Euler's Eyes
2026-10-27
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.
What to say of him
Von Neumann was at once one of the most abstract mathematicians of the century and one of the most involved in the military applications of science, and the two things are inseparable in his story. The computers he helped create were born to calculate bombs and soon went on to calculate the weather, markets and proteins. Game theory was born in a paper about parlour games and ended up in manuals of nuclear strategy and in auctions of mobile-phone frequencies.
The lesson he left is that of a mathematician who did not respect the borders between fields. He treated physics, economics and computers as places where mathematics could be useful, and it almost always was.
Exercise: the IAS memory
How many IAS memories would fit on a 1-gigabyte memory card ($10^9$ bytes)?
Each IAS memory has $5\,120$ bytes. $\frac{10^9}{5\,120} \approx 195\,000$. A small card today holds almost two hundred thousand times the memory of von Neumann’s computer.
Exercise: the forecast that arrived late
In 1950 the 24-hour forecast took about 24 hours to calculate. If a computer today were a billion times faster at the same calculation, how long would it take?
$24$ hours is $86\,400$ seconds. Dividing by $10^9$ gives about $0.0000864$ seconds, less than a tenth of a millisecond. That is why today’s models can use grids billions of times finer and still forecast days ahead.
Sources: Stanisław Ulam, “John von Neumann 1903–1957”, Bulletin of the AMS (1958) · Norman Macrae, John von Neumann (1992) · William Poundstone, Prisoner’s Dilemma (1992) · von Neumann’s biography, MacTutor.
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