A Man from the Future: Why Johnny Can't Stop Abstracting

I have come to suspect that to most people thinking is painful. Some of us are addicted to thinking. Some of us find it a necessity. Johnny enjoyed it. I even have the suspicion that he enjoyed almost nothing else.

-- Edward Teller, eulogizing John von Neumann

I listened to Ananyo Bhattacharya's "The Man from the Future: The Visionary Life of John von Neumann" in my car during the many commute hours Bay Area traffic afforded me. I thought I should share the Johnny phenomena here.

I knew about von Neumann being a very influential mathematician/scientist/engineer but the full list still left me bedazzled. In his 53 years of life (1903-1957), von Neumann seeded so many fields. Every time there was an important discovery in physics, mathematics or computation, John von Neumann was there. Set theory and the foundations of mathematics. The mathematical formulation of quantum mechanics. Operator algebras (still called von Neumann algebras). Ergodic theory. Game theory. Expected utility theory (which is the foundation of most of modern mathematical economics). The Monte Carlo method. The stored-program computer architecture that every machine you have ever touched is built on. Numerical weather prediction. Fault-tolerant computing. Self-reproducing automata (cellular automata), and arguably, the logical structure of molecular biology, four years before Watson and Crick. There is also, unfortunately, the implosion lens for the Nagasaki bomb, and a decade of hawkish nuclear strategy.

At six, Janos (the OG hungarian name for Johnny) could divide eight-digit numbers in his head and hold a conversation in ancient Greek. He later added Latin, French, German, English, and Yiddish. The rumor was that he remembered everything he ever read. Many years later Herman Goldstine decided to test this by asking him to recite "A Tale of Two Cities." Von Neumann started on chapter one and had to be asked to stop fifteen minutes later.

Budapest around 1900 produced an absurd cluster of minds: von Neumann, Eugene Wigner, Edward Teller, Leo Szilard, Paul Erdős, Theodore von Kármán, George Pólya. When several of them turned up at Los Alamos and realized they had gone to high school together, Szilard joked that Hungary was a cover story for a Martian landing. But even inside the Martian cluster, von Neumann was the outlier. Wigner, a Nobel laureate himself, was asked at Los Alamos to explain the Hungarian phenomenon and said there is only one phenomenon that is worthy of explanation, and that phenomenon is John von Neumann. The cluster may be less mysterious than it looks: Budapest was one of the few genuinely welcoming places in Europe for Jewish families, and it pushed them toward the professions the nobility considered beneath them, mathematics among them. The schools then did the rest, with a nationwide math competition and top quality high school teachers.

As for the Johnny phenomenon, from the pedagogical perspective, the following stood out to me from Johnny's early years. The first is the dinner table. Von Neumann's father Miksa (Max), a lawyer turned banker so successful that the Emperor ennobled him for service to the national economy, ran the evening meal as a seminar. He would start by going through his own day: the cases he argued, the deals he negotiated, whatever was on his mind. Then each of the three boys would present on what they had been reading. The whole family would then argue about Heine's poetry, or antisemitism in Europe, or the paradoxes of omnipotence. Sometimes Max would bring home business clients, and the boys were allowed to grill them about whatever they wanted funded. Frequent guests included the director of Budapest's Theoretical Physics Institute and one of Freud's students.

The second is the languages. Max knew the good times could end, told his sons so explicitly, and wanted them able to survive anywhere in any language. He was preparing them to flee. He was right. When John was fifteen, the communists took over Hungary and targeted rich families like his. A few months later the counterrevolutionaries beat the communists and then massacred thousands of Jews for supposedly collaborating. The von Neumanns survived by leaving the country at the right moments, and possibly by being too rich.

Johnny's first ten years included no formal schooling at all, because organized primary schools were not a tradition there yet. He had governesses, private teachers, and his father's enormous library, and he apparently used it to memorize a 44-volume German history of the world, which he would later deploy to freak people out at parties.

Finally, when he did get to the Fasori Gymnasium at eleven, the math teacher, László Rátz, concluded fairly quickly that he could not teach this kid and arranged a private tutor. The tutor was Gábor Szegő, who went on to chair the math department at Stanford. Szegő is said to have come away from the first lessons in tears at what the boy could already do. The results of this tutelage showed up a couple years later. At seventeen Johnny gave a new definition of the ordinal numbers, where each number simply is the set of all the numbers before it. Zero is the empty set. One is the set containing zero. Two is the set containing zero and one. And so on up into the infinite ordinals, with nothing but the empty set and a pair of braces to build from. If you think about it, this is a very computer-science-y way to define numbers: an inductive datatype with one constructor.

He pushed this further in his doctoral thesis a few years later. Set theory at the time was still suffering from the paradoxes, Russell's set of all sets that do not contain themselves being the famous one. Johnny's fix was essentially a type system for sets: the ordinary ones you can put inside other sets, and the oversized ones you can talk about but never use as an element. The paradox then becomes a type error. Note that it is the same abstraction move in both times: take an object everyone thinks they understand, and replace it with the smallest thing that behaves the same way.

Johnny kept doing the same thing in field after field for the next thirty years: arrive, find the smallest set of objects and axioms the mess is actually about, write them down so that everyone can finally argue about the right thing, and then leave before the theorems fell out. Teller said Johnny enjoyed thinking, but I think the addiction was much more specialized than that. His real superpower was "the art of abstraction".

Look at the list from the introduction again with this in mind. Matrix mechanics and wave mechanics looked like rival theories until he noticed they were two coordinate systems on the same Hilbert space. Economists thought comparing preferences across people was hopeless until he asked what minimal consistency axioms a chooser has to satisfy, and derived a cardinal utility scale from them in an afternoon. Self-reproduction looked like it needed an infinite recursion of blueprints until he separated "read the tape as instructions" from "copy the tape as data."

Having finished his PhD at 21, Johnny arrived at Göttingen at 22 because it was the world capital of mathematics, under Hilbert. The timing couldn't have been better. Heisenberg, one year older, was down the hall having just invented matrix mechanics. A year later Schrödinger produced wave mechanics in Vienna. The two gave the same answers and implied completely different pictures of reality, and physicists were uneasy with both: matrices were unfamiliar mathematics then, and nobody could say what the waves were. Von Neumann's first move was to prove the two were the same theory in different forms. Then he wrote "Mathematical Foundations of Quantum Mechanics" (1932), which set the whole thing up rigorously in Hilbert space. The author of the book Bhattacharya, who did physics as an undergrad, points out that this is still the most watertight formulation we have.

Once the math was clean, the measurement problem became more visible. Where does the wave function actually collapse? At the apparatus, the detector, the retina, or the conscious observer? Von Neumann declined to take a position, but proved it does not matter mathematically where you draw the line because the answer comes out the same. That line is now called the von Neumann cut.

Nothing much came of game theory until von Neumann proved the minimax theorem in 1928, in his early twenties. Every two-player zero-sum game has a solution: either a strategy one player can pursue to win, or a mixed strategy with randomness in it (e.g., rock-paper-scissors). Then he lost interest and wandered off, as was his habit.

He came back when Oskar Morgenstern turned up at Princeton. Morgenstern, an economist, was stuck on what is essentially a liveness bug in economic modeling: you intervene in an economy, people adjust their behavior in response, which invalidates the model you used to design the intervention, so you build a new model, which they also adjust to. An infinite loop. Morgenstern had concluded that all attempts to model economies must fail, which made him unpopular right up until the Wall Street crash made him look prescient. Von Neumann had the tools to attack the problem. Over about eighteen months they produced the 600-page "Theory of Games and Economic Behavior" (1944) and founded the field. Johnny did this as a hobby. During those same years he was designing implosion lenses at Los Alamos and building the first programmable computers. He would drop in on Morgenstern for a long session and then vanish back into the rest of his life.

Two things from this work are still everywhere. The first is the axiomatization of expected utility. Putting human preferences on a cardinal scale and comparing them across people was considered more or less impossible, and von Neumann sat down one afternoon and did it. This relates uncomfortably with the longtermism material from the Cal Newport/Émile Torres conversation I wrote up recently. Expected value math is a hell of a drug.

The second is its limited model. His framework is built on zero-sum games and on coalitions: given n players, which coalitions are stable? Most of the book is about that. And most of the world is not zero-sum. Nash's non-cooperative equilibrium, which generalizes minimax, is what opened that up, and von Neumann rejected it essentially out of hand when Nash walked into his office. To him it was a trivial step, a fixed-point argument he could see through instantly. He misjudged the importance of the non-cooperative approach. Bhattacharya flags a nice irony here: von Neumann, the alleged Cold War hawk and the model for Dr. Strangelove, built a theory centered on people talking to each other and forming working coalitions. It was Nash, who had trouble with people, who formalized everyone optimizing alone. All models are wrong, but some are useful.

In 1944 Johnny ran into Herman Goldstine on a train platform at Aberdeen, found out about the ENIAC, and attached himself to the project. In 1945 he wrote the "First Draft of a Report on the EDVAC", laying out the stored-program architecture: one memory holding both instructions and data, a control unit, an arithmetic unit, I/O. That is now the architecture of virtually all computers, from the desktops to smartphones, eighty years later.

Von Neumann was adamant that the technical details go into the public domain, and circulating that draft is exactly what made the architecture unpatentable. So Johnny was also an early proponent of Open Source. He did the same with the IAS machine design, which got copied worldwide and produced a run of machines with goofy names: MANIAC at Los Alamos, JOHNNIAC at RAND, ILLIAC at Illinois, SILLIAC in Sydney, WEIZAC in Israel. For a man who gets painted as the cold rationalist hawk, putting the foundational design of the computing industry into the commons on purpose is a generous act.

He also, with Ulam, invented the Monte C