Since the first publication of this book it has become more likely, it seems to me, that there might be an infinite chain of descending structures. To paraphrase a well-known statement about fleas, large quarks have bigger quarks on their backs to bite them, big ones have bigger ones, and so on ad infinitum.
There is also much speculation about the identity of or similarity between the different forces of nature. Certainly there is a strong analogy between electromagnetic forces and so-called weak interactions. There may even be a mathematical analogy between these forces, nuclear forces, and gravitational forces.
Mathematics remains the tool for investigating problems such as these. Electronic computers have helped immensely in solving complex calculations, and a great many new results have appeared in pure mathematical disciplines such as number theory, algebra, and geometry. The broadening range of "constructive" mathematical methods, such as the Monte Carlo method, indicates that a theory of complexity may soon affect many branches of mathematics and stimulate new points of view. Some physical problems such as the study and interpretation of particle collision on the new, miles-long accelerators call for gigantic Monte Carlo modeling.
Presently in vogue is the study of nonlinear transformations and operations. These began in the Los Alamos Laboratory, which now has a special center devoted to nonlinear phenomena. This center recently held an international conference on chaos and order. For the most part this work concerns the behavior of iterations — repetitions of a given function or flow. These problems require guidance from what are essentially mathematical experiments. Trials on a computer can give a mathematician a feeling or intuition of the qualitative behavior of transformations. Some of this work continues a study mentioned in Chapter 12, and some follows work Paul Stein, I, and others have done in the intervening years.
While much of physics can be studied using linear equations in an infinite number of variables (as in quantum theory), many problems — hydrodynamics included — are not linear. It is becoming more and more likely that there may be nonlinear principles in the foundations of physics. As Enrico Fermi once said, "It does not say in the Bible that all laws of nature are expressible linearly!"
To an amateur physicist such as I am the increasing mathematical sophistication of theoretical physics appears to bring about a decrease in the real understanding of both the small- and the large-scale universe. The increasing fragmentation may be due in part to neglect in the teaching of the history of science and certainly to the growth of specialization and overspecialization in various branches of science, in mathematics in particular. Although I am supposed to be a fairly well-read mathematician, there are now hundreds of new books whose very titles I do not understand.
I would like to devote a few words to what is manifestly the age of biology. I believe these past sixteen years have seen more significant advances in biology than in other sciences. Each new discovery brings with it a different set of surprises. Genes that were supposed to be fixed and immutable now appear to move. The portion of the code defining a gene may "jump," changing its location on the chromosome.
We now know that some segments of the genetic code do not express formulae for the manufacture of proteins. These sometimes longish sequences, called introns, lie between chromosome segments that do carry instructions. What purpose introns serve is still unclear.
The success of gene splicing — the insertion or removal of specific genes from a chromosome — has opened a new world of experimentation. The application of gene manipulation to sciences such as agriculture, for example, may have almost limitless benefits. In medicine we can already produce human-type insulin from genetically altered bacteria. Scientists have agreed to take precautions against accidentally creating dangerous new substances in gene-splicing experiments. This seems to satisfy the professional biologists. Still, there is a great debate over whether to allow unregulated genetic engineering, with all its possible consequences.
My article "Some Ideas and Prospects in Biomathematics" (see Bibliography) is an example of some of my own theoretical work in this area. It concerns ways of comparing DNA codes for various specific proteins by considering distances between them. This leads to some interesting mathematics that, inter alia, may be used to outline possible shapes of the evolutionary tree of organisms. The idea of using the different codes for a cytochrome C was suggested and first investigated by the biologist Emanuel Margoliash.
At Los Alamos, a group led by George Bell, Walter Goad, and other biologists is using computers to study the vast number of DNA codes now experimentally available. The group was recently awarded a contract by the National Institute of Health to establish a library of such codes and their interrelations.
It is well known that gradual changes, no matter how extensive, are barely noticeable while they occur. Only after a certain amount of time does one become aware of any transformation. One morning in Los Alamos during the war, I was thinking about the imperceptible changes in my own life in the past years that had led to my coming to this strange place. I was looking at the blue New Mexico sky where a few white clouds were moving slowly, seemingly retaining their shape. When I looked away for a minute and back up again, I noticed that they now had completely different shapes. A couple of hours later I was discussing the changes in physical theories with Richard Feynman. Suddenly he said, "It is really like the shape of clouds; as one watches them they don't seem to change, but if you look back a minute later, it is all very different." It was a curious coincidence of thoughts.
Changes are still taking place in my personal life. In 1976 I retired from the University of Colorado to become professor emeritus, a sobering title. At the same time, I accepted a position as research professor at the University of Florida in Gainesville, where I still spend a few months every year, mostly during the winter when it is not too hot.
My wife, Françoise, and I sold our Boulder house and bought another one in Santa Fe, which has become our base. From Santa Fe I commute three or four times a week to the Los Alamos Laboratory. Its superb scientific library and computing facilities allow me to continue working in some of the areas of' science mentioned above. Françoise acts as my ''Home Secretary," as I call her, alluding to the title of the British Interior Secretary. We still travel quite extensively and I continue to lecture in various places.
We are fortunate that our daughter Claire also lives in Santa Fe with her husband, Steven Weiner, an orthopedic surgeon. Their daughter, now five, gives me occasion to wonder at how remarkable the learning processes of small children are, how a child learns to speak and use phrases analogous to and yet different from the ones it has heard. Observing Rebecca speak provides me with additional impulses and examples for describing a mathematical schema for analogy in general.
My collaborator, Dan Mauldin, a professor at North Texas State University, has recently edited an English version of The Scottish Book mentioned in Chapter 2. We are now collaborating on a collection of new unsolved problems. This book will have a different emphasis from that of my Collection of Mathematical Problems, published in 1960. The new collection will deal more with mathematical ideas connected to theoretical physics and biological schemata.
Many of the people mentioned in this book have since died, or left, as my friend Paul Erdös prefers to say: Kazimir Kuratowski, my former professor; Karol Borsuk and Stanislaw Mazur, my Polish colleagues; my cousins Julek Ulam in Paris and Marysia Harcourt-Smith; in Boulder, Jane Richtmyer, who helped with the first writing of this book; George Gamow and his wife Barbara; my collaborators John Pasta and Ed Cashwell of the Monte Carlo experiments; and here in Los Alamos (within a few months of each other) the British physicist Jim Tuck and his wife Elsie. As Horace said, "Omnes eadem idimur, omnium versatur urna … sors exitura…"