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This article comes from the official account of Wechat: ID:fanpu2019, author: Ding Jiu

Born in 1903, Russian mathematician Kolmogorov and Hungarian-American Feng Neumann are two dazzling mathematical superstars. They have performed brilliant plays on their respective stages of life, leaving endless memories for future generations. This year marks the 120 anniversary of the birth of these two mathematical "Gemini stars". Here we take a glimpse of their great achievements beyond ideology, and hope that the stars of mathematics will continue to shine in this century.

Written by Ding Jiu (Professor of Mathematics, University of Southern Mississippi)

1903 has been proved by history to be a lucky year for the international mathematical community, and it can even be slightly exaggerated to change the course of the history of mathematics in the 20th century. The warm spring breeze at the beginning of the year gave birth to Andre Kolmogorov (April 25, Andrey Kolmogorov,1903-October 20, 1987) to Russia in the east, while in the winter of that year, Hungary gave birth to John von Neumann (December 28, John von Neumann,1903-February 8, 1957).

Kolmogorov, Peter the Great of the Eastern Mathematical Kingdom and King Solomon of Feng Neumann, the King of Western Mathematics, at the height of their respective academic careers, one opened up the direction of mathematics, proved profound theorems, trained generals for the Soviet Union, the head of the socialist camp, and designed computer models, solved natural mysteries and buried national defense technology for the United States in command of the capitalist alliance. They have a number of common interests, but also have many different characteristics. They all loved history when they were young, but they firmly went to the path of mathematicians; they all had aristocratic blood and temperament, one with the sturdy physique of a lumberjack from top to bottom, and the other famous for his fat figure of a banker from beginning to end. None of them lacked nutrition when they were alive. One was born to take students hiking and learn mathematics in nature, while the other gave exercise time to lively family parties and liked to drive around and tell jokes. They are all giants in the history of mathematics in the 20th century, but in the works of physicist Freeman Dyson,1923-2020, who portrays mathematicians, one may be called a "bird" flying in the mathematical sky to create a new field, and another has been classified as a "giant frog" who has dug deep into the mathematical treasure to solve difficult problems. In World War II, they all contributed extraordinary talents to the defeat of the Axis Powers. One worked out the best plan for the capital to avoid German air attacks, and the other devoted himself to advising the development of nuclear bombs. They were all gifted talents. One created a "garden full of spring" all his life to cultivate countless talents for the motherland, while the other was content to "blossom alone". After the flowers withered, there were few successors. They overloaded their unparalleled brains all their lives. One lived to the age of 84 without great regret, while the other lived 30 years less than the former and had no ambition before he died.

This year marks the 120 anniversary of their birth. Although they have different political beliefs and serve the Soviet Union and the United States, which have been in the Cold War with each other for decades, they deserve to be remembered as mathematical giants at the same time. Over the past hundred years, God has not generously let the world have a few more scientific great men like them, but we should follow their footsteps and understand their hearts, so that a new generation can be spurred and inspired.

Kolmogorov Bell (Eric Temple Bell,1883-1960), an American historian of mathematics, published Men of Mathematics in 1937 (a Chinese translation is "Masters of Mathematics") and influenced several generations of scientists. It is a pity that he wrote so early that among the more than 30 mathematicians carefully depicted in the book, he only wrote about the Frenchman Henri Poincar é (1854-1912), and even the Prussian Hilbert (David Hilbert,1862-1943), who was only eight years younger than Poincare, could not be included in his book. The title of the Poincare chapter is "The Last Universal Mathematician" (the last omnipotent mathematician). If the author had lived ten more years and revised this great work before he died, it would not only have added Hilbert, Kolmogorov and von Neumann, but also might have called Kolmogorov "the last omnipotent mathematician".

Kolmogorov can be regarded as the Poincare of Russia in the 20th century, but he was born far from being as lucky as the latter from a distinguished family. His father was an agronomist, but later joined the revolution that overthrew the czar and was exiled to the provinces by the St. Petersburg authorities and killed in the civil war in 1919; his unmarried mother died while giving birth to him. Fortunately, when he lost the company of his parents, he did not lose the love of his loved ones. with his two aunts growing up in the aristocratic manor of the rich grandfather, he received a good early education and entered Aunt Vera's country school. His earliest works of literature and mathematics were published in the school magazine Chunyan. At the age of five, he became the "math editor" of the school magazine. In the same year, he alone discovered the law of odd sum, which was published in the school journal: 1 = 12, 1 + 3 = 22, 1 + 3 + 5 = 32,.

In 1910, Kolmogorov's aunt formally adopted him, and the family settled in Moscow. Ten years later, Kolmogorov finished high school, was admitted to Moscow University, and also studied at Mendeleev Moscow Institute of Chemistry and Technology. "I came to Moscow University with a lot of mathematical knowledge," he later recalled. "I knew very well about the beginning of set theory. I studied many of the questions in the Brockhaus-Efron encyclopedia and filled in the too concise content for myself." Multi-volume encyclopedias are often "self-study textbooks" for prominent teenagers because they are never confined to school compulsory subjects that are only suitable for ordinary people. Since his college days, Kolmogorov has enjoyed a high reputation for his erudite talents.

Like Feng Neumann, a genius of the same age in other countries, Kolmogorov's extensive knowledge includes history. He attended a seminar by a famous professor of Russian history and wrote the first research paper of his life, which naturally discussed not the history of mathematics, but the practical history of land possession in the Novgorod Republic in the fifteenth and sixth centuries. When he finished it, he was quite proud and thought the professor would appreciate it. His unexpected answer was: "there is only one argument in your article, which is too little for history. There must be at least five arguments." The history professor should become the "doctoral mentor" of social scientists who only "make bold assumptions" and do not "carefully verify" to publish papers quickly. It was this disappointed historian who inadvertently pushed Kolmogorov to his other favorite, mathematics. He is committed to mathematics all his life, because only one proof of the theorem in mathematics is enough!

Before the age of 20, the young Kolmogorov proved several important results about set theory and trigonometric series. In 1922, he constructed a Fourier series which diverged almost everywhere and became a rising mathematical star. From then on, he decided to devote his whole life to mathematics, which was the luck of mathematics, but the loss of history. By his senior year, he had published eight papers and ten more when he got his doctorate in 1929, mentored by Nikolai Luzin,1883-1950, one of the leaders of the Moscow school of function theory, who was 20 years his senior. Kolmogorov's mathematical activities all his life have greatly promoted the development of many fields, including function theory, probability theory, functional analysis, topology, random process, mechanics, turbulence, ergodic theory and so on. In 1933, he established the axiom system of modern probability theory in metric language in the form of Foundations of the Theory of Probability (basis of probability Theory).

Kolmogorov became a professor at Moscow University at the age of 28 and was elected an academician of the Soviet Academy of Sciences eight years later. Like the Academician of the Academia Sinica of the Republic of China in 1948 or a member of the Department of the Chinese Academy of Sciences in 1955, the title of academician is completely "pure gold." In the scientific community of his home country, he won almost all the awards related to mathematics: the National Award, the Lenin Award, the Stalin Award, the Chebyshev Award, and the Robachevsky Award. On the international stage, he participated in four of the five International conferences of mathematicians from 1954 to 1970, and gave an one-hour plenary report on dynamical systems at the 12th International Congress of mathematicians held in Amsterdam, the Netherlands, in 1954. This report presents for the first time the original idea of "KAM Theorem" for solving divisor problems in classical perturbation theory, in which K is naturally he, M is German-American mathematician J ü ngen Moser,1928-1999 and An is Arnold (Vladimir Arnold,1937-2010). The latter two proved this famous theorem for smooth distortion mapping and analytic Hamiltonian system in 1962 and 1963 respectively.

Here is another outstanding contribution of Kolmogorov in the field of power systems, and a little explanation. In the mid-1950s, when considering the basic problem of "conjugate invariant" of ergodic theory, he created the concept of "measure entropy". This entropy, which is of basic significance in ergodic theory, is a conjugate invariant, which means that the conjugate test-preserving transformations share the same entropy value, just as eigenvalues are similar invariants of matrix similarity transformations.

As early as 1943, it has been known that the two test-preserving transformations "Bernoulli (1par 2,1ax 2)-bilateral shift" defined on the two-way sequential symbol space composed of 0 and 1 symbols and "Bernoulli (1Accord 3, 1Accord 3)-bilateral shift" defined on the two-way sequential symbol space composed of 0, 1 and 2 symbols have countable infinite Lebesgue spectral points. So they are spectral isomorphic. However, mathematicians do not know whether they are also conjugate, that is, whether there is a test-preserving isomorphism between the two symbolic spaces, so that the composition of one shift and its composition is equal to its composition with another shift. In 1958, Kolmogorov calculated that the two bilateral shifts have different measure entropy, one is ln 2, the other is ln 3, so they are not conjugate to each other.

The final perfection of measure entropy was completed by Kolmogorov's student Yakov Sinai,1935- in 1959, so this entropy is often called "Kolmogorov-Sinai entropy" in the literature. it measures the degree of confusion of the final behavior of the infinite iterative process of general test transformation. Kolmogorov is a top-notch original mathematician whose brains water the flowers of his thoughts, but the fruit sometimes depends on the ripening of the diligent gardener. Arnold, another of his disciples, once recalled in writing that the teacher would sometimes "hang the blackboard" in front of the podium of the discussion class because he had forgotten some of the details of the proof. It has also been counted that Kolmogorov has fifteen groundbreaking papers without any references.

Alexandrov (Pavel Alexandrov,1896-1982), a lifelong friend and doctoral brother of Kolmogorov, said that mathematical geniuses can be divided into "fast" and "slow" geniuses. "Hilbert belongs to the genius of 'slow', while Kolmogorov is undoubtedly the genius of 'fast'." In a commemorative article written by his teacher on his 100th birthday, Arnold also recalled the speed with which his guide set up a new discipline: a big problem could be solved in a week or two. But the depth of his rapid prototyping ideas, the breadth of accumulated knowledge and the creative energy of pushing through the old and bringing forth the new are almost unmatched in his contemporaries. Creative mathematicians are really great mathematicians, highly respected in life, famous in history, and achievements are often mentioned. Just like the three Chinese in S.T. Yau's speech to mathematicians in Beijing more than a decade ago: Shiing-Shen Chern (1911-2004) who created the "Chen-characteristic class" with basic uses in geometric topology, Hua Luogeng (1910-1985) who kept pace with the West in the Theory of functions of Multivariate complex variables, and Feng Kang (1920-1993) who put forward the theoretical framework of "finite element method" independently of the West.

Kolmogorov, like Fermi (Enrico Fermi,1901-1954), an Italian-American physicist, has brought out or influenced a large number of outstanding students. In the international mathematical and physical circles, it can be described as "I have never seen the saint of the previous generation, and when will the enlightened lord of future generations wait?" especially the former. According to statistics, Kolmogorov directly mentored as many as 67 students, which almost reached the record of Confucius'"virtuous disciple 72", of whom 14 were selected as academicians or communication academicians of the Soviet Academy of Sciences. The list can be found on page 368 of the book "Mathematical latitude and longitude in the 20th century" written by Zhang Dianzhou (1933-2018). In addition to the aforementioned Arnold and Sinai, Kolmogorov has a more famous disciple, Israel Gelfand,1913-2009, who became a doctoral student without attending high school or college. The legendary student and C.S. Wu (1912-1997) stood on the podium of the first Wolf Award to receive the award from the President of Israel, even two years earlier than his teacher.

Kolmogorov, because of his high academic status at home and abroad, even during the reign of the paranoid and cold Stalin (Joseph Stalin,1878-1953), life was better than that of most Soviet scientists without much political trouble. Interestingly, in 1930, von Neumann, who was eight months younger, went from Eastern Europe to the United States as a visiting professor at Princeton University, while Kolmogorov went to the mathematical cities of Western Europe: Gottingen, Munich and Paris, where he met great mathematicians such as Hilbert and Lebesgue (Henri Lebesgue,1875-1941). The nine-month academic visit was unforgettable for the rest of his life. If Kolmogorov stayed in Western Europe or was invited to live in the United States as Feng Neumann did all his life in the United States, during the decades of the Cold War between the United States and the Soviet Union, his exchanges with first-class mathematicians in the West will undoubtedly be more in-depth and extensive, and will have a more powerful and comprehensive impact on the international mathematical community. But just as Hua Luogeng's return from the United States in 1950 partially turned the sick man's Chinese mathematics in East Asia, Kolmogorov's existence took Russian mathematics, which has been admired by the West, to the next level.

Kolmogorov has been involved in the cultivation of gifted young people for a long time, and in his later years he is more keen on the reform of mathematics education in middle school. he not only teaches with his own hands, but also leads his children to go hiking and go camping to train mathematics from the training of their physique. But like Russell Russell (Bertrand Russel,1872-1970), a British philosopher who was enthusiastic about the experiment of youth education decades before him, he was not particularly successful. Kolmogorov, who was psychologically hit by the unsatisfactory reform, finally suffered from Parkinson's disease and died at the age of 84. Russell, who failed to live to nearly 100 like the best part of his life, rose and fell, but always maintained the "three passions" all his life.

Arnold said to his mentor: "Kolmogorov, Poincare, Gauss, Euler, Newton, only these five lives separate us from the source of science."

Von Neumann von Neumann was born into a wealthy Jewish family in Budapest, Hungary, and seems to be "born with knowledge" in the Analects of Confucius. At the age of six, he could calculate the multiplication and division of eight digits strangely. As an adult, when he used his mind to calculate complex problems, he often looked at the ceiling and chanted words like a monk, and the answer rolled out from his mouth. A widely circulated story says that when he was asked, between two people walking at a constant speed, there was a fly flying back and forth in a straight line at a constant speed, so how far did the fly fly when they met? Von Neumann told the answer in the blink of an eye, using the surprising summation of infinite series, rather than the usual elementary method.

Von Neumann is indeed a child prodigy who is familiar with calculus at the age of eight, reads the Theory of functions by É mile Borel,1871-1956 at the age of 12, and studies advanced calculus with the famous analyst G á bor Szeg engineer (1895-1985) at the age of 15. When they first met, Seg was so shocked by his mathematical talent that he burst into tears. But the young Feng Neumann was also interested in history and read through a 46-volume series of world history, General History in monographs, in his library. In his elite high school, except for geometry painting, writing and music, he got a B, but sports was the worst: C.

Feng Carmen (Theodore von K á RM á n, 1881-1963), the first winner of the US National Science Award, is also a Hungarian prodigy. In his autobiography, he recalled that Feng Neumann's 17-year-old banker father had specifically consulted the already famous Feng Carmen to ask his son what his major was in college. As a compromise between father and son, Feng Neumann studied chemical engineering and his favorite mathematics at the University of Berlin in Germany and the University of Budapest in his native country. Von Neumann's first mathematical paper was published in 1922, when he was less than 19 years old. When he transferred to Switzerland as a graduate student in chemical engineering at the Federal Institute of Technology in Zurich the following year, he was mainly doing math. Feng Neumann once asked Professor George P é cor lya,1887-1985, who was also born in Budapest, a math question in class, and the latter commented in his later years: "I've been afraid of him ever since."

Von Neumann was probably the smartest mathematician in the world in the 20th century, which made Hans Bethe,1906 (2015), a German-born American physicist and winner of the 1967 Nobel Prize in physics, sigh:

"does the existence of a brain like Feng Neumann mean that there are biological species at a higher level than humans?"

Eugene Wigner,1902-1995, a Nobel laureate in physics who was one year old and graduated from the same elite high school in Hungary, has an almost inferiority complex for him all his life:

"No matter how smart a person is, growing up with Feng Neumann is bound to feel frustrated."

Although von Neumann is recognized as the father of electronic computers and game theory, but he thinks his proudest academic contributions are in mathematics: the theory of self-conjugate operators in Hilbert space, the mathematical basis of quantum mechanics and the von Neumann mean ergodic theorem named after him. These three mathematical achievements are groundbreaking. The first item constitutes a major part of the functional analysis of the new discipline which was conceived at the beginning of the last century. The second item strengthens the logical foundation of quantum mechanics with the language of infinite dimensional linear operators. The third item gives a mathematical examination of the basic Boltzmann ergodic hypothesis in statistical mechanics. It is the first major theorem proved by ergodic theory in a comprehensive discipline destined to be more developed in this century.

Von Neumann's main mathematical achievements are, of course, more than the above three. 1940 marked a watershed in his research career. Before that, he showed his skill in the ivory tower of pure mathematics, left a deep footprint in the field of mathematical logic, and soon put forward a new axiomatic system in set theory. In 1933, after completing the three major mathematical achievements in a short time, Feng Neumann, who had not yet reached his 30th birthday, solved Hilbert's fifth problem with a tight group tool. then he went all the way to "sweep thousands of troops like mats" from the fields of measurement theory, lattice theory, continuous group, operator ring and so on, leaving behind a lot of results.

The big mathematician with a stocky head who often smiles and looks more like the president of Wells Fargo may not have the astonishing originality of Albert Einstein,1879-1955 in his life, but like China's fast-growing high-speed trains a decade ago, he was able to spread out the original ideas he had seen and seen with astonishing speed. Before the age of 30, he became one of the first six permanent professors, including Albert Einstein, hired by the Princeton Institute for Advanced Studies.

After the United States joined World War II, Feng Neumann changed his role with the times, like Sun WuKong, from a pure mathematician to an applied mathematician, establishing theories for practical disciplines such as shock waves of gas kinematics and turbulence of fluid mechanics, contributed basic concepts and methods to emerging computational mathematics, and played an important role in thermonuclear fusion research. In addition to his busyness, Game Theory and Economic behavior (Theory of Games and Economic Behavior), co-authored with Princeton University economics professor Morgan Stern (Oskar Morgenstern,1902-1977), became the foundation of game theory. In his final years after learning that he was suffering from an incurable disease, he worked hard to explore similar mechanisms between computers and the human brain, and his unfinished speech for the Slimane Memorial Lecture, established by Yale University in 1888, was published by the university's publishing house under the title "computer and Human brain" (The Computer and the Brain) after his death.

In the middle and late 1940s, with the emergence of the first electronic computer that Feng Neumann helped to develop successfully, the convenient and fast numerical calculation greatly helped the brain thinking of creative mathematicians and became a good helper for them to ask and solve problems. As the first mathematician to come into contact with modern computers, in the process of working with great physicists like Fermi to solve physical problems, mathematicians such as Feng Neumann became the pioneers in the scientific field of "nonlinear analysis", which integrates mathematics, physics, computer and other disciplines.

Although Feng Neumann is famous as a great mathematician, politicians and strategists always pay homage to him like scientists. In the ward of Reed Hospital in Washington, where he was on his deathbed, "the deputy secretary of defense, the commander of the armed forces, and other military and political dignitaries, who knew little about modern mathematics, gathered around his sickbed. listen to his final advice and extraordinary insights. It seems that no other mathematician in history received such a tearful farewell from so many military generals before he died.

However, some contemporary or later scientists feel that the originality of Feng Neumann's large head is a little less coquettish than its extraordinary function in quickly solving difficult problems. For example, Dyson, his colleague at the Princeton Institute of Advanced Studies and a famous mathematical physicist who started as a pure mathematician, classified Feng Neumann as a collection of "frog mathematicians" in his American article "Birds and Frogs" published in the second issue of the American Mathematical Society (Notices of the American Mathematical Society) in 2009. And put Yang Zhenning (1922 -) into the ranks of "bird mathematicians" who "soar in the vast sky".

According to Dyson's account in Bird and Frog, participants at the 1954 International Congress of mathematicians in Amsterdam, the Netherlands, witnessed an unpredictable and embarrassing scene revealed by chaos, which is a portrayal that the author does not appreciate very much. Fifty years after Hilbert's century report on 23 mathematical problems, it is hoped that a new and far-sighted Hilbert will ask new big questions to guide the development of mathematics in the next fifty years. The man who is widely expected seems to be the most popular von Neumann. As a result, the invitation to the conference was sent, and the title of the report, "outstanding mathematical problems", was also sent to the delegates. At 3: 00 p.m. on September 2, 1954, Feng Neumann, who arrived in a hurry, spit out from his mouth the "operator ring" of his early work in the 1930s, rather than an open question. This is a blow to many mathematicians who are waiting for it.

As a result, Dyson observes that von Neumann "is actually a frog, but everyone expects him to be like a He was really a frog but everyone expected him to fly like a bird."

Von Neumann, however, there must be some scientists of the same tonnage who disagree with the above assertion. In 1954, Feng Neumann was keen on the grand cause that he deserved the title of "father of modern computers". He may be too busy to neglect the great trust placed on him by the International Congress of mathematicians. The conclusion that "he is not a bird mathematician" as a result of this episode is probably not very good. Hilbert may be more creative than von Neumann, but according to the famous mathematician biography Hilbert: Hilbert of Mathematics, he was helped by his lifelong friend Minkowski (Hermann Minkowski,1864-1909) when he carefully prepared his speech on "Mathematical problems" at the 1900 Congress of mathematicians, so he was able to ask a pile of 23 questions that had far-reaching implications. It is also the crystallization of the "thinking collision" of two mathematical giants.

Dieudonne (Jean Dieudonn é, 1906-1992), one of the founders of the French Bulbaki school of mathematics and the first master of writing, called von Neumann "the last great mathematician". Halmus (Paul Halmos,1916-2006), a Hungarian-American mathematician who worked as an assistant to von Neumann in the 1940s and wrote several well-known mathematical masterpieces, commented on him in only one sentence: "the great von Neumann benefits all mankind." Another Hungarian-American mathematician, Peter Lax,1926-, who won the National Science Award, Wolf Award and Abel Prize, described him as a great scholar of the 20th century with "the most awesome technical ability" and "sparks of high intelligence", saying that he had the "strongest brain" (the most powerful brain). In an interview commemorating the sage of his motherland, he even thought: "if Feng Neumann had lived longer, he would certainly have won the Abel Prize in Mathematics, the Nobel Prize in Economics, the Grand Prize in computer, the Nobel Prize in Physics in Quantum Mechanics, and so on."

Von Neumann lives shorter than most of us, but the world is very different because of him. Just read McRae (Norman Macrae,1923-2010)'s excellent work, the Genius Pioneer-Biography of von Neumann (John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More), and take a look at one of the parties, Herman Goldstine. (1913-2004) A careful review of the development of modern computers in the United States in the early 1950s in his book computer: from Pascal to von Neumann (The Computer from Pascal to von Neumann), it is conceivable that von Neumann, who lived just 42 days after his 53rd birthday, would have lived to the first National Science Award in the early 1960s. Perhaps it is really difficult for the award committee to decide whether to award the highest honor to the American scientist or to Feng Carmen, a Hungarian who has also made great contributions to the United States. One of the reasons for this suspense is that the year before Feng Neumann's death, US President Eisenhower (Dwight Eisenhower,1890-1969) awarded him the Presidential Medal of Freedom.

After a brief review of the scientific careers of Kolmogorov and von Neumann, we admire the fact that these two shining planets once shone in the east and west skies of the mathematical firmament in the 20th century. it also regrets the fact that the turbulent turbulence caused by ideological disputes makes it impossible for them to "comrades and brothers, swim side by side" in the sea of mathematics. However, according to the universal belief that "science knows no national boundaries", their great achievements in the whole cause of mathematics and for the glory of all mankind will be "as long as heaven and earth, and the three lights will last forever."

The draft was completed on Saturday, February 4, 2023

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