Kolmogorov's mathematical View and performance

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Ito

When I learned of the great Soviet mathematician, 84-year-old Andreyii Nikolaevich Kolmogorov, who died on October 20, 1987, I felt as sad and lonely as the pillar that had been lost. When I was a student (1937) after reading his famous book "The Basic concept of probability theory", he determined to delve into probability theory, and lasted for 50 years. For me, Kolmogorov is the basis of my mathematics.

I have only been with Professor Kolmogorov 3 times. The first time was the 1962 International Conference of Mathematicians (Stockholm), I strolled in the hall before the opening ceremony. When you hear "ito? I was pleasantly surprised when kolmogorov."'s gracious greeting. He asked in German, "How old are you?" "I replied:" "seiben und vierzig." he asked again: "dreibig?" (more than 30?) Probably the Japanese look young, I may be 10 years younger. Another 23rd, Professor H. Cramer (a Swedish university president (Chancelor), an expert in probability theory, analytic number theory) held a dinner party at home to entertain about 10 scholars of probability in attendance at the meeting. Kolmogorov, J.L Doob and I are in it.

The second was in 1978, after attending the International Congress of Mathematicians (Helsinki) and attending the International Symposium on Probability Statistics (Vilnius, Lihtuania, USSR), on his way back to Moscow, Kolmogorov hosted Varadhan (NYU ), Prokhorov (Soviet Academy of Sciences) and I had lunch at an elegant restaurant next to the Kremlin. At that time, I heard that Kolomogorov was very enthusiastic about the math education in high school, enrolled some excellent students and taught in person. I inquired about the content, for example, to show the students a simple vector field (velocity field) diagram, and ask them to draw the integral curve (track), as well as students to consider the specific branching process problems and so on, in order to cultivate students ' mathematical intuition ability.

The third was held at the Tbilisi (Georgia, USSR, 1983) symposium on the Probability and statistics of the day and the Soviet Union. At that time, his health was not good, he still gave lectures, and at the party to create a lively atmosphere. Obviously the younger generation is very respectful of him.

Kolmogorov in almost all fields of mathematics, has proposed the original idea, has introduced the new method, his performance is very splendid. However, the impression I made when I met him was the image of a slovenly, well-natured gentleman, perhaps the image of a great mathematician.

Kolmogorov's thesis I thought I had read it all, and in writing, I made a direct or indirect investigation of his entire research. The breadth and depth of its research has to be amazed. Due to the limitations of time and space, I only talk to the reader about some of my own feelings that are not comprehensive.

Guizechemmine (Kyoto University), Chitang (Osaka University), two professors and mathematical Research Library of Kyoto University help me to find the information, here I express my heartfelt thanks.

Kolmogorov Resume

According to B. V. Gnedenko's speech at the 70 birthday of Kolmogorov, Kolmogorov was born in 1903 in the Russian village (now the city) Tambov. Father is a agronomy, mother in the birth of Kolmogorov soon after she died, he was aunt and so raised. 1920 (17 years old) before entering the University of Moscow, he was a train attendant, in his spare time wrote a paper on Newton's Law of mechanics, the manuscript of the paper has not been preserved, but we can imagine how premature he is genius. At that time, the Russian Revolution (1917) had erupted, and I would like to know the circumstances in which he was at that time and regretted that no information was available.

He entered the University of Moscow in 1920, initially interested in the history of Russia, and investigated the registration of the property of the 15~16 century. Later participated in V.V. Stepanov's Fourier series (trigonometric series) discussion class, and in 1922 (19 years old) wrote about the Fourier series, analytic set of famous papers, shook the academic community. As a result, many important studies have been published in succession. 1925 graduated from Moscow University, 1931 as a university professor, 1933 as director of the Institute of Mathematics, and in 1937 as a member of the Soviet Academy of Sciences. Until the death of 1987, many significant contributions have been made to the research education of mathematics.

Kolmogorov's view of mathematics

The best information about Kolmogorov's mathematical view is probably the "math" part he wrote in the Dictionary of the Soviet Encyclopedia. Already out of the English version, I read the English version, and the original (Russian) comparison, the English version slightly smaller slightly, in this article, he first elaborated his mathematical view, and then described the history of mathematics since ancient times, and from his mathematical view, detailed description of the history of the various stages, it can be said to be for mathematicians, The history of mathematics that scientists have written. I read the entire passage in an interesting way. To illustrate Kolmogorov's mathematical view, we should not only look at the beginning of this article, but also take into account the majority of the history of mathematics, but due to the limitations of the length of time to ask, I will only briefly introduce the beginning of the article as follows.

According to Kolmogorov's point of view, mathematics is the science of quantitative relationships and spatial forms in the real world.

(1) So the study of mathematics is like being produced in reality. However, as a study of mathematics, we must leave the material of reality (the abstraction of mathematics).

(2) However, the abstraction of mathematics does not mean that it is completely divorced from the material of reality. The quantity relationship and the type of space form that need to be studied by mathematics are constantly increasing with the request of science and technology. So the mathematical content defined above is constantly being enriched.

Mathematics and various sciences: the application of mathematics is diverse, from the principle, the application of mathematical methods is boundless, that is, all types of material movement can be studied mathematically. But the role and significance of mathematical methods are different in different situations. It is impossible to use a single pattern to encompass all aspects of the phenomenon. The process of knowing the specific thing (phenomenon) always has the following two intertwined tendencies.

(1) Only the form of the research object (phenomenon) is separated, and the form is logically analyzed.

(2) to ascertain the "aspect of the phenomenon" which is inconsistent with the established form, and to have more plasticity and to include the new form of "phenomenon" more completely.

If in the process of research must always examine the nature of the phenomenon of the new side, so the difficulties in the study is mainly reflected in the above (2) words. In the study of such phenomena (such as biology, economics, humanities, etc.), mathematical methods are not the main. At such times, the dialectical analysis of all aspects of the phenomenon becomes ambiguous due to the mathematical form.

On the contrary, if the object of study (phenomenon) can be grasped in a relatively simple and stable form, and in the scope of this form arises a difficult and complex question which needs to be studied mathematically (especially when new notation and calculation methods need to be created), The study of this phenomenon (e.g. physics) is in the sphere of domination of mathematical methods.

Having done these general discourses, the first detail shows that the planetary movement is entirely within the control circle of the mathematical method, where the mathematical form is Newton's ordinary differential equation for the finite particle system.

From mechanics to physics, the function of mathematical method almost does not decrease, but the difficulty of application is increased obviously. In physics, there is almost no need to use advanced mathematical techniques (such as partial differential equation theory, functional analysis) of the field. However, the difficulties in the study are often not in the derivation of mathematical theory, but in the "choice of assumptions for the use of mathematics" and "the interpretation of results obtained by mathematical means".

Mathematical methods have the ability to transfer such a process from a level of investigation to a higher, essentially new level. This example can be seen in the theory of physics: The phenomenon of diffusion is a good example of classical. From the macroscopic theory of diffusion (parabolic partial differential equation) to the higher microscopic level theory (using independent stochastic process to describe the statistical mechanics of particle stochastic motion in solution) transfer, from the latter, using the law of large numbers, can be derived to grasp the former differential equation, Kolmogorov to this kind of situation is more detailed specific description.

Mathematics is more subordinate in biology than in physics. In economics and the humanities, this is more pronounced, and the use of mathematical methods in biology and in the science of Du Pont is mainly in the form of cybernetics. In these disciplines, the importance of mathematics is retained in the form of auxiliary science, mathematical statistics, but in the precise analysis of the phenomena of the duke, the essential differences in each historical phase are dominated, and mathematical methods are often sidelined.

The principles of mathematics and technology, arithmetic, and elementary geometry, as the history of ancient mathematics have shown, arise from the needs of everyday life. Subsequent new mathematical methods or ideas are also influenced by astronomy, mechanics, physics and other disciplines that meet practical needs, but the direct link between mathematics and technology (engineering) is often achieved through the application of existing mathematical theories in technology. It is also necessary to point out that there are examples of general theories that produce new mathematics directly based on technical requirements (for example, the least squares (geodesic), the operand method (electrical engineering). Information theory as a new branch of probability theory (communication Engineering), a new branch of mathematical logic, an approximate solution to a differential equation, a numerical solution, etc.).

The high mathematical theory makes the method of calculator science develop rapidly. Calculator Science plays a major role in solving a lot of practical problems, such as the use of atomic energy and problems in the development of the universe.

Kolmnogorov also always attaches importance to the relationship between mathematics and other disciplines in the later narrative of mathematical history, and also highly evaluates the development of pure mathematics driven by the internal requirements of mathematics. For example, in the application of practical problems, ancient Greece is lagging behind Babylon, but in mathematical theory, Greece is far ahead of Babylon. In particular, he praised "the existence of infinite number of prime numbers", "isosceles right triangle the hypotenuse and the other side of the non-existence of the Convention" and other great discoveries. He explained in detail how the practical doctrine of Babylonian mathematics and idealistic Greek mathematics, which had evolved from medieval Arabic mathematics to modern mathematics in Europe, was very interesting. I learned a lot of historical facts from this history. For example, I previously knew that the concept of transformation group was effectively used by Lagrange (analysis) and Galois (equation theory) in the second half of the 18th century to the beginning of 19th century. But I also want to know who is giving the definition of the (abstract) group that is being taught in college now. According to Kolmogorov's mathematical history, this definition was given by a. Cayley in the middle of the 19th century century.

In short, Kolmogorov's mathematical view is inspired by his mathematical ingenuity, his passion for mathematical applications and his insight into the history of mathematical development. These aspects of the composition, it is difficult to use a word to summarize. If it must be summed up in a sentence, perhaps it can be said: Kolmogorov mathematics to become an unrestricted growth of the "organism."

Kolmogorov's mathematical performance

Kolmogorov wrote hundreds of papers, from which it can be seen that its characteristics are: "A wide range of research areas", "the introduction of new ideas of originality" and "crisp narrative", its research areas include real variable function theory, mathematical basis theory, topological space theory, functional analysis, probability theory, dynamic system, statistical mechanics, Mathematical statistics, Information theory and many other branches. Here's a summary of these studies with a background.

On the theory of real variable function

Kolmogorov attended the Stepanov Fourier series discussion class at Moscow University, and since then (1921) He has been interested in mathematics. At that time, the main research on the continuous function of the calculus is to study the measurable function of the real variable function theory development. This new field of mathematics has received great attention. Kolmogorov in 1922 (19 years old), by introducing set calculus, the new theorem of the existence theorem (Suslin) of the "borel analytic set was proved to be understood. In the same year, he also successfully studied the "(formal) Fourier series on almost all points (later studied at all points) on the composition of the integrable function on the divergence". These results were published as papers in the Mat. Sbornik ", 1925 and" Fund. " Math. ", 1923 (Doklady, 1925). He also wrote several papers on the Fourier series and the expansion of the direct-intersection function. He also tried to promote the Lebesgue integral, involving the study of Denjoy integrals. These are largely research work before 1930.

The basis of probability theory
One of the great feats of Kolmogorov in the plane of probability theory is that the probability theory is established as a field of modern mathematics with the language of measure theory. In the past, accidental events, accidental quantities are not defined and used. Kolmogorov sees the isomorphism of probability and measure, and on the probability measure space (ω,f,p), the accidental event is defined as the F-measurable subset of Ω, the probability of accidental event is defined as the P-measure of this subset, and the accidental quantity is defined as the F-measurable function on Ω, whose mean is defined by integral. Thus, the theoretical expansion of probability theory becomes clear and easy.

So the probability as a measure to grasp the method, for the special Problem E. Borel (The above example), N. Wiener (Brownian motion) has tried. But this method of dealing with all the problems is Kolmogorov's "basic concept of probability theory." And Kolmogorov proved that in this case purposefully constructs the theorem of P, which is the well-known Kolmogorov theorem of expansion.

In the past, as a concrete measure, only lebesgue-stieltjes measure and invariant measure on Lie group are considered. Because of the probability theory of Kolmogorov's measure theory, new probability measure and related new problems have been produced in the mathematical study of accidental phenomena.

Probability theory

Kolmogorov was affected by A.y.khinchin, and the convergence of the series of independent random variables and the order of divergence were studied before and after 1925. According to the study of the Wiener process, in these studies, Kolmogorov introduced several new ideas and methods, Kolmogorov 0-1 Law, Kolmogorov inequality, Khinchin-kolmogorov three-degree theorem, Kolmogorov Strong number law, Kolmogorov discriminant method, Kolmogorov spectrum (turbulence), etc. are particularly well-known. 1939 He also solved the problem of interpolation and extrapolation of weak stationary process to Fourier analysis.

Kolmogorov also divides dynamic systems into deterministic (classical) dynamical systems and dynamic systems of probability theory (Markov processes), describing the former as ordinary differential equations, while the latter is the parabolic partial differential equation, the forward-side program and the backward equation introduced by Kolmogorov ( On the analytic method in probability theory, Math. Ann. 1931). Before that, probability theory (functional analysis) began to be applied, and the content of probability theory became extremely rich. The significant development of the Markov process in the 50 's was the source of Kolmogorov's study. I was inspired by the idea in the preface of this paper by Kolmogorov to introduce a stochastic differential equation for the orbits of the Markov process. This also determines the direction of my future research. Kolmogorov's "Basic concepts" and "analytical methods". It's a treasure for me.

Mathematical Statistics

In Japan it is regrettable that the communication between probability theory and mathematical statistics is not very active, while the Kolmogorov and other Soviet probability experts attach great importance to the relationship between them. Probability theory is based on probabilistic space, when applied to real problems, we need to consider some probability space, and then decide which is the most suitable for the real problem probability mode. This decision can be said to be a goal of mathematical statistics. Kolmogorov also wrote a number of mathematical and statistical papers. The Kolmogorov-smirnov theorem used in nonparametric tests is well known.

Basic Theory of Mathematics

From a young age, Kolmogorov has a strong interest in mathematical fundamentals, especially Brouwer (limited stance) (e.g. math. Zeit., 35 (1932), 58-65), the algorithm is also studied.

On spatial theory of spatial theory of topological function

Kolmogorov and J.W Alexander jointly pioneered the theory of coherence, which is well known. Kolmogorov is also one of the pioneers of spatial theory (linear topological space, topological ring) with topological structure and algebraic structure at the same time.

He also studied the characteristics of the smallest possible number of points in the ε network of all-bounded distance space E, and introduced the concept of ε entropy and ε capacity as the characteristic quantity of E. Apply it to spaces where e is a continuous function space (with V. M. Tikhomirov, Uspehi (1959)). This is a new perspective on functional analysis.

Dynamic Systems

Kolmogorov has a deep knowledge of classical dynamic systems, and he has written several important papers (Proc. ICM, 1954, Amsterdam, 1, 315-333). He also studied the general dynamic system (single-parameter guaranteed transformation Group flow), introduced the concept of "kolmogorov flow. As the characteristics of the flow, you know the spectral type (Hellinger-hahn). Kolmogorov also introduces the new entropy (Dokl., 124 (1959), 754-755). There is no doubt that this has opened the way for a new ergodic theory.

In other respects, Kolmogorov has also done a lot of famous research work. For example, the negative resolution of Hilbert's 13th question (see Hilbert of the Rock Wave "Mathematical Dictionary"), the study of the random tables (Sankhya, A25, 1963), the research on information theory, etc.

Kolmogorov's View on mathematics education

Kolmogorov has cultivated a number of mathematicians at Moscow University, many of whom have become internationally renowned scholars, and this is widely known. He is also enthusiastic in high school mathematics education, himself to write the handout, the mathematics education should have a profound attitude of thinking. Kolmogorov's 60 Birthday (1963), P.s Alexandrov and B.V Gnedenko made a speech entitled "Educator kolmogoro" ". Here's a look at Kolmogorov's mathematical education. The Soviet education system differs slightly from that of Japan, for Primary (7~10岁), Junior High (11~14岁), high School (15~17岁), university (18 years old), and in the University of Mathematics major and physics in a department (called the Department of Mathematical Physics). High School is equivalent to the Japanese high school 2 grade to the university 1 grade, the university equivalent to Japan's University 2 grade to master degree. Some are similar to Japan's legacy high schools and universities, where they graduate with a dissertation to obtain a degree, equivalent to a master's degree in Japan. The doctoral degree was awarded to a number of particularly outstanding scholars who had written many papers after graduating from university.

Kolmogorov that some parents and teachers are trying to dig up children from the age of 10 to students with math skills, which will harm the child, but when the child is 14~16岁, the situation is different. Their interest in mathematical physics has been clearly demonstrated, and according to Kolmogorov's experience in teaching mathematics and physics in high school, about half of the students think that mathematical physics has only a small effect on themselves. For these students should arrange simple content of the course. In this way, the other half of the students (not necessarily they all have to engage in mathematics and physics) of the mathematics education can be more effective.

In high school, the Department of Mathematics and Physics, engineering, biology and Agriculture, the Department of Economics, and other major departments to separate as well. The main disciplines of the departments of the Professor time can be slightly increased (such as mathematics 1 hours, physical 1 hours, etc.), even if this effect is very significant. The education of professional departments can enhance students ' sense of purpose without affecting the width of general education. The slogan of "Unified Labor School", which was proposed at the beginning of the revolution, does not negate the development and special training of individual abilities, but merely means abolishing class-conscious schools and removing obstacles before the poor.

The argument that mathematics requires a special talent is exaggerated in many cases. Mathematics is a particularly difficult subject. This impression may be the result of clumsy and extremely dogmatic teaching methods. If there are good teachers and good textbooks, the normal average level of people's ability to digest high school mathematics, and further understanding of the preliminary knowledge of calculus.

However, when choosing mathematics as a major in college, high school students should naturally test their adaptability to mathematics. In fact, the speed, ease, and success of Understanding (mathematical) inferences, solving problems, or making new discoveries vary from person to person. In mathematics professional education, we should choose the young people who have great possibility of achievement in mathematics field.

What is the adaptability to mathematics? The Kolmogorov summarizes the following three points:

(1) Algorithmic ability: that is, for complex formulas to make a clever deformation, for the standard method can not solve the equation of the ability to skillfully solve (just remember a lot of theorems, formulas are not).

(2) Intuitive geometry: For abstract things, can be depicted in the mind like painting and thinking.
(3) A step-by-step logical reasoning capability: for example, the mathematical induction can be applied correctly.

Only these abilities, but not a strong interest in the research topic, do not make lasting research activities, still does not play any role.

In the University mathematics education, what kind of good teacher is?

(i) a brilliant lecture. Use other examples in the field of science to attract students.

(ii) to attract students with clear explanations and broad mathematical knowledge.

(iii) good at individual guidance. The ability of each student is clearly understood and the learning content is arranged within the scope of his/her ability to enhance their self-confidence.

Each of these is valuable, and the ideal teacher should be a teacher of type (iii).

In addition to the regular curriculum, Kolmogorov has highlighted the following two points in mathematics education for students of the Department of Mathematical Physics:

(i) Enable students to use functional analysis as a daily tool.

(ii) Attach importance to practical work.

At first I was not quite clear about this meaning, and recently saw a gentleman who had been Kolmogorov at Moscow University and asked for it, for example, given the specific coefficients and boundary conditions for the differential equation (different for each student), and then let the students examine the nature of the solution of the equation.

When students begin to study, they must first establish their self-confidence that they can make a point of doing something. Therefore, in arranging the research project, we should consider not only "the importance of such a topic", but also "whether the study can improve the student's level, whether it is within the competence of the students, and it needs to be solved with the greatest effort".

The above is a summary of Kolmogorov's mathematics education theory. Kolmogorov is not only a great mathematician, but also a great educator, perhaps a great thinker more appropriate.

Kolmogorov's mathematical View and performance