November 16, 2021

In 1956, shortly before his early death from bone cancer, John von Neumann received a letter from Kurt Gödel, the Austrian logician. After a paragraph of half-hearted inquiries into von Neumann’s health, Gödel finally got to the point: he had found an interesting new mathematical puzzle. And in the Fifties, if you found an interesting new mathematical puzzle, you sent it to John von Neumann.

The puzzle that Gödel was describing would come to be known as P vs NP. To oversimplify, it asks: can every mathematical question which can be checked quickly also be solved quickly? For instance: you are given a half-complete Sudoku puzzle. Is there a legal solution? If someone were to show you a solution, you could quickly verify whether it was legal. If you used a larger grid, the solution would take longer to check, but not exponentially so1.

But establishing that there is a legal solution is much slower. There might be quintillions of possible ways of filling it out; the number grows exponentially with the size of the grid. Checking them all one by one might take millions of years even on a powerful computer, if the grid is large enough.

What Gödel wanted to know was: is there some algorithm that could solve the Sudoku (or similar problems) as quickly as we could check a solution? P vs NP is one of the great outstanding questions of mathematics: it has profound implications, but no one has been able to prove it, one way or the other.

The Man from the Future, Ananyo Bhattacharya’s fascinating, fast-moving intellectual biography of von Neumann, made me think of P vs NP. Not because von Neumann solved it; but because von Neumann, in Bhattacharya’s telling, provided solutions to many other previously unsolved problems, in dozens of different fields; others simply had to check them, and expand on them. There is, I think, some discomfort about calling people “geniuses” these days, or in admitting that intelligence is a real thing or that it shapes history – but von Neumann was a genius, and his extraordinary intelligence shaped the modern world.

He was not an economist, but he developed the use of fixed-point theorems in economics in a paper which the historian Roy Weintraub calls “the single most important article in mathematical economics”, and which inspired “half a dozen” Nobel laureates.

His work on game theory – he invented the field, and coined the term “zero-sum game” – inspired at least half a dozen more. Game theory also transformed the study of evolution, inspiring the work of Bill Hamilton, John Maynard Smith, and Richard Dawkins.

He developed utility theory, the basis of modern economics. In 2011 Daniel Kahneman, another economics Nobel laureate (who won his Nobel partly for building on von Neumann’s game-theory ideas), called it“ the most important theory in the social sciences”.

Some of his last work, with Stanislaw Ulam on “cellular automata” – grids of squares that turn on and off according to simple rules – shaped modern computer science in thousands of ways, notably inspiring John McCarthy, who would go on to coin the term “artificial intelligence”.

Von Neumann’s genius was apparent early. In 1915, at the age of 11, he had gone to the famous gymnasium school in his native Budapest; the “legendary” maths teacher, László Rátz, immediately realised that von Neumann was beyond his ability to teach, and sent him for extra tuition at the local university. There he was mentored by Gábor Szegö, later head of Stanford’s maths department, who was “moved to tears” by his brilliance.

At 17, still at high school, he partly rescued Cantor’s set theory, the basis of much mathematical theory, from a crippling paradox. A couple of years later, he helped reconcile Werner Heisenberg and Erwin Schrödinger’s rival models of quantum mechanics. In the early Thirties, he met the astronomer Subrahmanyan Chandrasekhar, and worked with him on general relativity and the behaviour of stellar clusters. Chandrasekhar would later tell an interviewer, “If I say, ‘He reminds me of von Neumann,’ that’s about the best compliment I can give anyone.”

Von Neumann read some Alan Turing research which imagined a hypothetical computing machine, and saw how to build a working computer. The paper he produced building on Turing’s ideas is considered “the birth certificate of modern computers”, according to the computer scientist Wolfgang Coy. With his wife Kläri, and Ulam, he pioneered Monte Carlo simulations, vital now in climate modelling and a million other fields.

In almost every sphere of scientific inquiry – physics, biology, maths, economics, the social sciences, computing – you find von Neumann’s fingerprints. There is a Wikipedia page of “List of things named after John von Neumann.” Were it not for him, our understanding of the world would be decades behind where it is.

What created this genius? Bhattacharya does not speculate a great deal, but there are things worth considering. First, simple genetics: his family was high-achieving. His father was a doctor of law and an economic adviser to the Hungarian government; his uneducated maternal grandfather apparently could “add or multiply numbers into the millions” in his head instantly, a trick von Neumann emulated. The family was “puzzled” by their son’s inability to play the piano properly at the age of five, suggesting rather higher expectations than most. But it turned out to be because he “had taken to propping up books on his music stand so he could read while ‘practising’”.

He also grew up in a fertile environment. Around the turn of the 20th century, the Budapest Jewish community of which he was part produced an astonishing number of great thinkers. Near-contemporaries included Dennis Gabor, “who won the Nobel Prize in physics in 1971 for inventing the hologram”; Theodore von Kármán, after whom the “Kármán line” is named, denoting the boundary between the Earth’s atmosphere and space; and Eugene Wigner, Edward Teller, and Leo Szilard, three of the greatest minds behind the Manhattan Project. The atomic bomb has been described as a “Hungarian high school science fair project”.

The Hungarians who worked on America’s atomic weapons programme in the Thirties and Forties were known as “the Martians” by the other physicists – the joke being that the only way of explaining them was that super-intelligent aliens must have come to Budapest in the late 19th century and had babies with the locals. Von Neumann was the most alien of the lot.

But there was some accident of history that meant that European university departments at that time were disproportionately Jewish, and Belle Epoque Budapest, which was going through a less than usually antisemitic period, had a large and well-integrated Jewish population. Von Neumann himself speculated that insecurity drove this Jewish success – they recognised that Hungary’s tolerance might evaporate at any moment, and that they faced “the necessity to produce the unusual or face extinction”.

The tolerance did evaporate, in Hungary and elsewhere. Von Neumann, along with Teller, Wigner and the rest, had already left for Princeton, but Nazi persecution of the Jews in Germany devastated their universities: 15% of physicists and 19% of mathematicians were dismissed, including 20 who had won or would win Nobel prizes.

Ironically, this may have lost Germany the war: analysis suggests that the loss of Jewish scientists damaged German science for decades. Werner Heisenberg, a German quantum physicist, was branded a “white Jew” for believing in Einstein’s theories, despite being a nationalist. He later said that it was not worth Germany pursuing nuclear weapons, because they wouldn’t be ready in time to affect the war: he believed this because he thought Germany’s nuclear research was well ahead of other nations. “As it was,” says Bhattacharya. “Until 1933.”

So von Neumann, along with Wigner and others, ended up in Princeton — and then at the next-door Institute for Advanced Study, a sort of intellectual all-star team, where great brains were enticed from around the world with vast salaries, no undergrads to teach, and the promise that they could just think big thoughts. Einstein was there, along with Gödel, Robert Oppenheimer, Freeman Dyson, Ulam, and a host of others. It was an environment made for a certain kind of hard-to-pigeonhole genius, able to wander from subject to subject simply by walking around campus, surrounded by brilliant weirdos.

Von Neumann is compelling evidence, I think, that individual genius is important and influential. Yes, he worked in a series of collaborations; yes, he built on the work of others. But he seems to have pushed on, or sometimes simply created, entire subfields of science, into areas that no one else realised could exist. The economist Oskar Morgernstern remembers him rapidly devising utility theory, immediately overturning economic orthodoxy: “But didn’t anyone see that?”, von Neumann asked.

It’s fashionable to say that intelligence isn’t real, or that we can’t define it, or that it’s a Western colonial construct. But the word points to a real thing: there is some quality which rocks don’t have, and which mice have a bit of, and which chimpanzees have more of, and humans have a lot of; and which is something like problem-solving ability or ability to achieve goals. Calling it intelligence seems as good as anything. It is this ability which has allowed humanity to shape the world, and it is this ability that some people – von Neumann among them – seem to have in unusually large measure.

Via scientific and technological progress, intelligence has made human life better. But in itself, intelligence is morally neutral: it can serve any end, good or ill, to which it is put. Von Neumann is a case in point. He developed computers partly to better predict the behaviour of explosive shockwaves and ballistic shells; he designed two different kinds of nuclear weapon, including the plutonium implosion bomb that was dropped on Nagasaki; his game-theoretic ideas led him to suggest using atom bombs in a first strike on Russia, and he was part of the inspiration for Doctor Strangelove. He believed that all this was in the interest of America, his adopted country, and no doubt of humanity; but not everyone would agree with him. First and foremost he wanted to solve puzzles.

I put the book down wondering if it is still possible to encourage and harness genius. Perhaps it’s as simple as putting lots of clever people together and letting them think weird thoughts — and Von Neumann and his colleagues were often weird people. Or perhaps it is a true accident, and the only lesson is randomness. Perhaps the proposed new field of research into “progress studies” will yield some ideas as to how to recreate that environment in which Von Neumann and his fellow weirdos flourished.

I wondered, too, if John von Neumann was well enough to understand the P vs NP puzzle when he received that letter from Gödel. For it is a wonderful metaphor for genius. I can dimly understand, for instance, Turing’s solution to the “Halting Problem”, or Gödel’s incompleteness theorem, or Russell’s set paradox that undermined mathematics. (They’re all based on the “liar paradox” – the statement “this statement is a lie”, which is false if true or true if false.) But it often takes no great brilliance to understand an idea once it has been brought forth: checking the Sudoku solution is relatively straightforward. Finding that idea in the space of possible ideas, though — solving the great sprawling Sudokus of science and maths, as Von Neumann did time and again, that takes genius.

  1.  In fact the length of time would grow polynomially – hence P.