Genius in turbulent times

Forgot about Andrey

The other day on Twitter, the old “who is a stupid person’s idea of a clever person?” question was doing the rounds. It’s kind of a fun insult, even if it was likely first coined for Aldous Huxley, which seems pretty harsh. But someone turned it on its head and asked “who is a clever person’s idea of a clever person?”, which feels much more interesting.

I think there’s an implied level of hipsterness here. We might be looking for someone who only clever people know about, so I don’t think the answer can be Shakespeare, or Newton, or Einstein, or Galileo for example. I think even people like Shannon, Turing, Lovelace, Feynman or von Neumann might be ruled out, geniuses though they undoubtedly were, by virtue of being the subject of recent (excellent) biographies. So I’d like to suggest a different name instead: my great-grandfather Andrey Nikolayevich Kolmogorov. I don’t think he has anything like the level of popular name recognition that he deserves, and it’s time to do something about that.

Of course, when I say great-grandfather, I mean in the academic sense. That is, Kolmogorov supervised the PhD of the Abel Prize winner Yakov Sinai, Sinai supervised Yurii Suhov, Yurii Suhov supervised me. (Sadly I fear that there’s been a reversion to mean, quality-wise, along that family tree). Being Kolmogorov’s academic descendant doesn’t give me any rare distinction: in fact, one of his great contributions was to act as supervisor and mentor for many generations of Soviet mathematicians, meaning that (according to the fascinating Math Genealogy website) at least 4,000 people can make similar claims.

Indeed, Kolmogorov’s longevity meant that he lived through a turbulent period of Russian history: the fact that he was born in 1903 and died in 1987 meant that he lived to see two Russian revolutions, Stalin’s purges, the Great Patriotic War, the Khrushchev Thaw, Sputnik, the Cuban Missile Crisis, the Afghanistan invasion, Chernobyl, glasnost and perestroika. But of course, he’s not unique in that. However, his mathematical contributions are much more remarkable.

In 1900, the German mathematician David Hilbert presented a list of 23 unsolved problems which he believed would shape 20th Century mathematics. These included such notorious challenges as the Riemann Hypothesis and finding the densest packing of spheres. Solving any of Hilbert’s problems is enough to guarantee mathematical immortality, and even partial answers are very hard to come by - several of these problems remain unresolved to this day. Remarkably, Kolmogorov made significant contributions towards the solutions of two of Hilbert’s problems.

First, in 1933, he developed a framework of axioms that put probability theory on a rigorous footing, making a big step towards resolving Hilbert’s sixth problem. This perhaps doesn’t seem like a big deal: it might feel obvious what probability and randomness mean, but it’s worth remembering that the work of Kurt Gödel showed that it wasn’t possible to find similar consistent axioms for arithmetic. In particular, when you start to deal with infinities, apparently paradoxical things can sometimes happen. However, Kolmogorov’s axioms are now generally accepted as the standard framework, and he made many other important contributions in probability theory, such as the zero-one law which shows that a certain type of event must have probability zero or probability one - this is kind of surprising!

My copy of Kolmogorov’s “Foundations of the Theory of Probability”, setting out his axioms. This is the English translation, though it was first published in German as “Grundbegriffe der Wahrscheinlichkeitsrechnung” (fab compound word!), even though it was written by a Russian. Maths is very international.

In 1957, Kolmogorov’s student Vladimir Arnold resolved a version of Hilbert’s thirteenth problem, building directly on his supervisor’s work of the previous year. This led to what is now known as the Kolmogorov-Arnold theorem, which proves that complicated functions can be built up out of simple parts. This may feel like a technical result, but in fact it can be convincingly argued that this property underpins the way that neural networks work. On some level, some of the power of ChatGPT and all the other hot AI algorithms of 2023 goes back to Kolmogorov’s work over 60 years before.

It may not be obvious to non-mathematicians, not only are these both extremely significant contributions, they are in very different areas of maths. Most mathematicians tend to find a small niche, a subfield of pure or applied mathematics, and focus most of their energies there throughout their lives. The fact that this wasn’t Kolmogorov’s style might be indicated by the fact that his first published paper wasn’t in maths at all, but rather history. He had a deep and lifelong interest in the poetry of Pushkin and in his late 60s chose to join an oceanographic expedition. As a sign of “a clever person”, this kind of polymathy seems quite suggestive.

Indeed, throughout his life, Kolmogorov made vital contributions in a eclectic range of mathematical topics, far broader than the average researcher. These range from the purest of pure topics (mathematical logic, topology, algorithmic complexity and set theory) to extremely applied topics (including fundamental contributions in turbulence, in genetics and ecology and in statistics).

Indeed, it’s often the case that different groups of mathematicians will regard Kolmogorov as a genius simply on the basis of what he did in their own particular field, without even realising the value of his work elsewhere. In fact, there may well be people reading this who will complain that I haven’t talked about Kolmogorov’s contributions in information theory or dynamical systems (chaos) - but there’s just too much to talk about, and I don’t want to go on too long!

However, even just describing these scientific contributions is inadequate, without considering the context in which Kolmogorov was living and working. As I have described, his entire working life was under the rule of the Soviet communist party, and much of his rise to prominence came under Stalin’s rule, when the slightest issues could cause people to “disappear” overnight.

As Scott Aaronson describes, Kolmogorov was adept in navigating the challenges of scientific life under the Soviet system. He was a member of the Russian Academy of Science, winner of the Lenin and Stalin Prizes, and used probability theory to design the layout of barrage balloons to protect Moscow from German air attacks in the Second World War.

But equally, Kolmogorov was not simply a yes man of the regime. For example, his work on genetics led him into direct conflict with Lysenkoism, then the state-approved theory of biology, requiring a high degree of personal courage to oppose Stalin’s protégé.

Indeed, the remarkable aspects of Kolmogorov go beyond that. Despite the fact that male homosexuality had been criminalized by Stalin in 1933, and could be punished by five years of hard labour (which could often equate to a death sentence given the harshness of Soviet prisons), he came as close to living as an out gay man as it was possible to do at the time. While for obvious reasons there is limited documentary evidence that might confirm this for sure, it is generally accepted (as described in Masha Gessen’s book) that Kolmogorov and his fellow mathematician Pavel Alexandrov were lifelong partners, owning a house together from 1935. Indeed, it is even rumoured that Stalin was aware of the situation, and chose to ignore it because of the value of Kolmogorov’s scientific contributions.

I’ve only given a flavour of the depth and importance of Kolmogorov’s work, as well as the challenges of his personal life. But hopefully it has convinced you that he is indeed “a clever person’s idea of a clever person” — and perhaps deserves to be the subject of a popular biography or even a Hollywood movie, even if that means we have to look for our next hipster figure to celebrate.

Housekeeping notice: rate limits and other problems mean that Twitter is in trouble again, but if you found this interesting then do please share it on the social network of your choice (BlueSky, Mastodon, Threads also gladly accepted!). Do please subscribe if you didn’t already so that I can send you more articles directly (something like one a week). And do please let me know in the comments or on Substack Notes who your idea of a clever person is.