how long would it take you, an adult, to go crazy from such a life? And our children live like this for eleven years! And nothing. And - it seems that’s how it should be. Children very quickly understand that school is a world that must be fought with: most simply cannot exist in school. And then the child begins to think: whose side is the parent on? Is he for him or for the teacher? Do mom and dad also think that you should happily do what you don’t like? Are mom and dad also convinced that the teacher is always right and the child is always guilty? In our relationship with children, school is a test of whether parents can protect their child or not. Yes, I am absolutely convinced: protecting a child is the main parental job. Protect, not educate. Protect, not force, do homework. Protect, and not endlessly scold and criticize, because if you want, there will always be something you can scold and criticize a child for. There is a lot of nonsense and nonsense going on at school. It’s terrible when parents don’t seem to see this. It’s terrible when a student knows that he will be scolded and humiliated at school, and then the same thing will continue at home. And where is the way out for him then? School is a serious test that parents and children must go through together. Together. A schoolchild must understand: he has a home where he will always be understood and will not be offended. The main task of a parent is not to make the child an excellent student, but to ensure that he finds his calling and receives as much knowledge as possible necessary to fulfill this calling. This is what we should be focusing on. It's stupid to tell a child who dreams of being an artist that he needs algebra. It is not true. It is also not true that a boy can grow into a mathematician if the boy does not know what age Natasha Rostova went to the ball. But the truth is that in mathematics and literature you need to have at least a C in order to move to another class. You shouldn’t scold a “humanitarian” child for falling from a D to a C in math. One should feel sorry for him - after all, he is forced to do something that is neither interesting to him nor necessary. And help as much as possible. If a child does not have a good relationship with a teacher because the teacher is, say, an unintelligent person, you need to discuss this with him. And explain that in life you will often have to establish relationships with stupid people. You have a chance to learn this. Why not take advantage of this? If a child gets a bad grade for unfinished homework, that’s bad. He gets a bad mark not for lack of understanding, but for laziness. I could easily have not received it, but I did. This is worth talking about. If a child is endlessly reprimanded for bad behavior in class, you shouldn’t keep telling him that learning is very important. If a child is bored in class, it means they can’t teach him anything. However, we can clarify: despite the fact that you should try to do only what is interesting in life, alas, sometimes you have to do boring things. Learn - you can’t do without this skill in life. It’s right to scold a child for not studying subjects that will be useful to him in life. A little person must understand: if you have chosen a calling, you must do everything to fulfill it. Why don't you do it? In short: don’t lie to your child. We must try our best to help him find meaning even in such school situations when this meaning is completely unclear. Andrey Maksimov (from the book “How not to become your child’s enemy”).

Vladimir Igorevich Arnold, mathematician and fighter

Information sources - http://pedsovet.org/forum/index.php?autocom=blog&blogid=74&showentry=6105, http://www.svobodanews.ru/content/article/2061358.html(Published 06/03/2010 20:23).

Alexandra Egorova

On June 3, the outstanding Russian mathematician Vladimir Arnold passed away. In a few days he would have turned 73 years old. Friends and colleagues - academicians of the Russian Academy of Sciences Yuri Ryzhov and Viktor Maslov - remember him.

Vladimir Igorevich Arnold was born on June 12, 1937 in Odessa. He graduated from the Faculty of Mechanics and Mathematics of Moscow State University, where he studied with the famous Soviet mathematician Andrei Kolmogorov. At the age of twenty, he solved Hilbert's thirteenth problem, proving that any continuous function of several variables can be represented as a combination of a finite number of functions of two variables. Subsequently, Vladimir Arnold published many scientific papers, where he paid special attention to the geometric approach in mathematics. He worked at the Moscow Mathematical Institute. V.A. Steklov and at the University of Paris-Dauphine.

Vladimir Arnold was an academician of the Russian Academy of Sciences, a foreign member of the US National Academy of Sciences, the French Academy of Sciences, the Royal and Mathematical Society of London, and an honorary doctor of the Pierre and Marie Curie University. Winner of many awards, including the Lenin Prize, the Lobachevsky Prize of the Russian Academy of Sciences, the Crafoord Prize of the Royal Swedish Academy of Sciences, the Harvey Prize, the Wolf Prize, and the Danny Heinemann Prize in the field of mathematical physics. He was awarded the Order of Merit for the Fatherland, IV degree, and the Russian State Prize for his outstanding contribution to the development of mathematics.

In recent years, Vladimir Igorevich Arnold often visited Paris - he taught and went for treatment, as he was very ill. He died on June 3 in Paris. The relatives of Vladimir Arnold told a Radio Liberty correspondent about this.

Academician of the Russian Academy of Sciences Yuri Ryzhov calls Vladimir Arnold a “fighter for mathematical education.”

We studied at the same school - Moscow school No. 59,” recalls academician Yuri Ryzhov. - This school can be called a “white hole”: I sat at the same desk with another famous mathematician, academician Viktor Maslov. Vladimir Arnold graduated 6-7 years later than us. A couple more academicians of the Russian Academy, corresponding members, graduated from the same school... The character of Vladimir Igorevich Arnold is the character of a fighter for truth, for science, for education. At one time, apparently, he was not even very convenient in academic circles, because being a corresponding member of the Soviet Academy, he first became an academician of the French Academy and only then was elected academician of the RSFSR.

He was an irreconcilable fighter against all sorts of school reforms that would disfigure education, primarily in secondary schools, but also in higher education. He stood for the need for mathematical education for any people, not just those going into natural sciences. He apparently believed that without a decent knowledge and understanding of mathematics, logical thinking cannot be developed, and logic is needed in any field of activity if you want to do something,” said Yuri Ryzhov.

Doctor of Physical and Mathematical Sciences, Academician of the Russian Academy of Sciences Viktor Maslov, with whom Yuri Ryzhov sat at the same desk, met Vladimir Arnold in 1965. He is sure that his friend was “the best lecturer in the world”:

He was busy with science like no one else. He quickly grasped ideas and presented them brilliantly,” recalls Viktor Maslov.

The article is presented on the website in abbreviated form.

Vladimir Igorevich Arnold

The age of ignorance is coming

Conversation with an academician about educational problems

Our outstanding scientist, academician Vladimir Igorevich Arnold, is facing an alarming time, and he speaks about it frankly, moreover, sometimes even harshly - after all, we are talking about his favorite mathematics, to which the scientist devoted his entire life.

- What worries you most?

— Most of all, things are very bad with education in the world. In Russia, however, surprisingly, it’s a little better, but still bad! I will start with a statement made at one of the meetings in Paris, where the French Minister of Science, Education and Technology spoke. What he said applies to France, but it is just as relevant for the USA, England and Russia. It’s just that in France the catastrophe came a little earlier; in other countries it is still ahead. School education began to die as a result of the reforms that were carried out intensively in the second half of the twentieth century. And what is especially sad is that some outstanding mathematicians, for example, Academician Kolmogorov, whom I respect, are directly related to them... The French Minister noted that mathematics is gradually being squeezed out of school education. By the way, the minister is not a mathematician, but a geophysicist. So he talked about his experiment. He asked the schoolboy: “What is two plus three?” And this schoolboy, a smart boy, an excellent student, did not answer, because he did not know how to count... He had a computer, and the teacher at school taught him how to use it, but he could not add up “two plus three.” True, he was a capable boy and he answered: “Two plus three will be the same as three plus two, because addition is commutative...” The minister was shocked by the answer and proposed removing mathematics teachers from all schools who teach children this way.

— And what do you see as the main reason for what happened?

— Empty chatter flourishes, and it replaces genuine science. I can demonstrate this with another example. Several years ago, the so-called “California Wars” took place in America. The state of California suddenly declared that high school students were not prepared enough to attend college. Children coming to America, for example, from China, turn out to be much better prepared than American ones. And not only in mathematics, but also in physics, chemistry and other sciences. Americans are superior to their foreign counterparts in all sorts of “related” subjects—what I call “cooking” and “knitting”—but they are far behind in the basic sciences. Thus, when entering a university, Americans cannot compete with the Chinese, Koreans, Japanese...

— And how did the super-patriotic American society react to such an observation?

- Stormy. The Americans immediately created a commission that determined the range of problems, questions and tasks that a high school student should know when entering university. The Mathematics Committee was chaired by Nobel laureate Glenn Seaborg. He drew up the requirements for a student graduating from school. The main one is the ability to divide 111 by three!

- Are you joking?

- Not at all! By the age of 17, a student must perform this arithmetic operation without a computer. It turns out that Americans don't know how to do this... 80 percent of modern math teachers in America have no idea about fractions. They can't add half to third. Among students, this figure is already 95 percent!

However, Congress and senators condemned the state of California for daring to question the quality of American education. One of the senators in his speech said that he received 41.3 percent of the votes, this indicates the people’s trust in him, and he always fought in education only for what he himself understands. If not, then this should not be taught. Other speeches were similar. Moreover, they tried to give both “racial” and “political” overtones to California’s initiative. This battle lasted for two years. And yet, the state of California won, since a very meticulous lawyer found a precedent in US history in which State Law became superior to Federal Law in the event of a conflict. Thus, education in the USA temporarily won...

I tried to get to the bottom of the problem and discovered it - it turns out that it all started with Thomas Jefferson, the second President of the United States, the Founding Father of America, the creator of the Constitution, the ideologist of independence, and so on. In his letters from Virginia he has the following passage: “I know for sure that no Negro will ever be able to understand Euclid and understand his geometry.” Americans are accustomed to rejecting Euclid, mathematics and geometry. Reflections and the thought process are replaced by mechanical action, knowledge only of which button to press. And this, in addition, is presented as a fight... against racism!

- Or maybe it’s easier for them to buy those who know fractions than to learn it themselves?

- They buy it! American scientists are mainly emigrants from Europe, and graduate students are Chinese and Japanese.

—But you can’t deny the successes of American science?

“I’m not talking now about the state of science in the USA or about the American “way of life.” I'm talking about the state of mathematics teaching in US schools, and the situation here is dire. I discussed this problem with eminent mathematicians in America, many of them my friends, whose achievements I am proud of. I asked them the following question: “How did you manage to achieve such a high level in science with such a low school education?” And one of them answered me like this: “The fact is that I learned “double thinking” early on, that is, I had one understanding of the subject for myself, and another for the teachers at school. My teacher demanded that I answer him that two times three is eight, but I myself knew that it was six... I studied a lot in libraries, fortunately, there are wonderful books...”

- But today many mathematicians go into business...

- And this is quite understandable. Mathematics is mental gymnastics; oligarchs also need it. But, in my opinion, it does not determine the choice here - there are simply people who have a special talent for making money.

—Have you ever wanted to get into economics and business yourself?

“This is strictly contraindicated for me.” Not mine. But the threat of the onset of an age of ignorance seems completely real...

— Sometimes they say that mathematics is an art.

- I completely disagree! Mathematics is a science. She always was, is and will be! I also believe that there is no “theoretical” science and “applied” science. I completely agree with the great Pasteur, who said: “There have never been, are not, and never will be applied sciences, because there is science and there are its applications.”

— You spend more and more time in Paris, where you teach. Don't feel like an expat?

- Not at all! Moreover, my Parisian students often come to Moscow, and Moscow students often come to Paris. France is funding this project. For world science, this kind of relationship is the norm. My French colleagues lead a similar life; they spend half their time in Germany, America, and England. This has always been the case all over the world. And in Russia before the revolution too. And even after the revolution, some prominent scientists worked abroad for a long time. I repeat, for science and scientists this is normal life, and it cannot be otherwise!

— Let's go back to school education. If the trend of emasculating mathematics from the educational process continues, what does this threaten Russia with?

- It will turn into America, with which we started talking!

The fact that we still have actively working mathematicians is partly explained by the traditional idealism of the Russian intelligentsia (from the point of view of most of our foreign colleagues, simply stupidity), and partly by the great help provided by the Western mathematical community.

The significance of the Russian mathematical school for world science has always been determined by the originality of Russian research and its independence from Western fashion. The feeling of being involved in a field that will be fashionable in twenty years is extremely stimulating.

March 13, 2008The conversation was conducted by Vladimir Gubarev. The interview was published on the website of the information agency “Century”.

Vladimir Igorevich Arnold

What awaits Russian schools?

Analytic note

A source of information - http://scepsis.ru/library/id_653.html

December 2001

The following brief analysis is an abbreviated retelling of the plan for modernizing education in Russia (2001 project). His assessment is given after point 4 of the description of “strategy”.

1. The main goals of education are declared to be “cultivating independence, legal culture, the ability to cooperate and communicate with others, tolerance, knowledge of economics, law, management, sociology and political science, and proficiency in a foreign language.” No science is included in the “learning objectives”.

2. The main means for achieving these goals are declared to be “unloading the general education core”, “rejection of scienistic (i.e. scientific - V.A.) and subject-centric approaches” (i.e. from teaching the multiplication table - V.A.) , “a significant reduction in the volume of education” (see below, paragraph 4). Specialists need to be excluded from discussing the programs of “their specialties” (who would agree with obscurantism? - V.A.)

3. The assessment system “should” be changed, “providing for a grade-free education system”, “evaluate not students, but teams”, “give up academic subjects” (they are very “narrow”: literature lessons, geography, algebra...), “rejection of the demands of secondary school in relation to primary school” (why know the Russian alphabet and be able to count on fingers when there are computers! - V.A.), “transition to objectification of assessment procedures taking into account international experience” (that is, with a test instead exams - V.A.), refusal to “consider the mandatory minimum content of education” (this consideration allegedly “overloads the standards” - some are beginning to demand that schoolchildren understand why it is cold in winter and warm in summer).

4. In secondary school, per week there “should be”: three hours of Russian, three hours of mathematics, three of a foreign language, three of social studies, three of natural science; that’s the whole program, which abolishes the “dead-end subject-oriented approach” and allows “the inclusion of additional modules,” namely “humanization and humanitarianization,” “reflection of the culture of local peoples,” “integration of ideas about the world,” “reduction of homework,” “ differentiation”, “teaching communication technology and computer science”, “using general learning theories”. This is the plan for the “modernization” of the school.

In short, the plan is to abolish the training of all factual knowledge and subjects (“literature”, “physics”, for example, are completely thrown out even from those lists where different types of military training, called “differentiation”, have now appeared: Kalashnikov instead Shakespeare).

Instead of knowing that the capital of France is Paris (as Manilov told Chichikov), our schoolchildren will now be taught that “the capital of America is New York” and that the Sun revolves around the Earth (lowering the level of knowledge below that required under the Tsar in a parochial school ).

This triumph of obscurantism is an amazing feature of the new millennium, and for Russia it is a suicidal trend that will lead to a fall first in the intellectual and industrial level, and subsequently - and quite quickly - in the defense and military level of the country.

The only thing that gives us hope is that attempts (similar to those being made now) to destroy the high level of education in Russia, which were marked in the twenties and thirties by the “brigade-stream method” and destroyed both gymnasiums and real schools, were not crowned with success: the level of education in modern schools Russia remains high (which is recognized even by the authors of the document under discussion, who find this level “excessive”).

Vladimir Igorevich Arnold

Is mathematics necessary in school?

A source of information- http://scepsis.ru/library/id_649.html

Report at the All-Russian conference “Mathematics and Society. Mathematics education at the turn of the century” in Dubna on September 21, 2000.

I'm going to talk today about the rather sad circumstances surrounding the state of mathematics education around the world. I know the situation most of all, naturally, in Russia, but also in France and the United States. But the processes that I will talk about are occurring approximately simultaneously throughout the world. They are somewhat incredible, but what I will tell, no matter how incredible it may be, is the pure truth.

I would call the main process that I now notice, which is now underway and which inspires the main concern - I would call this process Americanization. Americanization consists in the fact that the world's population, those billions who live on the globe, all want McDonald's in every home, and, accordingly, they want to have such a "culture" as in America. But what is American “culture”? I’ll probably tell you an example so as not to be unfounded. At Harvard, I saw a student who majored in European art in her French class. There she had to speak French, and the teacher asked her in French: “Have you been to Europe?” - "Was." - “Have you visited France?” - “I stopped by.” - “Have you seen Paris?” - “I saw it.” - “Have you seen Notre-Dame de Paris there (i.e., Notre-Dame Cathedral)?” - “I saw it.” - "Did you like it?" - "No!" - “Why is this so?” - “He’s so old!”

The American point of view is that everything old should be thrown away. If the car is old, it needs to be replaced with a new one, Notre Dame Cathedral needs to be destroyed, and so on. So mathematics must be eliminated from education. Let me give you another example.

I recently read a text that belongs to Thomas Jefferson, the third President of the United States, the author of the Declaration of Independence, one of the “Fathers of the Nation.” And he already spoke out about mathematical education in his “Letters from Georgia.” He says this (and this statement, in my opinion, is defining for mathematics education in the United States today): “no black will ever understand a word of Euclid, and no teacher (or textbook) will explain Euclidean to him. geometry, he will never understand.” This means that all geometry must be excluded from school education, because democratic evolution must make everything understandable to minorities; “who needs it, this mathematics...”

French example. The Minister of Education and Science of France told (at a meeting of the Parisian meeting of mathematicians at the Palais des Discoveries) arguments that showed that teaching mathematics in school should be stopped altogether. This is a fairly intelligent person, Claude Allegret, a geophysicist, engaged in the navigation of continents, applies mathematics, the theory of dynamic systems. His argument was this. A French schoolboy, a boy of about eight years old, was asked how much 2 + 3 is. He was an excellent student in mathematics, but did not know how to count, because that is how mathematics is taught there. He didn’t know that it would be a five, but he answered like an excellent student, so that he would get a five: “2 + 3 will be 3 + 2, because addition is commutative.” French education is all organized according to this scheme. They learn such things and as a result they know nothing. And the minister believes that rather than teach like this, it is better not to teach at all. When they need something for business, when they need it, they will learn it themselves, and learning this pseudoscience is a waste of time. Here is the French point of view today. It's very sad, but that's how it is.

Americanization is also taking place in France now. In particular, I received a letter from their Academy of Sciences in April that they were revising the Academy's charter. One of the important points on how to change the charter of the French Academy of Sciences was that it was necessary that there should be no corresponding members, all corresponding members should be considered academicians, and in the new elections no one should be elected as a corresponding member, but only academicians. And then - twenty pages of justification of this theological nature, it says that France is like the eldest daughter of the Catholic Church, and so on... There are not necessarily religious justifications, there are all kinds, but I couldn’t understand anything, it was very difficult for me until I did not reach the last line on some distant page, and then I realized that I had already heard this line many times over the twenty years that I have been hearing this discussion. France is probably ahead, but we will also get to this point, and this argument, and this reasoning - all this will be found in our Russian Academy of Sciences, I believe. The argument that, in my opinion, is the only significant one in all these justifications and which, apparently, is the main one for them, is this: there are no corresponding members at the US National Academy of Sciences in Washington.

The next project was that modern humanity is faced with a large number of problems, and the academies of sciences are national, each country has its own academy that solves its own problems. This is a relic, this is not good. It is necessary to create a super-bureaucratic organization, a super-academy, which will be worldwide and whose relationship to ordinary academies of science will be the same as the relationship of the prefect of police to ordinary ordinary police officers. It will decide what the main problems of humanity are, for example, global warming of the atmosphere, the Malthusian problem of overpopulation, ozone holes, and others, several dozen such basic, fundamental problems are listed: there are too many cars, and they pollute the air with lead, and so on, I I don’t remember this whole list anymore. So, we need to decide which problems are of priority in order for humanity to survive, which country will solve which problem.

And further on this list it was written what problem the eldest daughter of the Catholic Church, France, is taking upon herself to propose, and what the problem is, and what the French method of solving this problem is. This problem is directly related to the topic of our conference today. The problem is this: the level of education is falling catastrophically all over the world. A new generation of children comes who know nothing: neither the multiplication table, nor Euclidean geometry - they don’t know anything, don’t understand and don’t want to know. They only want to press computer buttons and nothing else. What to do, how to be here? Ministers everywhere, in all countries, are people who do not understand anything, and it is clear that they need to destroy all civilization and culture, simply in order to survive, in order to remain among a higher cultural level environment, these people need to destroy all culture and all education. How to do it? (I'm talking about France.)

So, the French project: how to improve the situation with education. The French Academy of Sciences proposes: women must be educated. Well, this is again an American idea - this is feminism, which exists in France, and probably exists here too. It is possible to foresee that we will soon adopt the same project.

Now, after these sad words, I want to say a few words regarding how we came to this life, how it was formed, how it turned out over many thousands of years of development of mathematics, how we came to this situation. I must say that I have been a little interested in this history in recent years and found out that everything that is written in textbooks about the history of science, most of these things are gross mistakes, completely incorrect statements. And now I’ll tell you a little about the history of the development of mathematics, what I learned, things I didn’t know about.

Historians, of course, knew this; there are even books by historians in which all this is written. But if we look at what mathematicians write, what teachers write, what is written in the books that were given to me at this conference, in which even my friends write about what great mathematicians were, what great discoveries they made, when, what, how — a lot was different. Other people discovered, discoveries should appear under other names...

I will now tell you a number of these truths, which are generally known to historians, but unknown to mathematicians, as a rule. I learned very recently about the great discoveries of such a great mathematician, whose name is unknown, he was the chief surveyor of the Pharaoh in Egypt and was declared a god after his death, and his divine name is known, but I, in any case, do not know his original name. As an Egyptian god he was called Thoth. The Greeks then began to spread his theories under the name Hermes Trismegistus, and in the Middle Ages there was a book “The Emerald Tablet”, which was published several times every year, and there were many editions of this book, for example, in the library of Newton, who carefully studied it. And many of the things that are attributed to Newton were in fact already contained there. What did Thoth discover? I will list some small number of discoveries. In my opinion, every cultured person should know that there was such a Thoth, and what he discovered, and what his great inventions were. The fact that I didn't know about this until this year is a shame.

The first thing he came up with was numbers, the natural series. Before him, of course, there were numbers: 2, 3,... before the number that expressed the amount of the entire tax that was paid to the Egyptian pharaoh - the number that expressed the entire annual tax existed, but there were no large numbers. The idea that numbers can be continued indefinitely, that there is no greatest number, that you can always add one, that you can build a number system in which numbers can be written as large as you like - this is Thoth’s idea, this is his first idea. Today we call it the idea of ​​actual infinity.

The second discovery, which is also very significant, is the alphabet. Before him, there were hieroglyphs in which words were depicted as signs, for example “dog”. And he came up with the idea that phonemes and sounds should be written down, setting instead of thousands of hieroglyphs that were for words, only a few dozen hieroglyphs, for example, with a simplified “dog” to represent the sound “s” always, “s” in any word - it will look like this very “dog”, such a simplified “dog”. He invented the Egyptian alphabet. All our European alphabets came from him. We have such a legend, which can be found in all textbooks, that Champollion allegedly discovered the “Rosetta Stone”, as if Champollion, who took this “Rosetta Stone”, the trilingual that was there, found a match, read the hieroglyphs, and so on. So, this is all untrue. In fact, I’m going a little aside from mathematics, this is the history of a different science, it’s still not true. In fact, the story with Champollion was this: Champollion really solved this alphabet, he really read it, but without any “Rosetta stone”. This “Rosetta Stone” was found after Champollion had already published his theory. When - about twenty years later - the “Rosetta Stone” was found, he took this stone and showed on this stone what his theory gave, and compared it with the Greek translation that was on the stone, and everything agreed. Thus, this was proof, but the theory had long been published by this time. Champollion discovered the Egyptian alphabet in a completely different way. The main discovery, by the way, that Champollion took advantage of, which he took from Plutarch, and the main thing that allowed him to read hieroglyphs, hieroglyphic texts, this alphabet, was a very strange discovery that no one before him for some reason understood. It turns out that hieroglyphic texts were written not from left to right, like ours, but from right to left. Plutarch knew this, how it was written, Champollion understood this, and he began to read in the other direction, and then it worked. Then he came up with a decryption. But I will not go into details of the decryption theory.

Thoth's third discovery is geometry. Geometry in the literal sense is land surveying. Thoth was entrusted by the pharaoh, he had to know, a plot of land, fenced, of such and such size, what harvest it would bring. It depends on the area, he had to measure these areas, draw boundaries, separate water from the Nile, drain water and all this practical work. And he learned. For this, he came up with geometry, everything that we are now teaching, Euclidean geometry, all this geometry is actually Thoth. In particular, Thoth and subsequently his students measured the radius of the Earth using their geometric methods. The radius of the Earth that they measured was obtained with an error of one percent relative to modern data, this is colossal accuracy. Caravans of camels walked along the Nile, from Thebes to Memphis, they walked almost along the meridian and counted the camel steps, thereby knowing the distance. At the same time, you can, by observing the polar star, measure the latitudes of cities and, knowing the difference in latitudes and the distance along the meridian, you can measure the radius of the Earth, and they did this very well and found the radius with an accuracy of 1%.

And finally, his last discovery, which I will mention, is a relatively minor, but still interesting thing that he came up with: checkers. The Indians had chess, chess was known, but it is a complex and not a popular game, he democratized chess and invented checkers. The checkers also come from him.

There are dozens more of his discoveries and inventions in the history textbook; for the sake of brevity, of course, I will not list them now.

How did we know all this? Now we know Euclidean geometry. Where does Euclidean geometry come from, where did all this come from? It turns out that the study of the science that was created by Thoth was a trade secret of Egypt. In Alexandria there was a library (musium) in which seven million volumes were stored, in which all science was written down, but one had to have special permission in order to become familiar with this material, and one had to have permission from the priests of the pyramids so that everything study this. There are at least four great Greek scientists (industrial spies) who stole this science from the Egyptians, which was not all invented by the Egyptians, they borrowed a lot - from the Chaldeans, from the Babylonians, from the Hindus - but, in any case, it was kept secret.

The first of them, apparently, was Pythagoras. Some say that he lived among these priests for fourteen years, some say twenty. He received clearance, familiarized himself, learned all this science, all Euclidean geometry, algebra, arithmetic, and declared that he would never declassify this secret information. And indeed, not a single line from Pythagoras has survived; he never wrote anything down. The teachings of Pythagoras, when he returned to Greece, were spread orally by his disciples. There were no books of Pythagoras. The texts of Euclid several generations later were produced by various students of Pythagoras, who wrote everything down later. Pythagoras did not write anything himself because he swore that he would not. But he spread this knowledge to Greece - axioms, except, perhaps, for the fifth postulate, which, apparently, belongs to Euclid himself. In particular, the Pythagorean theorem was obviously published two thousand years before him in Babylon, in cuneiform, and in addition to the theorem, Pythagorean triplets were also known (I was recently given a book in which Tikhomirov seems to claim that these triplets were found by someone else another). But all this was known a long time ago, a thousand years before Pythagoras, and the Egyptian priests knew all this and used triangles (3, 4, 5), (12, 13, 5) and others when building pyramids, and they knew the general formula, how to construct all these triangles. All this was well known, but is attributed to Pythagoras (together with the theory of transmigration of souls).

I once received a letter from the English physicist Michael Berry (of the famous “Berry phases”), who wrote me a letter as a consequence of our discussion of priority issues. And he wrote that these discussions can be summarized by the following principle of Arnold: if any object has a personal name (for example, Pythagorean triplets or the Pythagorean theorem; America, for example), then it is never the name of the discoverer. It is always the name of some other person. America is not called Colombia, although Columbus discovered it.

By the way, why did Columbus discover America? This is closely related to what I just told you. When Columbus went to the Spanish Queen Isabella to ask for an expedition (he was not going to discover America, he was going to open a route across the Atlantic Ocean to India), the queen told him: no, it’s impossible. And here's the thing. Two hundred years after the Egyptians, the question of the size of the Earth was considered by the Greeks. The Greeks, using the information stolen by Pythagoras, knew about the Egyptian measurements, but did not believe the Egyptians (what kind of measurements are these, some camels, what are they...). And they took the measurements again. They took a trireme, a ship that crossed the Mediterranean Sea from south to north, from Alexandria to the island of Rhodes, measured the path, knowing the speed of the ship in a strong wind, the difference in latitude can also be measured, and received a new size (radius) of the Earth. But since, of course, the Egyptian method was reliable, because camels are a good measure of distances, and the speed of a ship in a strong wind is something so uncertain, the Greek estimate was twice as different from the Egyptian one. And the Greeks published this and said that the Egyptians had already measured it, but since they were an underdeveloped people, they could not measure it well and received an Earth that was half the size of the real one; in fact, they have erroneous data, and the correct size of the Earth is twice as large.

And since all Greek science - Euclid, Pythagoras, all this - then spread everywhere, as they taught in school, Queen Isabella also thought that the Earth was twice as large as it is, and she said to Columbus: “You will not sail to India, because that no ship can fit as many barrels of water as it takes to sail such a long distance.” Because it’s very far away, and there’s nothing on the road (America wasn’t supposed to be). Columbus went to her six times and in the end somehow avoided these prohibitions and still got there.

Of course, undoubtedly, scientific discoveries are stolen, they have always been stolen and continue to be stolen.

(From the audience: And they will steal!)

Maybe they will steal, or maybe not, because they will no longer be interested in science, because there will be no one to pay for this stolen property. Maybe they will stop stealing science simply because there will be no more customers, that’s the point.

I will list a few more discoveries that are very striking and which are attributed not to the discoverers, but to completely different people. Plato stole logic from Egypt - the art of reasoning, something that later passed to Europe through Aristotle, Aristotelian logic, sophisms, sorites (long chains of syllogisms) - all this science was among the Egyptian priests, was well known to them. It was stolen by Plato, who was also a spy. There was also such a famous man Orpheus, who stole music: harmony, scales, octaves, fifths, thirds... Pythagoras also studied music and knew how long the strings should be in order to obtain the appropriate frequency ratio, and what tension on the strings should be applied - This was all completely standard among the Egyptians, just for ritual music, they knew this with absolute certainty, and the Greeks borrowed all this. All our music is borrowed from the Egyptians through the Greeks. And finally, the last discovery I want to mention is a strange case. This name is perhaps less well known, although the author is a man who very much deserves our deep gratitude - Eudoxus. Eudoxus' theory is now called number theory. Eudox discovered the following. The Pythagoreans already knew (although who first discovered it is not very clear, maybe Pythagoras, maybe also Pythagoras’ students) that the diagonal of a square is incommensurate with its side and therefore there are irrational numbers. This discovery was immediately classified by the Greeks themselves, because what were the numbers used for? There were only rational numbers, and they served for measurement. But this discovery shows that numbers, i.e. rational fractions, are not enough for measurement, because the diagonal of a square cannot be measured. Consequently, arithmetic is a science unsuitable for practical life, for physics, for all applications. Consequently, if consumers - pharaohs, people in general - find out about this kind of thing, then they will drive away all the mathematicians, because they study proportions, fractions - some kind of nonsense that no one needs. So, Eudox overcame this difficulty. Because of this difficulty, the theory of rational numbers was banned, and he created it. He created what is now called Dedekind's theory of sections or Grothendieck's ring, which is the same thing. This theory was in fact completely created by Eudoxus and expounded by Euclid in the theory of proportions, in, in my opinion, the fifth book of Euclid. This is how irrational numbers entered mathematics.

Now I will allow myself to deviate a little from mathematics and talk about discoveries close to mathematics (even, strictly speaking, I would include this in mathematics, but some of my contemporaries do not, I will talk about this too). These are astronomical theories. Astronomy and celestial mechanics played a huge role in the development of mathematics and analysis - Newton and Kepler are well known. Kepler's laws, the fact that the force of gravity is inversely proportional to the square of the distance - we teach all this to our students, we explain what great discoveries Newton made, and so on. So, Newton himself had a completely different point of view on the history of these issues. In his unpublished works, alchemical and theological, which are ten times larger than the published mathematical and physical works, he recognizes the priority of the Egyptians, who knew all this a couple of thousand years before him. In fact, it was well known in Egypt - it is not very clear who first discovered this, but, in any case, the Egyptian priests were already aware of, firstly, the inverse square law, secondly, Kepler's laws and, thirdly, that Kepler's laws follow from the inverse square law. Newton writes that, unfortunately, the conclusion of one from the other was written down in those books, those millions of volumes that were burned in a fire in the library in Alexandria, and therefore for several centuries this wonderful ancient reasoning was lost, and he is proud of the fact that that he deserves the credit for restoring this evidence. The proof now again explains why Kepler's laws follow from the inverse square law. But in fact all this was well known. In the 7th century BC, the Roman king Numa Pompilius, who reigned shortly after Romulus, built the Temple of Vesta in Rome, which included a planetarium, which was built according to the Copernican heliocentric system. Copernicus, by the way, also quotes these ancients and says that the heliocentric system was not his discovery, but was known for a long time, but he simply drew the attention of people of modern times to what was known in old times. In the temple of Vesta, in the center, there was a fire that represented the Sun. Around him, the priests carried an image of Mercury at the required speed in the required elliptical orbit, then an image of Venus, then an image of the Earth, then an image of Mars, and, of course, Jupiter and Saturn. On any day you could stand at the place where the priests were holding the Earth at that time, and look, for example, in the direction of the place where the priests were holding Mars, and then go outside and look in the evening, and then see Mars in that direction.

Thus, this whole whirlwind of celestial-mechanical discoveries - all of this existed two thousand years before Newton. You won't find this in textbooks. Newton refers, in particular, to Vitruvius's textbook on architecture, which cites, but again without proof, the ellipticity of orbits, Kepler's laws, everything is quoted, everything was known, but everything was destroyed. Everything was destroyed because it was considered useless by pure science. Who needs this astronomy, celestial mechanics, planets... No one was interested in this, except perhaps astrologers. But architecture and construction are a different matter. Therefore, copies of books on military affairs, navigation and architecture were preserved from ancient books. And only in them can some traces be found when it is quoted that somewhere in Alexandria there is a book in which this and that is proven. Newton read, used, found evidence.

Here I would also like to quote one statement that I recently read in Hardy’s book “Apology for a Mathematician,” just published in Izhevsk. A terrible book by a completely, horribly illiterate person who writes, in particular, the following things. He writes in praise of Gauss that Gauss worked a lot on number theory and that number theory is rightly called the queen of mathematics (I would even say the queen of mathematics, but I think he says “queen”). Hardy explains why number theory is the queen of mathematics. This is Hardy’s explanation, which was recently repeated by Yuri Ivanovich Manin, in a slightly distorted form, but he said almost the same thing. Hardy's remarkable explanation is this: number theory is, he says, the queen of mathematics due to its complete uselessness. But Yuri Ivanovich is a little different, he explains something else: that mathematics in general is an extremely useful science, not because, as some say - this is actually me - that mathematics contributes to the progress of technology, humanity, and so on, no; because it hinders this progress, that is its merit, this is the main problem of modern science - to hinder progress, and mathematics primarily does this, because if the Fermatists, instead of proving Fermat's theorem, built airplanes, cars , they would have caused much more harm. And so mathematics distracts you, distracts you with some stupid tasks that no one needs, and then everything is fine. Hardy, by the way, also has this idea, in a slightly different form - it’s amazing how naive you can be in the 20th century! - Hardy writes this: the terrible attractiveness of mathematics, especially in comparison with physics and chemistry, is that it is “absolutely unsuitable for any military applications.” Now, of course, we have different points of view; maybe Yuri Ivanovich agrees with him, but I don’t. As for the military, they also have completely different points of view, and it must be said that Hardy somehow managed to work with Littlewood, who did a lot of applied mathematics, and applied it seriously to military affairs, and Littlewood, of course, would never signed up for such stupid words.

Manin argues that mathematics is a kind of linguistics with a slightly expanded list of grammatical rules, including, say, 1 + 2 = 3, and teaching mathematics is teaching fraud, since nothing new can be discovered by identical transformations, which are the only things mathematicians deal with.

The most complete modern embodiment of the idea of ​​the uselessness of mathematics is the activity of the Bourbakist sect.

In fact, Bourbaki's principles were formulated partly by Montaigne and partly by Descartes in the 16th-17th centuries. Montaigne formulated two principles of all French science, by which French science differs from the sciences of other countries and by which it is still guided. First principle. In order to succeed, a French scientist must adhere to the following rule in his publications: not a single word of what he publishes should be understandable to anyone, because if something is clear to anyone, then everyone will say that it is It was already known, so you didn’t discover anything. Therefore, it is necessary to write in such a way that it is unclear. Montaigne refers to Tacitus, who pointed out that “the human mind is inclined to believe the incomprehensible.” Descartes was his student in this sense, and Bourbaki followed him. Changing all texts to make them completely inaccessible is the first principle.

I will give a few of Montaigne’s arguments with which he justifies the need to write incomprehensibly (emphasis added throughout):

“I hate learning even more than complete ignorance.” (“Experiments”, book III, chapter VIII)

“Whoever sits astride the epicycle of Mercury - it seems to me that he is pulling out my tooth. After all, they themselves do not know the reasons for the movement of the eighth celestial sphere, nor the time of the flood on the Nile.” (Book II, Chapter XVII)

“It would be easier to understand the root causes of phenomena, but I don’t know how to explain them. I don't strive for simplicity. My recommendations are the most vulgar.” (Book II, Chapter XVII)

“Sciences deliver theories that are too subtle and artificial. When I write, I try to forget everything written in books so that these memories do not spoil the form of my composition.” (book III, chapter V)

“Our ordinary understandable language is of no use in practical life, because it becomes incomprehensible and full of contradictions when we try to apply it to the formulation of a contract or a will.” (Book III, Chapter XIII)

Quintilian (Inst. Orat., X, 3) noted long ago that “the difficulty of understanding is created by doctrines.” (Book III, Chapter XIII) And Montaigne wanted to instill doctrines in the reader.

According to Seneca (Epist., 89), “every object divided into parts like specks of dust becomes dark and incomprehensible” (Book III, Chapter XIII). Seneca noted (Epist., 118) that “Miramur ex intervallo fallentia” (i.e. “it is the deceptive that delights us, due to its remoteness”). (Bk. III, Ch. XI) To arouse admiration, it is necessary to introduce fog into your writings.

“The main conclusion of all my research is the conviction of universal human stupidity, the most reliable feature of all schools in the world.” (Book III, Chapter XIII) This principle of Montaigne applies to his school.

It is clear that Montaigne did not want to clearly describe the achievements of these schools. Pascal noted that it is difficult to understand what is correct in Montaigne. The Encyclopedia Britannica (1897) writes that Montaigne was misunderstood because this humorist and satirist appealed to readers without a sense of humor. Montaigne's experience is infectious. He wrote: “it is among scientists that we often see mentally poor people” (Book III, Chapter VIII) and “learning may be useful for the pocket, but it rarely gives anything to the soul.” “Science is not an easy business, it is often overwhelming.”

Montaigne's second principle is to completely avoid foreign terminology. All terminology should be yours, your own. You must introduce new concepts, you can refer to your previous works where these terms were introduced, so that you cannot read your next works without memorizing the previous ones. And no works of other authors should be quoted, and it is especially strictly forbidden to quote foreigners. This is the principle that is still followed today. In April, the French Ministry of Science, as well as the security authorities, sent me an invitation to take part in the work of their commission, which is very important (and because they know that I am busy, if I cannot come, then send a student who would I presented my opinion there, because it is very important for them to know my opinion), that’s what the commission is like. Commission for the protection of the heritage of French science from foreigners.

(Laughter in the audience.)

The fight against cosmopolitanism, which we had in the late forties, reached France, but for some reason only now. Although they, of course, have a lot of all kinds of xenophobia and find everywhere that any thing was necessarily discovered by a Frenchman, for example, they have their own inventor of the radio - neither Popov nor Marconi admits - they have their own monument near the Luxembourg station in Paris to the man who “invented radar,” and so on—everything was done by the French. By the way, I also want to quote one Frenchman whose statement I, on the contrary, really like, is Pasteur. Pasteur spoke about science in general and made a remarkable statement, which I would like to refer to, because, in my opinion, it is very important for us. Pasteur's statement is: “There never has been, is not, and never will be any applied science. There are sciences and their applications.” There is a scientific discovery, and then it is attached to something - yes, but applied mathematics, applied physics, applied chemistry, applied biology - all this is a deception in order to siphon money from taxpayers or businessmen - nothing more. There is no applied science, there is only science - just ordinary science.

By the way, this idea can also be found in Mayakovsky, who said that the man who discovered that two and two equals four was a great mathematician, even if he was counting cigarette butts. And anyone who now uses this same formula to calculate much larger objects, such as locomotives, is not a mathematician at all. This is what applied mathematics is. There is no applied mathematics; teaching “applied mathematics” is a lie. There is just mathematics, there is science, and in this science there is a multiplication table, for example, that two and two are four, there is Euclidean geometry, all this must be taught. If we stop—what does this Americanization or Bourbakization lead to—we stop teaching, then what will happen? One Chernobyl after another will happen, and, accordingly, submarines will sink, and, accordingly, towers like the Pisan and Ostankino towers will fall... I recently read in the Bulletin of the Academy of Sciences that Moscow will face a catastrophe similar to the one in Ulyanovsk, which may Perhaps, even in the coming winter, just a million people should die from the cold, because the heating systems, thermal power plants cannot cope, Moscow’s heating is not adapted, is not ready to withstand the cold, which is typical for our climate. If science is stopped, then all these misfortunes of an apocalyptic nature will fall on all of humanity, including Russia. According to American data, today some countries, including Russia and China, remain an oasis in which there is still some hope that these processes of educational degradation are progressing more slowly. They determined that in America, 80% of school mathematics teachers have no idea about fractions: they cannot add half and third, they don’t even know that there is more, half or third, they don’t understand anything. They didn't teach. And schoolchildren's knowledge is even worse. While in Japan, China and even Korea the situation is much better. These schoolchildren understand perfectly well what half is, what a third is, they can add half with a third... We, as always, lag behind advanced humanity. The destruction of science, the destruction of culture is happening everywhere, but in our country more slowly than in other places, which means that there is still some hope that we will maintain our traditional level of culture longer than the so-called more advanced countries.
* * *

George Malaty, university professor in Finland. I am very glad to listen to your report, and I can say frankly, from my heart, that I came here specifically to support your ideas, because if a culture falls, it is very difficult to stop it back, in the West we know well that you too It's very easy to break up a culture. And now we know that, naturally, logically, it is very difficult to stop it back. I thank you and hope that we all listen both here and abroad to you. Thanks again.

From the audience: In your opinion, should Euclidean geometry be taught at school?

- In my opinion, we haven’t come up with anything better (and whether to call it Euclidean or something else - there are different options, of course). I know one case of a person who did not study Euclidean geometry at school. This man is Newton. Newton read Euclid already at university. He learned geometry according to Descartes, using the Cartesian coordinate system, and learned Euclidean later, and was grateful to both. Although it must be said that Newton did not like Descartes, because Descartes, he says, said so many stupid things in both physics and mathematics that he was simply harmful to science. How Newton could nevertheless learn anything from him amazes me. Descartes' theory - I prepared it, but did not have time to tell it - was this. (It is still adopted in France; the Bourbakis follow it.) There are four basic principles. Descartes' first principle: whether the original axioms correspond to any reality does not matter. These experimental questions concern applications and some special sciences. According to Descartes, science is the derivation of consequences from arbitrarily taken axioms that have no relation to any experiment or to any reality. (Hilbert repeated this many times later.) The second principle: the correspondence of the final conclusions to any experiment matters just as little. We do some kind of reasoning, like multiplying multi-digit numbers, we deduce some new consequences from the original axioms, and comparing what we get with some kind of experiment is pure nonsense, which can only be done by some petty people like Newton (Descartes did not say the last phrase; Newton was not known to him). Third principle: mathematics is not science. In order for mathematics to become a science, it is first necessary to expel from it all traces of experiment, which appear in it in the form of drawings. When we draw straight lines, circles, and engage in Euclidean geometry, then, according to Descartes, we perform unnecessary activities that have nothing to do with science. Therefore, it is necessary to replace all straight lines, circles, and so on with ideals, modules, rings, and leave only what is now called algebraic geometry. But no geometry (in such an ordinary sense) is needed, according to Descartes. In fact, it is necessary to banish from all sciences all places where imagination plays any role. But in geometry it plays a huge role, so it must be excluded. And finally, the last, fourth, principle of Descartes, which applies directly to the Ministry of Education: “It is necessary to immediately prohibit all other methods of teaching except mine, because my method of education is the only truly democratic method. The democratic character of my method of education lies in the fact that among those studying according to my method, the stupidest, most mediocre mind will achieve the same success as the most brilliant.”

For example, Descartes “discovered” that the speed of light in water is 30% greater than in air (contrary to Fermat’s principle and Huygens’ theory of envelope waves). But there was no need to refer to predecessors.

When Pascal reported to Descartes his work on hydrostatics and barometric measurements based on experiments with Torricelli voids. Descartes contemptuously kicked out the young experimenter for ignorance of Aristotle's axiom (“nature abhors a vacuum”) and for violating his first two (anti-experimental) principles. He wrote about this to the President of the Academy of Sciences, Huygens: “personally, I don’t see emptiness anywhere in nature, except in Pascal’s head.” Six months later, Pascal’s theory became generally accepted, and Descartes already said that Pascal came to tell him about it, but he himself did not understand anything at that time; and now that he, Descartes, has explained everything to him, Pascal tells how his (Descartes’) theory is his.

It is interesting that Leonardo da Vinci’s attitude to experiment was completely different: in his hydrodynamic studies (where even turbulence is already analyzed), he insists on the need in this area to be guided primarily by experiments, and only then by reasoning. Following which he discusses the laws of similarity and self-similarity.

S.G. Shekhovtsov: You were talking about the supposedly existing principles of Montaigne... But the fact is that in Russian, at least twice, and now a lot of “Experiments” have begun to be published... Montaigne in these “Experiences” continuously quotes the ancients authors. How does this even relate? Maybe it was just a provocation?

- No, this is not a provocation. And the point is this. Montaigne was especially critical of French culture after his travels abroad. He writes about this many times. He writes that if we compare science in France with science in other countries: with science in Germany, in England, in Rome, in Spain, in the Netherlands - in all these countries, then those principles that are typical French do not apply there , and it's much better. Montaigne criticizes France, and these phrases that I read are not correct statements for Montaigne, but this is his criticism of a specifically French way of thinking. About the teachings of Bourbaki, Montaigne said: “Tout jugements universels sont laches et dangereux” (“all universal judgments are cowardly and dangerous”) - in the Essays in book III, ch. VIII, page 35 of the 1588 edition. In the Essays, much is said about the style of presentation in Chapter XII of Book II, Chapters VIII and IX of Book III. In book I ch. XXVI is especially devoted to education: “The main thing is to stimulate the appetite and feelings: otherwise you will raise a donkey loaded with books, blows of the whip and filling your pocket with science, which you should not only settle in your home, but which you should marry.” Therefore, you are absolutely right that he himself adhered to the opposite point of view expressed by the principles, this is true, but he emphasized that in France this point of view is dominant. By the way, it is interesting that the French point of view was like this much earlier. If you take the notes about Caesar’s Gallic War, then there is already severe criticism of the French, well, the Gauls at that time, of course, but the Celtic character remained in many ways among today’s French, and the characteristics of France that were given by Julius Caesar largely remain today is faithful. Caesar doesn't talk much about science, although he talks about that too. He says that the French (the Gauls) are characterized by theatricality and the desire to put on a theatrical performance where they cannot do anything for real. They cannot achieve anything, but they can pretend. The ability to pretend and pass off as supposedly perfect what they have not achieved is their extremely characteristic feature. They, he says, signed an agreement with Rome that they would not let a single German through and that Rome was completely protected from the Germans, because France would become a wall and would stop the German attack (not France, but Gaul). But, says Caesar, this is not true. If they (the French soldiers) are not fed such food, which is generally impossible to buy, and are not given such wonderful wine, which we cannot supply them, then they will not be able to fight at all, nor climb the Alps, much less , stop the Germans. As soon as the first German regiment crosses the Rhine, all the French will lie down simply so as not to be noticed, and will let the German legions pass, which will crush Rome. Therefore, the only way for Rome to defend itself against the Germans was to conquer this Gaul, and it began the Gallic War.

D. V. Anosov: It’s a great idea to conquer a country for protection from a third country.

From the audience: You have outlined your views on the history of the development of mathematics. How do you feel about theory, about Academician Fomenko’s views on history?

— There is a large book “History and Antihistory”, recently published by the publishing house “Languages ​​of Russian Culture” (Moscow, 2000), in which specialists, historians, astronomers and all sorts of others wrote about this in great detail. I will quote from there one small piece written by Andrei Zaliznyak, the main specialist on Novgorod birch bark documents. According to his description, Fomenko explains the origin of the Scots, who are called Scots in English. Two thousand years ago, Scythian tribes lived north of the Black Sea. The Scythians were pastoralists, and they had a lot of livestock. In addition, they had boats on which they sailed along various rivers; they loved to swim. They loaded their cattle onto boats, sailed up the Dnieper, along the Don, climbed the Oka, the Dvina, crossed the Baltic Sea, to Denmark, to the North Sea, to England, to Scotland, found empty places there, built villages, settled there. But they didn’t like it because the climate was bad, it rained all the time, it was cold. And they decided to return. But since the Aeroflot did not work well in those days, they realized that they would not be able to load all their livestock and return with their livestock quickly. Therefore, they had to leave the cattle there, and the cattle have lived there ever since, this is Scots.

Another of the authors of this book points out that from the experience of the commercial success of Fomenko’s theory, it clearly follows that the important conclusion for historical science is that the cultural and educational level of our population in the field of history is extremely low.

M.A. Tsfasman: Vladimir Igorevich, if in this audience there were several madmen who would like to preserve culture, including the culture of mathematics, what would you recommend them to do?

- You know, this is a very difficult question. I would recommend returning to Kiselev when teaching at school. But that's my personal opinion. My teacher, Andrei Nikolaevich Kolmogorov, really convinced me when he began his reform, to take part in this reform and rewrite all the textbooks, make them in a new way and present them the way he wanted, to Bourbakize school mathematics and so on. I categorically refused, almost quarreled with him, because when he began to tell me his idea, it was such nonsense that it was absolutely obvious to me that he should not be allowed to see the schoolchildren. Unfortunately, after him, several more academicians were missed, and they did even worse than he did. I’m afraid to do this, now I’m not taking on this business, in particular, taking advantage of all this experience. Dear people, A.D. Aleksandrov, Pogorelov, Tikhonov, Pontryagin - all took part and all wrote poorly. I can say for sure that Kolmogorov wrote poorly, for example, and I know about others too; I can criticize the textbooks they offered, but I cannot offer my own textbook...

I myself taught at a school (however, at a boarding school - however, this is not an ordinary school, but I also happened to teach at an ordinary school) - at the boarding school I gave lectures, about which even a book was published by Alekseev, who is present here, based on my lectures. He was one of the listeners, schoolchildren, who recorded these very lectures, exercises, and a good book “Abel’s Theorem in Problems and Solutions.” There is a proof of the theorem that an equation of the fifth degree is unsolvable in radicals. At the same time, complex numbers, Riemann surfaces, covering theory, group theory, solvable groups and much more are presented along the way (for schoolchildren!). I have repeatedly expressed my experience of how, in my opinion, mathematics should be taught, in a concrete way about specific things. I gave various lectures, recorded, published, and so on. I can do this. But to become the head of some large such project would be scary, because, in my opinion, there needs to be some kind of competition, in which the experience of the best teachers is allowed to rise to the top, as happened with Kiselev himself, who was not the best at all mathematician of Russia and who achieved his greatest success by repeatedly reworking his initially not so successful book. It needs good teachers, good teachers need to do it, and they need to do it well.

M.A. Tsfasman: What to do in higher and postgraduate education?

— I have a lot of experience, of course, in this too. The first thesis that has caused enormous damage in higher mathematics education is a thesis that also comes mainly from the French. I learned it from my friend Jean-Pierre Serres, a French mathematician, and the argument is as follows. Serres asserts: you, he says, write incorrectly in many places that mathematics is a part of physics. In fact, mathematics has nothing to do with physics (according to Serres), these are completely orthogonal sciences. Then Serre writes a phrase that I call a boomerang, that is, self-dangerous. This phrase is: “However, we mathematicians should not speak out on such philosophical questions, because even the best of us - well, it is clear that when we talked to him, it was he - even the best of us are capable of speaking on such issues is saying complete nonsense.” Hilbert published the article “Mathematics and Natural Sciences” in 1930, in which he wrote that geometry is part of physics. In this regard, I should have said at some point that the two great algebraists, Hilbert and Serres, act here in a contradictory manner. But my friends, in particular Dmitry Viktorovich Anosov, and others too, told me that this statement of mine is simply based on the fact that I am bad with formal logic, I have not read Aristotle. In fact, the conclusion from these two statements is not a contradiction at all, but by reasoning logically, as schoolchildren are taught, one can draw a logically strict conclusion from these two statements. It is as follows: geometry has nothing to do with mathematics. This is the logic of the French. They decided so, and they excluded geometry from their education. In university education, and in school education too, geometry textbooks are thrown out, and ask some student at Ecole Normale Superiore in Paris, for example, something about the surface xy = z(2) or about a plane curve parametrically defined by the equations x = t( 3) - 3t, y = t(4) - 2t(2) is hopeless, they don’t teach anything about it. The textbooks of L'Hopital, Goursat, Jordan - all these wonderful textbooks, books by Klein, Poincaré - have all been thrown out of student libraries.

D.V. Anosov: Hadamara...

- Hadamara too... Everything is thrown out! Everything was thrown out simply because, as they explained to me, these are old books, they contain a virus that is causing the entire library to rot, including Bourbaki’s books. Is this possible?

E.V. Yurchenko: I wanted to say a few words about the study of geometry and Kiselev’s textbook, what you said. I think that recently teachers have a great opportunity to use different textbooks, and there is a very interesting question about the early study of geometry, even to the point of starting to study it from the first grade, because it does a lot for the development of imagination in children, and based on my work experience, I would not insist only on returning to Kiselev’s textbook.

— I don’t argue, maybe there are better textbooks than Kiselev’s textbook, it’s quite possible. But, in any case, we need a textbook without these general scientific tricks, without Bourbakism, that’s what I mean.

A.Yu. Ovchinnikov: Very small question. In your wonderful book on ordinary differential equations there are an unusually large number of all kinds of beautiful pictures, overall a wonderful book, very interesting and pleasant to read. But, as you can easily verify with a very simple experiment, the vast majority of your students, thanks to this book, cannot solve even very simple differential equations. In your opinion, how does this relate to the seemingly somewhat applied approach that you are now promoting?

- Well, as applied to my students personally, this is simply not true, I have a lot of experience... At the end of the textbook, in the latest edition, there are almost a hundred problems with quite serious equations, and I have a lot of exam experience, written exams in which students in both Moscow and Paris perfectly solve equations that students cannot solve in other courses. And these equations are completely standard at the same time; These are not difficult equations, you know? I specifically dealt with this issue - about requirements, and several times I wrote lists of tasks that must be required in order to be able to solve them. For example, I have such a large article, not only on differential equations, on all mathematics, which I wrote for the Physics and Technology Institute, but it is also suitable for a mathematician, regarding what hundred problems make up the entire mathematics course. These hundred problems in Success have been published, and I highly recommend this article, Mathematical Trivium. These are easy tasks, there are many of them, a hundred, but they are easy. For example, the first task is: “Given a graph of a function. Draw a graph of the derivative." If a person does not know how to do this, then, even if he knew how to differentiate all polynomials and rational functions, he does not understand anything about derivatives. I taught differential equations in exactly the same way, and I have experience, I claim that if someone taught in my textbooks in such a way that students cannot solve the simplest equations, then this is a bad teacher.
* * *

Recently I had to face a task that five-year-old children can cope with, but which was not understood and distorted by the editors of one of the academic journals (“Advances in Physical Sciences”). There are two volumes of Pushkin on the shelf. The sheets of each volume are 2 cm, and each cover is 2 mm. The worm gnawed from the first page of the first volume to the last of the second. How far did he chew?

I'll say a few more words about the tasks.

Here is a typical example of a problem that French schoolchildren can easily cope with: “Prove that all RER trains on the planet Mars are red and blue.”

Here is a sample solution:

Let us denote by Xn(Y) the set of all trains of system Y on planet number n (counting from the Sun, if we are talking about the solar system).

According to the table published by CNRS there and then, the planet Mars has number 4 in the solar system. The set X4(RER) is empty. According to Theorem 999-в from the analysis course, all elements of an empty set have all predetermined properties.

Therefore, all RER trains on planet Mars are red and blue.

Teaching mathematics, as a kind of legal casuistry based on arbitrarily chosen laws, begins from a very early age: French schoolchildren are taught that any real number is greater than itself, that 0 is a natural number, that everything general and abstract is more important than the particular, concrete.

Instead of learning the simple and fundamental principles of science, French students are quickly specialized so that they become experts in some narrow field of their science, knowing nothing else.

Leonardo da Vinci already noted that any idiot, having studied exclusively one narrow topic, having practiced long enough, will achieve success in it. He wrote this in instructions for artists, but he himself was involved in many different areas of science. The adjacent sections of his notes contain detailed instructions for underwater saboteurs (including both the use of fire in underwater work and recommendations for toxic substances).

However, the American school test for decades included the task: find the area of ​​a right triangle with a hypotenuse of 10 inches and a height lowered on it, a length of 6 inches. May this cup pass us away.

Here are some more quotes from old sources explaining how the current sad situation in the field of education and the current illiteracy of the population came about.

Rousseau wrote in his Confessions that he did not believe the formula he himself had proven: “the square of the sum is equal to the sum of the squares of the terms with their double product” until he drew the corresponding partition of the square into four rectangles.

Leibniz explained to Queen Sophia-Charlotte, wanting to save her from the influence of the atheist Newton, that the existence of God is most easily proven by observing our own consciousness. For if our knowledge came only from external events, we could never know universal and absolutely necessary truths. The fact that we know them - and are thereby distinguished from animals - proves, according to Leibniz, our divine origin.

Reforming school education, the French wrote in 1880: “Every thing is worth what it is sold for. What will be the price of your free education?

Abel complained in 1820 that French mathematicians only wanted to teach, but did not want to learn anything. Later they wrote contemptuously that this poor man (whose essay was lost by the Academy of Sciences) “was returning from Paris to his part of Siberia, called Norway, on foot on the ice.”

Abel's schooling began with his father, who taught his son, in particular, that 0 + 1 = 0. The French still teach their schoolchildren and students that every real number is greater than itself and that 0 is a natural number (according to Bourbaki and Leibniz, all common concepts are more important than private ones).

Balzac mentions a “long and very narrow square.”

According to Marat, “the best of mathematicians are Laplace, Monge and Cousin: a kind of automata, accustomed to follow certain formulas, applying them blindly.” However, Napoleon later replaced Laplace as Minister of the Interior “for attempting to introduce the spirit of infinitesimals into administration” (I think that Laplace wanted the accounts to settle down to the penny).

American President Taft declared in 1912 that a spherical triangle with vertices at the North Pole, South Pole and the Panama Canal is equilateral. With American flags flying at the peaks, he considered “the entire hemisphere encompassed by this triangle” to be his.

A. Dumas the Son mentions the “strange architecture” of houses consisting of “half plaster, half brick, half wood” (1856). However, a Parisian newspaper wrote in 1911 that “Mahler’s fifth symphony lasts an hour and a quarter without a break, so that at the third minute listeners look at their watches and say to themselves: another hundred and twelve minutes!” That's probably what happened.

The next story is related to Dubna. Two years ago, the Lynch Academy in Rome celebrated the memory of Bruno Pontecorvo, who lived from 1950 until his death in 1996, either in Moscow or in Dubna. About thirty years before his death, he said that he once got lost (in the vicinity of Dubna?) and only got home by driving up on a tractor. The tractor driver, wanting to be polite, asked: “What are you doing there at the Institute in Dubna?” Pontecorvo answered honestly: “Neutrino physics.”

The tractor driver was very pleased with the conversation, but noted, praising the foreigner’s Russian language: “Still, you still have some accent: physics is not neutrino, but neutron!”

A speaker at the Lynch Academy, in the Proceedings of which I read the entire above incident, comments on it this way: “Now we can already say that Pontecorvo’s prediction has come true: now no one knows not only what a neutrino is, but also what a neutron is!”

Notes

Turaev B.A. God Thoth. - Leipzig, 1898.

. “Russian Champollion” N.A. Nevsky deciphered Tangut hieroglyphs and restored this forgotten language; he was shot in 1937 and posthumously rehabilitated in 1957. “Tangut Philology” was awarded the Lenin Prize in 1962.

The historian Diodorus Siculus writes: “Pythagoras learned from the Egyptians his teaching about the gods, his geometrical propositions and the theory of numbers, the orbit of the sun...” (The Library of History, Book I, 96-98).

For Thoth, apparently, the place of this postulate was taken by several axioms equivalent to it. The fact that they all follow from one of them was apparently proven by Euclid.

It was even alleged that Egyptian women publicly prostituted themselves to crocodiles (P.J. Proudhon, “De la celèbration du dimanche,” 1850). Alexander the Great claimed that the source of the Nile is the Indus River, since both of these rivers are full of crocodiles, and their banks are overgrown with lotuses. He also believed that the Amu Darya is the Tanais, flowing from the north into the Maeotian swamps (i.e., the Don, flowing into the Sea of ​​Azov) and that the Caspian Sea is connected by a strait to the Bay of Bengal of the Indian Ocean (and therefore did not go to China from India). Topology was poorly developed at that time.

Newton's original proof (1666?) was wrong, but he realized this many years later when, on the advice of Halley, he tried to use it to receive a prize of forty shillings promised in a pub by the great London architect Ren Hooke and Halley, who tried to prove ellipticity orbit.

. The "Cartesian" coordinate system was constantly used by the ancient Romans when setting up a military camp so that each legion could be easily located. Traces of this coordinate system are still visible in the topography of the Latin Quarter of Paris. Not far from the origin there is now a store called “Jeux Descartes” (“Descartes’ Games”). However, this name can hardly be considered an attempt to attribute the merits of Caesar to Descartes: after all, “jeux des cartes” are “card games” sold in the mentioned store.

Here is Montaigne's explicit formulation: “Il ne faudra jamais rencontrer quelque idiome du pays (Toscan, Napolitan, etc.) et de se joindre? quelqu"une des taut de formes. Ne faudra quelqu"un de dire "Voila d"o? il le print" ("Experiments", book II, chapter XII, page 274 of the 1588 edition). That is: "You should not use expressions of foreign languages ​​- Tuscan, Neapolitan, etc., nor follow any -or from numerous forms. There is no need for anyone to say: “That’s where he got it from!”" Montaigne was also surprised that “wherever my compatriots go, they always shun foreigners" (Book III, Ch. .ix).

Leibniz considered our innate propensity for deductive reasoning to be proof of the existence of God, who originally put this propensity into the structure of our brains. Literature on the issue of the struggle of Descartes and Leibniz against induction and Newton is given in the article “L"enfance de l"Homme”, Jacques Cheminade, in the journal Fusion, mars-avril 2000, Ed.Alcuin, Paris, p. 44.

. “For the French, deception and treachery are not a sin, but a way of life, a matter of honor, from the time of Emperor Valentinian to the present day.” (Book II, Chapter XVIII)

The French claim that geometry and the “trigonometric form” of complex numbers (moduli, arguments, etc.) were invented by Argan. But many years before him, all this was done in Denmark by Wessel (whose ideas influenced Abel). By the way, Wessel tried to apply hypercomplex numbers (essentially quaternions) to describe the rotations of three-dimensional space. Rotation by an angle around the axis bi + cj + dk (b2 + c2 + d2 = 1) corresponds to the quaternion cos(/2) + sin( /2). The half in this formula has enormous topological significance, and in physics it explains the so-called spin of particles.

The French Revolution obliged all citizens to address each other only as “you,” and violators could be guillotined. So in Paris this custom continues to this day.

According to information that has reached me, Phystech professors, on average, cope with a third of these tasks.

The word "Lynch" means "Lynx": the participants were supposed to have lynx-like vigilance and insight. Galileo, I remember, signed sixth in the thick folio where members of the Lynch Academy are registered (Newton’s number in the folio of the Royal Society of London is much higher).

Vladimir Igorevich Arnold

About the sad fate of “academic” textbooks

A source of information- http://scepsis.ru/library/id_652.html

I consider the experience of creating textbooks for secondary schools by mathematicians of the twentieth century to be tragic. My dear teacher, Andrei Nikolaevich Kolmogorov, for a long time convinced me of the need to finally give schoolchildren a “real” geometry textbook, criticizing all existing ones for the fact that in them such concepts as “an angle of 721 degrees” remain without an exact definition.

The definition of an angle he intended for ten-year-old schoolchildren took, it seems, about twenty pages, and I only remembered the simplified version: the definition of a half-plane.

It began with the “equivalence” of the points of the complement to a line on the plane (two points are equivalent if the segment connecting them does not intersect the line). Then - a rigorous proof that this relation satisfies the axioms of equivalence relations; A is equivalent to A and so on.

Several more theorems established successively that “the set of equivalence classes defined by the previous theorem is finite,” and then that “the cardinality of the finite set defined by the previous theorem is two.”

And in the end, a solemnly nonsensical “definition”: “Each of two elements of a finite set, the cardinality of which, according to the previous theorem, is equal to two, is called a half-plane.”

The hatred of schoolchildren who studied this “geometry” for both geometry and mathematics in general was easy to predict, which is what I tried to explain to Kolmogorov. But he responded with a reference to Bourbaki’s authority: in their book “History of Mathematics” (in the Russian translation of “Architecture of Mathematics” published under the editorship of Kolmogorov) it is said that “like all great mathematicians, according to Dirichlet, we always strive to replace transparent ideas with blind calculations.” .

In the French text, as in Dirichlet's original German statement, it was, of course, "replace blind calculations with transparent ideas." But Kolmogorov, according to him, considered the version introduced by the Russian translator to express the spirit of Bourbaki much more accurately than their own naive text, which goes back to Dirichlet.

Nevertheless, Andrei Nikolaevich forced or persuaded me to take part in his experiments, so in the early sixties I gave a course of lectures for schoolchildren (high school).

Starting with the geometry of complex numbers and Moavre's formula, I quickly moved on to algebraic curves and Riemann surfaces, the fundamental group and coverings, monodromy and regular polyhedra (including exact sequences, normal subgroups, transformation groups and solvable groups). The unsolvability of the symmetry group of the icosahedron is easily deduced from considering the five Kepler cubes inscribed in it. From this elementary geometry, by the end of the semester, I received a proof of Abel’s theorem on the unsolvability in radicals of equations of the fifth and higher powers.

My ideas about a truly modern school textbook can be understood from the text of this school course, subsequently published by one of my then schoolchildren, V.B. Alekseev, in the form of a book “Abel’s Theorem in Problems” (Moscow, Nauka, 1976), as well as in my recently published lecture for schoolchildren “Geometry of complex numbers, quaternions and spins”.

Most of both books are intended for the average student and explain real mathematics to him (although some of it may be unknown to most university mathematics professors).

I would mention here that the continuation of this theory by Abel (who will be 200 years old next year) includes remarkable theorems on the non-representability of integrals by elementary functions (for example, the square root of polynomials of the third degree).

Abel introduced topology into this theory (widely using Riemann surfaces to study his Abelian integrals of algebraic functions). He established the non-elementary nature of integrals in the case when the Riemann surface is not a sphere, but has “handles” (like a torus corresponding to “elliptic integrals” of roots of polynomials of degree three). I suppose that his considerations even lead to the “topological non-elementaryness” of integrals, meaning that neither the function expressing the integral from the upper limit (the so-called elliptic, or Abelian integral), nor its inverse function (the so-called “elliptic function”, like the elliptic sine, describing not too small oscillations of a pendulum without friction or the free rotation of a satellite around its center of gravity) - all these functions are not only non-elementary, but topologically nonequivalent to any elementary functions.

But, unfortunately, mathematicians of subsequent years poorly understood the topological nature of Abel’s reasoning (and did not include his theories in school courses).

For example, the obscurantist Hardy (who, however, was a foreign member of the Russian Academy of Sciences) wrote in his book “Apology for a Mathematician,” recently published in Russian in Izhevsk: “Without Abel, Riemann and Poincaré, mathematics would have lost nothing.”

As a result, the proofs of the two statements formulated above (about the topological non-elementaryness of elliptic, or Abelian, integrals and functions) remain, apparently, unpublished, and the topological theories of Abel, Riemann and Poincaré, which equally transformed both mathematics and physics, including those based on these theories first of all, quantum field theory - these topological sciences needlessly remain completely out of sight of modern schoolchildren, who are instead filled with either definitions of half-planes or specific features of computers from different companies.

The best, in my opinion, of the available mathematics textbooks is “Higher Mathematics for Beginning Physicists” by Ya.B. Zeldovich. Although he appears to be speaking to beginning students, this is, in my opinion, exactly how one should speak to schoolchildren.

And then in one of our best textbooks, written by a leading mathematician for schoolchildren ("Functions and Graphs" by I.M. Gelfand, E.I. Shnol and E.G. Glagoleva), I read that “the value of the function f(x) at point a is denoted by f(a).” After thinking like f(x) is a function and f(a) is a number, how are you supposed to perceive f(y) and f(b)? It is as impossible to teach, after such a beginning, what operators or functors are, as the position of the barber was difficult after the general ordered him to “shave all those who do not shave themselves.”

The distinction between different levels of mathematical objects: elements, sets, subsets, mappings, and so on to functors and even beyond is an absolutely necessary part of elementary mathematical culture, like the distinction between price and bill, or Uzi and hitman.

At one time, Kiselev’s mathematics textbooks conquered Russia with their undeniable merits, although he was not at all a great scientist. Moreover, the first ten editions of these textbooks were still far from the level that was subsequently achieved as a result of repeated revisions caused by comments from teachers who practically used these textbooks. Therefore, I think that in our current or even tomorrow’s conditions, the best textbook will be written not by the greatest scientist and not by me at all, but by the most experienced teacher, and even then not immediately, but after a long trial run in many schools by his equally experienced colleagues.

I would only like to warn against uncritical borrowing of foreign experience, especially American (where simple fractions were abolished, limiting themselves to decimal computer ones) and French (where they stopped teaching counting altogether, again referring to calculators, and drawings were banished on the advice of Descartes).

Recently I encountered the great joy of Parisian mathematics teachers when they elected their representative to the section of mathematical education for schoolchildren of the International Mathematical Union. They explained to me that they “pushed her up” so that she would not disturb her colleagues in Paris with her ideas for “introducing computer didactics into teaching schoolchildren the basics of mathematical analysis.”

This “didactics” consists of replacing traditional exercises like “draw graphs of the functions sin2(x) and sin(x)2” by cramming the rules for pressing computer buttons and accessing the “Mathematics” (and similar) systems of standard computer training.

On the other hand, my students in Paris explained to me that their military training included teaching reading, writing and arithmetic to recruit soldiers, of whom about twenty percent are now completely illiterate (and can send missiles on written orders that they could not understand, not in that side!).

An attempt to bring to us “modern” teaching methods from “advanced” countries would lead our school education system to precisely this state. May this cup pass us away!

Vladimir Igorevich Arnold

New obscurantism and Russian enlightenment

A source of information- http://scepsis.ru/library/id_650.html

I dedicate to my Teacher - Andrei Nikolaevich Kolmogorov

Reference: obscurantism is a hostile attitude towards education and science.

“Don’t touch my circles,” Archimedes said to the Roman soldier who was killing him. This prophetic phrase came to mind in the State Duma, when the chairman of the meeting of the Education Committee (October 22, 2002) interrupted me with the words: “We don’t have an Academy of Sciences, where we can defend the truth, but a State Duma, where everything is based on what we have.” Different people have different opinions on different issues.”

The view I advocated was that three times seven is twenty-one, and that teaching our children both the multiplication tables and the addition of single-digit numbers and even fractions is a national necessity. I mentioned the recent introduction in the state of California (on the initiative of Nobel laureate, transuranium physicist Glen Seaborg) of a new requirement for schoolchildren entering universities: you need to be able to independently divide the number 111 by 3 (without a computer).

The listeners in the Duma, apparently, could not separate, and therefore did not understand either me or Seaborg: in Izvestia, with a friendly presentation of my phrase, the number “one hundred eleven” was replaced by “eleven” (which makes the question much more difficult, since eleven is not divisible by three).

I came across the triumph of obscurantism when I read in Nezavisimaya Gazeta an article “Retrogrades and Charlatans” glorifying the newly built pyramids near Moscow, where the Russian Academy of Sciences was declared to be a collection of retrogrades inhibiting the development of science (trying in vain to explain everything with their “laws of nature”). I must say that I, apparently, am also a retrograde, since I still believe in the laws of nature and believe that the Earth rotates around its axis and around the Sun, and that younger schoolchildren need to continue to explain why it is cold in winter and warm in summer, without allowing the level of our school education to fall below what was achieved in parochial schools before the revolution (namely, our current reformers are striving for a similar decline in the level of education, citing the truly low American school level).

American colleagues explained to me that the low level of general culture and school education in their country is a deliberate achievement for economic purposes. The fact is that, after reading books, an educated person becomes a worse buyer: he buys less washing machines and cars, and begins to prefer Mozart or Van Gogh, Shakespeare or theorems to them. The economy of a consumer society suffers from this and, above all, the income of the owners of life - so they strive to prevent culture and education (which, in addition, prevent them from manipulating the population as a herd devoid of intelligence).

Faced with anti-scientific propaganda in Russia, I decided to look at the pyramid, recently built about twenty kilometers from my house, and rode there on a bicycle through the centuries-old pine forests between the Istra and Moscow rivers. Here I encountered a difficulty: although Peter the Great forbade cutting down forests closer than two hundred miles from Moscow, several of the best square kilometers of pine forest on my way had recently been fenced off and mutilated (as the local villagers explained to me, this was done by “a person known [to everyone except me!] V.A.] bandit Pashka"). But even twenty years ago, when I was picking up a bucket of raspberries in this now built-up clearing, a whole herd of wild boars walking along the clearing passed me, making a semicircle with a radius of ten meters.

Similar developments are happening everywhere now. Not far from my house, at one time the population did not allow (even using television protests) the development of a forest by Mongolian and other officials. But since then the situation has changed: the former government-party villages are seizing new square kilometers of ancient forest in front of everyone, and no one is protesting anymore (in medieval England, “fencing” caused uprisings!).

True, in the village of Soloslov, next to me, one member of the village council tried to object to the development of the forest. And then, in broad daylight, a car with armed bandits arrived, who shot him right in the village, at home. And the development took place as a result.

In another neighboring village, Daryin, an entire field has been rebuilt with mansions. The attitude of the people to these events is clear from the name that they in the village gave to this built-up field (a name, unfortunately, not yet reflected on the maps): “thieves’ field.”

The new motorized residents of this field have turned the highway leading from us to the Perkhushkovo station into their opposite. Buses have almost stopped running along it in recent years. At first, new residents-motorists collected money at the terminal station for the bus driver so that he would declare the bus “out of order” and passengers would pay private traders. Cars of new residents of the “field” are now rushing along this highway at great speed (and often in someone else’s lane). And I, walking five miles to the station, risk being knocked over, like my many pedestrian predecessors, whose places of death were recently marked on the roadsides with wreaths. Electric trains, however, now also sometimes do not stop at the stations provided for by the schedule.

Previously, the police tried to measure the speed of murderous motorists and prevent them, but after a policeman measuring the speed with a radar was shot by a guard of a passing person, no one dares stop cars anymore. From time to time I find spent cartridges right on the highway, but it’s not clear who was shot at. As for the wreaths over the places where pedestrians died, all of them have recently been replaced with notices “Dumping of garbage is prohibited”, hung on the same trees where there were previously wreaths with the names of those dumped.

Along the ancient path from Aksinin to Chesnokov, using the roads laid by Catherine II, I reached the pyramid and saw inside it “shelves for charging bottles and other objects with occult intellectual energy.” The instructions, several square meters in size, listed the benefits of a several-hour stay of an object or a patient with hepatitis A or B in the pyramid (I read in the newspaper that someone even sent a multi-kilogram load of stones “charged” by the pyramid to the space station for public money).

But the compilers of this instruction also showed honesty that was unexpected for me: they wrote that it is not worth crowding in line at the shelves inside the pyramid, since “tens of meters from the pyramid, outside, the effect will be the same.” This, I think, is absolutely true.

So, as a true “retrograde,” I consider this whole pyramidal enterprise to be a harmful, anti-scientific advertisement for a store selling “loading objects.”

But obscurantism has always followed scientific achievements, starting from antiquity. Aristotle's student, Alexander Philipovich of Macedon, made a number of "scientific" discoveries (described by his companion, Arian, in Anabasis). For example, he discovered the source of the Nile River: according to him, it is the Indus. The “scientific” evidence was: “These are the only two large rivers that are infested with crocodiles” (and confirmation: “In addition, the banks of both rivers are overgrown with lotuses”).

However, this is not his only discovery: he also “discovered” that the Oxus River (today called the Amu Darya) “flows - from the north, turning near the Urals - into the Meotian swamp of the Euxine Pontus, where it is called Tanais” (“Tanais " is the Don, and the "Meotian swamp" is the Sea of ​​​​Azov). The influence of obscurantist ideas on events is not always negligible:

Alexander from Sogdiana (that is, Samarkand) did not go further to the East, to China, as he first wanted, but to the south, to India, fearing a water barrier connecting, according to his third theory, the Caspian (“Hyrcanian”) Sea with the Indian Ocean (in the Bay of Bengal region). For he believed that seas, “by definition,” are bays of the ocean. This is the kind of “science” we are being led to.

I would like to express the hope that our military will not be so strongly influenced by obscurantists (they even helped me save geometry from the attempts of the “reformers” to expel it from school). But today’s attempts to lower the level of schooling in Russia to American standards are extremely dangerous both for the country and for the world.

In today's France, 20% of army recruits are completely illiterate, do not understand written orders from officers (and can send their missiles with warheads in the wrong direction). May this cup pass from us! Our people are still reading, but the “reformers” want to stop this: “Both Pushkin and Tolstoy are too much!” - they write.

It would be too easy for me, as a mathematician, to describe how they plan to eliminate our traditionally high-quality mathematics education in schools. Instead, I will list several similar obscurantist ideas regarding the teaching of other subjects: economics, law, social studies, literature (subjects, however, they propose to abolish everything in school).

The two-volume project “Standards of General Education” published by the Russian Ministry of Education contains a large list of topics whose knowledge it is proposed to stop requiring students to know. It is this list that gives the clearest idea of ​​the ideas of the “reformers” and what kind of “excessive” knowledge they seek to “protect” the next generations from.

I will refrain from political comments, but here are typical examples of supposedly “unnecessary” information extracted from the four-hundred-page Standards project:

Constitution of the USSR;
fascist “new order” in the occupied territories;
Trotsky and Trotskyism;
major political parties;
Christian democracy;
inflation;
profit;
currency;
securities;
multi-party system;
guarantees of rights and freedoms;
law enforcement agencies;
money and other securities;
forms of state-territorial structure of the Russian Federation;
Ermak and the annexation of Siberia;
foreign policy of Russia (XVII, XVIII, XIX and XX centuries);
Polish question;
Confucius and Buddha;
Cicero and Caesar;
Joan of Arc and Robin Hood;
Individuals and legal entities;
the legal status of a person in a democratic state governed by the rule of law;
separation of powers;
judicial system;
autocracy, Orthodoxy and nationality (Uvarov’s theory);
peoples of Russia;
Christian and Islamic world;
Louis XIV;
Luther;
Loyola;
Bismarck;
The State Duma;
unemployment;
sovereignty;
stock market (exchange);
state revenues;
family income.

“Social studies”, “history”, “economics” and “law”, devoid of discussion of all these concepts, are simply formal worship services, useless for students. In France, I recognize this kind of theological chatter on abstract topics by the key set of words: “France, as the eldest daughter of the Catholic Church...” (this can be followed by anything, for example: “... does not need to spend on science, since We already had scientists and still have them”), as I heard at a meeting of the National Committee of the Republic of France for Science and Research, of which the Minister of Science, Research and Technology of the Republic of France appointed me as a member.

In order not to be one-sided, I will also give a list of “undesirable” (in the same sense of “inadmissibility” of their serious study) authors and works mentioned in this capacity by the shameful “Standard”:

Glinka;
Chaikovsky;
Beethoven;
Mozart;
Grieg;
Raphael;
Leonardo da Vinci;
Rembrandt;
Van Gogh;
Omar Khayyam;
"Tom Sawyer";
"Oliver Twist";
Shakespeare's Sonnets;
“Journey from St. Petersburg to Moscow” by Radishchev;
"The Steadfast Tin Soldier";
"Gobsek";
"Père Goriot"
"Les Miserables";
"White Fang";
"Belkin's Tales";
"Boris Godunov";
"Poltava";
"Dubrovsky";
"Ruslan and Ludmila";
"Pig under the Oak";
"Evenings on a Farm Near Dikanka";
"Horse surname";
"Pantry of the Sun";
"Meshcherskaya side";
"Quiet Don";
"Pygmalion";
"Hamlet";
"Faust";
"A Farewell to Arms";
"Noble Nest";
"Lady with a dog";
"Jumper";
"A cloud in pants";
"Black man";
"Run";
"Cancer Ward";
"Vanity Fair";
"For whom the Bell Tolls";
"Three Comrades";
"In the first circle";
"The Death of Ivan Ilyich."

In other words, they propose to abolish Russian Culture as such. They try to “protect” schoolchildren from the influence of “excessive,” according to “Standards,” cultural centers; these turned out to be undesirable, according to the compilers of the “Standards”, for mention by teachers at school:

Hermitage Museum;
Russian Museum;
Tretyakov Gallery;
Pushkin Museum of Fine Arts in Moscow.

The bell is ringing for us!

It is still difficult to resist and not mention at all what exactly it is proposed to make “optional for learning” in the exact sciences (in any case, the “Standards” recommend “not requiring schoolchildren to master these sections”):

Structure of atoms;
concept of long-range action;
structure of the human eye;
uncertainty relation of quantum mechanics;
fundamental interactions;
starry sky;
The sun is like one of the stars;
cellular structure of organisms;
reflexes;
genetics;
origin of life on Earth;
evolution of the living world;
the theories of Copernicus, Galileo and Giordano Bruno;
theories of Mendeleev, Lomonosov, Butlerov;
the merits of Pasteur and Koch;
sodium, calcium, carbon and nitrogen (their role in metabolism);
oil;
polymers.

In mathematics, the same discrimination was applied to topics in the Standards, which no teacher can do without (and without a full understanding of which schoolchildren will be completely helpless in physics, technology, and a huge number of other applications of science, including both military and humanitarian):

Necessity and sufficiency;
locus of points;
sines of angles at 30o, 45o, 60o;
constructing the angle bisector;
dividing a segment into equal parts;
measuring the angle;
concept of length of a segment;
the sum of the terms of an arithmetic progression;
sector area;
inverse trigonometric functions;
simple trigonometric inequalities;
equalities of polynomials and their roots;
geometry of complex numbers (necessary for physics)
alternating current, and for radio engineering, and for quantum mechanics);
construction tasks;
plane angles of a trihedral angle;
derivative of a complex function;
converting simple fractions to decimals.

The only hope is that the existing thousands of well-trained teachers will continue to fulfill their duty and teach all this to new generations of schoolchildren, despite any orders from the Ministry. Common sense is stronger than bureaucratic discipline. We just need to remember to pay our wonderful teachers adequately for their feat.

Representatives of the Duma explained to me that the situation could be greatly improved if attention was paid to the implementation of the laws on education that have already been adopted.

The following description of the state of affairs was stated by Deputy I.I. Melnikov in his report at the Mathematical Institute. V.A. Steklov of the Russian Academy of Sciences in Moscow in the fall of 2002.

For example, one of the laws provides for an annual increase in the budget contribution to training by approximately 20% per year. But the minister said that “there is no need to worry about the implementation of this law, since the almost annual increase occurs by more than 40%.” Shortly after this speech by the minister, an increase (by a much smaller percentage) that was practically feasible for the next year (it was 2002) was announced. And if we also take into account inflation, then it turns out that a decision was made to reduce the real annual contribution to education.

Another law specifies the percentage of budget expenditures that must be spent on education. In reality, much less is spent (I was not able to find out exactly how many times). But spending on “defense against an internal enemy” increased from a third to half of spending on defense against an external enemy.

It’s natural to stop teaching children fractions, otherwise, God forbid, they’ll understand!

Apparently, it was precisely in anticipation of the reaction of teachers that the compilers of the “Standard” provided a number of names of writers in their list of recommended reading (like the names of Pushkin, Krylov, Lermontov, Chekhov and the like) with an “asterisk” sign, which they deciphered as: “At will teacher can introduce students to one or two more works by the same author” (and not just the “Monument” they recommended in the case of Pushkin).

The higher level of our traditional mathematical education compared to foreign countries became obvious to me only after I was able to compare this level with foreign ones, having worked many semesters at universities and colleges in Paris and New York, Oxford and Cambridge, Pisa and Bologna, Bonn and Berkeley, Stanford and Boston, Hong Kong and Kyoto, Madrid and Toronto, Marseille and Strasbourg, Utrecht and Rio de Janeiro, Conakry and Stockholm.

“There is no way we can follow your principle of choosing candidates based on their scientific achievements,” my colleagues told me on the commission for inviting new professors to one of the best universities in Paris. “After all, in this case we would have to choose only Russians - their scientific superiority is so clear to us all!” (I also spoke about selection among the French).

At the risk of being understood only by mathematicians, I will still give examples of responses from the best candidates for a professorship in mathematics at a university in Paris in the spring of 2002 (200 people applied for each position).

The candidate has been teaching linear algebra at various universities for several years, defended his dissertation and published a dozen articles in the best mathematical journals in France.

The selection includes an interview, where the candidate is always asked elementary but important questions (at the level of the question “Name the capital of Sweden” if the subject were geography).

So I asked, "What is the signature of the quadratic form xy?"

The candidate demanded the 15 minutes allotted to him to think, after which he said: “In my computer in Toulouse, I have a routine (program) that in an hour or two could find out how many pluses and how many minuses there will be in normal form. The difference between these two numbers will be the signature - but you only give 15 minutes, and without a computer, so I can’t answer, this form of xy is too complicated.”

For non-specialists, I will explain that if we were talking about zoology, then this answer would be similar to this: “Linnaeus listed all the animals, but whether the birch is a mammal or not, I cannot answer without a book.”

The next candidate turned out to be a specialist in “systems of elliptic partial differential equations” (a decade and a half after defending his dissertation and more than twenty published works).

I asked this one: “What is the Laplacian of the function 1/r in three-dimensional Euclidean space?”

The response (within the usual 15 minutes) was amazing to me; “If r were in the numerator, and not in the denominator, and the first derivative was required, and not the second, then I would be able to calculate it in half an hour, but otherwise the question is too difficult.”

Let me explain that the question was from the theory of elliptic equations, like the question “Who is the author of Hamlet?” in the English Literature exam. Trying to help, I asked a series of leading questions (similar to questions about Othello and Ophelia): “Do you know what the law of gravity is? Coulomb's law? How are they related to the Laplacian? What is the fundamental solution of Laplace’s equation?”

But nothing helped: neither Macbeth nor King Lear were known to the candidate if we were talking about literature.

Finally, the chairman of the examination committee explained to me what was going on: “After all, the candidate was studying not just one elliptic equation, but their systems, and you ask him about Laplace’s equation, which is only one - it’s clear that he has never encountered it!”

In a literary analogy, this “justification” would correspond to the phrase: “The candidate studied English poets, how can he know Shakespeare, he is a playwright!”

The third candidate (and dozens of them were interviewed) was working on “holomorphic differential forms,” and I asked him: “What is the Riemann surface of the tangent?” (I was afraid to ask about the arctangent).

Answer: “The Riemannian metric is the quadratic form of coordinate differentials, but what form is associated with the tangent function is not at all clear to me.”

I will explain again with a sample of a similar answer, this time replacing mathematics with history (to which the Mitrofans are more inclined). Here the question would be: “Who is Julius Caesar?”, and the answer: “The rulers of Byzantium were called Caesars, but I don’t know Julius among them.”

Finally, a candidate probabilist appeared, talking interestingly about his dissertation. He proved in it that the statement “A and B are true together” is false (statements A and B themselves are formulated at length, so I will not reproduce them here).

Question: “But what about statement A by itself, without B: is it true or false?”

Answer: “After all, I said that the statement “A and B” is false. This means that A is also false." That is: “Since it is not true that “Petya and Misha got cholera,” then Petya did not get cholera.”

Here my bewilderment was again dispelled by the chairman of the commission: he explained that the candidate was not a probabilist, as I thought, but a statistician (in the biography, called CV, there is not “proba”, but “stat”).

“Probabilists,” our experienced chairman explained to me, “have normal logic, the same as that of mathematicians, Aristotelian. For statisticians, it’s completely different: it’s not for nothing that they say “there are lies, blatant lies, and statistics.” All their reasoning is unsubstantiated, all their conclusions are erroneous. But they are always very necessary and useful, these conclusions. We definitely need to accept this statistician!”

At Moscow University, such an ignoramus would not be able to complete the third year of the Faculty of Mechanics and Mathematics. Riemann surfaces were considered the pinnacle of mathematics by the founder of the Moscow Mathematical Society, N. Bugaev (father of Andrei Bely). He, however, believed that in contemporary mathematics at the end of the 19th century, objects that did not fit into the mainstream of this old theory began to appear - non-holomorphic functions of real variables, which, in his opinion, were the mathematical embodiment of the idea of ​​free will to the same extent as Riemann surfaces and holomorphic functions embody the idea of ​​fatalism and predetermination.

As a result of these reflections, Bugaev sent young Muscovites to Paris to learn there the new “mathematics of free will” (from Borel and Lebesgue). This program was brilliantly carried out by N.N. Luzin, who upon his return to Moscow created a brilliant school, including all the main Moscow mathematicians of many decades: Kolmogorov and Petrovsky, Aleksandrov and Pontryagin, Menshov and Keldysh, Novikov and Lavrentiev, Gelfand and Lyusternik.

By the way, Kolmogorov recommended to me the Parisiana Hotel (on Tournefort Street, not far from the Pantheon) that Luzin subsequently chose for himself in the Latin Quarter of Paris. During the First European Mathematical Congress in Paris (1992) I stayed in this inexpensive hotel (with 19th-century amenities, no telephone, etc.). And the elderly owner of this hotel, having learned that I had come from Moscow, immediately asked me: “How is my old guest, Luzin, doing there? It’s a pity that he hasn’t visited us for a long time.”

A couple of years later, the hotel was closed for renovation (the owner probably died) and they began to rebuild it in an American way, so now you can no longer see this 19th-century island in Paris.

Returning to the choice of professors in 2002, I note that all the ignoramuses listed above received (from everyone except me) the best grades. On the contrary, the only, in my opinion, worthy candidate was almost unanimously rejected. He discovered (with the help of “Gröbner bases” and computer algebra) several dozen new completely integrable systems of Hamiltonian equations of mathematical physics (at the same time, but not including in the list of new ones, the famous Korteweg-de Vries, Sayn-Gordon, and the like equations).

As a future project, the candidate also proposed a new computer method for modeling diabetes treatment. To my question about the assessment of his method by doctors, he answered quite reasonably: “The method is now being tested in such and such centers and hospitals, and in six months they will give their conclusions, comparing the results with other methods and with control groups of patients, but for now this examination is not has been carried out, and there are only preliminary assessments, albeit good ones.”

They rejected him with the following explanation: “On every page of his dissertation, either Lie groups or Lie algebras are mentioned, but no one here understands this, so he will not fit into our team at all.” True, it would have been possible to reject both me and all my students, but some colleagues think that the reason for the rejection was different: unlike all the previous candidates, this one was not French (he was a student of a famous American professor from Minnesota).

The whole picture described leads to sad thoughts about the future of French science, in particular mathematics. Although the “French National Committee for Science” was inclined to not finance new scientific research at all, but to spend money (provided by Parliament for the development of science) on the purchase of ready-made American recipes, I sharply opposed this suicidal policy and still achieved at least some subsidizing new research.

However, a difficulty was caused by the division of money. Medicine, nuclear energy, polymer chemistry, virology, genetics, ecology, environmental protection, radioactive waste disposal and much more were consistently voted unworthy of subsidies by voting (during a five-hour meeting). In the end, they chose three “sciences” that allegedly deserved funding for their new research. These three “sciences” are:

2) psychoanalysis;

3) a complex branch of pharmaceutical chemistry, the scientific name of which I am unable to reproduce, but which is engaged in the development of psychotropic drugs, similar to lacrimogenic gas, turning the rebellious crowd into an obedient herd.

So now France is saved!

Of all Luzin’s students, the most remarkable contribution to science was made, in my opinion, by Andrei Nikolaevich Kolmogorov. Having grown up in a village with his grandfather near Yaroslavl, Andrei Nikolaevich proudly referred to Gogol’s words as “an efficient Roslavl peasant.”

He had no intention of becoming a mathematician, even having already entered Moscow University, where he immediately began studying history (in Professor Bakhrushin’s seminar) and, before he was even twenty years old, wrote his first scientific work.

This work was devoted to the study of land economic relations in medieval Novgorod. Tax documents have been preserved here, and the analysis of a huge number of these documents using statistical methods led the young historian to unexpected conclusions, which he spoke about at the Bakhrushin meeting.

The report was very successful, and the speaker was much praised. But he insisted on another approval: he wanted his conclusions to be recognized as correct.

In the end, Bakhrushin told him: “This report must be published; he is very interesting. But as for conclusions, we historians always need not one piece of evidence, but at least five to recognize any conclusion!”

The next day, Kolmogorov changed history to mathematics, where proof alone is enough. He did not publish the report, and this text remained in his archive until, after the death of Andrei Nikolaevich, it was shown to modern historians, who recognized it not only as very new and interesting, but also quite conclusive. Now this Kolmogorov report has been published, and is considered by the community of historians as an outstanding contribution to their science.

Having become a professional mathematician, Kolmogorov remained, unlike most of them, first of all a natural scientist and thinker, and not at all a multiplier of multi-digit numbers (which mainly appears when analyzing the activities of mathematicians to people unfamiliar with mathematics, including even L.D. Landau, who valued mathematics is precisely the continuation of counting skill: five five - twenty-five, six six - thirty-six, seven seven - forty-seven, as I read in a parody of Landau compiled by his Physics and Technology students; however, in Landau’s letters to me, who was then a student, mathematics no more logical than in this parody).

Mayakovsky wrote: “After all, he can extract the square root every second” (about the professor who “doesn’t get bored that under the window the students are actively going to the gymnasium”).

But he perfectly described what a mathematical discovery was, saying that “Whoever discovered that two and two equals four was a great mathematician, even if he discovered it by counting cigarette butts. And anyone who today calculates much larger objects, such as locomotives, using the same formula, is not a mathematician at all!”

Kolmogorov, unlike many others, was never intimidated by applied, “locomotive” mathematics, and he joyfully applied mathematical considerations to a variety of areas of human activity: from hydrodynamics to artillery, from celestial mechanics to poetry, from the miniaturization of computers to the theory of Brownian motion, from the divergence of Fourier series to the theory of information transmission and to intuitionistic logic. He laughed at the fact that the French write “Celestial Mechanics” with a capital letter, and “applied” with a small letter.

When I first arrived in Paris in 1965, the elderly Professor Fréchet warmly greeted me with the following words: “After all, you are a student of Kolmogorov, the young man who constructed an example of a Fourier series that is almost everywhere divergent!”

The work mentioned here by Kolmogorov was completed by him at the age of nineteen, solved a classical problem and immediately promoted this student to the rank of first-class mathematicians of world significance. Forty years later, this achievement still remained more significant for Frechet than all subsequent and much more important fundamental works of Kolmogorov, which revolutionized the theory of probability, the theory of functions, hydrodynamics, celestial mechanics, the theory of approximations, and the theory of algorithmic complexity, and the theory of cohomology in topology, and the theory of control of dynamical systems (where Kolmogorov’s inequalities between derivatives of different orders remain one of the highest achievements today, although control theorists rarely understand this).

But Kolmogorov himself was always somewhat skeptical about his beloved mathematics, perceiving it as a small part of natural science and easily abandoning the logical restrictions that the shackles of the axiomatic-deductive method impose on true mathematicians.

“It would be in vain,” he told me, “to look for mathematical content in my works on turbulence. I speak here as a physicist and am not at all concerned with mathematical proofs or derivations of my conclusions from initial premises, such as the Navier-Stokes equations. Even if these conclusions have not been proven, they are true and open, and this is much more important than proving them!”

Many of Kolmogorov’s discoveries were not only not proven (neither by himself nor by his followers), but were not even published. But nevertheless, they have already had and continue to have a decisive influence on a number of departments of science (and not only mathematics).

I will give just one famous example (from the theory of turbulence).

A mathematical model of hydrodynamics is a dynamic system in the space of fluid velocity fields, which describes the evolution of the initial velocity field of fluid particles under the influence of their interaction: pressure and viscosity (as well as under the possible influence of external forces, for example, weight force in the case of a river or water pressure in a water pipe).
Under the influence of this evolution, a dynamic system can come to an equilibrium (stationary) state, when the flow speed at each point in the flow region does not change with time (although everything flows, and each particle moves and changes its speed over time).

Such stationary flows (for example, laminar flows in terms of classical hydrodynamics) are attracting points of a dynamic system. They are therefore called (point) attractors.

Other sets that attract neighbors are also possible, for example, closed curves depicting flows that periodically change over time in the functional space of velocity fields. Such a curve is an attractor when the neighboring initial conditions, depicted by “perturbed” points of the functional space of velocity fields close to the indicated closed curve, begin, although not periodically changing with time, a flow that approaches it (namely, the perturbed flow tends to the one described earlier periodically over time).

Poincaré, who first discovered this phenomenon, called such closed attractor curves “stable limit cycles.” From a physical point of view, they can be called periodic steady flow regimes: the disturbance gradually fades during the transition process caused by the disturbance of the initial condition, and after some time the difference between the motion and the undisturbed periodic one becomes barely noticeable.

After Poincaré, such limit cycles were studied extensively by A.A. Andronov, who based on this mathematical model the study and calculation of radio wave generators, that is, radio transmitters.

It is instructive that the theory of the birth of limit cycles from unstable equilibrium positions, discovered by Poincaré and developed by Andronov, is today usually called (even in Russia) Hopf bifurcation. E. Hopf published part of this theory a couple of decades after Andronov’s publication and more than half a century after Poincaré, but unlike them he lived in America, so the well-known eponymic principle worked: if any object bears someone’s name , then this is not the name of the discoverer (for example, America is not named after Columbus).

The English physicist M. Berry called this eponymous principle “Arnold’s principle,” adding a second one to it. Berry's principle: Arnold's principle applies to itself (that is, it was known before).

I completely agree with Berry on this. I told him the eponymous principle in response to a preprint about the “Berry phase,” examples of which, in no way inferior to the general theory, were published by S.M. decades before Berry. Rytov (under the name “inertia of polarization direction”) and A.Yu. Ishlinsky (under the title “the departure of the submarine’s gyroscope due to a discrepancy between the path of returning to the base and the path of leaving it”),

Let us return, however, to attractors. An attractor, or attracting set, is a steady state of motion, which, however, does not have to be periodic. Mathematicians have also studied much more complex movements, which can also attract disturbed neighboring movements, but which themselves can be extremely unstable: small causes sometimes cause large consequences, Poincaré said. The state, or “phase,” of such a limiting regime (that is, a point on the surface of the attractor) can move along the surface of the attractor in a bizarre “chaotic” manner, and a slight deviation of the starting point on the attractor can greatly change the course of movement without changing the limiting regime at all. Averages over long times from all possible observable quantities will be close in the original and in the perturbed motion, but the details at a fixed moment in time will, as a rule, be completely different.

In meteorological terms, the “limit regime” (attractor) can be likened to climate, and the phase to weather. A small change in initial conditions can have a big impact on tomorrow's weather (and even more on the weather a week and a month from now). But such a change will not make the tundra a tropical forest: just a thunderstorm may break out on Friday instead of Tuesday, which may not change the average for the year (or even for the month).

In hydrodynamics, the degree of attenuation of initial disturbances is usually characterized by viscosity (so to speak, the mutual friction of fluid particles as they move relative to one another), or by the inverse viscosity, a value called the “Reynolds number.” Large values ​​of the Reynolds number correspond to weak attenuation of disturbances, and large values ​​of viscosity (that is, small Reynolds numbers), on the contrary, regularize the flow, preventing disturbances and their development. In economics, the role of “viscosity” is often played by bribes and corruption.

Due to high viscosity, at low Reynolds numbers, a stable stationary (laminar) flow is usually established, represented in the space of velocity fields by a point attractor.

The main question is how the flow pattern will change with increasing Reynolds number. In water supply, this corresponds, for example, to an increase in water pressure, which makes a smooth (laminar) stream from a tap unstable, but mathematically, to increase the Reynolds number, it is more convenient to reduce the particle friction coefficient expressing viscosity (which in an experiment would require a technically complex fluid replacement). However, sometimes to change the Reynolds number it is enough to change the temperature in the laboratory. I saw such an installation in Novosibirsk at the Institute of Precision Measurements, where the Reynolds number changed (in the fourth digit) when I brought my hand closer to the cylinder where the flow occurred (precisely due to a change in temperature), and on the computer screen processing the experiment, this change in the Reynolds number immediately indicated by electronic automation.

Thinking about these phenomena of transition from a laminar (stable stationary) flow to a stormy turbulent one, Kolmogorov long ago expressed a number of hypotheses (which to this day remain unproven). I think that these hypotheses date back to the time (1943) of his dispute with Landau about the nature of turbulence. In any case, he clearly formulated them at his seminar (on hydrodynamics and the theory of dynamical systems) at Moscow University in 1959, where they were even part of the announcement about the seminar that he posted at that time. But I don’t know of any formal publication of these hypotheses by Kolmogorov, and in the West they are usually attributed to their epigones of Kolmogorov, who learned about them and published them dozens of years later.

The essence of these Kolmogorov hypotheses is that as the Reynolds number increases, the attractor corresponding to the steady flow regime becomes more and more complex, namely, that its dimension increases.

First it is a point (zero-dimensional attractor), then a circle (Poincaré limit cycle, one-dimensional attractor). And Kolmogorov’s hypothesis about attractors in hydrodynamics consists of two statements: as the Reynolds number increases, 1) attractors of increasingly larger dimensions appear; 2) all low-dimensional attractors disappear.

From 1 and 2 together it follows that when the Reynolds number is sufficiently large, the steady state certainly has many degrees of freedom, so that to describe its phase (points on the attractor) you need to set many parameters, which then, when moving along the attractor, will be whimsical and non-periodic change in a “chaotic” way, and a small change in the starting point on the attractor leads, as a rule, to a large (after a long time) change in the “weather” (the current point on the attractor), although it does not change the attractor itself (that is, it will not cause a change in the “climate” ").

Statement 1 in itself is not sufficient here, since different attractors can coexist, including attractors of different dimensions in one system (which, thus, can perform a calm “laminar” movement under some initial conditions and a stormy “turbulent” one under others, depending on its initial state).

The experimental observation of such effects of “prolonged loss of stability” surprised physicists for a long time, but Kolmogorov added that even if the low-dimensional attractor does not disappear, it may not change the observed turbulence in the case when the size of its zone of attraction decreases significantly with increasing Reynolds number. In this case, the laminar regime, although possible in principle (and even stable), is practically not observed due to the extreme smallness of its area of ​​attraction: already small, but always present in the experiment, disturbances can lead the system from the area of ​​attraction of this attractor to the area of ​​attraction another, already turbulent, steady state, which will be observed.

This discussion may also explain this strange observation: some famous hydrodynamic experiments of the 19th century could not be repeated in the second half of the 20th century, although they tried to use the same equipment in the same laboratory. It turned out, however, that the old experiment (with its prolongation of the loss of stability) can be repeated if it is done not in the old laboratory, but in a deep underground mine.

The fact is that modern street traffic has greatly increased the magnitude of “imperceptible” disturbances, which began to have an effect (due to the smallness of the zone of attraction of the remaining “laminar” attractor).

Numerous attempts by many mathematicians to confirm Kolmogorov's hypotheses 1 and 2 (or at least the first) with evidence have so far only led to estimates of the dimensions of attractors in terms of Reynolds numbers from above: this dimension cannot become too large as long as viscosity prevents this.

The dimensionality is estimated in these works by a power function of the Reynolds number (that is, a negative degree of viscosity), and the exponent depends on the dimension of the space where the flow occurs (in a three-dimensional flow, turbulence is stronger than in plane problems).

As for the most interesting part of the problem, that is, estimating the dimension from below (at least for some attractors, as in Hypothesis 1, or even for all, as in Hypothesis 2, about which Kolmogorov expressed more doubts), here the mathematicians were not able to height, because, according to their habit, they replaced the real natural scientific problem with their formal axiomatic abstract formulation with its precise but treacherous definitions.

The fact is that the axiomatic concept of an attractor was formulated by mathematicians with the loss of some properties of the physical limiting mode of motion, which (not strictly defined) concept of mathematics they tried to axiomatize by introducing the term “attractor”.

Let us consider, for example, an attractor that is a circle (to which all close dynamics trajectories spirally approach).
On this very circle attracting neighbors, let the dynamics be arranged as follows: two opposite points (at the ends of the same diameter) are motionless, but one of them is an attractor (attracts neighbors), and the other is a repulsor (repels them).

For example, one can imagine a vertically standing circle, the dynamics of which shift down any point along the circle, except for the remaining fixed poles: the attractor at the bottom and the repulsor at the top.

In this case, despite the existence of a one-dimensional circle attractor in the system, the physically steady state will only be a stable stationary position (the lower attractor in the above “vertical” model).

Under an arbitrary small perturbation, the motion will first evolve towards the attractor-circle. But then the internal dynamics on this attractor will play a role, and the state of the system will ultimately approach a “laminar” zero-dimensional attractor, while a one-dimensional attractor, although it exists mathematically, is not suitable for the role of a “steady state”.

One way to avoid such troubles is to consider only minimal attractors as attractors, that is, attractors that do not contain smaller attractors. Kolmogorov's hypotheses refer precisely to such attractors, if we want to give them a precise formulation.

But then nothing has been proven about estimates of dimensions from below, despite numerous publications named as such.

The danger of the deductive-axiomatic approach to mathematics was clearly understood by many thinkers even before Kolmogorov. The first American mathematician J. Sylvester wrote that mathematical ideas should never be petrified, since they lose their power and application when trying to axiomatize the desired properties. He said that ideas should be perceived as water in a river: we never enter exactly the same water, although the ford is the same. Likewise, an idea can give rise to many different and non-equivalent axiomatics, each of which does not reflect the idea entirely.

Sylvester came to all these conclusions by thinking through, in his words, “the strange intellectual phenomenon that the proof of a more general statement often turns out to be simpler than the proof of the particular cases it contains.” As an example, he compared the geometry of vector space with (not yet established at that time) functional analysis.

This idea of ​​Sylvester was used a lot in the future. For example, it is precisely this that explains Bourbaki’s desire to make all concepts as general as possible. They even use the word “more” in France in a sense that in other countries (which they contemptuously call “Anglo-Saxon”) they express with the words “greater than or equal to,” since in France they considered the more general concept “>=” to be primary, and the more specific “ >" - an "unimportant" example. Because of this, they teach students that zero is a positive number (as well as negative, non-positive, non-negative and natural), which is not recognized elsewhere.

But they apparently did not get to Sylvester’s conclusion about the inadmissibility of fossilization of theories (at least in Paris, in the library of the Ecole Normale Superieure, these pages of his Collected Works were uncut when I recently got to them).

I am unable to convince mathematical “specialists” to correctly interpret the hypotheses about the growth of the dimensions of attractors, since they, like lawyers, object to me with formal references to the existing dogmatic codes of laws containing the “exact formal definition” of attractors of the ignorant.

Kolmogorov, on the contrary, never cared about the letter of someone’s definition, but thought about the essence of the matter.

He once explained to me that he came up with his topological cohomology theory not at all combinatorially or algebraically, as it looks like, but by thinking either about fluid flows in hydrodynamics or about magnetic fields: he wanted to model this physics in the combinatorial situation of an abstract complex and did so.

In those years, I naively tried to explain to Kolmogorov what happened in topology during those decades during which he drew all his knowledge about it only from P.S. Alexandrova. Because of this isolation, Kolmogorov knew nothing about homotopy topology; he convinced me that “spectral sequences were contained in the Kazan work of Pavel Sergeevich in 1942,” and attempts to explain to him what the exact sequence was were no more successful than my naive attempts to put him on water skis or put him on a bicycle, this great traveler and skier.

What was surprising to me, however, was the high assessment of Kolmogorov’s words about cohomology given by a strict expert, Vladimir Abramovich Rokhlin. He explained to me, not at all critically, that these words of Kolmogorov contained, firstly, a deeply correct assessment of the relationship between his two achievements (especially difficult in the case when, as here, both achievements are remarkable), and secondly, a shrewd foresight of a huge meanings of cohomology operations.

Of all the achievements of modern topology, Kolmogorov valued Milnor's spheres the most, which the latter spoke about in 1961 at the All-Union Mathematical Congress in Leningrad. Kolmogorov even persuaded me (then a beginning graduate student) to include these spheres in my graduate plan, which forced me to start studying differential topology from Rokhlin, Fuchs and Novikov (as a result of which I was even soon an opponent of the latter’s Ph.D. thesis on differentiable structures on products of spheres).

Kolmogorov’s idea was to use Milnor spheres to prove that a function of several variables cannot be represented by superpositions in Hilbert’s 13th problem (probably for algebraic functions), but I don’t know any of his publications on this topic or the formulation of his hypotheses.

Another little-known circle of Kolmogorov’s ideas relates to the optimal control of dynamic systems.

The simplest task of this circle is to maximize at some point the first derivative of a function defined on an interval or on a circle, knowing the upper bounds for the moduli of the function itself and its second derivative. The second derivative prevents the first from being quickly extinguished, and if the first is too large, the function outgrows the given limitation.

Probably, Hadamard was the first to publish the solution to this problem on the second derivative, and subsequently Littlewood rediscovered it while working on artillery trajectories. Kolmogorov, it seems, did not know the publications of either one or the other, and solved the problem of estimating from above any intermediate derivative through the maximum values ​​of the moduli of the differentiable function and its high (fixed) order derivative.

Kolmogorov's great idea was to explicitly indicate extremal functions, like Chebyshev polynomials (on which the inequality being proved becomes an equality). And in order for the function to be extremal, he naturally guessed that the value of the highest derivative must always be chosen to be maximal in absolute value, changing only its sign.

This led him to a remarkable series of special features. The zero function of this series is the signum of the sine of the argument (everywhere having a maximum modulus). The next, first, function is an antiderivative of zero (that is, a continuous “saw”, the derivative of which has a maximum modulus everywhere). Further functions are obtained each from the previous one by the same integration (increasing the number of derivatives by one). You just need to choose the integration constant so that the integral of the resulting antiderivative function over the period is equal to zero each time (then all the constructed functions will be periodic).

Explicit formulas for the resulting piecewise polynomial functions are quite complex (the integrations are introduced by rational constants associated even with Bernoulli numbers).

The values ​​of the constructed functions and their derivatives are given by constants in Kolmogorov’s power estimates (estimating the modulus of the intermediate derivative from above through the product of rational powers of the maxima of the modulus of the function and the highest derivative). The indicated rational exponents are easy to guess from the consideration of similarity, going back to the laws of similarity of Leonardo da Vinci and to Kolmogorov’s theory of turbulence, that the combination should turn out to be dimensionless, since it is clear (at least from Leibniz’s notation) how derivatives of different orders behave when units change Argument and function measurements. For example, for the Hadamard problem, both rational exponents are equal to half, so the square of the first derivative is estimated from above by the product of the maxima of the modulus of the function itself and its second derivative (with a coefficient depending on the length of the segment or circle where the function is considered).

It is easier to prove all these estimates than to come up with the extremal functions described above (and which deliver, among other things, Gauss’s theorem: the probability of irreducibility of a fraction p/q with integer numerator and denominator is equal to 6/P(2), that is, about 2/3).

In terms of modern control theory, the strategy chosen by Kolmogorov is called “big bang”: the control parameter must always be chosen to have an extreme value, any moderation only harms.

As for Hamilton’s differential equation for changing with time the choice of this extreme value from many possible ones, Kolmogorov knew it very well, calling it, however, Huygens’ principle (which is really equivalent to this equation and from which Hamilton obtained his equation by moving from envelopes to differentials) . Kolmogorov even pointed out to me, who was then a student, that the best description of this geometry of Huygens's principle is contained in Whittaker's mechanics textbook, where I learned it, and that in a more complicated algebraic form it is in the theory of "Berührung Transformation" of Sophus Lie (instead of which I learned the theory canonical transformations according to Birkhoff’s “Dynamical Systems” and which today is called contact geometry).

Tracing the origins of modern mathematics in classical works is usually not easy, especially due to the changing terminology that is accepted as a new science. For example, almost no one notices that the so-called theory of Poisson manifolds was already developed by Jacobi. The fact is that Jacobi followed the path of algebraic varieties - varieties, and not smooth varieties - manifolds. Namely, he was interested in the variety of orbits of a Hamiltonian dynamical system. As a topological or smooth object, it has features and even more unpleasant pathologies (“non-Hausdorffness” and the like) due to the entanglement of orbits (phase curves of a complex dynamic system).

But the algebra of functions on this (possibly bad) “manifold” is perfectly defined: it is simply the algebra of first integrals of the original system. By Poisson's theorem, the Poisson bracket of the first two integrals is again the first integral. Therefore, in the algebra of integrals, in addition to multiplication, there is another bilinear operation - the Poisson bracket.

The interaction of these operations (multiplication and parentheses) in the space of functions on a given smooth manifold is what makes it a Poisson manifold. I skip the formal details of its definition (they are not complicated), especially since they are not all fulfilled in the example that interested Jacobi, where the Poisson manifold is neither smooth nor Hausdorff.

Thus, Jacobi's theory contains the study of more general varieties with singularities than modern Poisson smooth varieties, and moreover, this theory was constructed by him in the style of algebraic geometry of rings and ideals, rather than differential geometry of submanifolds.

Following Sylvester's advice, specialists in Poisson manifolds should, not limiting themselves to their axiomatics, return to a more general and more interesting case, already considered by Jacobi. But Sylvester did not do this (being late, as he said, for the ship leaving for Baltimore), and mathematicians of more recent times are completely subordinate to the dictates of the axiomatists.

Kolmogorov himself, having solved the problem of upper estimates for intermediate derivatives, understood that he could solve many other optimization problems using the same techniques of Huygens and Hamilton, but he did not do this, especially when Pontryagin, whom he always tried to help, published his “principle maximum”, which is essentially a special case of the same Huygens principle of forgotten contact geometry, applied, however, to a not very general problem.

Kolmogorov correctly thought that Pontryagin did not understand either these connections with Huygens' principle, or the connection of his theory with Kolmogorov's much earlier work on estimates of derivatives. And therefore, not wanting to disturb Pontryagin, he did not write anywhere about this connection, which was well known to him.

But now, I think, this can already be said, in the hope that someone will be able to use these connections to discover new results.

It is instructive that Kolmogorov's inequalities between derivatives served as the basis for the remarkable achievements of Yu. Moser in the so-called KAM theory (Kolmogorov, Arnold, Moser), which allowed him to transfer Kolmogorov's 1954 results on invariant tori of analytic Hamiltonian systems to only three hundred and thirty-three times differentiable systems . This was the case in 1962, with Moser's invention of his remarkable combination of Nash smoothing and Kolmogorov's accelerated convergence method.

Now the number of derivatives needed for the proof has been significantly reduced (primarily by J. Mather), so that the three hundred and thirty-three derivatives needed in the two-dimensional problem of ring mappings have been reduced to three (while counterexamples have been found for two derivatives).

It is interesting that after the appearance of Moser’s work, American “mathematicians” tried to publish their “generalization of Moser’s theorem to analytical systems” (which generalization was simply Kolmogorov’s theorem published ten years earlier, which Moser managed to generalize). Moser, however, decisively put an end to these attempts to attribute to others Kolmogorov's classical result (correctly noting, however, that Kolmogorov never published a detailed presentation of his proof).

It seemed to me then that the proof published by Kolmogorov in a note in DAN was quite clear (although he wrote more for Poincaré than for Hilbert), in contrast to Moser’s proof, where I did not understand one place. I even revised it in my 1963 review of Moser's remarkable theory. Moser subsequently explained to me what he meant in this unclear place, but I am still not sure whether these explanations were properly published (in my revision I have to choose

I dedicate to my Teacher - Andrei Nikolaevich Kolmogorov

“Don’t touch my circles,” Archimedes said to the Roman soldier who was killing him. This prophetic phrase came to mind in the State Duma, when the chairman of the meeting of the Education Committee (October 22, 2002) interrupted me with the words: “We don’t have an Academy of Sciences, where we can defend the truth, but a State Duma, where everything is based on what we have.” Different people have different opinions on different issues.”
The view I advocated was that three times seven is twenty-one, and that teaching our children both the multiplication tables and the addition of single-digit numbers and even fractions is a national necessity. I mentioned the recent introduction in the state of California (on the initiative of Nobel laureate, transuranium physicist Glen Seaborg) of a new requirement for schoolchildren entering universities: you need to be able to independently divide the number 111 by 3 (without a computer).
The listeners in the Duma, apparently, could not separate, and therefore did not understand either me or Seaborg: in Izvestia, with a friendly presentation of my phrase, the number “one hundred eleven” was replaced by “eleven” (which makes the question much more difficult, since eleven is not divisible by three).
I came across the triumph of obscurantism when I read in Nezavisimaya Gazeta an article “Retrogrades and Charlatans” glorifying the newly built pyramids near Moscow, where the Russian Academy of Sciences was declared to be a collection of retrogrades inhibiting the development of science (trying in vain to explain everything with their “laws of nature”). I must say that I, apparently, am also a retrograde, since I still believe in the laws of nature and believe that the Earth rotates around its axis and around the Sun, and that younger schoolchildren need to continue to explain why it is cold in winter and warm in summer, without allowing the level of our school education to fall below what was achieved in parochial schools before the revolution (namely, our current reformers are striving for a similar decline in the level of education, citing the truly low American school level).
American colleagues explained to me that the low level of general culture and school education in their country is a deliberate achievement for economic purposes. The fact is that, after reading books, an educated person becomes a worse buyer: he buys less washing machines and cars, and begins to prefer Mozart or Van Gogh, Shakespeare or theorems to them. The economy of a consumer society suffers from this and, above all, the income of the owners of life - so they strive to prevent culture and education (which, in addition, prevent them from manipulating the population as a herd devoid of intelligence).
Faced with anti-scientific propaganda in Russia, I decided to look at the pyramid, recently built about twenty kilometers from my house, and rode there on a bicycle through the centuries-old pine forests between the Istra and Moscow rivers. Here I encountered a difficulty: although Peter the Great forbade cutting down forests closer than two hundred miles from Moscow, several of the best square kilometers of pine forest on my way had recently been fenced off and mutilated (as the local villagers explained to me, this was done by “a person known [to everyone except me!] V.A.] bandit Pashka"). But even twenty years ago, when I was picking up a bucket of raspberries in this now built-up clearing, a whole herd of wild boars walking along the clearing passed me, making a semicircle with a radius of ten meters.
Similar developments are happening everywhere now. Not far from my house, at one time the population did not allow (even using television protests) the development of a forest by Mongolian and other officials. But since then the situation has changed: the former government-party villages are seizing new square kilometers of ancient forest in front of everyone, and no one is protesting anymore (in medieval England, “fencing” caused uprisings!).
True, in the village of Soloslov, next to me, one member of the village council tried to object to the development of the forest. And then, in broad daylight, a car with armed bandits arrived, who shot him right in the village, at home. And the development took place as a result.
In another neighboring village, Daryin, an entire field has been rebuilt with mansions. The attitude of the people to these events is clear from the name that they in the village gave to this built-up field (a name, unfortunately, not yet reflected on the maps): “thieves’ field.”
The new motorized residents of this field have turned the highway leading from us to the Perkhushkovo station into their opposite. Buses have almost stopped running along it in recent years. At first, new residents-motorists collected money at the terminal station for the bus driver so that he would declare the bus “out of order” and passengers would pay private traders. Cars of new residents of the “field” are now rushing along this highway at great speed (and often in someone else’s lane). And I, walking five miles to the station, risk being knocked over, like my many pedestrian predecessors, whose places of death were recently marked on the roadsides with wreaths. Electric trains, however, now also sometimes do not stop at the stations provided for by the schedule.
Previously, the police tried to measure the speed of murderous motorists and prevent them, but after a policeman measuring the speed with a radar was shot by a guard of a passing person, no one dares stop cars anymore. From time to time I find spent cartridges right on the highway, but it’s not clear who was shot at. As for the wreaths over the places where pedestrians died, all of them have recently been replaced with notices “Dumping of garbage is prohibited”, hung on the same trees where there were previously wreaths with the names of those dumped.
Along the ancient path from Aksinin to Chesnokov, using the roads laid by Catherine II, I reached the pyramid and saw inside it “shelves for charging bottles and other objects with occult intellectual energy.” The instructions, several square meters in size, listed the benefits of a several-hour stay of an object or a patient with hepatitis A or B in the pyramid (I read in the newspaper that someone even sent a multi-kilogram load of stones “charged” by the pyramid to the space station for public money).
But the compilers of this instruction also showed honesty that was unexpected for me: they wrote that it is not worth crowding in line at the shelves inside the pyramid, since “tens of meters from the pyramid, outside, the effect will be the same.” This, I think, is absolutely true.
So, as a true “retrograde,” I consider this whole pyramidal enterprise to be a harmful, anti-scientific advertisement for a store selling “loading objects.”
But obscurantism has always followed scientific achievements, starting from antiquity. Aristotle's student, Alexander Philipovich of Macedon, made a number of "scientific" discoveries (described by his companion, Arian, in Anabasis). For example, he discovered the source of the Nile River: according to him, it is the Indus. The “scientific” evidence was: “These are the only two large rivers that are infested with crocodiles” (and confirmation: “In addition, the banks of both rivers are overgrown with lotuses”).
However, this is not his only discovery: he also “discovered” that the Oxus River (today called the Amu Darya) “flows - from the north, turning near the Urals - into the Meotian swamp of the Euxine Pontus, where it is called Tanais” (“Tanais " is the Don, and the "Meotian swamp" is the Sea of ​​​​Azov). The influence of obscurantist ideas on events is not always negligible:
Alexander from Sogdiana (that is, Samarkand) did not go further to the East, to China, as he first wanted, but to the south, to India, fearing a water barrier connecting, according to his third theory, the Caspian (“Hyrcanian”) Sea with the Indian Ocean (in the Bay of Bengal region). For he believed that seas, “by definition,” are bays of the ocean. This is the kind of “science” we are being led to.
I would like to express the hope that our military will not be so strongly influenced by obscurantists (they even helped me save geometry from the attempts of the “reformers” to expel it from school). But today’s attempts to lower the level of schooling in Russia to American standards are extremely dangerous both for the country and for the world.
In today's France, 20% of army recruits are completely illiterate, do not understand written orders from officers (and can send their missiles with warheads in the wrong direction). May this cup pass from us! Our people are still reading, but the “reformers” want to stop this: “Both Pushkin and Tolstoy are too much!” - they write.
It would be too easy for me, as a mathematician, to describe how they plan to eliminate our traditionally high-quality mathematics education in schools. Instead, I will list several similar obscurantist ideas regarding the teaching of other subjects: economics, law, social studies, literature (subjects, however, they propose to abolish everything in school).
The two-volume project “Standards of General Education” published by the Russian Ministry of Education contains a large list of topics whose knowledge it is proposed to stop requiring students to know. It is this list that gives the clearest idea of ​​the ideas of the “reformers” and what kind of “excessive” knowledge they seek to “protect” the next generations from.
I will refrain from political comments, but here are typical examples of supposedly “unnecessary” information extracted from the four-hundred-page Standards project:
· Constitution of the USSR;
· fascist “new order” in the occupied territories;
· Trotsky and Trotskyism;
· main political parties;
· Christian democracy;
· inflation;
· profit;
· currency;
· securities;
· multi-party system;
· guarantees of rights and freedoms;
· law enforcement agencies;
· money and other securities;
· forms of state-territorial structure of the Russian Federation;
· Ermak and the annexation of Siberia;
· Russian foreign policy (XVII, XVIII, XIX and XX centuries);
· Polish question;
· Confucius and Buddha;
· Cicero and Caesar;
· Joan of Arc and Robin Hood;
· Individuals and legal entities;
· the legal status of a person in a democratic state governed by the rule of law;
· separation of powers;
· judicial system;
· autocracy, Orthodoxy and nationality (Uvarov’s theory);
· peoples of Russia;
· Christian and Islamic world;
· Louis XIV;
· Luther;
· Loyola;
· Bismarck;
· The State Duma;
· unemployment;
· sovereignty;
· stock market (exchange);
· state revenues;
· family income.
“Social studies”, “history”, “economics” and “law”, devoid of discussion of all these concepts, are simply formal worship services, useless for students. In France, I recognize this kind of theological chatter on abstract topics by the key set of words: “France, as the eldest daughter of the Catholic Church...” (this can be followed by anything, for example: “... does not need to spend on science, since We already had scientists and still have them”), as I heard at a meeting of the National Committee of the Republic of France for Science and Research, of which the Minister of Science, Research and Technology of the Republic of France appointed me as a member.
In order not to be one-sided, I will also give a list of “undesirable” (in the same sense of “inadmissibility” of their serious study) authors and works mentioned in this capacity by the shameful “Standard”:
· Glinka;
· Chaikovsky;
· Beethoven;
· Mozart;
· Grieg;
· Rafael;
· Leonardo da Vinci;
· Rembrandt;
· Van Gogh;
· Omar Khayyam;
· "Tom Sawyer";
· "Oliver Twist";
· Shakespeare's sonnets;
· “Journey from St. Petersburg to Moscow” by Radishchev;
· "The Steadfast Tin Soldier";
· "Gobsek";
· “Father Goriot”;
· “Les Miserables”;
· "White Fang";
· "Belkin's Tales";
· "Boris Godunov";
· "Poltava";
· "Dubrovsky";
· "Ruslan and Ludmila";
· “Pig under the oak tree”;
· "Evenings on a Farm Near Dikanka";
· “Horse surname”;
· “Pantry of the sun”;
· “Meshcherskaya side”;
· "Quiet Don";
· "Pygmalion";
· “Hamlet”;
· "Faust";
· "A Farewell to Arms";
· "Noble Nest";
· "Lady with a dog";
· "Jumper";
· "A cloud in pants";
· "Black man";
· "Run";
· “Cancer building”;
· "Vanity Fair";
· "For whom the Bell Tolls";
· “Three Comrades”;
· “In the first circle”;
· “The Death of Ivan Ilyich.”
In other words, they propose to abolish Russian Culture as such. They try to “protect” schoolchildren from the influence of “excessive,” according to “Standards,” cultural centers; these turned out to be undesirable, according to the compilers of the “Standards”, for mention by teachers at school:
· Hermitage Museum;
· Russian Museum;
· Tretyakov Gallery;
· Pushkin Museum of Fine Arts in Moscow.
The bell is ringing for us!
It is still difficult to resist and not mention at all what exactly it is proposed to make “optional for learning” in the exact sciences (in any case, the “Standards” recommend “not requiring schoolchildren to master these sections”):
· structure of atoms;
· concept of long-range action;
structure of the human eye;
· uncertainty relation of quantum mechanics;
· fundamental interactions;
· starry sky;
· The sun is like one of the stars;
· cellular structure of organisms;
· reflexes;
· genetics;
· origin of life on Earth;
· evolution of the living world;
· theories of Copernicus, Galileo and Giordano Bruno;
· theories of Mendeleev, Lomonosov, Butlerov;
· merits of Pasteur and Koch;
· sodium, calcium, carbon and nitrogen (their role in metabolism);
· oil;
· polymers.
In mathematics, the same discrimination was applied to topics in the Standards, which no teacher can do without (and without a full understanding of which schoolchildren will be completely helpless in physics, technology, and a huge number of other applications of science, including both military and humanitarian):
necessity and sufficiency;
· geometric locus of points;
· sines of angles at 30o, 45o, 60o;
· construction of the angle bisector;
· dividing a segment into equal parts;
· measuring the angle;
· concept of length of a segment;
· sum of terms of an arithmetic progression;
· sector area;
· inverse trigonometric functions;
· simple trigonometric inequalities;
· equalities of polynomials and their roots;
· geometry of complex numbers (necessary for physics
alternating current, and for radio engineering, and for quantum mechanics);
· construction tasks;
· flat angles of a trihedral angle;
derivative of a complex function;
Converting simple fractions to decimals.
The only hope is that the existing thousands of well-trained teachers will continue to fulfill their duty and teach all this to new generations of schoolchildren, despite any orders from the Ministry. Common sense is stronger than bureaucratic discipline. We just need to remember to pay our wonderful teachers adequately for their feat.

American colleagues explained to me that the low level of general culture and school education in their country is a deliberate achievement for economic purposes. The fact is that, after reading books, an educated person becomes a worse buyer: he buys less washing machines and cars, and begins to prefer Mozart or Van Gogh, Shakespeare or theorems to them. The economy of a consumer society suffers from this and, above all, the income of the owners of life - so they strive to prevent culture and education (which, in addition, prevent them from manipulating the population as a herd devoid of intelligence).

© V.I. Arnold, academician of the Russian Academy of Sciences. One of the greatest mathematicians of the 20th century. (From the article “New Obscurantism and Russian Enlightenment”)

Vladimir Igorevich Arnold

New obscurantism
and Russian education

I dedicate to my Teacher - Andrei Nikolaevich Kolmogorov

“Don’t touch my circles,” Archimedes said to the Roman soldier who was killing him. This prophetic phrase came to mind in the State Duma, when the chairman of the meeting of the Education Committee (October 22, 2002) interrupted me with the words: “I have not the Academy of Sciences, where one can defend the truth, but the State Duma, where everything is based on the fact that different people have different opinions on different issues.”

The view I advocated was that three times seven is twenty-one, and that teaching our children both the multiplication table and the addition of single-digit numbers and even fractions is a national necessity. I mentioned the recent introduction in the state of California (on the initiative of Nobel laureate, transuranium physicist Glen Seaborg) of a new requirement for schoolchildren entering universities: you need to be able to independently divide the number 111 by 3 (without a computer).

The listeners in the Duma, apparently, could not separate, and therefore did not understand either me or Seaborg: in Izvestia, with a friendly presentation of my phrase, the number “one hundred eleven” was replaced by “eleven” (which makes the question much more difficult, since eleven is not divisible by three).

I came across the triumph of obscurantism when I read in Nezavisimaya Gazeta an article glorifying the newly built pyramids near Moscow, “Retrogrades and Charlatans,” where

The Russian Academy of Sciences was declared to be a meeting of retrogrades inhibiting the development of science (trying in vain to explain everything with their “laws of nature”). I must say that I am apparently also a retrograde, since I still believe in the laws of nature and believe that the Earth rotates around its axis and around the Sun, and that younger schoolchildren need to continue to explain why it is cold in winter and warm in summer, not allowing the level of our school education to fall below what was achieved in parochial schools before the revolution (namely, it is precisely this reduction in the level of education that our current reformers are striving for, citing the truly low American school level).

American colleagues explained to me that the low level of general culture and school education in their country is a deliberate achievement for economic purposes. The fact is that, after reading books, an educated person becomes a worse buyer: he buys less washing machines and cars, and begins to prefer Mozart or Van Gogh, Shakespeare or theorems to them. The economy of the consumer society suffers from this and, above all, the income of the owners of life - so they strive prevent culture and education(which, in addition, prevents them from manipulating the population like a herd devoid of intelligence).

Faced with anti-scientific propaganda in Russia, I decided to look at the pyramid, recently built about twenty kilometers from my house, and rode there on a bicycle through the centuries-old pine forests between the Istra and Moscow rivers. Here I encountered a difficulty: although Peter the Great forbade cutting down forests closer than two hundred miles from Moscow, several of the best square kilometers of pine forest on my way were recently fenced off and mutilated (as the local villagers explained to me, this was done by “a person known [to everyone except me! - V.A.] bandit Pashka"). But even twenty years ago, when I was getting a bucket from this now built-up clearing

raspberries, a whole herd of wild boars walking along the clearing passed me, making a semicircle with a radius of about ten meters.

Similar developments are happening everywhere now. Not far from my house, at one time the population did not allow (even using television protests) the development of a forest by Mongolian and other officials. But since then the situation has changed: the former government-party villages are seizing new square kilometers of ancient forest in front of everyone, and no one is protesting anymore (in medieval England, “fencing” caused uprisings!).

True, in the village of Soloslov, next to me, one member of the village council tried to object to the development of the forest. And then in broad daylight a car arrived with armed bandits who right in the village, at home, and shot. And the development took place as a result.

In another neighboring village, Daryin, an entire field has been rebuilt with mansions. The attitude of the people to these events is clear from the name that they in the village gave to this built-up field (a name, unfortunately, not yet reflected on the maps): “thieves’ field.”

The new motorized residents of this field have turned the highway leading from us to the Perkhushkovo station into their opposite. Buses have almost stopped running along it in recent years. At first, new residents-motorists collected money at the terminal station for the bus driver so that he would declare the bus “out of order” and passengers would pay private traders. Cars of new residents of the “field” are now rushing along this highway at great speed (and often in someone else’s lane). And I, walking five miles to the station, risk being knocked over, like my many pedestrian predecessors, whose places of death were recently marked on the roadsides with wreaths. Electric trains, however, now also sometimes do not stop at the stations provided for by the schedule.

Previously, the police tried to measure the speed of murderous motorists and prevent them, but after a policeman measuring the speed with a radar was shot by a guard of a passing person, no one dares stop cars anymore. From time to time I find spent cartridges right on the highway, but it is not clear who was shot at. As for the wreaths over the places where pedestrians died, all of them have recently been replaced with notices “Dumping of garbage is prohibited”, hung on the same trees where there were previously wreaths with the names of those dumped.

Along the ancient path from Aksinin to Chesnokov, using the roads laid by Catherine II, I reached the pyramid and saw inside it “shelves for charging bottles and other objects with occult intellectual energy.” Instructions V several square meters in size listed the benefits of a several-hour stay of an object or a patient with hepatitis A or B in the pyramid (I read in the newspaper that someone even sent a multi-kilogram load of stones “charged” by the pyramid to the space station for public money).

But the compilers of this instruction also showed honesty that was unexpected for me: they wrote that there is no point in crowding in line at the shelves inside the pyramid, since<в десятках метров от пирамиды, снаружи, эффект будет таким же». This, I think, is absolutely true.

So, as a true “retrograde,” I consider this whole pyramidal enterprise to be a harmful, anti-scientific advertisement for a store selling “loading objects.”

But obscurantism has always followed scientific achievements, starting from antiquity. Aristotle's student, Alexander Philipovich of Macedon, made a number of "scientific" discoveries (described by his companion, Arian, in Anabasis). For example, he discovered the source of the Nile River: according to him, it is the Indus. The "scientific" evidence was: " These are the only two large rivers that are infested with crocodiles."(and confirmation: “In addition, the banks of both rivers are overgrown with lotuses”).

However, this is not his only discovery: he also “discovered” that the Oxus River (today called the Amu Darya) “flows - from the north, turning near the Urals - into the Meotian swamp of the Pontus Euxine, where it is called Tanais”(“Ta-nais” is the Don, and “Meotian swamp” is the Sea of ​​Azov). The influence of obscurantist ideas on events is not always negligible:

Alexander from Sogdiana (that is, Samarkand) did not go further to the East, to China, as he first wanted, but to the south, to India, fearing a water barrier connecting, according to his third theory, the Caspian (“Hyrcanian”) Sea with the Indian Ocean(V Bay of Bengal region). For he believed that seas, “by definition,” are bays of the ocean. This is the kind of “science” we are being led to.

I would like to express the hope that our military will not be so strongly influenced by obscurantists (they even helped me save geometry from the attempts of the “reformers” to expel it from school). But today’s attempts to lower the level of schooling in Russia to American standards are extremely dangerous both for the country and for the world.

In today's France, 20% of army recruits are completely illiterate, do not understand written orders from officers (and can send their missiles with warheads in the wrong direction). May this cup pass from us! Our people are still reading, but the “reformers” want to stop this: “Both Pushkin and Tolstoy are too much!” - they write.

It would be too easy for me, as a mathematician, to describe how they plan to eliminate our traditionally high-quality mathematics education in schools. Instead, I will list several similar obscurantist ideas regarding the teaching of other subjects: economics, law, social studies, literature (subjects, however, they propose to abolish everything in school).

The two-volume project “Standards of General Education” published by the Ministry of Education of Russia contains a large list of topics knowledge of which it is proposed to stop demanding from trainees. It is this list that gives the clearest idea of ​​the ideas of the “reformers” and what kind of “excessive” knowledge they seek to “protect” the next generations from.

I will refrain from political comments, but here are typical examples of supposedly “unnecessary” information extracted from the four-hundred-page Standards project:

• Constitution of the USSR;
• fascist “new order” in the occupied territories;
• Trotsky and Trotskyism;
• major political parties;
• Christian democracy;
• inflation;
• profit;
• currency;
• securities;
• multi-party system;
• guarantees of rights and freedoms;
• law enforcement agencies;
• money and other securities;
• forms of state-territorial structure of the Russian Federation;
• Ermak and the annexation of Siberia;
• foreign policy of Russia (XVII, XVIII, XIX and XX centuries);
• Polish question;
• Confucius and Buddha;
• Cicero and Caesar;
• Joan of Arc and Robin Hood;
• Individuals and legal entities;
• the legal status of a person in a democratic state governed by the rule of law;
• separation of powers;
• judicial system;
• autocracy, Orthodoxy and nationality (Uvarov’s theory);
• peoples of Russia;
• Christian and Islamic world;
• Louis XIV;
• Luther;
• Loyola;
• Bismarck;
• The State Duma;
• unemployment;
• sovereignty;
• stock market (exchange);
• state revenues;
• family income.

“Social studies”, “history”, “economics” and “law”, devoid of discussion of all these concepts, are simply formal worship services, useless for students. In France, I recognize this kind of theological chatter on abstract topics by a key set of words: "France is like the eldest daughter of the Catholic Church..." (anything can follow, for example: "... does not need spending on science, since we already had and still have scientists"), as I heard at a meeting of the National Committee of the Republic of France for Science and Research, of which I am a member I was appointed by the Minister of Science, Research and Technology of the Republic of France.

In order not to be one-sided, I will also give a list of “undesirable” (in the same sense of “inadmissibility” of their serious study) authors and works mentioned in this capacity by the shameful “Standard”:

• Glinka;
• Chaikovsky;
• Beethoven;
• Mozart;
• Grieg;
• Raphael;
• Leonardo daVinci;
• Rembrandt;
• Van Togh;
• Omar Khayyam;
• "Tom Sawyer";
• "Oliver Twist";
• Shakespeare's Sonnets;
• “Journey from St. Petersburg to Moscow” by Radishchev;
• "The Steadfast Tin Soldier";
• "Gobsek";
• "Père Goriot"
• "Les Miserables";
• "White Fang";
• "Belkin's Tales";
• "Boris Godunov";
• "Poltava";
• "Dubrovsky";
• "Ruslan and Ludmila";
• "Pig under the Oak";
• "Evenings on a Farm Near Dikanka";
• "Horse surname";
• "Pantry of the Sun";
• "Meshcherskaya side";
• "Quiet Don";
• "Pygmalion";
• "Hamlet";
• "Faust";
• "A Farewell to Arms";
• "Noble Nest";
• "Lady with a dog";
• "Jumper";
• "A cloud in pants";
• "Black man";
• "Run";
• "Cancer Ward";
• "Vanity Fair";
• "For whom the Bell Tolls";
• "Three Comrades";
• "In the first circle";
• "The Death of Ivan Ilyich."

In other words, they propose to abolish Russian Culture as such. They try to “protect” schoolchildren from the influence of “excessive,” according to “Standards,” cultural centers; that's how they turned out to be here undesirable, according to the compilers of the Standards, for mention by teachers at school:

• Hermitage Museum;
• Russian Museum;
• Tretyakov Gallery;
• Pushkin Museum of Fine Arts in Moscow.

The bell is ringing for us!

It is still difficult to refrain from mentioning at all what exactly it is proposed to make “optional for training” in the exact sciences (in any case, “Standards” recommend “not requiring students to master these sections”):

• structure of atoms;
• concept of long-range action;
• structure of the human eye;
• uncertainty relation of quantum mechanics;
• fundamental interactions;
• starry sky;
• The sun is like one of the stars;
• cellular structure of organisms;
• reflexes;
• genetics;
• origin of life on Earth;
• evolution of the living world;
• the theories of Copernicus, Galileo and Giordano Bruno;
• theories of Mendeleev, Lomonosov, Butlerov;
• the merits of Pasteur and Koch;
• sodium, calcium, carbon and nitrogen (their role in metabolism);
• oil;
• polymers.

In mathematics, the same discrimination was applied to topics in the Standards, which no teacher can do without (and without a full understanding of which schoolchildren will be completely helpless in physics, technology, and a huge number of other applications of science, including both military and humanitarian):

• necessity and sufficiency;
• locus of points;
• sines of angles at 30 o, 45 o, 60 o;
• constructing the angle bisector;
• dividing a segment into equal parts;
• measuring the angle;
• concept of length of a segment;
• the sum of the terms of an arithmetic progression;
• sector area;
• inverse trigonometric functions;
• simple trigonometric inequalities;
• equalities of polynomials and their roots;
• geometry of complex numbers (necessary for alternating current physics, radio engineering, and quantum mechanics);
• construction tasks;
• plane angles of a trihedral angle;
• derivative of a complex function;
• converting simple fractions to decimals.

The only thing that gives me hope is that The existing thousands of well-trained teachers will continue to fulfill their duty and teach all this to new generations of schoolchildren, despite any orders from the Ministry. Common sense is stronger than bureaucratic discipline. We just need to remember to pay our wonderful teachers adequately for their feat.

Representatives of the Duma explained to me that the situation could be greatly improved if care was taken to implement the laws on education that have already been adopted.

The following description of the state of affairs was presented by Deputy I. I. Melnikov in his report at the Mathematical Institute. V. A. Steklov of the Russian Academy of Sciences in Moscow in the fall of 2002.

For example, one of the laws provides for an annual increase in the budget contribution to training by approximately 20% per year. But the minister said that “there is no need to worry about the implementation of this law, since the almost annual increase occurs by more than 40%.” Shortly after this speech by the minister, an increase (by a much smaller percentage) that was practically feasible for the next year (it was 2002) was announced. And if we also take into account inflation, it turns out that a decision was made to reduce the real annual contribution to education.

Another law specifies the percentage of budget expenditures that must be spent on education. In reality, much less is spent (I was not able to find out exactly how many times). But spending on “defense against an internal enemy” increased from a third to half of spending on defense against an external enemy.

It’s natural to stop teaching children fractions, otherwise, God forbid, they’ll understand!

Apparently, it was precisely in anticipation of the reaction of teachers that the compilers of the “Standard” provided a number of names of writers in their list of recommended reading (like the names of Pushkin, Krylov, Lermontov, Chekhov and the like) with an “asterisk” sign, which they deciphered as: “At his discretion, the teacher can introduce students to one or two more works by the same author.”(and not just with the “Monument” they recommended in the case of Pushkin).

The higher level of our traditional mathematical education compared to foreign countries became obvious to me only after I was able to compare this level with foreign ones, having worked many semesters at universities and colleges in Paris and New York, Oxford and Cambridge, Pisa and Bologna, Bonn and Berkeley, Stanford and Boston, Hong Kong and Kyoto, Madrid and Toronto, Marseille and Strasbourg, Utrecht and Rio de Janeiro, Conakry and Stockholm.

“We can’t possibly follow your principle of choosing candidates based on their scientific achievements,” my colleagues on the commission for inviting new professors to one of the best universities in Paris told me. - “After all, in this case we would have to choose only Russians - such is their scientific superiority to us all It's clear!" (I also spoke about selection among the French).

At the risk of being understood only by mathematicians, I will still give examples of responses from the best candidates for a professorship in mathematics at a university in Paris in the spring of 2002 (200 people applied for each position).

The candidate has been teaching linear algebra at various universities for several years, defended his dissertation and published a dozen articles in the best mathematical journals in France.

Selection includes an interview, where the candidate is always asked elementary but important questions (question level "Name the capital of Sweden" if the subject was geography).

So I asked, "What is the signature of the quadratic form xy?»

The candidate demanded the 15 minutes allotted to him to think, after which he said: “In my computer in Toulouse, I have a routine (program) that in an hour or two could find out how many pluses and how many minuses there will be in normal form. The difference between these two numbers will be the signature - but you only give 15 minutes, and without a computer, so I can’t answer, this form xy it’s too complicated.”

For non-specialists, let me explain that if we were talking about zoology, then this answer would be similar to this: “Linnaeus listed all the animals, but whether the birch is a mammal or not, I cannot answer without a book.”

The next candidate turned out to be a specialist in “systems of elliptic partial differential equations” (a decade and a half after defending his dissertation and more than twenty published works).

I asked this one: “What is the Laplacian of the function 1/r in three-dimensional Euclidean space?

The response (within the usual 15 minutes) was amazing to me; "If r stood in the numerator, and not in the denominator, and the first derivative would have been required, and not the second, then I would have been able to calculate it in half an hour, but otherwise the question is too difficult.”

Let me explain that the question was from the theory of elliptic equations, like the question “Who is the author of Hamlet?” in the English Literature exam. Trying to help, I asked a series of leading questions (similar to questions about Othello and Ophelia): “Do you know what the law of gravity is? Coulomb's law? How are they related to the Laplacian? What is the fundamental solution of Laplace’s equation?”

But nothing helped: neither Macbeth nor King Lear were known to the candidate if we were talking about literature.

Finally, the chairman of the examination committee explained to me what was going on: “After all, the candidate studied not just one elliptic equation, but their systems, and you ask him about the Laplace equation, which Total one thing is clear that he has never encountered it!”

In a literary analogy, this “justification” would correspond to the phrase: “The candidate studied English poets, how could he know Shakespeare, after all, he is a playwright!”

The third candidate (and dozens of them were interviewed) was working on “holomorphic differential forms,” and I asked him: “What is the Riemann surface of the tangent?” (I was afraid to ask about the arctangent).

Answer: “The Riemannian metric is the quadratic form of coordinate differentials, but what form is associated with the tangent function is not at all clear to me.”

I will explain again with a sample of a similar answer, this time replacing mathematics with history (to which the Mitrofans are more inclined). Here the question would be: "Who is Julius Caesar?" and the answer is: “The rulers of Byzantium were called Caesars, but I don’t know Julia among them.”

Finally, a candidate probabilist appeared, talking interestingly about his dissertation. He proved in it that the statement “A and B are fair together” is false(the statements themselves A And IN are formulated at length, so I will not reproduce them here).

Question: “And yet, what is the situation with the statement A on their own, without IN: is it true or not?

Answer: “After all, I said that the statement “A and B” is incorrect. This means that A is also false." That is: “Since it is not true that “Petya and Misha got cholera,” then Petya did not get cholera.”

Here my bewilderment was again dispelled by the chairman of the commission: he explained that the candidate was not a probabilist, as I thought, but a statistician (in the biography, called CV, there is not “proba”, but “stat”).

“The probabilists,” our experienced chairman explained to me, “have normal logic, the same as that of mathematicians, Aristotelian. For statisticians, it’s completely different: it’s not for nothing that they say “there are lies, blatant lies, and statistics.” All their reasoning is unsubstantiated, all their conclusions are erroneous. But they are always very necessary and useful, these conclusions. We definitely need to accept this statistician!”

At Moscow University, such an ignoramus would not be able to complete the third year of the Faculty of Mechanics and Mathematics. Riemann surfaces were considered the pinnacle of mathematics by the founder of the Moscow Mathematical Society, N. Bugaev (father of Andrei Bely). He, however, believed that in contemporary mathematics at the end of the 19th century, objects began to appear that did not fit into the mainstream of this old theory - non-holomorphic functions of real variables, which, in his opinion, are the mathematical embodiment of the idea of ​​free will to the same extent that Riemann surfaces and holomorphic functions embody the idea of ​​fatalism and predetermination.

As a result of these reflections, Bugaev sent young Muscovites to Paris to learn there the new “mathematics of free will” (from Borel and Lebesgue). This program was brilliantly carried out by N. N. Luzin, who upon his return to Moscow created a brilliant school, including all the main Moscow mathematicians of many decades: Kolmogorov and Petrovsky, Aleksandrov and Pontryagin, Menshov and Keldysh, Novikov and Lavrentiev, Gelfand and Lyusternik.

By the way, Kolmogorov recommended to me the Parisiana Hotel (on Tournefort Street, not far from the Pantheon) that Luzin subsequently chose for himself in the Latin Quarter of Paris. During the First European Mathematical Congress in Paris (1992) I stayed in this inexpensive hotel (with amenities at the level of the 19th century, without a telephone, and so on). And the elderly owner of this hotel, having learned that I had come from Moscow, immediately asked me: “ How is my old guest, Luzin, doing there? It’s a pity that he hasn’t visited us for a long time.”

A couple of years later, the hotel was closed for renovation (the owner probably died) and they began to rebuild it in an American way, so now you can no longer see this 19th-century island in Paris.

Returning to the choice of professors in 2002, I note that all the ignoramuses listed above received (from everyone except me) the best grades. On the contrary, the only, in my opinion, worthy candidate was almost unanimously rejected. He discovered (with the help of “Gröbner bases” and computer algebra) several dozen new completely integrable systems of Hamiltonian equations of mathematical physics (at the same time, but not including in the list of new ones, the famous Korteweg-de Vries, Sayn-Gordon, and the like equations).

As a future project, the candidate also proposed a new computer method for modeling diabetes treatment. To my question about the assessment of his method by doctors, he answered quite reasonably: “The method is now being tested in such and such centers and hospitals, and in six months they will give their conclusions, comparing the results with other methods and with control groups of patients, but for now this examination is not has been carried out, and there are only preliminary assessments, although they are good.”

They rejected it with this explanation: “On every page of his dissertation, either Lie groups or Lie algebras are mentioned, but no one here understands this, so he will not fit into our team at all.” True, it would have been possible to reject both me and all my students, but some colleagues think that the reason for the rejection was different: unlike all the previous candidates, this one was not French (he was a student of a famous American professor from Minnesota).

The whole picture described leads to sad thoughts about the future of French science, in particular mathematics. Although the “French National Committee for Science” was inclined to not finance new scientific research at all, but to spend money (provided by Parliament for the development of science) on the purchase of ready-made American recipes, I sharply opposed this suicidal policy and still achieved at least some subsidizing new research. However, a difficulty was caused by the division of money. Medicine, nuclear energy, polymer chemistry, virology, genetics, ecology, environmental protection, radioactive waste disposal and much more were consistently voted unworthy of subsidies by voting (during a five-hour meeting). In the end, they chose three “sciences” that allegedly deserved funding for their new research. These three “sciences” are: 1) AIDS; 2) psychoanalysis; 3) a complex branch of pharmaceutical chemistry, the scientific name of which I am unable to reproduce, but which deals the development of psychotropic drugs, similar to lacrimogenic gas, turning the rebellious crowd into an obedient herd.

So now France is saved!

Of all Luzin’s students, the most remarkable contribution to science was made, in my opinion, by Andrei Nikolaevich Kolmogorov. Having grown up in a village with his grandfather near Yaroslavl, Andrei Nikolaevich proudly referred to Gogol’s words as “an efficient Roslavl peasant.”

He had no intention of becoming a mathematician, even having already entered Moscow University, where he immediately began studying history (in Professor Bakhrushin’s seminar) and, before he was even twenty years old, wrote his first scientific work.

This work was devoted to the study of land economic relations in medieval Novgorod. Tax documents have been preserved here, and the analysis of a huge number of these documents using statistical methods led the young historian to unexpected conclusions, which he spoke about at the Bakhrushin meeting.

The report was very successful, and the speaker was much praised. But he insisted on another approval: he wanted his conclusions to be recognized as correct.

In the end, Bakhrushin told him: “This report must be published; he is very interesting. But as for the conclusions, then For us historians, to recognize any conclusion we always need not one piece of evidence, but at least five!«

The next day, Kolmogorov changed history to mathematics, where proof alone is enough. He did not publish the report, and this text remained in his archive until, after the death of Andrei Nikolaevich, it was shown to modern historians, who recognized it not only as very new and interesting, but also quite conclusive. Now this Kolmogorov report has been published, and is considered by the community of historians as an outstanding contribution to their science.

Having become a professional mathematician, Kolmogorov remained, unlike most of them, first of all a natural scientist and thinker, and not at all a multiplier of multidigit numbers (which mainly appears when analyzing the activities of mathematicians to people unfamiliar with mathematics, including even L.D. Landau, who valued mathematics is precisely the continuation of counting skill: five five - twenty-five, six six - thirty-six, seven seven - forty-seven, as I read in a parody of Landau compiled by his Physics and Technology students; however, in Landau’s letters to me, who was then a student, mathematics no more logical than in this parody).

Mayakovsky wrote: “After all, he can extract the square root every second” (about the professor who “doesn’t get bored that under the window the students are actively going to the gymnasium”).

But he perfectly described what a mathematical discovery is, saying that " Whoever discovered that two and two equals four was a great mathematician, even if he discovered it by counting cigarette butts. And anyone who today calculates much larger objects, such as locomotives, using the same formula, is not a mathematician at all!”

Kolmogorov, unlike many others, was never intimidated by applied, “locomotive” mathematics, and he joyfully applied mathematical considerations to a variety of areas of human activity: from hydrodynamics to artillery, from celestial mechanics to poetry, from the miniaturization of computers to the theory of Brownian motion, from the divergence of Fourier series to the theory of information transmission and to intuitionistic logic. He laughed at the fact that the French write “Celestial mechanics” with a capital letter, and “applied” with a small letter.

When I first arrived in Paris in 1965, I was warmly greeted by the elderly Professor Fréchet, with the following words: “After all, you are a student of Kolmogorov, that young man who constructed an example of a Fourier series that diverges almost everywhere!”

The work mentioned here by Kolmogorov was completed by him at the age of nineteen, solved a classical problem and immediately promoted this student to the rank of first-class mathematicians of world significance. Forty years later, this achievement still remained more significant for Frechet than all subsequent and much more important fundamental works of Kolmogorov, which revolutionized the theory of probability, the theory of functions, hydrodynamics, celestial mechanics, the theory of approximations, and the theory of algorithmic complexity, and the theory of cohomology in topology, and the theory of control of dynamical systems (where Kolmogorov’s inequalities between derivatives of different orders remain one of the highest achievements today, although control theory specialists rarely understand this).

But Kolmogorov himself was always somewhat skeptical about his favorite mathematics, perceiving it as a small part of natural science and easily abandoning those logical restrictions that the shackles of the axiomatic-deductive method impose on true mathematicians.

“It would be in vain,” he told me, “to look for mathematical content in my works on turbulence. I speak here as a physicist and am not at all concerned with mathematical proofs or derivations of my conclusions from initial premises, such as the Navier-Stokes equations. Even if these conclusions have not been proven, they are true and open, and this is much more important than proving them!”

Many of Kolmogorov’s discoveries were not only not proven (neither by himself nor by his followers), but were not even published. But nevertheless, they have already had and continue to have a decisive influence on a number of departments of science (and not only mathematics).

I will give just one famous example (from the theory of turbulence).

A mathematical model of hydrodynamics is a dynamic system in the space of fluid velocity fields, which describes the evolution of the initial velocity field of fluid particles under the influence of their interaction: pressure and viscosity (as well as under the possible influence of external forces, for example, weight force in the case of a river or water pressure in a water pipe).

Under the influence of this evolution, a dynamic system can come to equilibrium (stationary) state, when the flow velocity at each point of the flow region does not change with time(although everything flows, and each particle moves and changes its speed over time).

Such stationary flows (for example, laminar flows in terms of classical hydrodynamics) are attracting points of a dynamic system. They are therefore called (point) attractors.

Other sets that attract neighbors are also possible, for example, closed curves depicting currents that periodically change over time in the functional space of velocity fields. Such a curve is an attractor when the neighboring initial conditions, depicted by “perturbed” points of the functional space of velocity fields close to the indicated closed curve, begin, although not periodically changing with time, a flow that approaches it (namely, the perturbed flow tends to the one described earlier periodically over time).

Poincaré, who first discovered this phenomenon, called such closed attractor curves "stable limit cycles". From a physical point of view, they can be called periodic steady flow regimes: the disturbance gradually fades during the transition process caused by the disturbance of the initial condition, and after some time the difference between the movement and the unperturbed periodic one becomes barely noticeable.

After Poincaré, such limit cycles were extensively studied by A. A. Andronov, who based the study and calculation of radio wave generators, that is, radio transmitters, on this mathematical model.

It is instructive that Poincaré’s discovery and development by Andronov theory of the birth of limit cycles from unstable equilibrium positions Today it is usually called (even in Russia) the Hopf bifurcation. E. Hopf published part of this theory a couple of decades after Andronov’s publication and more than half a century after Poincaré, but unlike them, he lived in America, so the well-known eponymic principle worked: if any object bears someone else's name, then this is not the name of the discoverer(for example, America is not named after Columbus).

The English physicist M. Berry called this eponymous principle “Arnold’s principle,” adding a second one to it. Berry's principle: Arnold's principle applies to oneself(that is, it was known before).

I completely agree with Berry on this. I told him the eponymous principle in response to a preprint about the “Berry phase”, examples of which, in no way inferior to the general theory, were published decades before Berry by S. M. Rytov (under the name “inertia of polarization direction”) and A. Yu .Ishlinsky (under the name “the departure of the submarine’s gyroscope due to a discrepancy between the path of returning to the base and the path of leaving it”),

Let us return, however, to attractors. An attractor, or attracting set, is a steady state of motion, which, however, does not have to be periodic. Mathematicians have also studied much more complex movements, which can also attract perturbed neighboring movements, but which themselves can be extremely unstable: small causes sometimes cause big consequences, Poincaré said. The state, or “phase,” of such a limiting regime (that is, a point on the surface of the attractor) can move along the surface of the attractor in a bizarre “chaotic” manner, and a slight deviation of the starting point on the attractor can greatly change the course of movement without changing the limiting regime at all. Averages over long times from all possible observable quantities will be close in the original and in the perturbed motion, but the details at a fixed moment in time will, as a rule, be completely different.

In meteorological terms, the “limit regime” (attractor) can be likened to climate, and the phase - weather. A small change in initial conditions can have a big impact on tomorrow's weather (and even more on the weather a week and a month from now). But such a change will not make the tundra a tropical forest: just a thunderstorm may break out on Friday instead of Tuesday, which may not change the average for the year (or even for the month).

In hydrodynamics, the degree of attenuation of initial disturbances is usually characterized by viscosity (so to speak, mutual friction of liquid particles as they move one relative to another), or inverse viscosity, a value called the “Reynolds number”. Large values ​​of the Reynolds number correspond to weak attenuation of disturbances, and large values ​​of viscosity (that is, small Reynolds numbers) - on the contrary, regularize the flow, preventing disturbances and their development. In economics, the role of “viscosity” is often played by bribes and corruption 1 .

1 Multi-stage production management is unstable if the number of stages (worker, foreman, shop manager, plant director, chief executive officer, etc.) is more than two, but can be implemented in a sustainable manner if at least some of the managers are rewarded not only from above (for following orders ), but also from below (for the benefit of the cause, for decisions that contribute to production). Corruption is used for the latter encouragement. For details, see the article: V. I. Arnold. Mathematics and mathematics education in the modern world. In the book: Mathematics in education and upbringing. - M.: FAZIS, 2000, p. 195-205.

Due to high viscosity, at low Reynolds numbers, a stable stationary (laminar) flow is usually established, represented in the space of velocity fields by a point attractor.

The main question is how the flow pattern will change with increasing Reynolds number. In water supply, this corresponds, for example, to an increase in water pressure, which makes a smooth (laminar) stream from a tap unstable, but mathematically, to increase the Reynolds number, it is more convenient to reduce the particle friction coefficient expressing viscosity (which in an experiment would require a technically complex fluid replacement). However, sometimes to change the Reynolds number it is enough to change the temperature in the laboratory. I saw such an installation in Novosibirsk at the Institute of Precision Measurements, where the Reynolds number changed (in the fourth digit) when I brought my hand closer to the cylinder where the flow occurred (precisely due to a change in temperature), and on the computer screen processing the experiment, this change in the Reynolds number immediately indicated by electronic automation.

Thinking about these phenomena of transition from a laminar (stable stationary) flow to a stormy turbulent one, Kolmogorov long ago expressed a number of hypotheses (which to this day remain unproven). I think that these hypotheses date back to the time (1943) of his dispute with Landau about the nature of turbulence. In any case, he clearly formulated them at his seminar (on hydrodynamics and the theory of dynamical systems) at Moscow University in 1959, where they were even part of the announcement about the seminar that he posted at that time. But I don’t know of any formal publication of these hypotheses by Kolmogorov, and in the West they are usually attributed to their epigones of Kolmogorov, who learned about them and published them dozens of years later.

The essence of these Kolmogorov hypotheses is that as the Reynolds number increases, the attractor corresponding to the steady flow regime becomes more and more complex, namely, that its dimension increases.

First it is a point (zero-dimensional attractor), then a circle (Poincaré limit cycle, one-dimensional attractor). And Kolmogorov’s hypothesis about attractors in hydrodynamics consists of two statements: with increasing Reynolds number 1) attractors of ever larger dimensions appear; 2) all low-dimensional attractors disappear.

From 1 and 2 together it follows that when the Reynolds number is sufficiently large, the steady state necessarily has many degrees of freedom, so that to describe its phase (point on the attractor) it is necessary to set many parameters, which then, when moving along the attractor, will change in a whimsical and non-periodic “chaotic” way, and a small change in the starting point on the attractor leads, as a rule, to a large (after a long time) change in the “weather” (the current point on the attractor), although it does not change the attractor itself (that is, it will not cause a change in the “climate”).

Statement 1 in itself is not sufficient here, since different attractors can coexist, including attractors of different dimensions in one system (which, thus, can perform a calm “laminar” movement under some initial conditions and a stormy “turbulent” one under others, depending on its initial state).

Experimental observation of such effects "prolonged loss of stability" surprised physicists for a long time, but Kolmogorov added that even if the low-dimensional attractor does not disappear, it may not change the observed turbulence in the case when the size of its zone of attraction decreases significantly with increasing Reynolds number. In this case, the laminar regime, although possible in principle (and even stable), is practically not observed due to the extreme smallness of its area of ​​attraction: Already small, but always present in the experiment, disturbances can take the system out of the zone of attraction of this attractor into the zone of attraction of another, already turbulent, steady state, which will be observed.

This discussion may also explain this strange observation: Some famous hydrodynamic experiments of the 19th century could not be repeated in the second half of the 20th century, although attempts were made to use the same equipment in the same laboratory. It turned out, however, that the old experiment (with its prolongation of the loss of stability) can be repeated if it is done not in the old laboratory, but in a deep underground mine.

The fact is that modern street traffic has greatly increased the magnitude of “imperceptible” disturbances, which began to have an effect (due to the smallness of the zone of attraction of the remaining “laminar” attractor).

Numerous attempts by many mathematicians to confirm Kolmogorov's hypotheses 1 and 2 (or at least the first) with evidence have so far only led to estimates of the dimensions of attractors in terms of Reynolds numbers from above: this dimension cannot become too large as long as viscosity prevents it.

The dimensionality is estimated in these works by a power function of the Reynolds number (that is, a negative degree of viscosity), and the exponent depends on the dimension of the space where the flow occurs (in a three-dimensional flow, turbulence is stronger than in plane problems).

As for the most interesting part of the problem, that is, estimating the dimension from below (at least for some attractors, as in Hypothesis 1, or even for all, as in Hypothesis 2, about which Kolmogorov expressed more doubts), here the mathematicians were not able to height, because, according to his habit, replaced the real natural science problem with their formal axiomatic abstract formulation with its precise but treacherous definitions.

The fact is that the axiomatic concept of an attractor was formulated by mathematicians with the loss of some properties of the physical limiting mode of motion, which (not strictly defined) concept of mathematics they tried to axiomatize by introducing the term “attractor”.

Let us consider, for example, an attractor that is a circle (to which all nearby dynamics trajectories spirally approach).

On this very circle attracting neighbors, let the dynamics be arranged as follows: two opposite points (at the ends of the same diameter) are motionless, but one of them is an attractor (attracts neighbors), and the other is a repulsor (repels them).

For example, one can imagine a vertically standing circle, the dynamics on which shift down any point along the circle, except for the remaining fixed poles:

an attractor at the bottom and a repulsor at the top.

In this case, despite the existence of a one-dimensional attractor-circle in the system, the physically steady state will only be a stable stationary position(the lower attractor in the above “vertical” model).

Under an arbitrary small perturbation, the motion will first evolve towards the attractor-circle. But then the internal dynamics on this attractor will play a role, and state of the system, will in the end, approach a “laminar” zero-dimensional attractor; a one-dimensional attractor, although it exists mathematically, is not suitable for the role of a “steady-state regime”.

One way to avoid such troubles is consider only minimal attractors as attractors, that is, attractors that do not contain smaller attractors. Kolmogorov's hypotheses refer precisely to such attractors, if we want to give them a precise formulation.

But then nothing has been proven about estimates of dimensions from below, despite numerous publications named as such.

The danger of the deductive-axiomatic approach to mathematics Many thinkers before Kolmogorov clearly understood this. The first American mathematician J. Sylvester wrote that In no case should mathematical ideas be petrified, since they lose their power and application when trying to axiomatize the desired properties. He said that ideas should be perceived as water in a river: we never enter exactly the same water, although the ford is the same. Likewise, an idea can give rise to many different and non-equivalent axiomatics, each of which does not reflect the idea entirely.

Sylvester came to all these conclusions by thinking through, in his words, “the strange intellectual phenomenon that the proof of a more general statement often turns out to be simpler than the proof of the particular cases it contains.” As an example, he compared the geometry of vector space with (not yet established at that time) functional analysis.

This idea of ​​Sylvester was used a lot in the future. For example, it is precisely this that explains Bourbaki’s desire to make all concepts as general as possible. They even use in In France, the word “more” in the sense that in other countries (which they contemptuously call “Anglo-Saxon”) is expressed by the words “greater than or equal to,” since in France the more general concept “>=” was considered primary, and the more specific “>” - “ unimportant" example. Because of this, they teach students that zero is a positive number (as well as negative, non-positive, non-negative and natural), which is not recognized elsewhere.

But they apparently did not get to Sylvester’s conclusion about the inadmissibility of fossilization of theories (at least in Paris, in the library of the Ecole Normale Superieure, these pages of his Collected Works were uncut when I recently got to them).

I am unable to convince mathematical “specialists” to correctly interpret the hypotheses about the growth of the dimensions of attractors, since they, like lawyers, object to me with formal references to the existing dogmatic codes of laws containing the “exact formal definition” of attractors of the ignorant.

Kolmogorov, on the contrary, never cared about the letter of someone’s definition, but thought about the essence of the matter 2.

2 Having solved Birkhoff's problem on the stability of fixed points of non-resonant systems in 1960, I published a solution to this very problem in 1961. A year later, Yu. Moser generalized my result, proving stability at resonances of order greater than four. Only then did I notice that my proof established this more general fact, but, being hypnotized by the formulation of Birkhoff's definition of non-resonance, I did not write that I had proved more than Birkhoff claimed.

One day he explained to me that he came up with his topological cohomology theory not at all combinatorially or algebraically, as it looks like, but by thinking about fluid flows in hydrodynamics, then about magnetic fields: he wanted to model this physics in the combinatorial situation of an abstract complex and did so.

In those years, I naively tried to explain to Kolmogorov what happened in topology during those decades during which he drew all his knowledge about it only from P. S. Aleksandrov. Because of this isolation, Kolmogorov knew nothing about homotopy topology; he convinced me that “spectral sequences were contained in the Kazan work of Pavel Sergeevich 1942 of the year", and attempts to explain to him what the exact sequence was were no more successful than my naive attempts to put him on water skis or put him on a bicycle, this great traveler and skier.

What was surprising to me, however, was the high assessment of Kolmogorov’s words about cohomology given by a strict expert, Vladimir Abramovich Rokhlin. He explained to me, not at all critically, that these words of Kolmogorov contained, firstly, a deeply correct assessment of the relationship between his two achievements (especially difficult in the case when, as here, both achievements are remarkable), and secondly, a shrewd foresight of a huge meanings of cohomology operations.

Of all the achievements of modern topology, Kolmogorov valued Milnor's spheres the most, which the latter spoke about in 1961 at the All-Union Mathematical Congress in Leningrad. Kolmogorov even persuaded me (then a beginning graduate student) to include these spheres in my graduate plan, which forced me to start studying differential topology from Rokhlin, Fuchs and Novikov (as a result of which I was even soon an opponent of the latter’s Ph.D. thesis on differentiable structures on products of spheres).

Kolmogorov’s idea was to use Milnor spheres to prove that a function of several variables cannot be represented by superpositions in Hilbert’s 13th problem (probably for algebraic functions), but I don’t know any of his publications on this topic or the formulation of his hypotheses.

Another little-known circle of Kolmogorov’s ideas relates to optimal control of dynamic systems.

The simplest task of this circle is to maximize at some point the first derivative of a function defined on an interval or on a circle, knowing the upper bounds for the moduli of the function itself and its second derivative. The second derivative prevents the first from being quickly extinguished, and if the first is too large, the function outgrows the given limitation.

Probably, Hadamard was the first to publish the solution to this problem on the second derivative, and subsequently Littlewood rediscovered it while working on artillery trajectories. Kolmogorov, it seems, did not know the publications of either one or the other, and decided the problem of estimating from above any intermediate derivative through the maximum values ​​of the moduli of the differentiable function and its high (fixed) order derivative.

Kolmogorov's wonderful idea was to explicitly indicate extremal functions, such as Chebyshev polynomials (on which the inequality being proved becomes an equality). And in order for the function to be extreme, he naturally guessed that the value of the highest derivative must always be chosen to be the maximum in absolute value, changing only its sign.

This led him to a remarkable series of special features. The zero function of this series is the signum of the sine of the argument (everywhere having a maximum modulus). The next, first, function is an antiderivative of zero (that is, already continuous “saw”, the derivative of which has a maximum modulus everywhere). Further functions are obtained each from the previous one by the same integration (increasing the number of derivatives by one). You just need to choose the integration constant so that the integral of the resulting antiderivative function over the period is equal to zero each time (then all the constructed functions will be periodic).

Explicit formulas for the resulting piecewise polynomial functions are quite complex (the integrations are introduced by rational constants associated even with Bernoulli numbers).

The values ​​of the constructed functions and their derivatives are given by constants in Kolmogorov’s power estimates (estimating the modulus of the intermediate derivative from above through the product of rational powers of the maxima of the modulus of the function and the highest derivative). The indicated rational exponents are easy to guess from the consideration of similarity, going back to the laws of similarity of Leonardo da Vinci and to Kolmogorov’s theory of turbulence, that the combination should turn out to be dimensionless, since it is clear (at least from Leibniz’s notation) how derivatives of different orders behave when units change Argument and function measurements. For example, for the Hadamard problem, both rational exponents are equal to half, so the square of the first derivative is estimated from above by the product of the maxima of the modulus of the function itself and its second derivative (with a coefficient depending on the length of the segment or circle where the function is considered).

It is easier to prove all these estimates than to come up with the extremal functions described above (and delivering, among other things, Gauss’s theorem: the probability of irreducibility of the fraction p/q with integer numerator and denominator is equal to 6/p 2, that is, about 2/3).

In terms of today's management theory, The strategy chosen by Kolmogorov is called “big bang”: the control parameter must always be chosen to have an extreme value, any moderation only harms.

As for Hamilton’s differential equation for changing with time the choice of this extreme value from many possible ones, Kolmogorov knew it very well, calling it, however, Huygens’ principle (which is really equivalent to this equation and from which Hamilton obtained his equation by moving from envelopes to differentials) . Kolmogorov even pointed out to me, who was then a student, that the best description of this geometry of Huygens' principle is contained in Whittaker's textbook of mechanics, where I learned it, and that in a more intricate algebraic form it is in the theory of “Berurung Transformation” by Sophus Lie (instead of which I learned the theory of canonical transformations from Birkhoff’s “Dynamical Systems” and which today is called contact geometry).

Tracing the origins of modern mathematics in classical works is usually not easy, especially due to the changing terminology that is accepted as a new science. For example, almost no one notices that the so-called theory of Poisson manifolds was already developed by Jacobi. The fact is that Jacobi followed the path of algebraic varieties - varieties, and not smooth varieties - manifolds. Namely, he was interested in the variety of orbits of a Hamiltonian dynamical system. As a topological or smooth object, it has features and even more unpleasant pathologies (“non-Hausdorffness” and the like) due to the entanglement of orbits (phase curves of a complex dynamic system).

But the algebra of functions on this (possibly bad) “manifold” is perfectly defined: it is simply the algebra of first integrals of the original system. By Poisson's theorem, the Poisson bracket of the first two integrals is again the first integral. Therefore, in the algebra of integrals, in addition to multiplication, there is another bilinear operation - the Poisson bracket.

The interaction of these operations (multiplication and parentheses) in the space of functions on a given smooth manifold is what makes it a Poisson manifold. I skip the formal details of its definition (they are not complicated), especially since they are not all fulfilled in the example that interested Jacobi, where the Poisson manifold is neither smooth nor Hausdorff.

Thus, Jacobi's theory contains the study of more general varieties with singularities than modern Poisson smooth varieties, and moreover, this theory was constructed by him in the style of algebraic geometry of rings and ideals, rather than differential geometry of submanifolds.

Following Sylvester's advice, specialists in Poisson manifolds should, not limiting themselves to their axiomatics, return to a more general and more interesting case, already considered by Jacobi. But Sylvester did not do this (being late, as he said, for the ship leaving for Baltimore), and mathematicians of more recent times are completely subordinate to the dictates of the axiomatists.

Kolmogorov himself, having solved the problem of upper estimates for intermediate derivatives, understood that he could solve many other optimization problems using the same techniques of Huygens and Hamilton, but he did not do this, especially when Pontryagin, whom he always tried to help, published his “principle maximum”, which is essentially a special case of the same Huygens principle of forgotten contact geometry, applied, however, to a not very general problem.

Kolmogorov correctly thought that Pontryagin did not understand either these connections with Huygens' principle, or the connection of his theory with Kolmogorov's much earlier work on estimates of derivatives. And therefore, not wanting to disturb Pontryagin, he did not write anywhere about this connection, which was well known to him.

But now, I think, this can already be said, in the hope that someone will be able to use these connections to discover new results.

It is instructive that Kolmogorov's inequalities between derivatives served as the basis for the remarkable achievements of Yu. Moser in the so-called KAM theory (Kolmogorov, Arnold, Moser), which allowed him to transfer Kolmogorov's 1954 results on invariant tori of analytic Hamiltonian systems to only three hundred and thirty-three times differentiable systems . This was the case in 1962, with Moser's invention of his remarkable combination of Nash smoothing and Kolmogorov's accelerated convergence method.

Now the number of derivatives needed for the proof has been significantly reduced (primarily by J. Mather), so that the three hundred and thirty-three derivatives needed in the two-dimensional problem of ring mappings have been reduced to three (while counterexamples have been found for two derivatives).

It is interesting that after the appearance of Moser’s work, American “mathematicians” tried to publish their “generalization of Moser’s theorem to analytical systems” (which generalization was simply Kolmogorov’s theorem published ten years earlier, which Moser managed to generalize). Moser, however, decisively put an end to these attempts to attribute to others Kolmogorov's classical result (correctly noting, however, that Kolmogorov never published a detailed presentation of his proof).

It seemed to me then that the proof published by Kolmogorov in a note in DAN was quite clear (although he wrote more for Poincaré than for Hilbert), in contrast to Moser’s proof, where I did not understand one place. I even revised it in my 1963 review of Moser's remarkable theory. Moser subsequently explained to me what he meant in this unclear place, but I am still not sure whether these explanations were properly published (in my revision I have to choose s < e /3, а не e /2, как указывалось в непонятном месте, вызвавшем затруднения не только у меня, но и у других читателей и допускающем неправильное истолкование неясно сказанного).

It is also instructive that "Kolmogorov's accelerated convergence method"(correctly attributed by Kolmogorov to Newton) was used for a similar purpose in solving a nonlinear equation by A. Cartan ten years before Kolmogorov, in proving what is now called the theorem A beam theory. Kolmogorov knew nothing about this, but Cartan pointed this out to me in 1965, and was convinced that Kolmogorov could have referred to Cartan (although his situation in the theory of beams was somewhat simpler, since when solving a linearized problem there was no fundamental in celestial mechanics is the difficulty of resonances and small denominators, present in Kolmogorov and Poincaré). Kolmogorov’s not mathematical, but broader approach to his research was clearly manifested in two of his works with co-authors: in an article with M.A. Leontovich on the area of ​​the neighborhood of a Brownian trajectory and in the article “KPP” (Kolmogorov, Petrovsky and Piskunov) on the speed of propagation of nonlinear waves

In both cases, the work contains both a clear physical formulation of a natural science problem and a complex and non-trivial mathematical technique for solving it.

And in both cases Kolmogorov performed not the mathematical, but the physical part of the work, associated, first of all, with the formulation of the problem and with the derivation of the necessary equations, while their research and proof of the corresponding theorems belong to the co-authors.

In the case of Brownian asymptotics, this difficult mathematical technique involves the study of integrals along deformable paths on Riemann surfaces, taking into account the complex deformations of integration contours required for this when changing parameters, that is, what is today called either “Picard-Lefschetz theory” or “connectivity theory” Gauss-Manin".

And this entire study of asymptotic integrals belongs to M. A. Leontovich, a remarkable physicist (by the way, who, together with his teacher L. I. Mandelstam, came up with a theory that provided an explanation of radioactive decay using the quantum tunneling effect of passage under a barrier, and the work they published was subsequently generalized by their student G. Gamow, who left for the USA, 3 under whose name it is now better known).

3 My fellow countryman, Odessa resident G. Gamow is most famous for the following three of his discoveries: the theory of alpha decay, the solution to the three-letter coding of amino acids by bases in DNA and the theory of the “big bang” in the formation of the Universe. Now his wonderful books are also available to the Russian reader (who for a long time did not have this opportunity due to Gamow’s non-return from the Solvay Congress).

The above-mentioned work on Brownian trajectory was published in the collected works of both Leontovich and Kolmogorov. And in both publications it is said that the physical part of the work belongs to the mathematician, and the mathematical part to the physicist. This explains many features of Russian mathematical culture.

The same situation is in the work of “KPP” on the speed of propagation of environmental waves. Kolmogorov told me that he was responsible for the formulation of a mathematical problem in it (invented by him while thinking about ecological situation of the movement of the front of the spread of a species or gene in the presence of migration and diffusion).

Mathematical solutions (as unconventional as the problem itself) were developed by I.G. Petrovsky (for whom this nonlinear work is also rather an exception). The article was mainly written by Piskunov, without whom it would not have existed either. Although this wonderful work on “intermediate asymptotics,” as Ya. B. Zeldovich called it, is widely known to applied scientists and is constantly used, it is little known to mathematicians, despite the completely original and brilliant ideas it contains about the competition of waves moving at different speeds.

I have long been waiting for a serious mathematician to continue this research, but so far I have only seen “applied scientists” applying ready-made results and not adding new ideas and methods.

The great applied scientist Pasteur said that There are no “applied sciences”, but only ordinary fundamental sciences, where new truths are discovered, and there are their applications, where these truths are used.

For the true continuation of the work of the “KPP”, it is precisely the advancement in fundamental science that is needed.

Marat wrote that “of all mathematicians, the best are Laplace, Monge and Cousin, who calculate everything using pre-prepared formulas.” This phrase is a sign of the revolutionaries’ complete misunderstanding of mathematics, the main thing in which is free thinking outside the framework of any pre-prepared schemes.

A little later, Marat Abel wrote from Paris, where he spent about a year, that “you can’t talk about anything with the local mathematicians, since each of them wants to teach everyone and doesn’t want to learn anything themselves. As a result, he wrote prophetically, each of them understands only one narrow area and understands nothing outside of it. There is a specialist in the theory of heat [Fourier], there is a specialist in the theory of elasticity [Poisson], there is a specialist in celestial mechanics [Laplace], and only Cauchy [Lagrange lived in Berlin] could understand something, but he is only interested in his own priority.” [for example, in the application of complex numbers to Lamé’s solution to Fermat’s problem by expanding the binomial x n +y n to complex factors].

Both Abel and (ten years later) Galois went far beyond the framework of “ready-made schemes” (having developed, in Abel’s case, the topology of Riemann surfaces and deducing from it both the impossibility of solving equations of the fifth degree in radicals and the inexpressibility in the form of elementary functions of “elliptic integrals", such as the integral of the square root of a polynomial of the third or fourth degree, expressing the length of the arc of the ellipse, and their inverse "elliptic functions").

Therefore, Cauchy “lost” the manuscripts of both Abel and Galois, so that Abel’s work on undecidability was published (by Liouville) only decades after, according to a Parisian newspaper of the time, “this poor man returned to his part of Siberia, called Norway , on foot - without money for a ship ticket - across the ice of the Atlantic Ocean."

Already in the 20th century, the famous English eccentric Hardy wrote that “Abel, Riemann and Poincaré lived their lives in vain, bringing nothing to humanity.”

Most of modern mathematics (and most of all the mathematics used by physicists) are repetitions or developments of the wonderful geometric ideas of Abel, Riemann, Poincaré, which permeate all modern mathematics as a single whole, where, according to Jacobi, “the same function solves both the question of representing numbers as a sum of squares, and the question of the law of large oscillations of a pendulum,” also solving the question of the length of an ellipse, which ellipse describes the movement of planets, the tumbling of satellites, and conic sections. A Riemannian surfaces, Abelian integrals, and Poincaré differential equations are the main keys to the amazing world of mathematics.

Kolmogorov perceived as a single whole not only all mathematics, but also all natural science. Here is an example of his thoughts on the miniaturization of a computer, as the simplest model of which he considered a graph (diagram, diagram) from P vertices (balls (fixed radius), each connected to no more than k others (using connections: “wires” of a fixed thickness). Most connections k he fixed each vertex, and the number of vertices P considered very large (there are about 10 10 neurons in the human brain). The question about miniaturization is: What is the smallest ball that can fit a given graph without self-intersections with the following properties: how does the radius of this minimal ball grow with the number of vertices n?

One limitation is obvious: the volume of the ball should grow no slower than that, since the total volume of the ball vertices grows at such a speed, and they all need to fit.

But will it be possible to fit the entire graph into a ball of radius proportional to the cube root of n. After all, in addition to the peaks, connections must also fit in! And although their number is also on the order of ta, the volume can be much larger, since with large ta long connections may be required.

Kolmogorov reasoned further, imagining the count as a brain. A very stupid brain (“worm”) consists of one chain of vertices connected in series. It is easy to fit such a brain like a “snake” into a “skull” with a radius on the order of the cube root of n.

At the same time, the evolution of animals should have tried to arrange the brain economically, reducing, if possible, the size of the skull. How is it with animals?

It is known that the brain consists of gray matter (the body of neuron-vertices) and white matter (connections: axons, dendrites). Gray matter is located along the surface of the brain, and white matter is located inside. With this arrangement on the surface, the radius of the skull should grow not like a cubic radius, but faster, like the square root of the number of vertices (the radius is much larger than the volume of the vertex balls dictates).

So Kolmogorov came to the mathematical hypothesis that the minimum radius must be of the order of the square root of the number of vertices(based on the fact that the arrangement of real brain cells has been brought by evolution to a state that minimizes the radius of the skull). In his publications, Kolmogorov deliberately avoided writing about these biological considerations and about the brain in general, although at first he did not have any arguments in favor of the square root, other than biological ones.

Prove that every graph from n vertices can be accommodated (subject to the limitation k by the number of connections of the vertex) into a ball of radius on the order of the square root of that, we succeeded (although it was not easy). This is already pure mathematics of rigorous proofs.

But the question of why the graph cannot be placed in a “skull” of smaller radius turned out to be more difficult (if only because “impossible” is not always: The “very stupid” worm’s brain fits into a skull with a radius on the order of the cube root of n, which is much less than the square root).

In the end, Kolmogorov managed to completely deal with this problem. Firstly, he proved that investments in a “skull” smaller than the square root of n radius are not allowed by most “brains” of n “neurons”: embeddables (like a “one-dimensional” brain in the form of a chain of vertices connected in series) constitute a tiny minority of the huge total number n-vertex graphs (with limited given constant k

Second, he established a remarkable criterion for complexity that prevents embedding in a smaller “skull”: the sign of complexity turned out to be universality. Namely, a graph with those vertices is called universal, if it contains as subgraphs (with a slightly smaller number of vertices) all the graphs from this smaller number of vertices (with a limited, of course, that same constant k the number of connections of each vertex).

The words “slightly fewer vertices” can be understood here in different ways: as an or how n a, Where A less than 1. With this correct understanding of universality, the following two facts are proven: firstly, for some c = const any universal graph with n vertices turns out to be non-embeddable in a ball of radius less than the square root of n, and secondly, non-universal graphs make up an insignificant minority(in a huge number of all n-vertex graphs with the above restriction k in touch).

In other words, Although stupid brains may be small, no sufficiently smart brain (or computer) can be contained in a small volume, and, in addition, the mere complexity of the system alone will overwhelmingly ensure the possibility of its good (“universal”) functioning, that is, its ability to replace (“model”) all other (almost as complex as itself) systems.

These achievements constituted one of Andrei Nikolaevich’s last works (the final inequalities were obtained by him together with his student Bardzin; Kolmogorov’s original inequalities contained extra logarithms, which Bardzin managed to remove).

Kolmogorov's attitude towards logarithms in asymptotics was very specific. He explained to the students that numbers are divided into the following four categories:

• small numbers: 1, 2, …, 10, 100;
• average numbers: 1000, 1000000;
• large numbers: 10 100, 10 1000;
• practically infinite numbers: 10 1010.

Taking logarithms moves a number to the previous category. That's why logarithms in asymptotics like n 3 ln n - these are just constants: n 3 ln at n= 10 - this is practically 2p 3, and the growth of the logarithm is so slow that it can be neglected as a first approximation, considering the logarithm to be “limited.”

Certainly, all this is completely wrong from the point of view of formal axiomatic mathematics. But this is much more useful for practical work than refined “rigorous reasoning” and estimates that begin with the words “consider the following auxiliary function of eighteen arguments” (followed by a one-and-a-half-page formula that came from nowhere).

Kolmogorov's approach to logarithms reminded me of Ya.B. Zeldovich's point of view on mathematical analysis. In his analysis textbook “for beginning physicists and technicians,” Zeldovich defined the derivative as the ratio of the increments of a function and its argument, assuming that the latter increment is not too large.

To the objections of true mathematicians that a limit is needed, Zeldovich replied that the “limit of the ratio” is unsuitable here, since too small (say, less than 10 -10 meters or seconds) increments of the argument cannot be taken, simply because in such scale, the properties of space and time become quantum, so their description using a mathematical one-dimensional continuum R becomes an excess of model accuracy.

Zeldovich perceived “mathematical derivatives” as convenient approximate asymptotic formulas to calculate the ratio of finite increments that really interests us, which is given by a more complex formula than the derivatives of mathematicians.

As for the “rigor” of mathematicians, Kolmogorov never overestimated its importance (although he tried to introduce a multi-page definition of the concept of angle into the school geometry course in order, in his words, to give a strict meaning to “an angle of 721 degrees”).

His lectures were difficult for students and schoolchildren to understand, not only because not a single phrase ended, and half did not have either a subject or a predicate. What’s even worse is that (as Andrei Nikolaevich explained to me when I started lecturing to students), in his deep conviction, “Students don’t care at all what they are told in lectures: they just memorize the answers to a few of the most common exam questions for the exam, without understanding anything at all.”

These words indicate a completely correct understanding of the situation by Kolmogorov: with his lectures, for most students, exactly what he described happened. But those who wanted to understand the essence of the matter could, if they wished, learn much more from them than from standard deductions like "X more y, therefore y is less than X". It was precisely the basic ideas and secret springs hidden behind “auxiliary functions of eighteen variables” that he tried to make understandable, and he willingly left the derivation of formal consequences from these basic ideas to his listeners. What made it especially difficult was that Kolmogorov thought during his lectures, and this was noticeable to the listeners.

I was always struck by Andrei Nikolaevich’s noble desire to see in each interlocutor at least an equal intellect (which is why it was so difficult to understand him). At the same time, he knew very well that in reality the level of most of his interlocutors was completely different. Andrei Nikolaevich once named me only two mathematicians, when talking with whom he “felt the presence of a higher mind” (one of them he named his student I.M. Gelfand).

At the anniversary of Andrei Nikolaevich, Gelfand said from the podium that he not only learned a lot from the teacher, but also visited him in Komarovka, a village on the banks of the Klyazma, near Bolshevo, where Kolmogorov lived most of the time (coming to Moscow only for one or two days in Week).

Pavel Sergeevich Alexandrov, who was present at this speech by Gelfand, who bought the Komarovsky house together with Kolmogorov (from the Alekseev, that is, Stanislavsky family) in the late 20s, readily confirmed: “Yes, Israel Moiseevich really visited Komarovka, and was even very useful, as he saved a cat from being burned in the stove.”

One of the listeners told me that Gelfand, already sitting in the anniversary hall, commented on these words to his neighbor as follows: “This cat had been meowing there in the oven for half an hour, and I had heard it for a long time, but I interpreted this meowing incorrectly, not knowing about the cat and attributing the sounds to another source.”

Andrei Nikolaevich’s diction was indeed not easy to understand; I, however, more often guessed what he wanted to say than understood the half-words he uttered, so this diction did not bother me.

Still, schoolchildren at the mathematical boarding school N18, organized by Andrei Nikolaevich in Moscow in 1963, learned a lot from him. Of course, these were not ordinary schoolchildren, but winners of mathematical Olympiads gathered from all over Russia and having attended a summer school in Krasnovidovo on the Mozhaisk Sea, and not only Andrei Nikolaevich himself taught them, but also many excellent teachers, for example, mathematician Vladimir Mikhailovich Alekseev, one one of the best school teachers in Moscow, Alexander Abramovich Shershevsky, and so on.

Special efforts were made to provide good food and interesting teaching not only in mathematics, but also in physics, literature, history, and English: Andrei Nikolaevich perceived the boarding school in many ways as his family. Of the first graduates, the majority entered the best mathematical and physics universities (with more successful admission to the Moscow Institute of Physics and Technology than to the Faculty of Physics of Moscow University, famous, as Kolmogorov said, for “its hostility” in exams).

Now many of these graduates have already become professors, heads of departments, and directors of institutes; I have no doubt that some of them are worthy of selection into the Russian Academy of Sciences and awards such as the Fields or Abel medals.

The theorem of Nekhoroshev, who was far ahead of Littlewood, has long become a classic result in celestial mechanics and the theory of Hamiltonian evolution of dynamical systems. Yu. Matiyasevich, who then moved to Leningrad, also began together with the first Moscow boarding mathematicians at the summer school organized by Kolmogorov in Krasnovidovo on the Mozhaisk Sea. A. Abramov for a long time headed an institute dedicated to improving the mathematical education of schoolchildren (but his fight against the attempts of the Ministry of Education to destroy a perfectly working system made him undesirable for the “reformers,” whose obscurantist ideas I described above, at the beginning of this article).

One of the students of the first graduating class of the boarding school, V.B. Alekseev, published in 1976 his notes of my lectures at the boarding school in 1963: “Abel’s theorem in problems.” In these lectures he told topological proof of Abel's theorem on the unsolvability in radicals (combinations of roots) of algebraic equations of the fifth degree (and higher degrees). At school they teach the case of degree 2, but equations of degrees 3 and 4 in radicals are also solved.

The purpose of these lectures was to convey an important (and difficult) mathematical result, connecting many areas of modern physics and mathematics, to completely unprepared (but intelligent) schoolchildren in the form of a long series of understandable and accessible problems that they could cope with themselves, but which would lead them , at the end of the semester, to Abel's theorem.

To do this, schoolchildren quickly became familiar with the geometric theory of complex numbers, including Moivre’s formulas (which current “reformers” are trying to exclude from new programs), then moving on to Riemann surfaces and topology, including the fundamental group of curves on a surface and the monodromy (multivalued) groups of coverings and branched coverings.

These most important geometric concepts (which could be compared with the atomic theory of the structure of matter in physics and chemistry or with the cellular structure of plants and animals in biology in terms of their fundamentality) then lead to algebraic equally important objects: transformation groups, their subgroups, normal divisors, exact sequences.

In particular, there appear symmetry and ornaments, and crystals, and regular polyhedra: tetrahedron, cube, octahedron, icosahedron and dodecahedron, including the constructions of embedding them into each other used by Kepler (to describe the radii of planetary orbits) (eight vertices of a cube can be divided into two quadruple vertices of two tetrahedrons “inscribed” in a cube, and five cubes can be “inscribed” in a dodecahedron, the vertices of each of which form part of the vertices of the dodecahedron (at of which there are twenty), and the edges of the cube turn out to be diagonals of the pentagonal faces of the dodecahedron, one on each of the twelve faces). “Dodeca” is just “twelve” in Greek, and the cube has twelve edges.

This remarkable geometric construction by Kepler relates the symmetry group of the dodecahedron to the group of all one hundred and twenty permutations of five objects (namely, cubes). It establishes, in algebraic terms, also the undecidability of both of these groups (that is, their irreducibility to commutative groups, which is the case, for example, for the symmetry groups of the tetrahedron, cube and octahedron and for groups of permutations of three or four objects, such as the four great diagonals cube and three diagonals of the octahedron). Commutative groups (where the product - the execution of consecutive transformations - do not depend on their order) are called Abelian in algebra due to the importance for his theory of the non-commutativity of permutations of cubes.

A From the unsolvability of the monodromy group of an equation of the fifth degree, the non-existence of a formula expressing its roots in terms of radicals is topologically deduced. The point is that the monodromy group, which measures the polysemy of each radical, is commutative, and the monodromy group of a combination of radicals is composed of their monodromy groups in the same way as a soluble group is composed of commutative ones. So all these topological considerations of the theory of Riemann surfaces lead to the proof of Abel’s algebraic theorem(which laid the foundations of the Galois theory, named after the young French mathematician who transferred Abel’s theory from complex geometry to the theory of numbers and died in a duel without publishing his theory).

Deep unity of all mathematics is very clearly manifested in this example of the interaction of topology, logic, algebra, analysis and number theory, which created a new fruitful method, with the help of which the physics of quantum theory and the theory of relativity was later far developed, and in mathematics the unsolvability of many other problems of analysis was also proven: for example , problems of integration using elementary functions and problems of explicitly solving differential equations using the operation of integration.

The fact that all these questions are topological is an absolutely amazing mathematical achievement, which, in my opinion, could be compared with the discoveries of the connection between electricity and magnetism in physics or between graphite and diamond in chemistry.

Perhaps the most famous result about impossibility in mathematics was the discovery Lobachevsky’s geometry, the central result of which is the impossibility of deducing the “parallel axiom” from the other axioms of Euclid’s geometry, its unprovability.

It is instructive that Lobachevsky did not establish this result about unprovability, but only proclaimed it as his hypothesis, confirmed by many pages of (unsuccessful) attempts to prove the axiom of parallels, that is, to come to a contradiction based on the statement opposite to the axiom of parallels: “ Through a point outside a line there pass several (many) lines that do not intersect with it.”

Proof that in the contradictions arising from this axiom of Lobachevsky geometry are no more than in Euclidean geometry (postulating the uniqueness of a parallel line), was found only after Lobachevsky (apparently, independently of each other by several authors, including Beltrami, Bogliai, Klein and Poincaré, or even Gauss, who highly appreciated Lobachevsky’s ideas).

This proof of the consistency of Lobachevsky geometry is not simple; it is carried out by presenting a model of Lobachevsky’s geometry, in which precisely his axioms are satisfied. One of these models (the “Klein model”) depicts the Lobachevsky plane as the interior of a circle, and the Lobachevsky lines as its chords. It is not difficult to draw many chords through a point on a circle that do not intersect any given chord that does not pass through this point. Checking the remaining geometry axioms in this model is also not very difficult, but it is time-consuming, since there are many of these axioms. For example, “any two points inside a circle can be connected by a Lobachevsky straight line (chord), and, moreover, only by one” and so on. All this is clearly done in textbooks and takes up many (boring) pages.

The continuation of Klein's model of the Lobachevsky plane beyond the circle that depicted the Lobachevsky plane in this model delivers the relativistic world of de Sitter, but, unfortunately, few people understand this fact (both among mathematicians and among relativists).

Modern “reformers” of the school mathematics course announced their desire to introduce Lobachevsky’s geometry there (which Kolmogorov did not dare to do). But they do not even mention its main result (most likely, without suspecting it) and do not plan to prove Lobachevsky’s thesis (without which this whole enterprise becomes just an advertising stunt, of a patriotic, however, hue).

Unlike these “reformers,” Kolmogorov tried to teach children mathematics for real. In his opinion, For this purpose, problem solving is best suited, for example, Olympiads, and he more than once organized mathematical Olympiads for schoolchildren, especially insisting that this enterprise should not only be in Moscow, but also cover all cities and even villages of the country (today the Olympiads have spread to the whole world, and the successes of our schoolchildren have them - indisputable evidence of the still high level of schools).

He told me with pleasure how happy the teacher, who was on the jury of one of the Moscow Olympiads with him, was when she presented a set of gift mathematical books to the tenth grader who received the first prize at the awards ceremony at Moscow State University: "So glad, - she said - that the prize was given to a simple village schoolboy from the village of Khotkovo!”

This lady from pedagogy did not know that the “simple village schoolboy” was the son of an academician who lived in the academic village of Abramtsevo, and Kolmogorov, although he laughed, did not explain this to her.

Now this “village schoolboy” (who was already my student at school) is an established independent mathematician who has published many works and long ago graduated from the Faculty of Mechanics and Mathematics of Moscow State University. By the way, he wrote an interesting commentary on A.D. Sakharov’s mathematical problem about chopping cabbage. Sakharov studied mathematics at the University with my father (about which A.D. writes warmly in his memoirs), and after the death of Andrei Dmitrievich, his colleagues asked me to comment on his mathematical manuscripts (containing several dozen interesting purely mathematical problems invented and thought out by him).

Cabbage chopping problem arose from Andrei Dmitrievich as a result of his wife’s request to chop it, which begins with dividing the head of cabbage into circular layers with a knife. Each layer is then divided by random knife blows into many convex "polygons".

While doing this work, Sakharov asked himself the question: How many sides do these polygons have? Some are triangles, some have many sides. The question was therefore posed mathematically as follows: What is the average number of sides of a piece?

Sakharov came by some (perhaps experimental?) route to the (correct) answer: four.

When commenting on his manuscript for its publication, my Italian student F. Aicardi came to the following generalization of this statement of Sakharov: when cutting an n-dimensional body with a large number of random hyperplanes (planes of dimension n- 1) onto convex n-dimensional polyhedra, the resulting pieces the average number of faces of any dimension will be the same as that of an n-dimensional cube. For example, in our ordinary three-dimensional space the average number of vertices of a piece is 8, the average number of edges is 12, and the average number of edges of a piece is 6.

In any case, even if it was sometimes difficult for schoolchildren in the boarding school, the benefits from the boarding school were and remain enormous, immeasurably, in my opinion, greater than from Kolmogorov’s attempts to modernize courses in mathematical sciences by replacing the classical textbooks of A. Kiselyov with new textbooks of the Bourbakist type ( with their modern terminology, which replaced the classical Euclidean “tests for the equality of triangles” with the obscure, although logically preferable, “tests of congruence”).

This reform undermined the authority of the school, teachers, and textbooks, creating a scientific illusion of pseudo-knowledge that covers up a complete misunderstanding of the simplest facts, such as the fact that 5 + 8 = 13. In the draft of the new reform, the same tendencies of fooling schoolchildren are noticeable, who are offered incomprehensible “geometry” Lobachevsky" instead of writing simple fractions into decimals and "text arithmetic problems" about crews moving from point A to point excluded from training IN, or about merchants selling cloth for axes, or about diggers and pipes filling reservoirs - problems on which previous generations learned to think.

The result of the “reform” will be pseudo-education, leading the ignorant to statements like the criticism of one political figure attributed to Stalin: “It’s not just a negative value, it’s a negative value squared!”

At one of the discussions of the school reform project by the Academic Council of the Mathematical Institute. Steklov RAS, I mentioned that it would be good to return to Kiselev’s excellent textbooks and problem books.

In response, I was praised for this by the head of some educational department who was at this meeting: “I am so glad that Kiselyov’s activities received the support of such qualified specialists!”

Later they explained to me that Kiselev is the name of one of the young subordinates of this leader, who manages school mathematics, having never heard of the wonderful textbooks of the outstanding gymnasium teacher Kiselev, which were reprinted many dozens of times. Kiselyov’s textbooks, by the way, were not so good from the very beginning. The first editions had many shortcomings, but the experience of dozens and hundreds of gymnasium teachers made it possible to correct and supplement these books, which (after about a dozen first editions) became monumental examples of school textbooks.

Andrei Nikolaevich Kolmogorov was also a school teacher from his youth (at a school on Potylikha), and so successful that he hoped that schoolchildren would elect him (it was common then to elect him) as their class teacher. But the physical education teacher won the election - this is closer to schoolchildren.

I wonder what Another great mathematician, K. Weierstrass, began his career as a physical education teacher at school. He, according to Poincaré, was especially successful in teaching his gymnasium students how to work on parallel bars. But Prussian rules required a gymnasium teacher to submit written work at the end of the year proving his professional suitability. And Weierstrass presented an essay on elliptic functions and integrals.

No one at the gymnasium could understand this essay, so it was sent to the university for evaluation. And very soon the author was transferred to where he quickly became one of the most outstanding and famous mathematicians of the century, both in Germany and in the world. Of the Russian mathematicians, his direct student was Sofya Kovalevskaya, whose main achievement, however, was not a confirmation, but a refutation of the teacher’s point of view (who asked her to prove the absence of new first integrals in the problem of the rotation of a rigid body around a fixed point, and she found these integrals, analyzing the reasons for the failure of his attempts to prove the assumption of his beloved teacher).

The preference shown by schoolchildren to the physical education teacher influenced Kolmogorov as follows: he began to play sports much more, skied a lot, sailed on boats on distant rivers, became an inveterate traveler (and achieved the approval, although not of his Potylikhin students, but of many generations of first MSU students, and then the schoolchildren of the Boarding School he created).

Kolmogorov’s usual daily ski trips were approximately forty kilometers long, along the banks of the Vori, approximately from Radonezh to the monastery in Berlyuki, and sometimes to Bryusovskie Glinki at the confluence of the Vori and the Klyazma. Kayak and boat routes included, for example, Zaonezhye with its wonderful Svyatukha, Lake Seremo with the rivers Granichnaya, Shlina, connecting this area with the Vyshnevolotsk reservoir, from which the Meta (to Ilmen, Volkhov, Svir) and Tvertsa (flowing to the Volga) flow. , with further sailing to the Moscow Sea and Dubna.

I remember Andrei Nikolaevich’s stories about a cart that frightened him in the middle of Ilmen, crossing a multi-kilometer bay ford, which caused difficulties for the kayak with its stormy waves. Probably his greatest journey began in the North from Kuloy, continuing further along Pechora and Shugor to the pass through the Urals, with a descent to the Ob and an ascent along it to Altai, where the end of this many-thousand-kilometer journey was either horseback riding or walking “barefoot along mountain paths."

Andrei Nikolaevich amazed me with his ability to quickly install a homemade oblique sail on a kayak from scrap materials: this little-known technology today probably dates back to the Volga robbers who preceded Stepan Razin.

Andrei Nikolaevich's geographical knowledge was diverse and unusual. Few Muscovites know why Rogozhskaya Zastava and Stromynka Street are called that, why the Tsaritsyno station was called (but is no longer called) Lenino, where the Moscow rivers Rachka and Khapilovka are located, but he knew. For those interested, I will provide some answers:

The Rogozhskaya outpost stands at the beginning of the road to the city of Rogozha, which Catherine II, for the sake of euphony, renamed (in 1781) Bogorodsk (but which has not yet been renamed back to Kitai-Gorod, although they got rid of the name “Bogorodsk” during the revolution).

The Stromyn road is now called Shchelkovsky Highway, but it led to the ancient city of Stromyn (the suburb of which is now called Chernogolovka), on the way from Moscow to Kirzhach, Suzdal and Vladimir. Tsaritsyno was built for the ruins that Catherine lacked in Russia and on which climbers now train.

The Clean Pond was formed on the Rachka River. As for Khapilovka, it is deeper than the Yauza on the first topographic plan of Moscow (1739), flowing into the Yauza just above the Elektrozavodsky Bridge. Now Cherkizovsky Pond is visible on it, but I could not understand how it flows to it through Golyanovo from its source between Balashikha and Reutov.

The name “Lenino” comes from the name of the daughter of Kantemir, from whom Catherine bought “Black Dirt”, which now became Tsaritsyn: he named several surrounding villages donated to him by the names of his daughters.

Andrei Nikolaevich Kolmogorov was characterized by good-naturedness towards clearly unscrupulous opponents. For example, he argued that T.D. Lysenko is a conscientiously misguided ignoramus, and sat at his table in the dining room of the Academy of Sciences (from where others, starting with the notorious session of the All-Union Academy of Agricultural Sciences in 1948, tried to move to other tables).

The fact is that Andrei Nikolaevich once analyzed the experimental work of one of Lysenko’s students to refute Mendel’s laws of character splitting [N.I. Ermolaeva, Vernalization, 1939, 2(23)]. In this experiment, I think 4,000 pea seeds were sown, and according to Mendel's laws, 1,000 peas of one (recessive) color and 3,000 of another (dominant) color were expected to emerge. In the experiment, instead of 1000, there were only, if my memory serves me right, 970 sunrises of the recessive color and 3030 of the dominant color.

The conclusion that Kolmogorov made from this article is this:

the experiment was carried out honestly, the observed deviation from the theoretical proportion is exactly the order of magnitude that should be expected with such a volume of statistics. If the agreement with the theory were better, then this would precisely indicate the dishonesty of the experiment and the manipulation of the results.

Andrei Nikolaevich told me that he did not publish his findings in full because the objections of classical geneticists had already appeared, claiming that they had repeated the experiment and obtained exact agreement with theory. So Kolmogorov, in order not to harm them, limited himself to the message (DAN USSR, 1940, 27(1), 38-42) that the experiment conducted by Lysenko’s student is not a refutation, but an excellent confirmation of Mendel’s laws.

This, however, did not stop T.D. Lysenko, who declared himself a “fighter against chance in science,” and thus with the entire theory of probability and statistics, and therefore with their patriarch, A. N. Kolmogorov. Andrei Nikolaevich, however, did not waste time arguing with Lysenko (following, apparently, Pushkin’s advice on the use of “sound thoughts” and “bloody paths”, which clearly protects all obscurantists - both Lysenko and the current “reformers” of the Russian school).

Kolmogorov's influence on the entire development of mathematics in Russia remains completely exceptional today. I’m talking not only about his theorems, which sometimes solve thousand-year-old problems, but also about his creation of a wonderful cult of science and enlightenment, reminiscent of Leonardo and Galileo. Andrei Nikolaevich opened up enormous opportunities for many people to use their intellectual efforts for fundamental discoveries of new laws of nature and society, and not only in the field of mathematics, but in all areas of human activity: from space flights to controlled thermonuclear reactions, from hydrodynamics to ecology, from theory dispersion of artillery shells to the theory of information transmission and the theory of algorithms, from poetry to the history of Novgorod, from Galileo's laws of similarity to Newton's three-body problem.

Newton, Euler, Gauss, Poincare, Kolmogorov -
only five lives separate us from the origins of our science.

Pushkin once said that he had more influence on youth and Russian literature than the entire Ministry of Public Education, despite the complete inequality of funds. Kolmogorov's influence on mathematics was the same.

I met Andrei Nikolaevich during my student years. Then he was the dean of the Faculty of Mechanics and Mathematics at Moscow University. These were the heydays of the faculty, the heyday of mathematics. The level that the faculty reached then, thanks primarily to Andrei Nikolaevich Kolmogorov and Ivan Georgievich Petrovsky, it has never reached again and is unlikely to ever reach.

Andrei Nikolaevich was a wonderful dean. He said that talented people should be forgiven for their talent, and I could name the now very famous mathematicians whom he then saved from expulsion from the university.

The last decade of Andrei Nikolaevich’s life was overshadowed by a serious illness. At first he began to complain about his eyesight, and the forty-kilometer ski routes had to be reduced to twenty kilometers.

Later, it became difficult for Andrei Nikolaevich to fight the sea waves, but he still ran behind the fence of the Uzkoye sanatorium from the strict supervision of Anna Dmitrievna and doctors to swim in the pond.

In recent years, Andrei Nikolaevich’s life was very difficult, sometimes he had to literally be carried in his arms. We are all deeply grateful to Anna Dmitrievna, Asa Aleksandrovna Bukanova, students of Andrei Nikolaevich and graduates of the physics and mathematics boarding school N18, which he created, for their round-the-clock duty for several years.

At times Andrei Nikolaevich could only speak a few words per hour. But still, it was always interesting with him - I remember how a few months ago Andrei Nikolaevich told how tracer shells flew slowly over Komarovka, how at the age of 70 he could not get out of the freezing Moscow River, how in Calcutta he swam for the first time in the Indian Ocean of his students there.

Vladimir Igorevich Arnold

I dedicate to my Teacher - Andrei Nikolaevich Kolmogorov

“Don’t touch my circles,” Archimedes said to the Roman soldier who was killing him. This prophetic phrase came to mind in the State Duma, when the chairman of the meeting of the Education Committee (October 22, 2002) interrupted me with the words: “I have not the Academy of Sciences, where one can defend the truth, but the State Duma, where everything is based on the fact that different people have different opinions on different issues.”

The view I advocated was that three times seven is twenty-one, and that teaching our children both the multiplication table and the addition of single-digit numbers and even fractions is a national necessity. I mentioned the recent introduction in the state of California (on the initiative of Nobel laureate, transuranium physicist Glen Seaborg) of a new requirement for schoolchildren entering universities: you need to be able to independently divide the number 111 by 3 (without a computer).

The listeners in the Duma, apparently, could not separate, and therefore did not understand either me or Seaborg: in Izvestia, with a friendly presentation of my phrase, the number “one hundred eleven” was replaced by “eleven” (which makes the question much more difficult, since eleven is not divisible by three).

I came across the triumph of obscurantism when I read in Nezavisimaya Gazeta an article glorifying the newly built pyramids near Moscow, “Retrogrades and Charlatans,” where

The Russian Academy of Sciences was declared to be a meeting of retrogrades inhibiting the development of science (trying in vain to explain everything with their “laws of nature”). I must say that I am apparently also a retrograde, since I still believe in the laws of nature and believe that the Earth rotates around its axis and around the Sun, and that younger schoolchildren need to continue to explain why it is cold in winter and warm in summer, not allowing the level of our school education to fall below what was achieved in parochial schools before the revolution (namely, it is precisely this reduction in the level of education that our current reformers are striving for, citing the truly low American school level).

American colleagues explained to me that the low level of general culture and school education in their country is a deliberate achievement for economic purposes. The fact is that, after reading books, an educated person becomes a worse buyer: he buys less washing machines and cars, and begins to prefer Mozart or Van Gogh, Shakespeare or theorems to them. The economy of the consumer society suffers from this and, above all, the income of the owners of life - so they strive prevent culture and education(which, in addition, prevents them from manipulating the population like a herd devoid of intelligence).

Faced with anti-scientific propaganda in Russia, I decided to look at the pyramid, recently built about twenty kilometers from my house, and rode there on a bicycle through the centuries-old pine forests between the Istra and Moscow rivers. Here I encountered a difficulty: although Peter the Great forbade cutting down forests closer than two hundred miles from Moscow, several of the best square kilometers of pine forest on my way were recently fenced off and mutilated (as the local villagers explained to me, this was done by “a person known [to everyone except me! - V.A.] bandit Pashka"). But even twenty years ago, when I was getting a bucket from this now built-up clearing

raspberries, a whole herd of wild boars walking along the clearing passed me, making a semicircle with a radius of about ten meters.

Similar developments are happening everywhere now. Not far from my house, at one time the population did not allow (even using television protests) the development of a forest by Mongolian and other officials. But since then the situation has changed: the former government-party villages are seizing new square kilometers of ancient forest in front of everyone, and no one is protesting anymore (in medieval England, “fencing” caused uprisings!).

True, in the village of Soloslov, next to me, one member of the village council tried to object to the development of the forest. And then in broad daylight a car arrived with armed bandits who right in the village, at home, and shot. And the development took place as a result.

In another neighboring village, Daryin, an entire field has been rebuilt with mansions. The attitude of the people to these events is clear from the name that they in the village gave to this built-up field (a name, unfortunately, not yet reflected on the maps): “thieves’ field.”

The new motorized residents of this field have turned the highway leading from us to the Perkhushkovo station into their opposite. Buses have almost stopped running along it in recent years. At first, new residents-motorists collected money at the final station for the bus driver so that he would declare the bus “out of order” and passengers would pay private traders. Cars of new residents of the “field” are now rushing along this highway at great speed (and often in someone else’s lane). And I, walking five miles to the station, risk being knocked over, like my many pedestrian predecessors, whose places of death were recently marked on the roadsides with wreaths. Electric trains, however, now also sometimes do not stop at the stations provided for by the schedule.

Previously, the police tried to measure the speed of murderous motorists and prevent them, but after a policeman measuring the speed with a radar was shot by a guard of a passing person, no one dares stop cars anymore. From time to time I find spent cartridges right on the highway, but it is not clear who was shot at. As for the wreaths over the places where pedestrians died, all of them have recently been replaced with notices “Dumping of garbage is prohibited”, hung on the same trees where there were previously wreaths with the names of those dumped.

Along the ancient path from Aksinin to Chesnokov, using the roads laid by Catherine II, I reached the pyramid and saw inside it “shelves for charging bottles and other objects with occult intellectual energy.” Instructions V several square meters in size listed the benefits of a several-hour stay of an object or a patient with hepatitis A or B in the pyramid (I read in the newspaper that someone even sent a multi-kilogram load of stones “charged” by the pyramid to the space station for public money).

But the compilers of this instruction also showed honesty that was unexpected for me: they wrote that there is no point in crowding in line at the shelves inside the pyramid, since<в десятках метров от пирамиды, снаружи, эффект будет таким же". This, I think, is absolutely true.

So, as a true “retrograde,” I consider this whole pyramidal enterprise to be a harmful, anti-scientific advertisement for a store selling “loading objects.”

But obscurantism has always followed scientific achievements, starting from antiquity. Aristotle's student, Alexander Philippovich of Macedon, made a number of "scientific" discoveries (described by his companion, Arian, in Anabasis). For example, he discovered the source of the Nile River: according to him, it is the Indus. The "scientific" evidence was: " These are the only two large rivers that are infested with crocodiles."(and confirmation: “In addition, the banks of both rivers are overgrown with lotuses”).

However, this is not his only discovery: he also “discovered” that the Oxus River (today called the Amu Darya) "flows - from the north, turning near the Urals - into the Meotian swamp of the Pontus Euxine, where it is called Tanais"(“Ta-nais” is the Don, and “Meotian swamp” is the Sea of ​​Azov). The influence of obscurantist ideas on events is not always negligible:

Alexander from Sogdiana (that is, Samarkand) did not go further to the East, to China, as he first wanted, but to the south, to India, fearing water barrier connecting, according to his third theory, the Caspian ("Hyrcanian") Sea with the Indian Ocean(V Bay of Bengal region). For he believed that seas, “by definition,” are ocean bays. This is the kind of “science” we are being led to.

I would like to express the hope that our military will not be so strongly influenced by obscurantists (they even helped me save geometry from the attempts of the “reformers” to expel it from school). But today’s attempts to lower the level of schooling in Russia to American standards are extremely dangerous both for the country and for the world.

In today's France, 20% of army recruits are completely illiterate, do not understand written orders from officers (and can send their missiles with warheads in the wrong direction). May this cup pass from us! Our people are still reading, but the “reformers” want to stop this: “Both Pushkin and Tolstoy are too much!” - they write.

It would be too easy for me, as a mathematician, to describe how they plan to eliminate our traditionally high-quality mathematics education in schools. Instead, I will list several similar obscurantist ideas regarding the teaching of other subjects: economics, law, social studies, literature (subjects, however, they propose to abolish everything in school).

The two-volume project “Standards of General Education” published by the Ministry of Education of Russia contains a large list of topics knowledge of which it is proposed to stop demanding from trainees. It is this list that gives the most vivid idea of ​​the ideas of the “reformers” and what “excessive” knowledge they seek to “protect” the next generations from.

I will refrain from political comments, but here are typical examples of supposedly “excessive” information extracted from the four-hundred-page Standards project:

• Constitution of the USSR;
• fascist “new order” in the occupied territories;
• Trotsky and Trotskyism;
• major political parties;
• Christian democracy;
• inflation;
• profit;
• currency;
• securities;
• multi-party system;
• guarantees of rights and freedoms;
• law enforcement agencies;
• money and other securities;
• forms of state-territorial structure of the Russian Federation;
• Ermak and the annexation of Siberia;
• foreign policy of Russia (XVII, XVIII, XIX and XX centuries);
• Polish question;
• Confucius and Buddha;
• Cicero and Caesar;
• Joan of Arc and Robin Hood;
• Individuals and legal entities;
• the legal status of a person in a democratic state governed by the rule of law;
• separation of powers;
• judicial system;
• autocracy, Orthodoxy and nationality (Uvarov’s theory);
• peoples of Russia;
• Christian and Islamic world;
• Louis XIV;
• Luther;
• Loyola;
• Bismarck;
• The State Duma;
• unemployment;
• sovereignty;
• stock market (exchange);
• state revenues;
• family income.

“Social studies”, “history”, “economics” and “law”, devoid of discussion of all these concepts, are simply formal worship services, useless for students. In France, I recognize this kind of theological chatter on abstract topics by a key set of words: "France is like the eldest daughter of the Catholic Church..." (anything can follow, for example: "... does not need spending on science, since we already had and still have scientists"), as I heard at a meeting of the National Committee of the Republic of France for Science and Research, of which I was appointed member by the Minister of Science, Research and Technology of the Republic of France.

In order not to be one-sided, I will also give a list of “undesirable” (in the same sense of “inadmissibility” of their serious study) authors and works mentioned in this capacity by the shameful “Standard”:

• Glinka;
• Chaikovsky;
• Beethoven;
• Mozart;
• Grieg;
• Raphael;
• Leonardo daVinci;
• Rembrandt;
• Van Togh;
• Omar Khayyam;
• "Tom Sawyer";
• "Oliver Twist";
• Shakespeare's Sonnets;
• "Journey from St. Petersburg to Moscow" by Radishchev;
• "The Steadfast Tin Soldier";
• "Gobsek";
• "Père Goriot"
• "Les Miserables";
• "White Fang";
• "Belkin's Tales";
• "Boris Godunov";
• "Poltava";
• "Dubrovsky";
• "Ruslan and Ludmila";
• "Pig under the oak tree";
• "Evenings on a Farm Near Dikanka";
• "Horse surname";
• "Pantry of the Sun";
• "Meshchera side";
• "Quiet Don";
• "Pygmalion";
• "Hamlet";
• "Faust";
• "A Farewell to Arms";
• "Noble Nest";
• "Lady with a dog";
• "Jumper";
• "A cloud in pants";
• "Black man";
• "Run";
• "Cancer Ward";
• "Vanity Fair";
• "For whom the Bell Tolls";
• "Three Comrades";
• "In the first circle";
• "The Death of Ivan Ilyich."

In other words, they propose to abolish Russian Culture as such. They try to “protect” schoolchildren from the influence of “excessive,” according to “Standards,” cultural centers; that's how they turned out to be here undesirable, according to the compilers of the Standards, for mention by teachers at school:

• Hermitage Museum;
• Russian Museum;
• Tretyakov Gallery;
• Pushkin Museum of Fine Arts in Moscow.

The bell is ringing for us!

It is still difficult to refrain from mentioning at all what exactly it is proposed to make “optional for training” in the exact sciences (in any case, "Standards" recommend "not requiring students to master these sections"):

• structure of atoms;
• concept of long-range action;
• structure of the human eye;
• uncertainty relation of quantum mechanics;
• fundamental interactions;
• starry sky;
• The sun is like one of the stars;
• cellular structure of organisms;
• reflexes;
• genetics;
• origin of life on Earth;
• evolution of the living world;
• the theories of Copernicus, Galileo and Giordano Bruno;
• theories of Mendeleev, Lomonosov, Butlerov;
• the merits of Pasteur and Koch;
• sodium, calcium, carbon and nitrogen (their role in metabolism);
• oil;
• polymers.

In mathematics, the same discrimination was applied to topics in the Standards, which no teacher can do without (and without a full understanding of which schoolchildren will be completely helpless in physics, technology, and a huge number of other applications of science, including both military and humanitarian):

• necessity and sufficiency;
• locus of points;
• sines of angles at 30 o, 45 o, 60 o;
• constructing the angle bisector;
• dividing a segment into equal parts;
• measuring the angle;
• concept of length of a segment;
• the sum of the terms of an arithmetic progression;
• sector area;
• inverse trigonometric functions;
• simple trigonometric inequalities;
• equalities of polynomials and their roots;
• geometry of complex numbers (necessary for alternating current physics, radio engineering, and quantum mechanics);
• construction tasks;
• plane angles of a trihedral angle;
• derivative of a complex function;
• converting simple fractions to decimals.

The only thing that gives me hope is that The existing thousands of well-trained teachers will continue to fulfill their duty and teach all this to new generations of schoolchildren, despite any orders from the Ministry. Common sense is stronger than bureaucratic discipline. We just need to remember to pay our wonderful teachers adequately for their feat.

Representatives of the Duma explained to me that the situation could be greatly improved if care was taken to implement the laws on education that have already been adopted.

The following description of the state of affairs was presented by Deputy I. I. Melnikov in his report at the Mathematical Institute. V. A. Steklov of the Russian Academy of Sciences in Moscow in the fall of 2002.

For example, one of the laws provides for an annual increase in the budget contribution to training by approximately 20% per year. But the minister said that “there is no need to worry about the implementation of this law, since the almost annual increase occurs by more than 40%.” Shortly after this speech by the minister, an increase (by a much smaller percentage) that was practically feasible for the next year (it was 2002) was announced. And if we also take into account inflation, it turns out that a decision was made to reduce the real annual contribution to education.

Another law specifies the percentage of budget expenditures that must be spent on education. In reality, much less is spent (I was not able to find out exactly how many times). But spending on “defense against an internal enemy” increased from a third to half of spending on defense against an external enemy.

It’s natural to stop teaching children fractions, otherwise, God forbid, they’ll understand!

Apparently, it was precisely in anticipation of the reaction of teachers that the compilers of the “Standard” provided a number of names of writers in their list of recommended reading (like the names of Pushkin, Krylov, Lermontov, Chekhov and the like) with an “asterisk” sign, which they deciphered as: “At his discretion, the teacher can introduce students to one or two more works by the same author.”(and not just with the “Monument” they recommended in the case of Pushkin).

The higher level of our traditional mathematical education compared to foreign countries became obvious to me only after I was able to compare this level with foreign ones, having worked many semesters at universities and colleges in Paris and New York, Oxford and Cambridge, Pisa and Bologna, Bonn and Berkeley, Stanford and Boston, Hong Kong and Kyoto, Madrid and Toronto, Marseille and Strasbourg, Utrecht and Rio de Janeiro, Conakry and Stockholm.

“We can’t possibly follow your principle of choosing candidates based on their scientific achievements,” my colleagues on the commission for inviting new professors to one of the best universities in Paris told me. - “After all, in this case we would have to choose only Russians - such is their scientific superiority to us all clear!” (I was talking about selection among the French).

At the risk of being understood only by mathematicians, I will still give examples of responses from the best candidates for a professorship in mathematics at a university in Paris in the spring of 2002 (200 people applied for each position).

The candidate has been teaching linear algebra at various universities for several years, defended his dissertation and published a dozen articles in the best mathematical journals in France.

Selection includes an interview, where the candidate is always asked elementary but important questions (question level "Name the capital of Sweden" if the subject was geography).

So I asked, "What is the signature of the quadratic form xy?"

The candidate demanded the 15 minutes allotted to him to think, after which he said: “On my computer in Toulouse, I have a routine (program) that in an hour or two could find out how many pluses and how many minuses there will be in normal form. The difference of these two numbers and it will be a signature - but you only give 15 minutes, and without a computer, so I can’t answer, this form xy It's too complicated."

For non-specialists, let me explain that if we were talking about zoology, then this answer would be similar to this: “Linnaeus listed all the animals, but whether the birch is a mammal or not, I cannot answer without a book.”

The next candidate turned out to be a specialist in “systems of elliptic partial differential equations” (a decade and a half after defending his dissertation and more than twenty published works).

I asked this one: “What is the Laplacian of the function 1/r in three-dimensional Euclidean space?"

The response (within the usual 15 minutes) was amazing to me; "If r stood in the numerator, and not in the denominator, and the first derivative would have been required, and not the second, then I would have been able to calculate it in half an hour, but otherwise the question is too difficult.”

Let me explain that the question was from the theory of elliptic equations, like the question “Who is the author of Hamlet?” in the English Literature exam. Trying to help, I asked a series of leading questions (similar to the questions about Othello and Ophelia): “Do you know what the law of universal gravitation is? Coulomb’s law? How are they related to the Laplacian? What is the fundamental solution of Laplace’s equation?”

But nothing helped: neither Macbeth nor King Lear were known to the candidate if we were talking about literature.

Finally, the chairman of the examination committee explained to me what was going on: “After all, the candidate studied not just one elliptic equation, but systems of them, and you ask him about Laplace’s equation, whichTotal one thing is clear that he has never encountered it!”

In a literary analogy, this "justification" would correspond to the phrase: “The candidate studied English poets, how could he know Shakespeare, because he is a playwright!”

The third candidate (and dozens of them were interviewed) was working on “holomorphic differential forms,” and I asked him: “What is the Riemann surface of the tangent?” (I was afraid to ask about the arctangent).

Answer: “The Riemannian metric is the quadratic form of coordinate differentials, but what form is associated with the tangent function is not at all clear to me.”

I will explain again with a sample of a similar answer, this time replacing mathematics with history (to which the Mitrofans are more inclined). Here the question would be: "Who is Julius Caesar?" and the answer is: “The rulers of Byzantium were called Caesars, but I don’t know Julia among them.”

Finally, a candidate probabilist appeared, talking interestingly about his dissertation. He proved in it that the statement “A and B are fair together” is false(the statements themselves A And IN are formulated at length, so I will not reproduce them here).

Question: “And yet, what is the situation with the statement A on their own, without IN: is it true or not?

Answer: "After all, I said that the statement 'A and B' is false. This means that A is also false." That is: “Since it is not true that “Petya and Misha got cholera,” then Petya did not get cholera.”

Here my bewilderment was again dispelled by the chairman of the commission: he explained that the candidate was not a probabilist, as I thought, but a statistician (in the biography, called CV, there is not “proba”, but “stat”).

“The probabilists,” our experienced chairman explained to me, “have a normal logic, the same as that of mathematicians, Aristotelian. But for statisticians it is completely different: it’s not for nothing that they say “there are lies, blatant lies and statistics.” All their reasoning is unsubstantiated, all their conclusions are erroneous. But they are always very necessary and useful, these conclusions. We definitely need to accept this statistician!”

At Moscow University, such an ignoramus would not be able to complete the third year of the Faculty of Mechanics and Mathematics. Riemann surfaces were considered the pinnacle of mathematics by the founder of the Moscow Mathematical Society, N. Bugaev (father of Andrei Bely). He, however, believed that in contemporary mathematics at the end of the 19th century, objects began to appear that did not fit into the mainstream of this old theory - non-holomorphic functions of real variables, which, in his opinion, are the mathematical embodiment of the idea of ​​free will to the same extent that Riemann surfaces and holomorphic functions embody the idea of ​​fatalism and predetermination.

As a result of these reflections, Bugaev sent young Muscovites to Paris to learn there the new “mathematics of free will” (from Borel and Lebesgue). This program was brilliantly carried out by N. N. Luzin, who upon his return to Moscow created a brilliant school, including all the main Moscow mathematicians of many decades: Kolmogorov and Petrovsky, Aleksandrov and Pontryagin, Menshov and Keldysh, Novikov and Lavrentiev, Gelfand and Lyusternik.

By the way, Kolmogorov recommended to me the Parisiana Hotel (on Tournefort Street, not far from the Pantheon) that Luzin subsequently chose for himself in the Latin Quarter of Paris. During the First European Mathematical Congress in Paris (1992) I stayed in this inexpensive hotel (with amenities at the level of the 19th century, without a telephone, and so on). And the elderly owner of this hotel, having learned that I had come from Moscow, immediately asked me: “ How is my old guest, Luzin, doing there? It's a pity that he hasn't visited us for a long time."

A couple of years later, the hotel was closed for renovation (the owner probably died) and they began to rebuild it in an American way, so now you can no longer see this 19th-century island in Paris.

Returning to the choice of professors in 2002, I note that all the ignoramuses listed above received (from everyone except me) the best grades. On the contrary, the only, in my opinion, worthy candidate was almost unanimously rejected. He discovered (with the help of “Gröbner bases” and computer algebra) several dozen new completely integrable systems of Hamiltonian equations of mathematical physics (at the same time, but not including in the list of new ones, the famous Korteweg-de Vries, Sayn-Gordon, and the like equations).

As a future project, the candidate also proposed a new computer method for modeling diabetes treatment. To my question about the evaluation of his method by doctors, he answered quite reasonably: “The method is now being tested in such and such centers and hospitals, and in six months they will give their conclusions, comparing the results with other methods and with control groups of patients, but for now this examination is not has been carried out, and there are only preliminary assessments, although they are good.”

They rejected it with this explanation: “On every page of his dissertation, either Lie groups or Lie algebras are mentioned, but no one here understands this, so he will not fit into our team at all.” True, it would have been possible to reject both me and all my students, but some colleagues think that the reason for the rejection was different: unlike all the previous candidates, this one was not French (he was a student of a famous American professor from Minnesota).

The whole picture described leads to sad thoughts about the future of French science, in particular mathematics. Although the “French National Committee for Science” was inclined to not finance new scientific research at all, but to spend money (provided by Parliament for the development of science) on the purchase of ready-made American recipes, I sharply opposed this suicidal policy and still achieved at least some subsidizing new research. However, a difficulty was caused by the division of money. Medicine, nuclear energy, polymer chemistry, virology, genetics, ecology, environmental protection, radioactive waste disposal and much more were consistently voted unworthy of subsidies by voting (during a five-hour meeting). In the end, they chose three “sciences” that allegedly deserved funding for their new research. These three “sciences” are: 1) AIDS; 2) psychoanalysis; 3) a complex branch of pharmaceutical chemistry, the scientific name of which I am unable to reproduce, but which deals the development of psychotropic drugs, similar to lacrimogenic gas, turning the rebellious crowd into an obedient herd.

So now France is saved!

Of all Luzin’s students, the most remarkable contribution to science was made, in my opinion, by Andrei Nikolaevich Kolmogorov. Having grown up in a village with his grandfather near Yaroslavl, Andrei Nikolaevich proudly referred to Gogol’s words “an efficient Roslavl man.”

He had no intention of becoming a mathematician, even having already entered Moscow University, where he immediately began studying history (in Professor Bakhrushin’s seminar) and, before he was even twenty years old, wrote his first scientific work.

This work was devoted to the study of land economic relations in medieval Novgorod. Tax documents have been preserved here, and the analysis of a huge number of these documents using statistical methods led the young historian to unexpected conclusions, which he spoke about at the Bakhrushin meeting.

The report was very successful, and the speaker was much praised. But he insisted on another approval: he wanted his conclusions to be recognized as correct.

In the end, Bakhrushin told him: “This report definitely needs to be published; it is very interesting. But as for the conclusions, For us historians, to recognize any conclusion we always need not one piece of evidence, but at least five!"

The next day, Kolmogorov changed history to mathematics, where proof alone is enough. He did not publish the report, and this text remained in his archive until, after the death of Andrei Nikolaevich, it was shown to modern historians, who recognized it not only as very new and interesting, but also quite conclusive. Now this Kolmogorov report has been published, and is considered by the community of historians as an outstanding contribution to their science.

Having become a professional mathematician, Kolmogorov remained, unlike most of them, first of all a natural scientist and thinker, and not at all a multiplier of multidigit numbers (which mainly appears when analyzing the activities of mathematicians to people unfamiliar with mathematics, including even L.D. Landau, who valued mathematics is precisely the continuation of counting skill: five five - twenty-five, six six - thirty-six, seven seven - forty-seven, as I read in a parody of Landau compiled by his Physics and Technology students; however, in Landau’s letters to me, who was then a student, mathematics no more logical than in this parody).

Mayakovsky wrote: “After all, he can extract the square root every second” (about the professor who “doesn’t get bored that the students outside the window are actively going to the gymnasium”).

But he perfectly described what a mathematical discovery is, saying that " Whoever discovered that two and two equals four was a great mathematician, even if he discovered it by counting cigarette butts. And anyone who today calculates much larger objects, such as locomotives, using the same formula, is not a mathematician at all!”

Kolmogorov, unlike many others, was never intimidated by applied, “locomotive” mathematics, and he joyfully applied mathematical considerations to a variety of areas of human activity: from hydrodynamics to artillery, from celestial mechanics to poetry, from the miniaturization of computers to the theory of Brownian motion, from the divergence of Fourier series to the theory of information transmission and to intuitionistic logic. He laughed at the fact that the French write “Celestial Mechanics” with a capital letter, and “applied” with a small letter.

When I first arrived in Paris in 1965, I was warmly greeted by the elderly Professor Fréchet, with the following words: “After all, you are a student of Kolmogorov, that young man who constructed an example of a Fourier series that diverges almost everywhere!”

The work mentioned here by Kolmogorov was completed by him at the age of nineteen, solved a classical problem and immediately promoted this student to the rank of first-class mathematicians of world significance. Forty years later, this achievement still remained more significant for Frechet than all subsequent and much more important fundamental works of Kolmogorov, which revolutionized the theory of probability, the theory of functions, hydrodynamics, celestial mechanics, the theory of approximations, and the theory of algorithmic complexity, and the theory of cohomology in topology, and the theory of control of dynamical systems (where Kolmogorov’s inequalities between derivatives of different orders remain one of the highest achievements today, although control theory specialists rarely understand this).

But Kolmogorov himself was always somewhat skeptical about his favorite mathematics, perceiving it as a small part of natural science and easily abandoning those logical restrictions that the shackles of the axiomatic-deductive method impose on true mathematicians.

“It would be in vain,” he told me, “to look for mathematical content in my works on turbulence. I speak here as a physicist and do not at all worry about mathematical proofs or derivations of my conclusions from initial premises, such as the Navier-Stokes equations. Even if these conclusions have not been proven, they are true and open, and this is much more important than proving them!”

Many of Kolmogorov’s discoveries were not only not proven (neither by himself nor by his followers), but were not even published. But nevertheless, they have already had and continue to have a decisive influence on a number of departments of science (and not only mathematics).

I will give just one famous example (from the theory of turbulence).

A mathematical model of hydrodynamics is a dynamic system in the space of fluid velocity fields, which describes the evolution of the initial velocity field of fluid particles under the influence of their interaction: pressure and viscosity (as well as under the possible influence of external forces, for example, weight force in the case of a river or water pressure in a water pipe).

Under the influence of this evolution, a dynamic system can come to equilibrium (stationary) state, when the flow velocity at each point of the flow region does not change with time(although everything flows, and each particle moves and changes its speed over time).

Such stationary flows (for example, laminar flows in terms of classical hydrodynamics) are attracting points of a dynamic system. They are therefore called (point) attractors.

Other sets that attract neighbors are also possible, for example, closed curves depicting currents that periodically change over time in the functional space of velocity fields. Such a curve is an attractor when the neighboring initial conditions, depicted by “perturbed” points of the functional space of velocity fields close to the indicated closed curve, begin, although not periodically changing with time, a flow that approaches it (namely, the perturbed flow tends to the one described earlier periodically over time).

Poincaré, who first discovered this phenomenon, called such closed attractor curves "stable limit cycles". From a physical point of view, they can be called periodic steady flow regimes: the disturbance gradually fades during the transition process caused by the disturbance of the initial condition, and after some time the difference between the movement and the unperturbed periodic one becomes barely noticeable.

After Poincaré, such limit cycles were extensively studied by A. A. Andronov, who based the study and calculation of radio wave generators, that is, radio transmitters, on this mathematical model.

It is instructive that Poincaré’s discovery and development by Andronov theory of the birth of limit cycles from unstable equilibrium positions Today it is usually called (even in Russia) the Hopf bifurcation. E. Hopf published part of this theory a couple of decades after Andronov’s publication and more than half a century after Poincaré, but unlike them, he lived in America, so the well-known eponymic principle worked: if any object bears someone else's name, then this is not the name of the discoverer(for example, America is not named after Columbus).

The English physicist M. Berry called this eponymous principle “Arnold’s principle,” adding a second one to it. Berry's principle: Arnold's principle applies to oneself(that is, it was known before).

I completely agree with Berry on this. I told him the eponymous principle in response to a preprint about the “Berry phase”, examples of which, in no way inferior to the general theory, were published decades before Berry by S. M. Rytov (under the name “inertia of polarization direction”) and A. Yu .Ishlinsky (under the title “the departure of the submarine’s gyroscope due to a discrepancy between the path of returning to the base and the path of leaving it”),

Let us return, however, to attractors. An attractor, or attracting set, is a steady state of motion, which, however, does not have to be periodic. Mathematicians have also studied much more complex movements, which can also attract perturbed neighboring movements, but which themselves can be extremely unstable: small causes sometimes cause big consequences, Poincaré said. The state, or “phase,” of such a limiting regime (that is, a point on the surface of the attractor) can move along the surface of the attractor in a bizarre “chaotic” manner, and a slight deviation of the starting point on the attractor can greatly change the course of movement without changing the limiting regime at all. Averages over long times from all possible observable quantities will be close in the original and in the perturbed motion, but the details at a fixed moment in time will, as a rule, be completely different.

In meteorological terms, the “limit regime” (attractor) can be likened to climate, and the phase - weather. A small change in initial conditions can have a big impact on tomorrow's weather (and even more on the weather a week and a month from now). But such a change will not make the tundra a tropical forest: just a thunderstorm may break out on Friday instead of Tuesday, which may not change the average for the year (or even for the month).

In hydrodynamics, the degree of attenuation of initial disturbances is usually characterized by viscosity (so to speak, the mutual friction of liquid particles as they move one relative to the other), or the inverse viscosity, a value called the “Reynolds number”. Large values ​​of the Reynolds number correspond to weak attenuation of disturbances, and large values ​​of viscosity (that is, small Reynolds numbers) - on the contrary, regularize the flow, preventing disturbances and their development. In economics, the role of “viscosity” is often played by bribes and corruption 1 .

1 Multi-stage production management is unstable if the number of stages (worker, foreman, shop manager, plant director, chief executive officer, etc.) is more than two, but can be implemented in a sustainable manner if at least some of the managers are rewarded not only from above (for following orders ), but also from below (for the benefit of the cause, for decisions that contribute to production). Corruption is used for the latter encouragement. For details, see the article: V. I. Arnold. Mathematics and mathematics education in the modern world. In the book: Mathematics in education and upbringing. - M.: FAZIS, 2000, p. 195-205.

Due to high viscosity, at low Reynolds numbers, a stable stationary (laminar) flow is usually established, represented in the space of velocity fields by a point attractor.

The main question is how the flow pattern will change with increasing Reynolds number. In water supply, this corresponds, for example, to an increase in water pressure, which makes a smooth (laminar) stream from a tap unstable, but mathematically, to increase the Reynolds number, it is more convenient to reduce the particle friction coefficient expressing viscosity (which in an experiment would require a technically complex fluid replacement). However, sometimes to change the Reynolds number it is enough to change the temperature in the laboratory. I saw such an installation in Novosibirsk at the Institute of Precision Measurements, where the Reynolds number changed (in the fourth digit) when I brought my hand closer to the cylinder where the flow occurred (precisely due to a change in temperature), and on the computer screen processing the experiment, this change in the Reynolds number immediately indicated by electronic automation.

Thinking about these phenomena of transition from a laminar (stable stationary) flow to a stormy turbulent one, Kolmogorov long ago expressed a number of hypotheses (which to this day remain unproven). I think that these hypotheses date back to the time (1943) of his dispute with Landau about the nature of turbulence. In any case, he clearly formulated them at his seminar (on hydrodynamics and the theory of dynamical systems) at Moscow University in 1959, where they were even part of the announcement about the seminar that he posted at that time. But I don’t know of any formal publication of these hypotheses by Kolmogorov, and in the West they are usually attributed to their epigones of Kolmogorov, who learned about them and published them dozens of years later.

The essence of these Kolmogorov hypotheses is that as the Reynolds number increases, the attractor corresponding to the steady flow regime becomes more and more complex, namely, that its dimension increases.

First it is a point (zero-dimensional attractor), then a circle (Poincaré limit cycle, one-dimensional attractor). And Kolmogorov’s hypothesis about attractors in hydrodynamics consists of two statements: with increasing Reynolds number 1) attractors of ever larger dimensions appear; 2) all low-dimensional attractors disappear.

From 1 and 2 together it follows that when the Reynolds number is sufficiently large, the steady state necessarily has many degrees of freedom, so that to describe its phase (point on the attractor) it is necessary to set many parameters, which then, when moving along the attractor, will change in a whimsical and non-periodic “chaotic” way, and a small change in the starting point on the attractor leads, as a rule, to a large (after a long time) change in the “weather” (the current point on the attractor), although it does not change the attractor itself (that is, it will not cause a change in the “climate”).

Statement 1 in itself is not sufficient here, since different attractors can coexist, including attractors of different dimensions in one system (which, thus, can perform a calm “laminar” movement under some initial conditions and a stormy “turbulent” one under others, depending on its initial state).

Experimental observation of such effects "prolonged loss of stability" surprised physicists for a long time, but Kolmogorov added that even if the low-dimensional attractor does not disappear, it may not change the observed turbulence in the case when the size of its zone of attraction decreases significantly with increasing Reynolds number. In this case, the laminar regime, although possible in principle (and even stable), is practically not observed due to the extreme smallness of its area of ​​attraction: Already small, but always present in the experiment, disturbances can take the system out of the zone of attraction of this attractor into the zone of attraction of another, already turbulent, steady state, which will be observed.

This discussion may also explain this strange observation: Some famous hydrodynamic experiments of the 19th century could not be repeated in the second half of the 20th century, although attempts were made to use the same equipment in the same laboratory. It turned out, however, that the old experiment (with its prolongation of the loss of stability) can be repeated if it is done not in the old laboratory, but in a deep underground mine.

The fact is that modern street traffic has greatly increased the magnitude of “imperceptible” disturbances, which began to have an effect (due to the smallness of the zone of attraction of the remaining “laminar” attractor).

Numerous attempts by many mathematicians to confirm Kolmogorov's hypotheses 1 and 2 (or at least the first) with evidence have so far only led to estimates of the dimensions of attractors in terms of Reynolds numbers from above: this dimension cannot become too large as long as viscosity prevents it.

The dimensionality is estimated in these works by a power function of the Reynolds number (that is, a negative degree of viscosity), and the exponent depends on the dimension of the space where the flow occurs (in a three-dimensional flow, turbulence is stronger than in plane problems).

As for the most interesting part of the problem, that is, estimating the dimension from below (at least for some attractors, as in Hypothesis 1, or even for all, as in Hypothesis 2, about which Kolmogorov expressed more doubts), here the mathematicians were not able to height, because, according to his habit, replaced the real natural science problem with their formal axiomatic abstract formulation with its precise but treacherous definitions.

The fact is that the axiomatic concept of an attractor was formulated by mathematicians with the loss of some properties of the physical limiting mode of motion, which (not strictly defined) concept of mathematics they tried to axiomatize by introducing the term “attractor”.

Let us consider, for example, an attractor that is a circle (to which all nearby dynamics trajectories spirally approach).

On this very circle attracting neighbors, let the dynamics be arranged as follows: two opposite points (at the ends of the same diameter) are motionless, but one of them is an attractor (attracts neighbors), and the other is a repulsor (repels them).

For example, one can imagine a vertically standing circle, the dynamics on which shift down any point along the circle, except for the remaining fixed poles:

an attractor at the bottom and a repulsor at the top.

In this case, despite the existence of a one-dimensional attractor-circle in the system, the physically steady state will only be a stable stationary position(the lower attractor in the above “vertical” model).

Under an arbitrary small perturbation, the motion will first evolve towards the attractor-circle. But then the internal dynamics on this attractor will play a role, and state of the system, will in the end, approach a “laminar” zero-dimensional attractor; a one-dimensional attractor, although it exists mathematically, is not suitable for the role of a “steady-state regime”.

One way to avoid such troubles is consider only minimal attractors as attractors, that is, attractors that do not contain smaller attractors. Kolmogorov's hypotheses refer precisely to such attractors, if we want to give them a precise formulation.

But then nothing has been proven about estimates of dimensions from below, despite numerous publications named as such.

The danger of the deductive-axiomatic approach to mathematics Many thinkers before Kolmogorov clearly understood this. The first American mathematician J. Sylvester wrote that In no case should mathematical ideas be petrified, since they lose their power and application when trying to axiomatize the desired properties. He said that ideas should be perceived as water in a river: we never enter exactly the same water, although the ford is the same. Likewise, an idea can give rise to many different and non-equivalent axiomatics, each of which does not reflect the idea entirely.

Sylvester came to all these conclusions by thinking through, in his words, “the strange intellectual phenomenon that the proof of a more general statement often turns out to be simpler than the proof of the particular cases it contains." As an example, he compared the geometry of vector space with (not yet established at that time) functional analysis.

This idea of ​​Sylvester was used a lot in the future. For example, it is precisely this that explains Bourbaki’s desire to make all concepts as general as possible. They even use in In France, the word “more” in the sense that in other countries (which they contemptuously call “Anglo-Saxon”) is expressed by the words “greater than or equal to,” since in France they considered the more general concept “>=" to be primary, and the more specific “>” - " unimportant" example. Because of this, they teach students that zero is a positive number (as well as negative, non-positive, non-negative and natural), which is not recognized elsewhere.

But they apparently did not get to Sylvester’s conclusion about the inadmissibility of fossilization of theories (at least in Paris, in the library of the Ecole Normale Superieure, these pages of his Collected Works were uncut when I recently got to them).

I am unable to convince mathematical “specialists” to correctly interpret the hypotheses about the growth of the dimensions of attractors, since they, like lawyers, object to me with formal references to the existing dogmatic codes of laws containing an “exact formal definition” of attractors of the ignorant.

Kolmogorov, on the contrary, never cared about the letter of someone’s definition, but thought about the essence of the matter 2.

2 Having solved Birkhoff's problem on the stability of fixed points of non-resonant systems in 1960, I published a solution to this very problem in 1961. A year later, Yu. Moser generalized my result, proving stability at resonances of order greater than four. Only then did I notice that my proof established this more general fact, but, being hypnotized by the formulation of Birkhoff's definition of non-resonance, I did not write that I had proved more than Birkhoff claimed.

One day he explained to me that he came up with his topological cohomology theory not at all combinatorially or algebraically, as it looks like, but by thinking about fluid flows in hydrodynamics, then about magnetic fields: he wanted to model this physics in the combinatorial situation of an abstract complex and did so.

In those years, I naively tried to explain to Kolmogorov what happened in topology during those decades during which he drew all his knowledge about it only from P. S. Aleksandrov. Because of this isolation, Kolmogorov knew nothing about homotopy topology; he convinced me that "spectral sequences were contained in the Kazan work of Pavel Sergeevich 1942 of the year", and attempts to explain to him what the exact sequence was were no more successful than my naive attempts to put him on water skis or put him on a bicycle, this great traveler and skier.

What was surprising to me, however, was the high assessment of Kolmogorov’s words about cohomology given by a strict expert, Vladimir Abramovich Rokhlin. He explained to me, not at all critically, that these words of Kolmogorov contained, firstly, a deeply correct assessment of the relationship between his two achievements (especially difficult in the case when, as here, both achievements are remarkable), and secondly, a shrewd foresight of a huge meanings of cohomology operations.

Of all the achievements of modern topology, Kolmogorov valued Milnor's spheres the most, which the latter spoke about in 1961 at the All-Union Mathematical Congress in Leningrad. Kolmogorov even persuaded me (then a beginning graduate student) to include these spheres in my graduate plan, which forced me to start studying differential topology from Rokhlin, Fuchs and Novikov (as a result of which I was even soon an opponent of the latter’s Ph.D. thesis on differentiable structures on products of spheres).

Kolmogorov’s idea was to use Milnor spheres to prove that a function of several variables cannot be represented by superpositions in Hilbert’s 13th problem (probably for algebraic functions), but I don’t know any of his publications on this topic or the formulation of his hypotheses.

Another little-known circle of Kolmogorov’s ideas relates to optimal control of dynamic systems.

The simplest task of this circle is to maximize at some point the first derivative of a function defined on an interval or on a circle, knowing the upper bounds for the moduli of the function itself and its second derivative. The second derivative prevents the first from being quickly extinguished, and if the first is too large, the function outgrows the given limitation.

Probably, Hadamard was the first to publish the solution to this problem on the second derivative, and subsequently Littlewood rediscovered it while working on artillery trajectories. Kolmogorov, it seems, did not know the publications of either one or the other, and decided the problem of estimating from above any intermediate derivative through the maximum values ​​of the moduli of the differentiable function and its high (fixed) order derivative.

Kolmogorov's wonderful idea was to explicitly indicate extremal functions, such as Chebyshev polynomials (on which the inequality being proved becomes an equality). And in order for the function to be extreme, he naturally guessed that the value of the highest derivative must always be chosen to be the maximum in absolute value, changing only its sign.

This led him to a remarkable series of special features. The zero function of this series is the signum of the sine of the argument (everywhere having a maximum modulus). The next, first, function is an antiderivative of zero (that is, already continuous "saw", the derivative of which has a maximum modulus everywhere). Further functions are obtained each from the previous one by the same integration (increasing the number of derivatives by one). You just need to choose the integration constant so that the integral of the resulting antiderivative function over the period is equal to zero each time (then all the constructed functions will be periodic).

Explicit formulas for the resulting piecewise polynomial functions are quite complex (the integrations are introduced by rational constants associated even with Bernoulli numbers).

The values ​​of the constructed functions and their derivatives are given by constants in Kolmogorov’s power estimates (estimating the modulus of the intermediate derivative from above through the product of rational powers of the maxima of the modulus of the function and the highest derivative). The indicated rational exponents are easy to guess from the consideration of similarity, going back to the laws of similarity of Leonardo da Vinci and to Kolmogorov’s theory of turbulence, that the combination should turn out to be dimensionless, since it is clear (at least from Leibniz’s notation) how derivatives of different orders behave when units change Argument and function measurements. For example, for the Hadamard problem, both rational exponents are equal to half, so the square of the first derivative is estimated from above by the product of the maxima of the modulus of the function itself and its second derivative (with a coefficient depending on the length of the segment or circle where the function is considered).

It is easier to prove all these estimates than to come up with the extremal functions described above (and delivering, among other things, Gauss’s theorem: the probability of irreducibility of the fraction p/q with integer numerator and denominator is equal to 6/p 2, that is, about 2/3).

In terms of today's management theory, The strategy chosen by Kolmogorov is called “big bang”: the control parameter must always be chosen to have an extreme value, any moderation only harms.

As for Hamilton’s differential equation for changing with time the choice of this extreme value from many possible ones, Kolmogorov knew it very well, calling it, however, Huygens’ principle (which is really equivalent to this equation and from which Hamilton obtained his equation by moving from envelopes to differentials) . Kolmogorov even pointed out to me, who was then a student, that the best description of this geometry of Huygens' principle is contained in Whittaker's textbook of mechanics, where I learned it, and that in a more intricate algebraic form it is in the theory of “Berurung Transformation” by Sophus Lie (instead of which I learned the theory of canonical transformations from Birkhoff’s “Dynamical Systems” and which today is called contact geometry).

Tracing the origins of modern mathematics in classical works is usually not easy, especially due to the changing terminology that is accepted as a new science. For example, almost no one notices that the so-called theory of Poisson manifolds was already developed by Jacobi. The fact is that Jacobi followed the path of algebraic varieties - varieties, and not smooth varieties - manifolds. Namely, he was interested in the variety of orbits of a Hamiltonian dynamical system. As a topological or smooth object, it has peculiarities and even more unpleasant pathologies (“non-Hausdorffity” and the like) with the entanglement of orbits (phase curves of a complex dynamic system).

But the algebra of functions on this (possibly bad) “manifold” is well defined: it is simply the algebra of first integrals of the original system. By Poisson's theorem, the Poisson bracket of the first two integrals is again the first integral. Therefore, in the algebra of integrals, in addition to multiplication, there is another bilinear operation - the Poisson bracket.

The interaction of these operations (multiplication and parentheses) in the space of functions on a given smooth manifold is what makes it a Poisson manifold. I skip the formal details of its definition (they are not complicated), especially since they are not all fulfilled in the example that interested Jacobi, where the Poisson manifold is neither smooth nor Hausdorff.

Thus, Jacobi's theory contains the study of more general varieties with singularities than modern Poisson smooth varieties, and moreover, this theory was constructed by him in the style of algebraic geometry of rings and ideals, rather than differential geometry of submanifolds.

Following Sylvester's advice, specialists in Poisson manifolds should, not limiting themselves to their axiomatics, return to a more general and more interesting case, already considered by Jacobi. But Sylvester did not do this (being late, as he said, for the ship leaving for Baltimore), and mathematicians of more recent times are completely subordinate to the dictates of the axiomatists.

Kolmogorov himself, having solved the problem of upper estimates for intermediate derivatives, understood that he could solve many other optimization problems using the same techniques of Huygens and Hamilton, but he did not do this, especially when Pontryagin, whom he always tried to help, published his “principle maximum", which is essentially a special case of the same Huygens principle of forgotten contact geometry, applied, however, to a not very general problem.

Kolmogorov correctly thought that Pontryagin did not understand either these connections with Huygens' principle, or the connection of his theory with Kolmogorov's much earlier work on estimates of derivatives. And therefore, not wanting to disturb Pontryagin, he did not write anywhere about this connection, which was well known to him.

But now, I think, this can already be said, in the hope that someone will be able to use these connections to discover new results.

It is instructive that Kolmogorov's inequalities between derivatives served as the basis for the remarkable achievements of Yu. Moser in the so-called KAM theory (Kolmogorov, Arnold, Moser), which allowed him to transfer Kolmogorov's 1954 results on invariant tori of analytic Hamiltonian systems to only three hundred and thirty-three times differentiable systems . This was the case in 1962, with Moser's invention of his remarkable combination of Nash smoothing and Kolmogorov's accelerated convergence method.

Now the number of derivatives needed for the proof has been significantly reduced (primarily by J. Mather), so that the three hundred and thirty-three derivatives needed in the two-dimensional problem of ring mappings have been reduced to three (while counterexamples have been found for two derivatives).

It is interesting that after the appearance of Moser’s work, American “mathematicians” tried to publish their “generalization of Moser’s theorem to analytical systems” (which generalization was simply Kolmogorov’s theorem published ten years earlier, which Moser managed to generalize). Moser, however, decisively put an end to these attempts to attribute to others Kolmogorov's classical result (correctly noting, however, that Kolmogorov never published a detailed presentation of his proof).

It seemed to me then that the proof published by Kolmogorov in a note in DAN was quite clear (although he wrote more for Poincaré than for Hilbert), in contrast to Moser’s proof, where I did not understand one place. I even revised it in my 1963 review of Moser's remarkable theory. Moser subsequently explained to me what he meant in this unclear place, but I am still not sure whether these explanations were properly published (in my revision I have to choose s < e/3, а не e/2, как указывалось в непонятном месте, вызвавшем затруднения не только у меня, но и у других читателей и допускающем неправильное истолкование неясно сказанного).

It is also instructive that "Kolmogorov's accelerated convergence method"(correctly attributed by Kolmogorov to Newton) was used for a similar purpose in solving a nonlinear equation by A. Cartan ten years before Kolmogorov, in proving what is now called the theorem A beam theory. Kolmogorov knew nothing about this, but Cartan pointed this out to me in 1965, and was convinced that Kolmogorov could have referred to Cartan (although his situation in the theory of beams was somewhat simpler, since when solving a linearized problem there was no fundamental in celestial mechanics is the difficulty of resonances and small denominators, present in Kolmogorov and Poincaré). Kolmogorov’s not mathematical, but broader approach to his research was clearly manifested in two of his works with co-authors: in an article with M.A. Leontovich on the area of ​​the neighborhood of a Brownian trajectory and in the article “KPP” (Kolmogorov, Petrovsky and Piskunov) on the speed of propagation of nonlinear waves

In both cases, the work contains both a clear physical formulation of a natural science problem and a complex and non-trivial mathematical technique for solving it.

And in both cases Kolmogorov performed not the mathematical, but the physical part of the work, associated, first of all, with the formulation of the problem and with the derivation of the necessary equations, while their research and proof of the corresponding theorems belong to the co-authors.

In the case of Brownian asymptotics, this difficult mathematical technique involves the study of integrals along deformable paths on Riemann surfaces, taking into account the complex deformations of the integration contours required for this when changing parameters, that is, what is today called either “Picard-Lefschetz theory” or “connectivity theory Gauss-Manin".