The Man Who Invented Modern Probability - Andrei Kolmogorov

The Paradox of the Great Circle was a major mathematical conundrum that Kolmogorov’s conception of probability finally put to rest. Assume aliens landed randomly on a perfectly spherical Earth and the probability of their landing was equally distributed. Does this mean that they would be equally likely to land anywhere along any circle that divides the sphere into two equal hemispheres, known as a “great circle?” It turns out that the landing probability is equally distributed along the equator, but is unevenly distributed along the meridians, with the probability increasing toward the equator and decreasing at the poles. In other words, the aliens would tend to land in hotter climates. This strange finding might be explained by the circles of latitude getting bigger as they get closer to the equator—yet this result seems absurd, since we can rotate the sphere and turn its equator into a meridian. Kolmogorov showed that the great circle has a measure zero, since it is a line segment and its area is zero. This explains the apparent contradiction in conditional landing probabilities by showing that these probabilities could not be rigorously calculated.

By Kolmogorov’s own measure, his life was a complex one.  By the time he died, in 1987 at the age of 84, he had not only weathered a revolution, two World Wars, and the Cold War, but his innovations left few mathematical fields untouched, and extended well beyond the confines of academe. Whether his random walk through life was of the inebriated or mushroom-picking variety, its twists and turns were neither particularly predictable nor easily described. His success at capturing and applying the unlikely had rehabilitated probability theory, and had created a terra firma for countless scientific and engineering projects. But his theory also amplified the tension between human intuition about unpredictability and the apparent power of the mathematical apparatus to describe it.

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