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This article comes from the official account of Wechat: ID:fanpu2019, author: Slava Gerovitch

Kolmogorov, one of the greatest mathematicians of the 20th century. In the previously pushed article "Masters of probability and Statistics-Mathematics and the Crystal Ball (part two)", it was briefly mentioned that what made probability theory a part of modern mathematics, in addition to the groundwork of some mathematical "gods" in early Europe, it was Kolmogorov who established the axiomatization of probability theory. This work earned him the reputation of "Euclid in probability theory". Academically, this talented mathematician has studied almost all fields of mathematics, and has made achievements in many fields, such as classical mechanics, ballistic calculation, crystallography, turbulence and so on. He was also an educator who not only brought out a large number of excellent mathematicians, but also devoted himself to basic education and founded schools.

As the founder of modern probability theory, his life is quite "random". It is impossible to understand his life and academic achievements in a few articles. This paper takes only one side to understand this great mathematician at the beginning of his academic career, under the political storm and in the study of art.

Slava Gerovitch (Lecturer, Department of Mathematics, Massachusetts Institute of Technology)

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Step into the world of mathematics if two statisticians get lost in the forest, the first thing they have to do is get themselves drunk, so that two drunken men wandering around may meet each other. But if they want to pick mushrooms with bamboo baskets on their back, they'd better drink less. After all, random walking aimlessly will take them back to the place where they have already been picked.

This is known in statistics as a random walk (random walk) or a drunken walk (drunkard's walk). This model indicates that the future state of the system depends only on the current state, not on the past. Today, this model has been widely used in stock price modeling, molecular diffusion, neural activity and population dynamics. It can also be used to explain how genetic "gene drift" leads to the prevalence of a gene (such as eye color) in the population.

Ironically, the theoretical model does not care about the past or history, but it itself has a long history. It is one of many theoretical achievements of the Soviet mathematician Andre Kolmogorov (Andrei Kolmogorov,1903-1987). The mathematician with wizards and talents dabbled in a wide range of skills, and while balancing political and academic life, he also changed the status of "impossible" in mathematics.

Kolmogorov: the cultural thought of Moscow in the post-yarwiki.ru era was active, when the atmosphere was full of experimental literature, avant-garde art and radical new scientific ideas. Young Kolmogorov was also affected by this. When Kolmogorov was a history student in the early 1920s, he submitted a paper at Moscow University that made an unconventional statistical analysis of the lives of medieval Russians. He found that the government's tax on the village is often a round number, and when it is distributed to each household, it becomes a score. Therefore, he believes that at that time, the tax policy was paid by the village and then distributed to the household, not by the household tax and then collected by the village. "the evidence is not standing alone," the history professor severely criticized his findings. "one evidence is not enough, you have to find at least five examples."

It is not surprising that Kolmogorov devoted himself to mathematics-it is enough to prove the mathematical theorem only once.

Kolmogorov in his youth: the political reality behind the academic theory of yarwiki.ru Kolmogorov's turning to the field of probability theory also stems from an accident. At that time, probability theory, a branch of mathematics, did not have a good reputation, because in the past, people used to regard probability as the embodiment of divine will. In ancient Egypt and classical Greece, people thought that rolling dice was a reliable method of divination and fortune telling. By the early 19th century, European mathematicians had mastered some correct methods to calculate probability, and defined probability as the ratio of the number of target events to the number of all possible events. But its disadvantage is that probability is defined according to possible events, so it is only applicable to systems with limited elements. When faced with infinite systems, such as dice with countless faces, or a continuous sphere, the theory of probability seemed stretched at that time.

However, Kolmogorov cherished fame and fame. After changing majors, Kolmogorov was first attracted by a mathematical circle at Moscow University. Their leader is a charismatic teacher, Nicola Luzin (Nikolai Luzin,1883-1950). Lujin's disciples nicknamed the organization Luzitania-linking Luzin's name to the famous British ocean liner RMS Lusitania, which was sunk in World War I. As Kolmogorov said, they are United by a "common heartbeat". They often criticize mathematical innovation together after class. In their words, the partial differential equation (partial differential equations) becomes the "partial disrespect equation" (partial irreverential equations), and the finite difference (finite differences) becomes the "dream difference" (fine night differences). At that time, because of its weak theoretical foundation and full of paradoxes, probability theory became "theory of misfortune" in their mouth.

Nicholas Luzin, one of the founders of set theory, has made outstanding contributions in the fields of trigonometric series, complex analysis, differential equations and numerical computation. Ru.wikipedia.org is influenced by Lujin Tania, and Kolmogorov's attitude towards probability theory has also changed. When Stalinist terrorism broke out in the 1930s, the secret police knocked on everyone's door in the middle of the night, and random randomness seemed to dominate people's lives. Under the deterrence of fear, many people report others in order to protect themselves. Among the Bolshevik activists in the mathematical circle are Lujin's students. They accused Lu Jin of political disloyalty and criticized him on the grounds that he had published his own research papers abroad. Kolmogorov, who has also published articles abroad, showed obvious compromises on political issues out of concerns about his academic career. When the director of the Institute of Mathematics at Moscow University was imprisoned for supporting sects, he took over the post. At this time, Kolmogorov also joined the crowd of critics of Lujin. Lu Jin received a trial from the Soviet Academy of Sciences and lost all his duties, but he escaped arrest and shooting from the government authorities.

Like the Lusitania, Lusitania was sunk. The difference is that Luzintana was sunk by his own crew.

Despite the big circle of "impossible" and not talking about Kolmogorov's moral issues, he did "win" and kept his research job. Different from his political obedience, Kolmogorov's research thought is more radical, he made a fundamental revision to the theory of probability. What he used was something called Measure theory introduced from France, which was a fashionable theory at the time. Measure theory generalizes the concepts of length, area, volume and so on, so that mathematical objects that can not be measured by conventional methods may be measured. For example, if a square with an infinite number of holes is cut into an infinite number of pieces and scattered in an infinite plane, we can still express the "area" (measure) of this piecemeal object with the help of measure theory.

Kolmogorov made an analogy between probability and measurement, resulting in five axioms, which are now usually expressed as six. his work makes probability theory a real part of mathematical analysis. The most basic concept in his theory is the "fundamental event", the result of a single experiment, such as a coin toss. All the basic events constitute a "sample space", that is, a collection of all possible results. For example, if lightning often occurs in an area, the sample space includes all possible locations of lightning in that area. A random event is defined as a "measurable set" in the sample space, and the probability of a random event is the "measure" of the measurable set. For example, the probability of lightning hitting a location depends only on the area of that location ("measure"). Two simultaneous events can be represented by the intersection of their measures; the conditional probability can be regarded as the division of the measure; the probability of one of the two incompatible events is expressed by the addition of the measure (that is, the probability of An or B being struck by lightning is equal to the sum of their areas).

The great circle paradox (The Paradox of the Great Circle) is solved through Kolmogorov's theory of probability. The Great Circle Paradox is that assuming that aliens will randomly land on a perfect spherical planet, and the probability of landing at each point is equal, does this mean that the landing probability on the great circle (the intersection of the plane and the sphere passing through the center of the sphere, which divides the sphere into two equal hemispheres) is the same? As a result, for the large circle where the equator is located, the probability of each point on the circle is equal. For the longitude, the probability of the point near the equator is large, and that of the point near the poles is small. This finding may be explained by the fact that the nearer the equatorial latitude circle is, the larger it is. However, this result is counterintuitive, because for a perfect sphere, the equator can become any meridian by rotation. Kolmogorov believes that the great circle is a line segment, the area is zero, so the measure is zero. The paradox of this paradox is that we cannot strictly calculate the relevant probabilities.

The definition of "big circle" is not strict. (source: en.wikipedia.org

Large circle paradox is called Borel-Kolmogorov paradox in probability theory. Random variables get different results under the condition of longitude and latitude. In fact, it is a conditional probability problem with measure zero. Map source: after Yarin Gal briefly evaded the "purge" in the world of conditional probability of zero measure, Kolmogorov is still perplexed by the problems of the real world. During World War II, the Soviet authorities asked Kolmogorov to study ways to improve the efficiency of artillery fire. Kolmogorov found that in some cases, instead of increasing the hit rate of each shell, it would be better to make continuous blows that deviated slightly from the target. This strategy is called "artificial dispersion". The probability Department of Moscow University, under the auspices of Kolmogorov, also calculated the ballistics of low-altitude and low-speed bombing. In recognition of Kolmogorov's contributions during World War II, the Soviet government awarded him two Lenin medals in 1944 and 1945. After World War II, he served as a mathematical adviser to the thermonuclear program.

The probability perspective of insight into the art world out of professional interest, Kolmogorov actually has a preference for philosophy. With a background in mathematics, he believes that this randomly determined world operates in an orderly manner, and there are laws behind it that probability theory can follow. He often thinks about the impact of "impossible" things on human life.

In 1929, Kolmogorov met mathematician Pavel Alexandrov (Pavel Alexandrov,1896-1982) on a canoe trip, and the two became lifelong friends ever since. In a long letter, Alexandrov bluntly accused Kolmogorov of talking to people on the train and hinted that the communication was too superficial to really understand a person. Kolmogorov objected, looking at social communication from a radical perspective of probability. In such interactions, the object of communication is a statistical sample of a larger group. "people will understand the true meaning from the environment and bring the developed lifestyle and worldview to anyone around them, not just specific friends." Kolmogorov said in his reply.

Music and literature are also very important to Kolmogorov, who believes that he can gain insight into the workings of the human mind from the perspective of probability. He is also a cultural elitist who believes that the value of art is divided into three or six or nine grades. At the top are the works of Goethe, Pushkin and Thomas Mann, as well as the works of Bach, Vivaldi, Mozart and Beethoven whose eternal value is similar to the eternal mathematical truth. Kolmogorov stressed that every real work of art is unique, something that is so-called "impossible" and something beyond the laws of statistics. In a 1965 article, he sarcastically asked, "is it possible to include Tolstoy's War and Peace in a reasonable way into the collection of all possible novels?" and further assume that there is a particular probability distribution in this set. "

Kolmogorov has a strong interest in painting, music, sculpture, architecture and other arts. At the same time, Kolmogorov is also eager to find the key to deciphering the nature of artistic creation. In 1960, Kolmogorov equipped a group of researchers with electromechanical calculators and assigned them to calculate the rhythmic structure of Russian poetry. Kolmogorov is particularly interested in the deviation between the actual rhythm and the classical rhythm. In traditional poetics, iambic consists of an unstressed syllable followed by an stressed syllable. But in the actual creation, this rule is rarely followed. Pushkin's Eugene Onegin is the most famous classical iambic poem in Russian. In the 5300 lines of the poem, almost 3/4 of the verses violate the definition of iambic, and more than 1/5 of the syllables are unstressed. Kolmogorov believes that the frequency at which stress deviates from the definition of classical prosody provides an objective "statistical portrait" for poets. In his view, an unlikely stress pattern reflects the creativity and expressiveness of art. Through the study of the works of Pushkin, Pasternak and other Russian poets, Kolmogorov believes that the poet's unique use of prosodic format has established the "tonality" of his own works.

In order to measure the artistic value of the text, Kolmogorov also used letter guessing to estimate the entropy of natural language (entropy). In information theory, entropy is a measure of uncertainty or unpredictability. For information, the greater the unpredictability of a piece of information, the more information it carries. In Kolmogorov's eyes, entropy has become an index to evaluate artistic originality. His team conducted a series of experiments: show volunteers a passage of Russian prose or poetry and ask them to guess a letter, guess another, and so on. Kolmogorov said privately that from the point of view of information theory, Soviet newspapers are not as informative as poetry. Because political discourse uses a large number of fixed phrases, the content is more predictable. As for poetry, although there are strict metrical requirements, the works of those great poets are difficult to predict. He believes that this is the unique symbol of the poet and the artistic impossibility, but the theory of probability helps to measure the value of art.

Although the idea of putting a novel such as War and Peace in a probabilistic sample space is scorned by Kolmogorov, he can express its unpredictability by calculating the complexity of War and Peace. Kolmogorov assumes that complexity is the shortest description length of an object, or the length of the algorithm that generates an object. The description of deterministic objects is simple. For example, it can be generated by a periodic sequence of zeros and ones. But uncertain, truly random objects are complex, and the length of any generation algorithm must be as long as the object itself. For example, irrational numbers, numbers after the decimal point have no rules to follow (recurring decimals can be expressed as a concise fraction). As a result, most irrational numbers are complex objects because they can only be written again as they are. This understanding of complexity is intuitive, that is, there is no way to predict and describe a random object. Today, this view is very important to measure the computing resources needed by an object, and it has been applied in network routing, sorting algorithm and data compression.

Kolmogorov, along with the students of the school he founded, internat.msu.ru understands that the impossible is the most likely, and by Kolmogorov's standards, his own life is complicated. Kolmogorov died in 1987 at the age of 84. He experienced the Russian Revolution, two World Wars and the Cold War in his life, and academically he touched almost all fields of mathematics, and his influence was far beyond that of academia. Whether his life course belongs to "drunken man wandering" or "mushroom picking trip without going back", it is difficult to predict and describe. He describes and applies the success of "impossible", which makes the theory of probability really "possible", thus opening up a new world for countless scientific and engineering applications. Of course, for unpredictability, his theory also widens the gap between human intuition and mathematical theory.

For Kolmogorov, his thoughts neither eliminate uncertainty nor affirm the fundamental uncertainty of our world. He just provides a rigorous set of language to discuss things that are uncertain. He once said that "absolutely random" is no more meaningful than "absolute necessity", and we cannot have an exact understanding of the unknowable.

But thanks to Andrei Kolmogorov, we can explain when and why we don't know.

This article is translated from The Man Who Invented Modern Probability

Https://nautil.us/issue/4/the-unlikely/the-man-who-invented-modern-probability

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