John Von Neumann

The life of John von Neumann

Timeline 1903Born on 28 December in Budapest, the son of a banker. 1913His father receives a title of nobility; the son becomes “von Neumann”. 1921Finishes the Lutheran gymnasium in Budapest, where he studied alongside Eugene Wigner. 1923Publishes, at 19, his definition of ordinal numbers. 1926Diploma in chemical engineering in Zurich and doctorate in mathematics in Budapest. 1927Becomes the youngest Privatdozent at the University of Berlin. 1928Publishes the minimax theorem, the basis of game theory. 1930Moves to Princeton. In 1933 he is one of the first professors of the Institute for Advanced Study. 1932Publishes the Mathematical Foundations of Quantum Mechanics. 1943Begins working at Los Alamos, on the atomic bomb project. 1944Publishes, with Oskar Morgenstern, the Theory of Games and Economic Behavior. 1945Writes the EDVAC report, which describes the stored-program computer. 1946With Stanisław Ulam, develops the Monte Carlo method. 1952The IAS computer comes into operation. 1955Appointed to the Atomic Energy Commission. Discovers he has cancer. 1957Dies on 8 February, in Washington, aged 53. ◀ Previous Follow the journey Next ▶ 1903–1921 BudapestHe is born on 28 December, the son of a banker. He studies at the Lutheran gymnasium, where Eugene Wigner is a schoolmate. 1921–1923 BerlinHe studies chemistry, at his father’s insistence, while enrolled for a doctorate in mathematics in Budapest. 1923–1926 ZurichHe graduates in chemical engineering at the Polytechnic. He attends classes by George Pólya, who would say he was the only student he was ever afraid of. 1926–1927 GöttingenHe works with David Hilbert on the foundations of mathematics and of the new quantum mechanics. 1927–1930 BerlinThe university’s youngest Privatdozent. In 1928 he publishes the minimax theorem. 1930–1957 PrincetonProfessor at the university and, from 1933, at the Institute for Advanced Study, alongside Einstein and Gödel. There he builds the IAS computer. 1943–1955 Los AlamosConsultant to the atomic bomb project. With Ulam, he creates the Monte Carlo method. 1955–1957 WashingtonMember of the Atomic Energy Commission. He dies of cancer on 8 February 1957, at Walter Reed Hospital. Enable JavaScript to see the map. The stages are listed below. From Budapest to Washington, by way of Zurich, Göttingen, Berlin, Princeton and Los Alamos. Map: © OpenStreetMap contributors

October 26, 2026 Read

Ordinals: numbers made of empty sets

At the beginning of the twentieth century mathematicians wanted to build the whole of mathematics from a single idea, that of a set. But what, in that language, is the number $3$? Georg Cantor had spoken of “ordinals”, numbers that measure positions in a queue: first, second, third, and after all the finite ones, a first infinite one, $\omega$. What was missing was to say what these numbers are, as sets.

October 26, 2026 Read

The minimax theorem: how to play without being second-guessed

In odds and evens there is no good move. If you always show one finger, your opponent learns and starts winning every time. The only defence is not to be predictable: to play at random, one half of the time, two the other half. Then, whatever your opponent does, you win half the games. In 1928, in a paper called On the theory of parlour games, von Neumann showed that this idea holds for every two-person zero-sum game, in which what one wins the other loses. And that it always leads to an exact number, the value of the game.

October 26, 2026 Read

Quantum mechanics in a Hilbert space

In 1925 and 1926 two versions of quantum mechanics were born almost at once. Werner Heisenberg’s used infinite matrices; Erwin Schrödinger’s, a wave equation. Both gave the same results in experiments, but they seemed to speak of different things, and physicists used tricks that mathematicians found unacceptable, such as Dirac’s delta “function”. In Göttingen, working with Hilbert, von Neumann showed that the two were the same theory written in different coordinates, and rewrote everything in a single language. The book Mathematical Foundations of Quantum Mechanics, of 1932, is still the reference for anyone who wants the theory with full rigour.

October 26, 2026 Read

The von Neumann architecture: the EDVAC report

The ENIAC, finished in 1945 at the University of Pennsylvania, was the fastest electronic computer in the world: it did five thousand additions a second with almost eighteen thousand valves. But to change calculation one had to rewire cables and switches on a panel, a job of days done by a team of programmers. Two programmers operating the ENIAC’s control panel, at the Moore School of the University of Pennsylvania, around 1946.United States Army · Wikimedia Commons · public domain The ENIAC’s engineers, J. Presper Eckert and John Mauchly, were already planning the next machine, the EDVAC, in which the program would be kept in memory. Von Neumann began attending the meetings and, in June 1945, wrote a text that organised the group’s ideas in an abstract, almost biological language, speaking of “organs” and “neurons”.

October 26, 2026 Read

Merge sort: sorting by joining

Sorting a list is one of the tasks computers do most: names in a register, prices in a search, election results. In 1945, when the EDVAC was still on paper, von Neumann wanted to show that the machine in his report was good for work that was not just arithmetic, and wrote by hand a program to sort and merge lists. It is one of the first programs written for a stored-program computer, and the method he used, now called merge sort, is still one of the most widely used.

October 26, 2026 Read

The Monte Carlo method: calculating by rolling dice

In 1946 the Polish mathematician Stanisław Ulam, von Neumann’s colleague at Los Alamos, was recovering from encephalitis and passing the time playing solitaire. He wondered what the probability was that a game would come out. Working it out by combinatorics was hopelessly complicated. But there was another way: play hundreds of games and count how many came out. With a computer, one could play thousands. Ulam told von Neumann the idea, and he saw its reach at once: the same trick served for the problem occupying them at Los Alamos, the path of neutrons in a bomb or a reactor, where each neutron is deflected, absorbed or multiplied at random. Instead of solving the equations, simulate thousands of neutrons, drawing lots for what happens to each one, and take averages. Nicholas Metropolis named the method after the Monte Carlo casino, where an uncle of Ulam’s used to gamble.

October 27, 2026 Read

The biased coin that becomes fair

Suppose you have a crooked coin. It lands heads with a probability $p$ that you do not know, perhaps $0.7$, perhaps $0.55$. Can you use it for a perfectly fair, fifty-fifty draw? In the same 1951 paper in which he spoke of the “state of sin” of those who manufacture random numbers, von Neumann gave a short and surprising answer. The trick Toss the coin twice: if it comes up heads, tails, say “heads”; if it comes up tails, heads, say “tails”; if it comes up heads, heads or tails, tails, forget it and toss twice more. Why is it fair? The two tosses are independent, so

October 27, 2026 Read

Self-reproducing automata

In the last years of his life von Neumann asked himself a question that sounded like philosophy: can a machine build a copy of itself? The difficulty is logical. To build a copy, the machine needs a description of itself; but the description is part of the machine, so it would have to describe itself, and so on, in an endless regress. Description, constructor and copier Von Neumann’s way out, presented in lectures from 1948 on, was to separate three pieces:

October 27, 2026 Read

The legacy of John von Neumann

I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man. — Hans Bethe, Nobel laureate in Physics, as quoted by Norman Macrae The eight texts in this trail show only part of what von Neumann did. The full list fills six volumes of collected works and runs through almost every branch of the mathematics of his time. Pure mathematics Operator algebras. Between 1929 and 1943, partly with Francis Murray, he created the theory of operator algebras now called von Neumann algebras, one of the most active fields of analysis and mathematical physics. The ergodic theorem. In 1932 he proved a version of the theorem that justifies a principle of statistical physics: the average of a quantity over time equals the average over all possible states. Hilbert’s fifth problem. In 1933 he solved the compact case of one of the problems Hilbert had posed in 1900, about when a continuous group is automatically “smooth”. Economics, weather and strategy Economic growth. In a paper published in 1937, presented years earlier at a seminar in Princeton, he built a model of an expanding economy and proved the existence of an equilibrium with the same mathematics as minimax. It is one of the founding texts of mathematical economics. Weather forecasting. In 1950 a group led by Jule Charney, with von Neumann, made the first numerical weather forecast on the ENIAC: 24 hours of forecast took about 24 hours of calculation. The models that forecast weather and climate today descend from that experiment. Nuclear strategy. In the Cold War, game theory was used to think about the arms race, and von Neumann took part in planning intercontinental missiles. The phrase “mutually assured destruction”, tied to that reasoning, still sums up an era. The word “singularity” In 1958 Stanisław Ulam wrote a piece in memory of his friend. In it he recalled a conversation about the ever-accelerating progress of technology, which seemed to be approaching “some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue”. It is the first recorded use of the word in this sense, which decades later became a theme of discussions about artificial intelligence.

October 27, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, with no editing beyond compression. Von Neumann’s Los Alamos badge photo, Los Alamos National Laboratory: source page · free use with attribution to the laboratory. Von Neumann in New Mexico, Los Alamos National Laboratory: source page · free use with attribution to the laboratory. The plaque on his birthplace in Budapest, photo by Misibacsi: source page · CC BY-SA 3.0. The Lutheran gymnasium of Budapest, photo by VinceB: source page · CC BY-SA 3.0. Fuld Hall, Princeton, photo by Zeete: source page · CC BY-SA 4.0. Two programmers at the ENIAC, United States Army: source page · public domain. The title page of the EDVAC report: source page · public domain. The EDVAC at Aberdeen, United States Army: source page · public domain. Oppenheimer and von Neumann in front of the IAS machine, Jeremy Norman Collection of Images: source page · CC BY 4.0. Ulam with the FERMIAC, Los Alamos National Laboratory: source page · public domain. Ulam’s Los Alamos badge: source page · public domain. Figures made for the blog The cover and the interactive figures for the minimax theorem, the Monte Carlo method and merge sort were made for this trail and work in all four languages of the site.

October 27, 2026 Read