“Russians alone are able to combine so many opposites in themselves at one and the same time.”
― Fyodor Dostoyevsky, The Gambler
Mathematics: Chance encounters in the life of Andrei Kolmogorov.
If two statisticians were to lose each other in an infinite forest, the first thing they would do is get drunk. That way, they would walk more or less randomly, which would give them the best chance of finding each other. However, the statisticians should stay sober if they want to pick mushrooms. Stumbling around drunk and without purpose would reduce the area of exploration, and make it more likely that the seekers would return to the same spot, where the mushrooms are already gone.
Such considerations belong to the statistical theory of “random walk” or “drunkard’s walk,” in which the future depends only on the present and not the past. Today, random walk is used to model share prices, molecular diffusion, neural activity, and population dynamics, among other processes. It is also thought to describe how “genetic drift” can result in a particular gene—say, for blue eye color—becoming prevalent in a population. Ironically, this theory, which ignores the past, has a rather rich history of its own. It is one of the many intellectual innovations dreamed up by Andrei Kolmogorov, a mathematician of startling breadth and ability who revolutionized the role of the unlikely in mathematics, while carefully negotiating the shifting probabilities of political and academic life in Soviet Russia.
As a young man, Kolmogorov was nourished by the intellectual ferment of post-revolutionary Moscow, where literary experimentation, the artistic avant-garde, and radical new scientific ideas were in the air. In the early 1920s, as a 17-year-old history student, he presented a paper to a group of his peers at Moscow University, offering an unconventional statistical analysis of the lives of medieval Russians. It found, for example, that the tax levied on villages was usually a whole number, while taxes on individual households were often expressed as fractions. The paper concluded, controversially for the time, that taxes were imposed on whole villages and then split among the households, rather than imposed on households and accumulated by village. “You have found only one proof,” was his professor’s acid observation. “That is not enough for a historian. You need at least five proofs.” At that moment, Kolmogorov decided to change his concentration to mathematics, where one proof would suffice.
It is oddly appropriate that a chance event drove Kolmogorov into the arms of probability theory, which at the time was a maligned sub-discipline of mathematics. Pre-modern societies often viewed chance as an expression of the gods’ will; in ancient Egypt and classical Greece, throwing dice was seen as a reliable method of divination and fortune telling. By the early 19th century, European mathematicians had developed techniques for calculating odds, and distilled probability to the ratio of the number of favorable cases to the number of all equally probable cases. But this approach suffered from circularity—probability was defined in terms of equally probable cases—and only worked for systems with a finite number of possible outcomes. It could not handle countable infinity (such as a game of dice with infinitely many faces) or a continuum (such as a game with a spherical die, where each point on the sphere represents a possible outcome). Attempts to grapple with such situations produced contradictory results, and earned probability a bad reputation....MORE