Musings of a Mathematician

Mathematics is both essential and frivolous, jubilant and despairing, lovely, and brutal, perfect and broken. MIT applied mathematician, Gian-Carlo Rota paints this multi-faceted aspect of mathematics in a memoir of his friend, Polish-born mathematician Stanislaw Ulam (1909 – 1984) who lived through one of the most intellectually fertile period in the history of mathematics which included the development of the atomic bomb.

Mathematics as an Escape

“Of all the escapes from reality, mathematics is the most successful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same reality we are trying to evade. All other escapes – love, drugs, hobbies, whatever – are ephemeral by comparison. The mathematician’s feeling of triumph, as he forces the world to obey the laws of his imagination has freely created, feeds on its own success. The world is permanently changed by the workings of his mind, and the certainty that his creations will endure renews his confidence as no other pursuit. The mathematician becomes totally committed, a monster like Nabokov’s chess player, who eventually sees all life as subordinate to the game of chess.”

Company of a Genius: von Neumann

John Von Neumann (right) stands with J. Robert Oppenheimer in front of an early computer.  GETTY IMAGES

“Many of us remember the feeling of ecstasy we experienced when we first read (John) von Neumann’s series of papers on rings of operators in Hilbert space. It is a paradise from which no one will ever dislodge us (as Hilbert said of Cantor’s set theory). But von Neumann’s achievements went far beyond the reaches of pure mathematics. Together with (Stanislaw) Ulam, he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first computer. No other mathematician of this century has had as deep and lasting an influence on the course of civilization.” [1]

The Los Alamos ‘Kids’

Los Alamos colloquium from 1946, featuring (foreground, from left-to-right), Norris Bradbury, J. Robert Oppenheimer, John Manley, Richard Feynman, and Enrico Fermi. Photo: Emilio Segre Visual Archives.

“The assembly of geniuses who roamed the corridors of the Los Alamos laboratory during World War II has not been matched in recorded history, with the possible exception of ancient Greece. In the hothouse of the Manhattan Project, Stan’s mind opened up as it hasn’t since the days of the Scottish Cafe (referring to a cafe in the city of Lwow, Poland, where some of the best minds in mathematics met regularly to brainstorm).

Los Alamos was a turning point in Stan Ulam’s career. From that time on, physics, not mathematics, became the center of his interest … He admired (Enrico) Fermi’s genius for solving physical problems with no more than the minimum amount of math…After watching Fermi and (Richard) Feynman at the blackboard, he discovered that he too had a knack for accurately estimating physical quantities by doing simple calculations..It is hard to overestimate how rare such an ability is in a mathematician.”

Scribbles on a Blackboard

Stanislaw Ulam (1909 – 1984)

“Stan Ulam’s best work is a game played in the farthest reaches of abstraction, where the cares of the world cannot intrude: in set theory, in measure theory, and in the foundations of mathematics. As a mathematician, his name is most likely to survive for his two problem books, which will remain bedside books for young mathematicians eager to make their mark by solving at least one of them. He also wanted to be remembered for those insights that found substantial practical applications, such as the Monte Carlo method, for which he will share the credit with (Nicholas) Metropolis and von Neumann, and the (atomic) bomb, for which he will be remembered alongside with (Edward) Teller. As Stan himself recalled, “It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.”


[1] von Neumann has a storied history about his towering genius. Here’s one recollection of him by the mathematician, George Polya who collected memorable photos of famous mathematicians, many of whom his  contemporaries. “Von Neumann is the only student of mine I was ever intimidated by. He was so quick. There was a seminar for advanced students at Zurich that I was teaching and von Neumann was in the class. I came to a certain theorem, and said it is not proved and it may be difficult. Von Neumann didn’t say anything, but after five minutes he raised his hand. When I called on him, he went to the blackboard and proceeded to write down the proof. After that, I was afraid of von Neumann.” Source: G.L. Alexanderson (ed), The Polya Picture Album: Encounters of a Mathematician, Birkhäuser; 1st Edition edition, 1987.

Further Reading

Gian Carlo-Rota, “The Lost Cafe”, in N.G. Cooper, From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam, Cambridge Univ. Press, 1989, 23-32.

Stanislaw Ulam, Adventures of a Mathematician, University of California Press, 1991. 

For a detailed biography of Stan Ulam, check out this site.

Leave a Reply