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. — Stanislaw Ulam.

This remark of Stanislaw Ulam’s is particularly appropriate to his own career. Our world is very different today because of Ulam’s contributions in mathematics, physics, computer science, and the design of nuclear weapons.

While still a schoolboy in Lwów, then a city in Poland, he signed his notebook “S. Ulam, astronomer, physicist and mathematician.”

Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam’s early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day. The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam’s highlyverbal and collaborative style early on. Ulam’s early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam’s lifelong, highly creative quest for new mathematical and scientific problems.

As conditions in prewar Poland deteriorated, Ulam welcomed opportunities to visit Princeton and Harvard, eventually accepting a faculty position at the University of Wisconsin. As United States involvement in World War II deepened, Ulam’s students and professional colleagues began to disappear into secret government laboratories. Following a failed attempt to contribute to the Allied war effort by enlisting in the U. S. military, Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. It was at Los Alamos that Ulam’s scientific interests underwent a metamorphosis and where he made some of his most far reaching contributions.

By virtue of his defense work at the Los Alamos Laboratory, Ulam enjoyed many advantages not available to academic scientists. Chief among these was his early access to the most powerful and fastest computers in existence. For several decades after the war, the computing facilities at the national weapons labs far exceeded those available to university scientists working on non classified research. This was an advantage that Ulam exploited in a variety of remarkable ways.

The growth of powerful computers was initially driven by the war effort. At the beginning of World War II there were no electronic computers in the modern sense,only a few electromechanical relay machines. During the war, scientists at the University of Pennsylvania and at the Aberdeen Proving Ground in Maryland developed the ENIAC, the Electronic Numerical Integrator and Computer, which had circuitry specifically designed for computing artillery firing tables for the Army. By modern standards, this early computer was extremely slow and elephantine: the ENIAC operating at the University of Pennsylvania in 1945 weighed thirty tons and contained about eighteen thousand vacuum tubes with 500,000 soldered connections. While on a visit to the University of Pennsylvania in 1944, John von Neumann was inspired to design an electronic computer that could be programmed in the modern sense, one which could be instructed to perform any calculation and would not be restricted to computing artillery tables. The new computer would have circuits that could perform sequences of fundamental arithmetic operations such as addition and multiplication. Von Neumann desired a more flexible computer to solve the mathematically difficult A bomb implosion problem being discussed at Los Alamos. The first electronic computer at Los Alamos, however, known as the MANIAC(Mathematical Analyzer, Numerical Integrator and Computer), was not available until 1952.

One of Ulam’s early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers, a method which has become appropriately known as the Monte Carlo method. It occurred to Ulam during a game of solitaire that the probability of various outcomes of the card game could be determined by programming a computer to simulate a large number of games. Newly selected cards could be chosen from the remaining deck at random, but weighted by the probability that such a card would be the next selected. The computer would use random numbers whenever an unbiased choice was necessary. When the computer had played thousands of games, the probabilty of winning could be accurately determined. In principle the probability of solitairesuccess could be rigorously calculated using probabilty theory rather than computers. However, this approach is impossible in practice since it would involve too many mathematical steps and exceedingly large numbers. The advantage of the Monte Carlo method is that the computer can be efficiently programmed to execute each step in a particular game according to known probabilities and the final outcome can be determined to any desired precision depending on the number of sample games computed. The game of solitaire is an example of how the Monte Carlo method can be used to solve otherwise intractable problems with brute computational power.

Stanislaw Ulam had formidable memory power and laser like concentration. He would do all the deep, complicated, esoteric math in his head not requiring the use of paper and pencil also. He was endowed with exceptional charisma. In his autobiography (Adventures of a Mathematician) (also, this article is based on this source), he mentions that as a child, he was mesmerized by pictures in an astronomy book, bought some more astronomy books, even bought a telescope, listened to public lectures on relativity, and believed that the thirst to go deeper into astronomy brought him to mathematics.

More later,

Nalin Pithwa