Andrey Kolmogorov

Before Andrey Kolmogorov (1903–1987), probability was a slightly shady subject — dice, card tables, and gamblers' hunches, not quite respectable mathematics. In 1933 he fixed that with three clean axioms built on measure theory, and overnight turned it into the real thing. Every probability you will ever compute sits on his foundation:

\mathbb{P}(A) \ge 0, \qquad \mathbb{P}(\Omega) = 1, \qquad \mathbb{P}\!\left(\bigsqcup_i A_i\right) = \sum_i \mathbb{P}(A_i).

That third line — countable additivity — is the quiet workhorse behind the whole probability space.

The polymath from nowhere

Orphaned as a baby and raised by an aunt, Kolmogorov went on to do landmark work in an almost comical range of fields: the turbulence of tumbling fluids, the complexity of information (Kolmogorov complexity — the length of the shortest program that could print a given string), dynamical systems, cohomology, and more. He treated entire branches of mathematics the way other people treat weekend hobbies.

The teacher

He was as proud of his schools for gifted children as of any theorem — personally designing their curricula and dragging students off on hikes and to concerts, on the theory that a mathematician should also be a whole human being. A startling fraction of the great Russian mathematicians of the next generation — including the people whose ideas Itô and Wiener would build alongside — simply called him "teacher".

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The full story (with far fewer jokes) is on Wikipedia: Andrey Kolmogorov — Wikipedia.