I gave a lot of "propaganda" talks for this method all over the United States. Interest and improvements in the theory came rapidly. Here is an easy example of this procedure which I often selected: One may choose a computation of a volume of a region defined by a number of equations or inequalities in spaces of a high number of dimensions. Instead of the classical method of approximating everything by a network of points or cells, which would involve billions of individual elements, one may merely select a few thousand points at random and obtain by sampling an idea of the value one seeks.
The first questions concerned the production of the random or pseudo-random numbers. Tricks were quickly devised to produce them internally in the machine itself without relying on any outside physical mechanism. (Clicks from a radioactive source or from cosmic rays would have been very good but too slow.) Beyond the literal or "true" imitation of a physical process on electronic computers, a whole technique began to develop on how to study mathematical equations which on their face seem to have nothing to do with probability processes, diffusion of particles, or chain processes. The question was how to change such operator equations or differential equations into a form that would allow the possibility of a probabilistic interpretation. This is one of the main theses behind the Monte Carlo method, and its possibilities are not yet exhausted. I felt that in a way one could invert a statement by Laplace. He asserts that the theory of probability is nothing but calculus applied to common sense. Monte Carlo is common sense applied to mathematical formulations of physical laws and processes.
Much more generally, electronic computers were to change the face of technology. We discussed the many possibilities endlessly. But not even von Neumann could foresee their full economic or technological impact. These aspects of their development were still in their infancy so far as industrial applications were concerned when he died in 1957. Little did we know in 1946 that computing would become a fifty-billion-dollar industry annually by 1970.
Almost immediately after the war Johnny and I also began to discuss the possibilities of using computers heuristically to try to obtain insights into questions of pure mathematics. By producing examples and by observing the properties of special mathematical objects one could hope to obtain clues as to the behavior of general statements which have been tested on examples. I remember proposing in 1946 a calculation of a very great number of primitive roots of integers so that by observing the distributions one obtained enough statistical material on their appearance and on the combinatorial behavior to perhaps get some ideas of how to state and prove some possible general regularities. I do not think that this particular program has been advanced much until now. (In mathematical exploratory work on computers my collaborators were especially Myron Stein and Robert Schrandt.) In the following years in a number of published papers, I have suggested — and in some cases solved — a variety of problems in pure mathematics by such experimenting or even merely "observing." The Gedanken Experimente, or Thought Experiments, of Einstein are possible and often useful in the purest part of mathematics. One of the papers outlining a field of exploration in "non-linear problems" was written in collaboration with Paul Stein. By now, a whole literature exists in this field.
Quite early, in fact only some months after the electronic computer called MANIAC became available in Los Alamos, I tried with a number of associates (Paul Stein, Mark Wells, James Kister, and William Walden) to code the machine to play chess. It was not so terribly difficult to code it to play correctly according to the rules. The real problem is that, even today, nobody knows how to put in its memory experiences of previous games and a general recognition of the quality of patterns and positions. Nevertheless it can be made to play so it can beat a rank amateur. We realized that the differences between playing poorly and playing well are much greater than teaching it to make legal moves and respond to obvious threats, and so on. This game was played on a six-by-six board without bishops (to shorten the time between moves). We wrote an article that appeared in the U.S. Chess Review and was soon reprinted by a Russian chess magazine. Stein, originally a physicist, "converted" to mathematics and became one of my closest collaborators.
Curiously the patterns of chess remind me of oriental rugs and also of something that laymen won't understand — very complicated non-measurable sets. I think I am a fair chess player. When I first came to this country I played with other mathematicians for relaxation. In Los Alamos during the years after the war, friends and younger colleagues organized a chess club, and I played many games. The Los Alamos chess team on which I played board number one several times beat Santa Fe and even Albuquerque with their populations respectively three and fifteen times that of Los Alamos.
It was in 1949, after Teller's return, that George Gamow, whom I had met briefly in Princeton before the war, came to Los Alamos for a lengthy visit. He was on a year's leave from George Washington University in Washington. In physical appearance he was quite an impressive man, six feet three inches tall, slim in 1937 (by 1949 heavy set), blond, blue-eyed, youthful looking, full of good humor. He had a very characteristic shuffling way of' walking with mincing steps. He was very different from the popular picture of the specialized, scholarly scientist — not at all the standard type of academic personage. There was nothing dry about him. A truly "three-dimensional" person, he was exuberant, full of life, interested in copious quantities of good food, fond of anecdotes, and inordinately given to practical jokes.
Almost at once we became friends and engaged in interminable discussions. In some ways our temperaments matched. He found something congenial in my way of thinking (or not thinking) about problems of physics along standard lines. He liked to approach different problems from many different directions in an unassuming, direct, and original way. He talked about himself a great deal. Generally he was one of the most egocentric persons I have known, yet paradoxically (because this combination is so rare) he was at the same time completely devoid of malice towards others.
It was he (and Edward U. Condon, independently and almost simultaneously), who started theoretical nuclear physics in a 1928 paper on the quantum theoretical explanation of radioactivity. In scientific research, he concentrated on a few given problems over a period of years, returning to the same questions time and again.
Banach once told me, "Good mathematicians see analogies between theorems or theories, the very best ones see analogies between analogies." Gamow possessed this ability to see analogies between models for physical theories to an almost uncanny degree. In our ever-more-complicated and perhaps oversophisticated uses of mathematics, it was wonderful to see how far he could go using intuitive pictures and analogies from historical or even artistic comparisons. Another quality of his work was the nature of the topics with which he dealt. He never allowed his facility to carry him away from the essence of his subject in pursuit of unimportant details and elaborations. It was along the great lines of the foundations of physics, in cosmology, and in the recent discoveries in molecular biology that his ideas played an important role. His pioneering work in explaining the radioactive decay of atoms was followed by his theory of the explosive beginning of the universe, the "big bang" theory (he disliked the term by the way), and the subsequent formation of galaxies. The recent discovery of the radiation pervading the universe, corresponding to a temperature of some three degrees absolute, seems to confirm his prediction in 1948 concerning residual radiation from the big bang about ten billion years ago. This discovery came after his death in 1968.