Adventures of a Mathematician

Preface to the 1983 edition

In writing a preface to another edition of this book I cannot resist the temptation to compare the present with the guesses and timid predictions I made about the future of science as it looked to me ten years ago. If anything, the present looks even more exciting than I had hoped. It is wonderful to observe how many unforeseen or unforeseeable facts and ideas have emerged. While I shall mention just a few of the many developments in recent science, it is important to realize that the rate at which we comprehend the universe is as vital as what we finally understand.

Progress in science and technology has proceeded at an ever-increasing pace, making the short period since I wrote this book as significant as any in the history of science. To see this one has only to think of the landings on the moon, the now commonplace launching of satellites, and the enormous discoveries made in astronomy and in the study of the earth itself.

Most notable has been the exponential growth in the technology of electronic computers, whose use pervades many aspects of daily life. Now elements of a “meta-theory” of computing are being outlined and problems of computability in the general sense, especially with respect to its limits, are being studied successfully.

I wonder what John von Neumann’s reaction would have been had he lived to see it all. He prophesied the growing importance of the computer’s role, but even he would probably have been amazed at the scope of the computer age and the rapidity of its appearance.

One could say that after the atomic age there came the computer age, which, in turn, made the space age possible. All space vehicles — rockets, satellites, projectiles, shuttles, and so on — depend on the feasibility of very fast calculations that must be instantly transmitted to them in outer space to correct their orbits. Before the advent of the fastest electronic computers this kind of remote control was not possible.

Recently a great wealth of observations in physics and astronomy has increased the perplexity of the description of the universe. The enigma of quasars is still unresolved. These quasi-stellar objects seem to be billions of light-years away with an intrinsic luminosity hundreds of times greater than that of the galaxies in their foreground. In the few years since I wrote this book, vast “empty regions” hundreds of millions of light-years wide have been found. These areas make us question the sameness and isotropy of the universe suggested by the apparent uniformity of the cosmic radiation remaining from the Big Bang. It is now widely believed that black holes do exist. They may explain the behavior of several observed astronomical objects. In addition, growing evidence supports the theory that violent processes cause gigantic explosions in starlike objects and galaxies.

To a mathematician like myself, the question, “Is the universe in space finite and bounded, or does it extend indefinitely?” remains the number-one problem of cosmogony and cosmology.

In physics, the number of new, fundamental, or primary particles is constantly increasing. Quarks seem more and more to represent real, not merely mathematical, constituents of matter, but their number and nature remain unverifiable, and scientists are considering the existence of subparticles, such as gluons.

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…”

A few weeks ago I was invited to give a Sunday talk at the Los Alamos Unitarian Church on the subject of “Pure Science in Los Alamos.” The discussion that followed centered on problems that are of growing concern nowadays: the relation of science to morality; the good and the bad in scientific discoveries. Around 1910, Henri Poincaré, the famous French mathematician, had considered such dilemmas in his Dernières Pensées. The questions were less disturbing then. Now, the release of nuclear energy and the possibility of’ gene manipulation have complicated the problems enormously.

I was asked what would have happened had the Los Alamos studies proved that it was impossible to build an atomic bomb. The world, of course, would be a less dangerous place in which to live, without the risk of suicidal war and total annihilation. Unfortunately, proofs of impossibility are almost nonexistent in physics. In mathematics, on the contrary, they provide some of the most beautiful examples of pure logic. (Think of the Greeks’ proof that the square root of two cannot be a rational number, the quotient of two integers!) Humanity, it seems, is not emotionally or mentally ready to deal with these enormous increases in knowledge, whether they involve the mastery of energy sources or the inanimate and primitive life processes.

Someone in the audience wondered if some of the current research on the human brain might not ultimately lead to a wiser and better world. I would like to think so, but this possibility lies too far in the future to even guess at.

In the short span of my life great changes have taken place in the sciences. Seventy years amounts to some 2 percent of the total recorded history of mankind. I mentioned this once to Robert Oppenheimer at Princeton. He replied, “Ah! but one-fiftieth is really a large number, except to mathematicians!”

Sometimes I feel that a more rational explanation for all that has happened during my lifetime is that I am still only thirteen years old, reading Jules Verne or H. G. Wells, and have fallen asleep.

S.M.U.

SANTA FE

AUGUST 1982