»  Chronicles

February 2023

  The Mathematics Behind the Man

The Man from the Future
        by Ananyo Bhattacharya

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It is a brave author who attempts a biography of Hungarian mathematician John von Neumann (1903-57). The business of a biographer is to give us both the man and his work. The man, for any author well-acquainted with human nature, should not present much difficulty. Von Neumann's work, however, was at the very highest levels of human intellection in math and physics, with direct consequences in fields as disparate as nuclear weaponry and economics.

To give a fair account of von Neumann's work to a lay reader is therefore a daunting challenge. So far as I know, Ananyo Bhattacharya's is the first attempt in English since Norman Macrae's thirty years ago. Macrae was an economist, in fact an editor at The Economist, well equipped to describe von Neumann's contributions to that discipline. He confessed in his book, however, that most of von Neumann's math and physics was beyond him, and he had been helped by friends qualified in those fields.

Bhattacharya is a physicist. He knows the relevant math and science. That is a great help in explaining von Neumann's work. However, it tempts the biographer to give us too much of the work, not enough of the man. To put it as positively as I am able, Bhattacharya has not skimped on describing the content, prior history, and subsequent development of von Neumann's main interests.

His last chapter, covering his subject's final decade, is much too long. Von Neumann's primary focus in those last years was summed up in the title of his posthumous book Theory of Self-reproducing Automata. It is hard not to be struck by how few practical results have come from these cogitations of seventy years ago by comparison with the swift and striking impact of von Neumann's earlier work in nuclear weapons and computing. Where are the self-building moonbases?

So who was John von Neumann?

Von Neumann was of Ashkenazi-Jewish lineage. He therefore belonged to that human stock which, of all that are known, has the highest mean IQ, for reasons plausibly explained by Cochran, Hardy, and Harpending in their 2006 paper "The Natural History of Ashkenazi Intelligence." To be a high-IQ outlier among the Ashkenazim is to be an outlier indeed.

Furthermore von Neumann had the good fortune to be raised in a time and place — the Hungarian belle époque of the years around 1900 — in which the Ashkenazim had unlimited freedom to express their talents. Hungary had been granted autonomy from Austria in the Compromise of 1867 and was a more-or-less equal partner in the "imperial and royal" Dual Monarchy. The ruling Emperor-King of Austria-Hungary was Franz Josef I, a philosemite. "I will not tolerate Jew-baiting in my empire," Franz Josef told his ministers more than once. Von Neumann's father Max, an investment banker, was granted a hereditary title by Franz Josef for services to the national economy: hence the "von," although Max did not use it.

John von Neumann was recognized as a prodigy as soon as he could speak. He breezed through his elementary and high-school education with top grades in all but physical education, music, and handwriting.

That staggering intellect aside, he was in adult life perfectly normal, with none of the geeky weirdness of Paul Erdős or the mental instability of Kurt Gödel. He was well-groomed and dapper. Bhattacharya's book contains nine photographs in which the adult John von Neumann can be identified: he is wearing a suit in every one, a tie in eight. He had the manners of a gentleman, and took criticism well. Uncommonly among mathematicians, he had no interest in music. Even more counter-intuitively, but consoling to us chess duffers, he was only an average player.

Von Neumann was in fact something of a bon vivant. Lecturing at the University of Berlin in 1927-29 he enjoyed that city's legendary night life. He first met his second wife in a Monte Carlo casino, von Neumann "seated at one of the more modestly priced roulette tables." (He had a system, of course.) The couple's parties at Princeton were famously lively. Von Neumann's automobile of choice was a Cadillac.

In politics he was conservative, and in the early Cold War years took a strong line against the Soviet Union in all the many defense-related consulting positions he held. As a teenager in Budapest he had lived through Béla Kun's brief but nasty Leninist regime, and the experience inoculated him for life against communism.

When his leftist friend Robert Oppenheimer was summoned to secret hearings of the Atomic Energy Commission in 1954 under suspicion of being a security risk, Von Neumann none the less defended Oppenheimer vigorously, although unsuccessfully. When, the following year, von Neumann himself accepted a position on the AEC — the body that had canceled Oppenheimer — his leftist colleagues were very critical. Oswald Veblen never forgave him.

Von Neumann's family were not observant Jews. When Max died in 1929 they converted to Catholic Christianity; but religion played little part in John von Neumann's life until his sad last days, when he was hospitalized with bone cancer. He then took instruction from the hospital's Catholic priest. With proper respect to a fellow mathematician three hundred years his senior, he also told his daughter that he conceded the point of Pascal's wager.

So much for the man. What about the work?

Mathematicians traditionally divide their subject into "pure" and "applied." When practicing pure mathematics they are taking delight in abstract ideas that appear in our minds "as if summoned from the void" (Grothendieck). Applied mathematicians use those ideas as tools with which to study the natural world and solve practical problems: the motions of the planets, the generation of electricity, the most effective way to fire an artillery shell.

The border between pure and applied is not always clearly marked, and there are flows from one side to the other. What is the purest of pure intellection to mathematicians of one generation may find practical relevance a hundred years later, as marveled at by von Neumann's lifelong friend Eugene Wigner in a famous 1960 essay. Contrariwise, pondering the motions of solid bodies may lead to concepts taken up and brought to full flower by pure mathematicians, as happened with calculus.

Von Neumann moved effortlessly between the pure and the applied realms. The Wikipedia page "List of things named after John von Neumann" lists 52 concepts, theorems, conjectures, and so on; 25 are indisputably pure-mathematical. The corresponding numbers for Albert Einstein are 50 and eight.

His doctoral thesis, submitted when he was 22, was on the axiomatization of set theory, firmly in the zone called "Foundations." What is the nature of mathematical truth? What are the most fundamental concepts? Can propositions always be rigorously proven either true or false? This, the purest of pure math, was a key area of research in the 1920s.

The frontier of physics at that time was Quantum Theory, which was badly in need of mathematical help. Turning to the applied side, Von Neumann dived in, at age 28 publishing Mathematical Foundations of Quantum Mechanics in which he aimed to show that quantum mechanics arises naturally from the properties of a highly abstract pure-mathematical object called Hilbert space.

(How abstract is Hilbert space? Well, if you think a space of four dimensions is a stretch, try imagining a space with infinitely many dimensions. It's not as hard as it sounds: Pythagoras' Theorem, for instance, can be put to useful work in Hilbert space. After explaining this to my own undergraduate class in Functional Analysis, Dr Kestelman broke off to chortle: "Pythagoras would have sacrificed a chicken if you'd shown him that!")

By this point — the early 1930s — von Neumann was known among mathematicians and physicists worldwide as a genius. In 1929 he had been hired to lecture at Princeton along with the aforementioned Eugene Wigner. In 1933, at age 29, von Neumann joined the new Institute for Advanced Study, along with 54-year-old Albert Einstein.

Now a U.S. citizen (1937), and convinced there would soon be a European war, von Neumann became a consultant to the U.S. Army's Ballistics Research Laboratory in Maryland. Other defense-related appointments followed. When, in 1942, the Manhattan Project was launched, he was inevitably drawn in, and made key contributions to the design of the first nuclear weapons.

Simultaneously with this military work, from mid-1941 on von Neumann had been collaborating with the economist Oskar Morgenstern on a book about the mathematics of games. Given the stakes, the odds, the rules, and the likely behavior of opponents, how can we determine an optimal strategy? Theory of Games and Economic Behavior came out in 1944. It is now considered a classic in its field.

The military work had involved great masses of numerical computation. That turned von Neumann's thoughts to computing machines. He got involved in plans for a machine more advanced than the best then available, to be called EDVAC, the Electronic Discrete Variable Computer. In June 1945 he issued a paper titled "First Draft of a Report on the EDVAC," the foundational document of modern computer science. The internal organization of computers has ever since been based on prescriptions in that paper — on the "von Neumann architecture."

In the postwar years von Neumann worked on issues of complexity and on machines that can replicate themselves, the latter leading to him being considered the father of the computer virus.

All of that, and more, is described by Bhattacharya in lay terms as well as it can be. I put down this book knowing much more about cellular automata than I had before. About John von Neumann the man, though, I was very little wiser.