Andrey Kolmogorov

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Lore & Background

Andrey Kolmogorov was born in Tambov, about 500 kilometers southeast of Moscow, in 1903. His unmarried mother, Maria Yakovlevna Kolmogorova, died giving birth to him. He was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do nobleman. Little is known about his father, supposedly named Nikolai Matveyevich Katayev, an agronomist exiled from Saint Petersburg for revolutionary activity; he disappeared in 1919, presumed killed in the Russian Civil War. Kolmogorov was educated in his aunt Vera's village school, and at age five became the 'editor' of the mathematical section of the school journal 'The Swallow of Spring'. His interest in mathematics was spurred at age six when he noticed the regularity in the sum of the series of odd numbers: 1=1²; 1+3=2²; 1+3+5=3², etc. In 1910, his aunt adopted him and they moved to Moscow, where he graduated from high school in 1920. That same year he began studying at Moscow State University and at the Mendeleev Moscow Institute of Chemistry and Technology. As an undergraduate, he attended seminars of historian S. V. Bakhrushin and published his first research paper on fifteenth and sixteenth centuries' landholding practices in the Novgorod Republic. During 1921–22, he worked out and proved several results in set theory and the theory of Fourier series. In 1922, he gained international recognition for constructing a Fourier series that diverges almost everywhere. He graduated in 1925 and began studying under Nikolai Luzin, forming a lifelong close friendship with fellow student Pavel Alexandrov. Also in 1925, he published work in intuitionistic logic, 'On the principle of the excluded middle', proving that under a certain interpretation, all statements of classical formal logic can be formulated as those of intuitionistic logic. He earned his Doctor of Philosophy degree from Moscow State University in 1929. In 1930, he traveled to Göttingen, Munich, and Paris, working with Richard Courant, Hermann Weyl, and Edmund Landau. His pioneering work 'About the Analytical Methods of Probability Theory' was published in German in 1931, and he became a professor at Moscow State University that same year. In 1933, he published 'Foundations of the Theory of Probability', laying the modern axiomatic foundations of probability theory. In 1935, he became the first chairman of the department of probability theory at Moscow State University. Around 1936, he contributed to ecology by generalizing the Lotka–Volterra model of predator–prey systems. During the Great Purge in 1936, his doctoral advisor Nikolai Luzin became a target in the 'Luzin Affair'; Kolmogorov and several other students testified against Luzin, accusing him of plagiarism, nepotism, and other misconduct. Historians debate whether they were coerced; Semën Samsonovich Kutateladze concluded in 2013 that the students initiated accusations out of personal acrimony, with no definitive evidence of coercion by the state. In a 1938 paper, Kolmogorov established basic theorems for smoothing and predicting stationary stochastic processes, with major military applications during the Cold War. In 1939, he was elected a full member of the USSR Academy of Sciences. During World War II, he applied statistical theory to artillery fire and developed a scheme of stochastic distribution of barrage balloons to protect Moscow from German bombers. He independently developed the Chapman–Kolmogorov equations with Sydney Chapman. He married Anna Dmitrievna Egorova in 1942. Later, he focused on turbulence, beginning publications in 1941, and is best known in classical mechanics for the Kolmogorov–Arnold–Moser theorem, first presented in 1954. In 1957, with student Vladimir Arnold, he solved a particular interpretation of Hilbert's thirteenth problem. Around this time, he began developing algorithmic complexity theory, often called Kolmogorov complexity. He pursued vigorous teaching throughout his life, including developing pedagogy for gifted children in literature, music, and mathematics. At Moscow State University, he headed several departments: probability, statistics, and random processes; mathematical logic; and served as Dean of the Department of Mechanics and Mathematics. In 1971, he joined an oceanographic expedition aboard the research vessel Dmitri Mendeleev. He wrote articles for the Great Soviet Encyclopedia. He died in Moscow in 1987 and was buried in the Novodevichy cemetery. A quotation attributed to him: 'Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people.' Vladimir Arnold once said: 'Kolmogorov – Poincaré – Gauss – Euler – Newton, are only five lives separating us from the source of our science.'

Reader's Guide

Kolmogorov's significance lies in his foundational role in modern probability theory, which he placed on rigorous axiomatic grounds in his 1933 book 'Foundations of the Theory of Probability'. This work underpins much of modern statistics, physics, and finance. His contributions to algorithmic information theory—now called Kolmogorov complexity—established a measure of randomness and information content central to theoretical computer science. The Chapman–Kolmogorov equations are essential in the study of stochastic processes, and the Kolmogorov–Arnold–Moser theorem is a landmark in classical mechanics and dynamical systems. His work on turbulence, beginning in 1941, and Fourier series also had lasting impact. Despite his scientific achievements, his involvement in the Luzin Affair during Stalin's purges remains a controversial aspect of his legacy, with historians divided on whether his testimony was coerced or voluntary. Kolmogorov's wide-ranging erudition and pedagogical efforts, including developing curricula for gifted children, further cement his reputation as one of the most influential mathematicians of the 20th century.

Did You Know?

• Kolmogorov noticed the pattern in sums of odd numbers (1=1², 1+3=2², 1+3+5=3²) at age six, sparking his interest in mathematics.
• At age five, he was the 'editor' of the mathematical section of his school journal 'The Swallow of Spring'.
• His first research paper, published while an undergraduate, was on fifteenth and sixteenth centuries' landholding practices in the Novgorod Republic.
• In 1922, he gained international recognition for constructing a Fourier series that diverges almost everywhere.
• During World War II, he developed a scheme of stochastic distribution of barrage balloons to help protect Moscow from German bombers.