Andrey Kolmogorov

Meet Andrey Nikolaevich Kolmogorov (1903–1987): the man who took probability — for three centuries a slightly shady subject of dice, card tables and gamblers' hunches, not quite respectable mathematics — and in 1933 turned it into the real thing with a handful of clean axioms built on measure theory. 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 statement — countable additivity — is the quiet workhorse behind the whole probability space. But pinning down chance was only one project among a dozen. Kolmogorov also wrote the equations of turbulence while a world war raged around him, proved deep results about whether the Solar System will fly apart, defined what it means for a string of symbols to contain information, and then spent his last decades doing the thing he said he'd always wanted to do anyway: teaching schoolchildren. He treated entire branches of mathematics the way other people treat weekend hobbies — and he was very serious about his actual weekend hobbies, which involved skiing enormous distances in shorts.

A childhood out of a Russian novel

Kolmogorov's life began with a tragedy so complete he never knew either parent. His mother, Maria Kolmogorova, died giving birth to him in April 1903, in the town of Tambov — she had been travelling home from Crimea and didn't make it. His father, an agronomist, played no part in his upbringing and vanished into the chaos of the Russian Civil War, dying in 1919. The baby was scooped up by his mother's sister, Aunt Vera, who raised him on his grandfather's country estate near Yaroslavl, formally adopted him, and gave him his mother's surname. By every account she was superb at the job: independent-minded, principled, and convinced that the boy in her care was capable of anything.

She may have been onto something. The village household ran a little handwritten journal for the children called Spring Swallows, and its mathematics column was edited by Andrey, aged five. His first published discovery appeared there: he had noticed that

1 = 1^2, \qquad 1 + 3 = 2^2, \qquad 1 + 3 + 5 = 3^2, \qquad 1 + 3 + 5 + 7 = 4^2, \; \ldots

— the sum of the first n odd numbers is always a perfect square. Mathematicians had known this since antiquity, but nobody had told him that; he found it the way he would find everything else, by looking hard at something until it confessed. A five-year-old rediscovering a classical theorem in a family newsletter is about as clear an omen as biography provides.

Curiously, mathematics nearly lost him — to history. Arriving at Moscow State University in 1920 (aged seventeen, in the hungry years just after the Revolution, when students earned bread rations and Kolmogorov earned extra money as a railway conductor), he threw himself into a serious study of medieval Novgorod's land records and presented a paper with a genuinely original thesis. The story goes that his history professor was impressed but unmoved: "You have supplied one proof of your thesis. In the mathematics that you study, one proof is considered sufficient. Here we require at least five." Kolmogorov reflected on this, decided he preferred a subject where one proof settles the matter forever, and went where the proofs were.

Famous at nineteen

In 1920s Moscow, the centre of the mathematical universe was the seminar of Nikolai Luzin — a charismatic, demanding analyst whose circle of brilliant students called itself Luzitania, half in jest, as if it were a country. Admission was by talent alone, and the young Kolmogorov walked in and detonated.

The question of the day concerned Fourier series — the trick of building complicated functions out of pure waves. Everyone "knew," or at least hoped, that a Fourier series of any reasonable function should converge almost everywhere; the field was inching toward a proof. In 1922, aged nineteen and still an undergraduate, Kolmogorov constructed a function whose Fourier series diverges almost everywhere — and then, for good measure, sharpened it a few years later to a series that diverges at every single point. It was the mathematical equivalent of a teenager wandering into a debate among professors and quietly setting the motion on fire. The example made him internationally famous before he had a degree, and it announced the signature Kolmogorov move: where others argued about what ought to be true, he would simply build the object that settled it.

He never really slowed down. By his mid-twenties he had proved landmark results on the law of large numbers, written a pioneering paper on intuitionistic logic, and begun the work on probability that would remake the subject. Asked much later about ambition among mathematicians, he offered one of the great deadpan lines of the profession: "Every mathematician believes that he is ahead of all the others. The reason none say this in public is that they are intelligent people."

1933: six pages that ended three centuries of hand-waving

For three hundred years, probability had worked without anyone knowing exactly what it was. Pascal and Fermat founded it in letters about gambling debts; Laplace built it into a calculating machine for astronomy and insurance; by 1900 it powered physics and statistics. And yet its basic notions were fog. What, precisely, is an "event"? What does it mean to say a probability is 1/2? Paradoxes lurked everywhere — you could set up the "same" random experiment two ways and get two different answers, with no principled way to say who was wrong. Hilbert found the situation embarrassing enough to list it in his famous 1900 catalogue of the great open problems: put probability on rigorous axiomatic foundations. For thirty-three years, nobody could.

Kolmogorov's 1933 monograph — the Grundbegriffe der Wahrscheinlichkeitsrechnung, the Foundations of the Theory of Probability — did it in a booklet so slim it barely qualifies as a book, with the axioms themselves dispatched in about six pages. The key insight was an act of radical modesty: stop trying to say what randomness is, and instead say how it must behave. Take the measure theory that Borel and Lebesgue had built for lengths, areas and integrals, and declare: a probability is simply a measure of total mass one. An event is a measurable set. Independence is a product. Conditional probability, expectation, the law of large numbers — all of it falls out as theorems, provable with the full rigour of analysis.

The effect was immediate and total. Overnight, the disreputable art of the card table became a branch of pure mathematics, and questions that had been philosophical shouting matches became exercises. Better still, the Grundbegriffe contained an extension theorem showing how to construct probability measures on infinite families of random quantities — which is exactly what you need to talk rigorously about a process unfolding in time: a stock price, a diffusing particle, a signal in noise. That single theorem is the legal foundation on which Markov chains, Wiener's Brownian motion and Itô's stochastic calculus all stand. Kolmogorov's own equations for diffusing processes — the "forward" and "backward" equations, plus the Chapman–Kolmogorov relation stitching a random journey together out of its legs — became, decades later, the mathematical backbone of quantitative finance, the machinery under every option-pricing model since Bachelier's lonely head start. Every bank on Earth now runs, unknowingly, on a Soviet monograph from 1933.

It's tempting to file Kolmogorov as "the inventor of probability." He wasn't, and the distinction he'd want you to draw is one of the most useful in all of mathematics.

People had been using chance expertly for centuries before him. Gamblers priced bets by instinct; Cardano wrote a gambler's manual in the 1500s; Pascal and Fermat solved the problem of fairly splitting an interrupted game's pot in 1654 and founded the calculus of chances; the Bernoullis proved the first law of large numbers; Laplace computed planetary uncertainties and jury verdicts; Maxwell and Boltzmann built the physics of gases on randomness. By 1900, probability worked — brilliantly.

What it lacked was a definition. Nobody could say what the theory was about with the precision that geometry or arithmetic enjoyed, and where there is no definition there can be no proof — only plausible reasoning that usually succeeds and occasionally, spectacularly, doesn't (ask two mathematicians to pick a "random chord of a circle" and you could get three different answers, each impeccably argued). Kolmogorov's contribution was not the subject but its rigour: axioms that turned three centuries of intuition into theorems, so that a probabilistic argument could be checked, line by line, like any other piece of mathematics. The difference between using chance and defining it is the difference between being a superb cook and writing the chemistry of cooking — and it's the second one that lets everyone after you build skyscrapers on the recipe.

The breadth explosion

Having rebuilt probability, Kolmogorov did something almost nobody of his stature does: he changed fields. Repeatedly. Deliberately. He said the ideas came in waves, and when a wave receded he went looking for the next one. Four crests deserve the tour.

Turbulence (1941). Swirling fluid — the chaos behind a boat, the gusts in a storm — was and largely remains physics' great unsolved embarrassment. In 1941, in wartime papers written as the German army drove toward Moscow, Kolmogorov cut into the problem statistically: in a turbulent flow, big eddies feed their energy to smaller eddies, those to smaller still, in a cascade that ends only when the swirls are tiny enough for friction to eat them. From two crisp similarity assumptions he deduced how the energy must be shared across eddy sizes — the celebrated 5/3 law, which says the energy at scale k falls off like k^{-5/3}. Measure a real gale, a wind tunnel, or the plasma between the stars, and there, again and again, is the Kolmogorov spectrum. It remains the single most important result ever obtained about turbulence — scribbled out with wartime data and dimensional analysis while the world burned.

Planetary stability — KAM theory (1954). Is the Solar System stable, or will the planets' accumulated tugs on one another eventually fling one into the void? Newton worried about it; Laplace thought he'd settled it; Poincaré showed the question was vastly harder than anyone dreamed. In 1954 Kolmogorov announced the decisive breakthrough: under small perturbations, most of the orderly orbits of a well-behaved system survive — deformed, but not destroyed. His student Vladimir Arnold and the German-American Jürgen Moser hammered the proof out in full, and the result is now called KAM theory after all three. It is the deepest thing known about the boundary between order and chaos: a mathematical explanation of why the sky, mostly, stays put.

The information in a single object (1960s). In his sixties, Kolmogorov asked a question that sounds like philosophy and answered it like an engineer: how much information does one particular object contain — not on average, as in Shannon's theory, but this string, this picture? His answer — measure a thing by the length of the shortest program that produces it — is now called Kolmogorov complexity, and it gives the first rigorous meaning to words like "simple," "complex" and "random." (More on this gem, and its beautiful built-in catch, in the vignette below.)

And the rest. Along the way he co-solved Hilbert's 13th problem with the 19-year-old Arnold, showing that any continuous function of many variables can be built by composing functions of one variable with addition — a theorem that resurfaces today in the mathematics of neural networks. He founded the theory of \varepsilon-entropy, reshaped mathematical statistics (the Kolmogorov–Smirnov test is his), wrote on intuitionistic logic, topology and classical mechanics, analysed the rhythms of Pushkin's poetry statistically, and — in 1940, at real personal risk, when the charlatan Lysenko was destroying Soviet biology with Stalin's blessing — published a paper politely demonstrating that a set of experimental results claimed to refute Mendelian genetics in fact confirmed it beautifully. A list of his distinct research fields runs past a dozen. It is genuinely difficult to name another twentieth-century mathematician with both his depth and his range.

Here is Kolmogorov complexity in one sitting. Compare these two strings of 24 digits:

111111111111111111111111 \qquad \text{vs.} \qquad 731205964012877340259118

Both are 24 characters long, so a file containing either takes the same space. But your intuition screams that the first is simpler — and Kolmogorov found the right way to say why. The first string has a very short recipe: "print 1 twenty-four times." The second (if it's genuinely random) has no recipe meaningfully shorter than the string itself — the only way to specify it is, essentially, to quote it. Define the complexity of a string as the length of the shortest computer program that outputs it, and suddenly "simple" means compressible and "random" means incompressible: a random string is precisely one that is its own shortest description. The first million digits of \pi look like noise, but they're deeply non-random — a five-line program prints them all.

Now the catch, and it's a beauty: Kolmogorov complexity is uncomputable. No algorithm can take an arbitrary string and return the length of its shortest program — this follows from the same wall of impossibility that Turing discovered with the halting problem. There's even a paradox-flavoured proof sketch: if a program could identify "the first string whose shortest description exceeds a billion characters," that program — far shorter than a billion characters — would itself be a description of the string, contradiction. (Compare the old riddle of "the smallest number not describable in under eleven words" — which that phrase just described in ten.) So mathematics now possesses a perfect, rigorous definition of randomness that provably can never be fully checked. You can prove a string is simple, by exhibiting the short program; you can almost never prove one is random. Kolmogorov, who had defined probability without defining randomness in 1933, returned thirty years later to define randomness itself — and found nature had locked the last door.

The teacher who kept a promise

Ask Russian mathematicians about Kolmogorov and many will speak of the theorems second. First they'll tell you he was a teacher — perhaps the most consequential mathematics teacher of the century. He supervised, by the usual count, some seventy-nine doctoral students, and the list reads like the index of a textbook on twentieth-century mathematics: Vladimir Arnold, Israel Gelfand, Yakov Sinai, Eugene Dynkin, Albert Shiryaev, and on and on. His method was the opposite of lecturing at people: he would take students hiking. Twenty, thirty, forty kilometres of walking, skiing or rowing, with mathematics discussed in bursts along the way, an icy swim in whatever river presented itself (students were encouraged, sometimes with alarm, to join), and music or poetry in the evening. He believed a mathematician should be a whole human being — that the same person ought to know Bach, snow, and the strong law of large numbers — and he manufactured mathematicians of exactly that kind by example.

Then he went further down the age range. In 1963 he founded a boarding school in Moscow for mathematically gifted children scooped up from the provinces — kids from towns with no advanced teachers at all. Now known simply as the Kolmogorov school, it was staffed at the top: one of the greatest living mathematicians wrote its curriculum and then stood in its classrooms, week after week for years, teaching teenagers — and not only mathematics but music, art history and Russian poetry, on the same whole-human-being principle. He also led a sweeping (and stormily debated) reform of the Soviet school mathematics syllabus, wrote textbooks for ordinary schools, and edited a mathematics magazine for children. He had said for years that he planned to spend his last working decades as a schoolteacher. Almost uniquely among people who say such things, he did it.

Komarovka: the good life, rigorously pursued

The centre of Kolmogorov's private world was a old wooden house in the village of Komarovka outside Moscow, which he shared for over fifty years with the topologist Pavel Aleksandrov — his closest friend and lifelong companion from their first long Volga boat expedition together in 1929 until Aleksandrov's death in 1982. The house ran on a schedule as considered as an axiom system: days of intense work; one day a week declared a walking day, in any weather; Sunday evenings given over to music, the two of them working through a record collection of Bach, Mozart and Vivaldi with invited students sprawled about the room. Between them they more or less ran Moscow mathematics from a house with a garden.

The physical regime was legendary and slightly terrifying. Kolmogorov swam in the Klyazma river practically the moment the ice permitted (and sometimes before), took cross-country ski runs of many kilometres wearing as little as the law allowed, and kept it all up into his eighties. Visitors invited for "a short walk" learned to bring provisions. He held that his best ideas arrived in motion — on skis, in the water, on the trail — and given the publication record, it is hard to argue with the method.

The end was cruel in a specific way: Parkinson's disease slowly took his speech and his sight, the two channels through which a teacher lives. His students organised themselves into a rota and cared for him around the clock in his final years — reading to him, taking dictation, sitting with him through the nights. It is its own kind of theorem about the man: you can measure a teacher by what his students do when he can no longer teach. He died in Moscow on 20 October 1987, aged 84.

The Luzin affair: a stain and a knot

Be honest about your heroes. In 1936, Kolmogorov's own teacher — Nikolai Luzin, the founder of Luzitania — was subjected to a Stalinist press campaign: denounced in Pravda as an "enemy in a Soviet mask," accused of servility to the West for publishing in foreign journals, of sabotage in his teaching, of plagiarism. In the Soviet Union of 1936, at the mouth of the Great Terror, such campaigns routinely ended in the Gulag or the execution cellar. An Academy commission was convened to try him — and among the colleagues who testified against Luzin were his own former students, Kolmogorov and Aleksandrov included.

The knot is that nothing about it is simple. The scientific grievances were partly real — Luzin had a documented habit of, let us say, under-crediting his students' results — and the generational revolt against him predated the politics. But the venue was a Stalinist show trial in all but name, and the young stars lent it their names. Historians still argue about the mixture of motives: genuine anger, career ambition, fear, and the impossible arithmetic of behaving decently inside a terror state, where refusing to testify carried risks we can barely price from the comfort of hindsight. Luzin, in the end, was censured but — unusually, perhaps on a signal from above — not arrested; he kept his Academy chair and died of natural causes in 1950. The wound between teacher and students never healed. A famous coda, ten years later: after a public insult from Luzin, Kolmogorov — a member of the Academy in good standing, in front of witnesses — slapped him across the face. The story goes that when the scandal reached Stalin he waved it off: "such things happen with us too." Kolmogorov's later moral ledger is genuinely mixed company for that entry — recall the real courage of the 1940 pro-genetics paper against Lysenko. Great mathematicians, it turns out, are exactly as complicated as everyone else, only with better-documented decisions.

A life in ten lines

The honours piled up — the Stalin, Lenin, Balzan and Wolf prizes, memberships of a couple of dozen academies — but the better summary is structural: open a modern textbook on probability, statistics, dynamical systems, information theory, turbulence or the theory of computation, and within a chapter you will hit his name. Few people have ever added so many load-bearing walls to the building.

Read more

The full story (with far fewer jokes) is on Wikipedia: Andrey Kolmogorov — Wikipedia. For the axioms, see the article on the probability axioms; for the shortest-program idea, Kolmogorov complexity; and for the darkest chapter, the Luzin affair.