Adventures of a Mathematician

Chapter 13. Government Science

It is more difficult to write about recent events; the perspective is poorer, the separation of the characteristic or the important from the fortuitous is more difficult. My story of the past fifteen years or so will therefore be contracted and will concern activities and people even more arbitrarily chosen than the reminiscences and reflections of the earlier chapters.

Returning to Los Alamos in 1957 after a year’s leave of absence at MIT, I was asked by Bradbury to accept one of the two newly created positions of research advisor to the director of the laboratory. The other advisor was to be John Manley, a physicist who had held important administrative posts at Los Alamos during the war and wanted to return to New Mexico after a long absence as professor at the University of Washington in Seattle. Administratively, research advisors were to be on the same level as division leaders, and their duties were to oversee the research activities through-

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out the laboratory, in the various divisions: theoretical, physics, chemistry and metallurgy, weapons, health, “Rover” (nuclear rocket), and others. Together we tried to influence the various programs of the lab. This was an arduous and many-sided task, and talking with many people about their research activities enabled me to broaden my own interests. I held this position until 1967 when I retired from Los Alamos and joined the mathematics department at the University of Colorado in Boulder. In laboratory jargon, Manley and I were respectively known as RAJM and RASU.

In my administrative role in Los Alamos and later as chairman of the mathematics department in Boulder, I came to understand and appreciate better and also commiserate with friends and acquaintances who had become fully occupied by administrative duties. In my younger years I had had the usual skeptical attitude towards most chairmen of departments, deans, presidents, directors, and the like. There had been exceptions, of course. One of these was J. Carson Mark, leader of the theoretical division in Los Alamos since the middle forties. Mark is a Canadian mathematician who came to Los Alamos towards the end of the war with the British Mission. He became an American citizen some years later. He bore the brunt of the difficulties with Teller with remarkable calm and objectivity; he is one of the few mathematicians I know who have an understanding for the problems of physics and associated technology in a broad sense. His direction of the theoretical division was an example of intelligent management of a scientific group without exercising undue pressures for programmatic work. He was able to encourage free scientific pursuits in areas which were only indirectly related to the tasks of the laboratory, and he supported theoretical physics and applied mathematics in the best sense. (Incidentally, he was also a regular participant of our poker sessions. From 1945 until now I cannot recall a single occasion when he refused to come or missed one of our games. These by now have considerably slowed in frequency. From weekly they became monthly and now occur only occasionally, mainly when I happen to be visiting Los Alamos.)

After the war it had become clear that science and technology had become so crucial for national affairs that the governments of the Western world had to devote enormous amounts of time and huge budgets to them. Famous scientists were called upon to enter the inner circles of government to help direct their countries’ scientific activities, not only for the arms race but for technological advancement. Churchill had had Lord Cherwell; De Gaulle, Francis Perrin; America, her Scientific Advisory Committees. Beginning with Bush and Conant, Oppenheimer, von Neumann, and many others became government “sages.” Government science peaked and committees proliferated under the Eisenhower, Kennedy, and Johnson administrations, and even I found myself called upon. Until then I had always resisted being drawn into any kind of organizational position; for years I could claim that my only administrative job had been on the Wine Tasting Committee of the Society of Fellows back in my early Harvard days.

A few years before Johnny’s death and increasingly so after, as a result of my work on the hydrogen bomb, I became drawn into a maze of involvements. These had to do precisely with government science and with work as a member of various Space and Air Force committees. Also, in some circles I became regarded as Teller’s opponent, and I suspect I was consulted as a sort of counterweight. Some of these political activities included my stand on the Test Ban Treaty and testimony in Washington on that subject. The cartoonist Herblock drew in the Washington Post a picture of the respective positions of Teller and me in which I fortunately appeared as the “good guy.”

Since I never have kept notes or diaries of any kind, I may not always be entirely correct about the chronology of events or how things and people were connected during these busy years of scientifico-technological activities.

Washington committees, I soon noticed, were often very envious of new ideas, with their members exhibiting the well-known ’’not invented here” syndrome of rejecting proposals or ideas only because of the vested interests of committee members. This feeling was a greater obstacle to the development of new projects than the concern about their cost and the amounts of money they would require. Decisions also seemed sometimes dictated, not so much by objective evaluation as by the usual academic rivalries and envy of scientific fame. Had I not been already quite old and cynical this would have made me leave governmental science altogether. I remembered how in Los Alamos Johnny remarked on several occasions that it was not easy to introduce new things; one had to persuade every janitor, he said. But once something was accepted it became a sort of bible and it was equally hard or even impossible to change it or get rid of it. This national situation has become even worse now, partly because of the recent spreading skepticism regarding the value of science and its benefits, and a kind of general passivity so removed from the traditional traits of American enterprise, energy, and spirit of cooperation.

The idea of nuclear propulsion of space vehicles was born as soon as nuclear energy became a reality. It was an obvious thought to try to use its more powerful concentration of energy to propel vehicles with a very large payload for ambitious space voyages of exploration or even for excursions to the moon. I think Feynman was the first in Los Alamos during the war to talk about using an atomic reactor which would heat hydrogen and expel the gas at high velocity. A simple calculation shows that this would be more efficient than expelling the products of chemical reactions.

I became involved with two such projects, one in an advisory capacity, with the other more directly. The first was Project Rover, a nuclear-reactor rocket which was being designed in Los Alamos already quite a few years before the Russian Sputnik, but with very limited funds. The second was a space vehicle, later named Orion. Around 1955 Everett and I wrote a paper about a space vehicle propelled by successive explosions of small nuclear charges. The idea has even been patented by the AEC in our names. This method could be much more powerful than Rover and is a very ambitious but efficient way to undertake space explorations with a vehicle able to travel at high speeds with high payloads and an extremely good ratio of payload to total initial weight. The spaceship could transport hundreds or thousands of people. When Kistiakowski was President Eisenhower’s Scientific Advisor I informed him about such possibilities, but his reception of it was not enthusiastic. But more about Orion later.

Soon after John Kennedy’s election in November 1960, I received a telephone call from Jerry Wiesner from Cambridge. I had met Wiesner the year I was at MIT as a visiting professor; we had seen each other several times and had had good conversations about science projects, national programs, education, and so on. We had talked also about the Teller business. Wiesner was wary of the Edwardian brand of politics. I was not too surprised when I received this call. It must have lasted more than half an hour and Jerry informed me that President Kennedy had appointed him chairman of a task force on science and technology. He asked what my ideas were about the nationally important scientific or technological projects the President should know about and consider for the country. “How about going to the moon?” I asked. I imagine dozens of other people had made the same kind of suggestion. In his inaugural address Kennedy proposed a national project to put a man on the moon. My involvement with the space effort began in earnest with that conversation. I became consultant to Wiesner’s Scientific Advisory Committee and visited Washington frequently.

Immediately after the war, Clinton P. Anderson, Senator from New Mexico, a former member of President Truman’s cabinet, became one of the most interested, knowledgeable, influential, and effective proponents of the uses of nuclear energy. He was instrumental in helping the Los Alamos Laboratory and the associated big installation in Albuquerque, the Sandia Laboratory.

I became acquainted with him during one of his early visits to Los Alamos, enjoying his confidence and — it seems to me — his trust in and reliance on my opinions, not only in the area of nuclear energy but also in the field of space activities. Several times he invited me to testify before Congress on specific space matters, such as the organization of NASA and whether it should be part of the military establishment or an independent organization.

When it was decided to do something in earnest about Project Rover, Wiesner named a Presidential Committee to look into the matter. I was one of its members. Among some members who were chemists I noticed a degree of skepticism about its worth and feasibility, again motivated in my opinion by their apprehension that it might compete with the already existing chemical rocket propulsion systems which were being developed. Some of the discussions reminded me of the big debates at the beginning of the century between lighter-than-air and heavier-than-air advocates, or even of the earlier competition of steamships versus sailing vessels. And indeed the committee wrote a report which by faint praise, essentially condemned Project Rover to a de facto death by proposing to make it a purely theoretical study without funds for experimental work or any investment in construction. The physicist Bernd Matthias was the only member of the committee who joined me in writing a dissenting opinion.

Senator Anderson was chairman of the congressional Space Committee. He knew my position on Rover. With his feeling for the psychological and political motivations of committees, and his vital interest in the new technology and its importance for the nation, he took me one day to the office of the then Vice-President Johnson. Together we walked to a nearby building to see Wiesner. Since Jerry and I were friends, it embarrassed me a little to be present at a meeting in which Johnson and Anderson were pressing him hard to change the attitudes of the Scientific Advisory Committee on questions of nuclear propulsion. They supported my views in opposition to his. Ultimately the minority opinion prevailed, and the Rover Project was saved. Funds were allotted for Los Alamos work and over the years it became an extremely successful venture. Unfortunately it was stopped again later by economies in the space program.

I was also invited to join an Air Force Committee on a similar subject: general problems of plans for space and the Air Force’s role in it. The Committee was chaired by Trevor Gardner, a former assistant secretary of the Air Force during Eisenhower’s presidency. Gardner was a very interesting person of whom I became very fond. His vigorous and lusty personality, his great energy, the wide scope of his imagination appealed to me very much. I found him very congenial.

The committee originally comprised a number of persons important in science and technology. The only other mathematician, Mark Kac, was present at a few sessions. Among the “big shot” members, I remember Harold Brown, director of the Livermore Laboratory, later Secretary of the Air Force, Charlie Townes, who received the Nobel Prize for the invention of masers, General Bernard Shriever, a frequent visitor. Vince Ford, an Air Force colonel who had been Johnny’s aide on the von Neumann ICBM Committee, was now Gardner’s assistant. He organized the meetings of a working subcommittee which met in Los Alamos. These meetings involved many people from the newly born aerospace industry. Sometimes we met in Los Angeles where the headquarters of the ballistics division of the Air Force under General Shriever were located. At other times we met in Washington where General Shriever, Gardner, Ford and I discussed among ourselves during restaurant lunches how to plan the exploration of space and more generally the problems of space study for the Air Force.

Early at one of these meetings, somebody from industry presented plans for retrieving rocket engines, which would save money by making them reusable. The real problem, as I and some others saw it, was to do something important in putting up the satellites and do it quickly rather than to start by saving money. Also it seemed to me that boosters, namely the engines, were a small part of the overall cost, and that it would be awkward, to say the least, to start by reusing second-hand engines, perhaps damaged. When the proponents droned on about their ideas, showing their plots and graphs, I whispered to Gardner, “This sounds to me like a proposal to use the same condom twice.” He burst out laughing and sent the remark around the table in repeated whispers. Perhaps this joke saved the United States some millions of dollars in expenditure for what would have been pointless and impractical work at the time.

Orion was also discussed by the Gardner Committee. At my suggestion Ted Taylor became the executive director of a group working on it. Starting in 1957, Taylor developed the Orion idea as a reaction to the Russian Sputnik, as he said. He assembled an impressive group of bright young men at the General Atomic Laboratory in La Jolla, Calif. The physicist Freeman Dyson became very interested and enthusiastic and took a leave of absence from the Princeton Institute to work for a year with Taylor. A few years later he wrote an eloquent article describing the project and how it was put on a shelf. It appeared under the title “La Vie et Mort d’Orion” in the Paris paper Le Monde.

Somehow the Gardner Committee report got lost at a high level in the Washington maze. Wiesner disagreed with Gardner about the role of the Air Force in space. Somebody in Washington managed to bury the report, and I don’t think President Kennedy ever saw it. The whole thing is still a mystery to me. After the Gardner Committee finished its work it was succeeded by another one called the Twining Committee. Its members included some hawkish types like Teller and Dave Griggs. General Doolittle of Tokyo air raid fame was also a member.

I became connected with Trevor Gardner later in a more private capacity. He asked me to join the scientific advisory board of the Hycon Corporation in California which he headed. The company manufactured highly secret military equipment including special cameras. Fowler, Lauritsen, Al Hill, a physicist from MIT, and Jesse Greenstein the Palomar astronomer, were the other members of the board. I learned that Wiesner and his group in Hycon East and Gardner in California had had some serious disagreements about financial problems concerning the corporation, and apparently Wiesner and Gardner were barely on speaking terms.

To some Gardner was a controversial person because of his quick temper and his strong opinions. He had great political ambitions (he would have liked to become Secretary of Defense), but he was at cross purposes with some members of the Kennedy Administration. He died of a heart attack shortly before Kennedy’s assassination. It was Gardner who had established the von Neumann ICBM Committee. This had been of immense importance for the U.S. space effort; I think it really got it off the ground. The military and national importance of this and other Gardner initiatives can hardly be exaggerated.

At the same time, I was continuing my own work. After Fermi’s death Pasta and I decided to continue exploratory heuristic experimental work on electronic computers in mathematical and physical problems. We felt that the combination of classical mechanics and astronomy problems lent itself to two kinds of studies: one, the behavior of large numbers of particles — call them stars — in a cluster or galaxy; the other, the history of a single mass of gas as it developed from initial conditions by contraction at first, perhaps giving rise to a double or multiple star, then generating more and more nuclear reactions, exhausting its nuclear material, and finally perhaps collapsing. Many calculations have been done since on this latter problem over the last twenty years, changing the whole understanding of astrophysics and of the development of the universe as far as individual stars are concerned.

The problem of clusters of stars was I think the first study of this nature using computers. We took a great number of mass points representing stars in a cluster. The idea was to see what would happen in the long-range time scale of thousands of years to the spherical-looking cluster whose initial conditions imitated the actual motions of such stars. This was a really pioneering calculation showing that this sort of investigation was possible. It gave very curious and unexpected observations in classical mechanics, formation of subgroups, and contractions. We made a film of these motions on an accelerated scale which showed these interesting phenomena. This work gave rise to other studies of this sort at Berkeley, in France, and elsewhere.

Another problem which I attacked but which is still not solved is an attempt to see what will happen when a mass of gas of very large dimensions, say of the whole solar system, at very low density and having initially a mild amount of turbulence starts to contract. How would it contract, and how would it finally form a star? What is interesting and the actual purpose of the problem was to see whether and how often it would form a double, triple, or multiple star. The reason for this curiosity is that many stars, in our neighborhood at least, are double. According to recent studies, at least one star out of three is multiple. It would be nice to see by brute-force calculations how a contraction of an irregularly shaped mass of gas develops. Beyond all this is the problem of the formation and development of galaxies — that is, the assemblies of billions of stars. On this, too, astrophysicists have accomplished much with the aid of computers.

While such astrophysical calculations were going on, I began in an amateurish way to work on some questions of biology. After reading about the new discoveries in molecular biology which were coming fast, I became curious about a conceptual role which mathematical ideas could play in biology. If I may paraphrase one of President Kennedy’s famous statements, I was interested in “not what mathematics can do for biology but what biology can do for mathematics.” I believe that new mathematical schemata, new systems of axioms, certainly new systems of mathematical structures will be suggested by the study of the living world. Its combinatorial arrangements may lead us in the future to a logic and mathematics of a different nature from what we know now. The reader is referred to one of my papers on mathematical biology. Too technical to be included here, it is listed in the bibliography at the end of this volume.

My interest in biology took a more tangible form when I engaged in discussions with James Tuck, and we talked to the biologists in the laboratory. Los Alamos had always had a division for the studies of biological effects of radiation. Radioactive damage was, of course, one of the first things to worry about from the beginning of the nuclear age. With Tuck, and Gordon Gould and Donald Peterson from the Health Division, we organized a seminar devoted to current problems of cellular biology and the new results in molecular biology. I really learned a lot about the elementary facts of biology there, the role of cells, their structure, and so on. The seminars, which had about twenty participants, have had important consequences, although they lasted only two years. Two of the participants, Los Alamos physicists Walter Goad and George Bell, both extremely brilliant and talented, and among the best young brains in these fields in the country, are doing a lot of biology research now. Goad is working in the field of biological mathematics, while Bell has some new ideas on immunology. Ted Puck from Colorado visited the seminar and gave some lectures.

I met Puck shortly after the war and found him to be full of new ideas, suggesting interesting experiments and methods for the study of the behavior of cells and problems in molecular biology in general. I think it was Ted Puck’s group which first succeeded in keeping mammalian cells alive and even multiplying in vitro. I always look forward to discussions with him; it was he who arranged for me to give seminars for the faculty and young researchers in the biophysics department and even succeeded in having me appointed a member of the professorial staff at the University of Colorado’s Medical School. I told him that, being a beginner and a layman in this field, I might be arrested for impersonating a doctor.

Almost every month there are fascinating new facts discovered in biology. It is now widely recognized that the discoveries of Crick and Watson have opened up a new era in the psychological attitudes in biology as well. Years ago at Harvard, when I talked to biologists and tried to ask about or propose even a mildly general statement, there would always be the retort: “It isn’t so because there is an exception in such and such an insect” or “such and such a fish is different.” There was a general distrust or at least a hesitation to formulate anything of even a slightly general nature. This attitude has drastically changed since the discovery of the role of DNA and the mechanism of replication of the cell and of the code which seem so universal.

During all these years I did not live continuously in Los Alamos. I spent periods of time as a visiting professor at Harvard, MIT, the University of California in La Jolla, the University of Colorado, plus innumerable visits to various universities, scientific meetings, and government or industrial laboratories, where I gave lectures and consultations. These latter were called business trips. If one adds our almost yearly vacations in Europe since 1950 (mainly in France where Françoise still has relatives and I have many scientific friends), it seems to me that about twenty-five percent of my time was spent away from Los Alamos.

It was in those periods that my friendship with Victor Weisskopf developed. I had met him in Los Alamos during the war when he was Bethe’s alternate as leader of the theoretical division. He left at the war’s end to become professor at MIT and our relationship deepened during my visits to Cambridge, Harvard and MIT.

Viki, as he is universally called, is a theoretical physicist. He made a name for himself as a young man with his important work on problems of radiation in quantum theory. He was for a time assistant to Pauli and also worked in Copenhagen at the famous Niels Bohr Institute. Viki was born in Vienna, a fact I take note of because he exhibits the best side of the Viennese temperament. This is contained in the following saying: In post-World-War-I Berlin people used to say, “The situation is desperate but not hopeless”; in Vienna they said, “The situation is hopeless but not serious.” This certain insouciance combined with the highest intelligence has enabled Viki to navigate not only through the usual difficulties of administrative and academic affairs — he has been, among other things, director-general of CERN (the European Center for Nuclear Research) near Geneva and chairman of the large physics department at MIT — but also in the more abstract realm of the intellectual and scientific difficulties of theoretical physics. I would say his intellectual stability is based on a real knowledge of and feeling for the spirit of the history of physics. This he has achieved through perspective and comprehension, sifting and evaluation of the quickly changing scene in the physical theories which concern the very foundations of this science. I should add here for the benefit of the reader who is not a professional physicist that the last thirty years or so have been a period of kaleidoscopically changing explanations of the increasingly strange world of elementary particles and of fields of force. A number of extremely talented theorists vie with each other in learned and clever attempts to explain and order the constant flow of experimental results which, or so it seems to me, almost perversely cast doubts about the just completed theoretical formulations. Through all this turmoil in the overly mathematical theoretical physics research, constant good progress has been made, but it takes a person like Viki (and really there are no more such than one can count on the fingers of one hand) to stabilize this flow and extract the gist of the new elaborations of the ideas of quantum theory and to be able to explain and describe it both to the physicists themselves and to the more general public.

His semipopular books on physics are uniquely interesting and successful in presenting his philosophy and the human side of the story. He was always and still is immensely concerned with the problems of man and world affairs. As a person he is affable and kind, gets along with everybody, and he loves to tell stories, and sometimes our exchanges of Jewish jokes can last an hour.

During his tenure at CERN where he still visits every summer and consults, the Weisskopfs built a modest summer house on the French side of the border in a small Jura village overlooking the Lake of Geneva. It is twenty minutes’ drive from CERN and they spend most of the summer there. On our own European trips of the last several years we have almost always included a brief stay with the Weisskopfs in their house in Vesancy. This village is just a few kilometers away from another one known as Ferney-Voltaire because Voltaire lived there for many years. In like manner I have dubbed Vesancy, Vesancy-Weisskopf. Viki likes that and it fits for he has become quite a personage in the village. He is known as Monsieur le Directeur, and the farmers tip their hats when they see his tall, lanky silhouette walking carefully across their fields.

In 1960 my book, Unsolved Problems of Mathematics, was published. Many years ago Françoise asked Steinhaus what it was that made me what people seemed to consider a fairly good mathematician. According to her, Steinhaus replied: ’’C’est l’homme du monde qui pose le mieux les problèmes.” Apparently my reputation, such as it is, is founded on my ability to pose problems and to ask the right kind of questions. This book presents my own unsolved problems. As a young man I liked the motto in front of George Cantor’s thesis which is a Latin quotation: “In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.”

Shortly after 1960 the book was translated into Russian. There is no copyright agreement between Russia and the West, and the Russians pay no royalties, but some Western authors discovered when they were in the Soviet Union that they could obtain some payment for the translations of their work. Hans Bethe and Bob Richtmyer successfully received compensation. So when I attended an International Mathematics Congress in Moscow in 1966 I remember that I could try too. The Russian language being close to Polish, I went to the publishing house to talk about this matter in my imitation of Russian. At the publishing house, which looked the same as everywhere else — girls typing and masses of files and papers — an elderly gentleman seemed to understand my request and asked me how I knew to come and see them. I gave him the names of my friends. He went to a back room, then returned. The reader should know that Russians do not pronounce H as in English. They say G. For instance Hitler is pronounced Gitler, Hamlet Gamlet, Hilbert Gilbert. The gentleman said to me in Russian with an engaging smile: “Come back tomorrow please with your passport, and we will give you” — I seemed to hear — “your gonorrhea.” Of course he had really said “gonorar’’ for “honorar” (honorarium or royalties). I wanted to say, “No, thank you,” but I understood what he meant. The next day when I returned he handed me an envelope which contained three hundred rubles in cash. One is not allowed to export rubles from Russia, so after I had bought some souvenirs, amber, fur hats, books and the like, I still had one hundred rubles left. I had to put them in a postal savings account which in Russia pays one or two percent interest. This makes me a Soviet Union capitalist.

Sometime in the early 1960s I met Gian-Carlo Rota, a mathematician who is almost a quarter-century younger than I and definitely representative of the next generation. Or maybe even several generations later, for academically in mathematics the generation gap may exist already between lecturers and their students where chronologically the difference is only a few years. Our relationship is not built on our age difference. Rota claims that he is greatly influenced by me. So I coined the expression “influencer and influencee.” Rota is one of my best influencees. Banach, for example, I consider as an influencer.

From the start I was impressed by Rota’s feeling for several different mathematical fields and his opinions in many areas of research where he exhibits both erudition and common sense. It is increasingly rare now — in fact, it has been for the last twenty years or more in this era of increasing specialization — to find a person with knowledge of the historical lines of mathematical development.

Rota impressed me by his knowledge of some half-forgotten fields, the work of Sylvester, Cayley and others on classical invariant theory, and by the way he managed to connect the work of Italian geometers to Grassmanian geometry and modernize much of this research which dates to the last century. His main field of work was in combinatorial analysis, where again he managed to update some classical ideas and adapt them to geometry.

I suggested that Rota be invited to Los Alamos for a visit as a consultant. He has been doing this periodically ever since and proved very useful in several ways, including numerical analysis which is important in many of the large computational problems worked on the electronic computers.

Rota’s personality is compatible with mine. His general education, active interest in philosophy (he is an expert on the work of Edmund Husserl and Martin Heidegger), and, above all, his knowledge of classical Latin and ancient history, have made him fill the gap left by the loss of von Neumann. Indeed we often vie in quoting from Horace, Ovid, and other authors in a good-humored display of boasting erudition. Rota is also a true bon vivant, exceedingly fond of good wines and foods, especially those from Italy. He is incredibly adept in the preparation of a great variety of pasta dishes. Italian born, he was brought to South America right after World War II, and at the age of eighteen he came to the United States. His college education took place here, but he has retained many European mannerisms in dress, tastes and habits. He is a Princeton graduate and now a professor at MIT.

Chapter 14. Professor Again

During the Los Alamos years I frequently took time off to return to academic life, and around 1965 I started visiting the University of Colorado on a more regular basis, so it was not a discontinuous change when in 1967 I decided to retire from Los Alamos and accept a professorship in Boulder. Nor was I going to a strange, new place; on the contrary, I was joining several of my good old friends who had also selected the Colorado Rockies as a place to live, David Hawkins, Bob Richtmyer and George Gamow. Hawkins had been a professor of philosophy in Boulder since he had left Los Alamos after the war; Richtmyer, the post-war leader of the theoretical division before Carson Mark, had given up the Courant Institute in New York for the cleaner air of Boulder; Gamow had become a professor in the physics department several years earlier. The University of Colorado was flourishing and expanding, especially in the sciences, and the mathematics department experienced an explosive growth in size and in quality. Besides, Boulder was sufficiently close to Los Alamos, an easy day’s drive through spectacular scenery, so I could continue as a consultant and visit frequently. The focus of my involvement, however, shifted from Los Alamos to Boulder.

In Boulder I saw a great deal of Gamow until his death in 1968. His health had been failing for a few years, his liver had weakened under the assaults of a lifetime of carefree drinking. He was quite aware of this and said to me on some occasion: “Finally my liver is presenting me with the bill.” This did not prevent him from working and writing till the very end. At his Russian funeral, when he lay in an open casket, I realized that he was only the second dead man I had ever seen in my life. Though I was not conscious of the shock this gave me, I had to hold onto the rail when we stood up for the chants so that my knees would not buckle under me.

Gamow’s autobiography, My World Line, was published posthumously from fragments of his unfinished manuscript.

By an incredible coincidence, Gamow and Edward Condon, who had discovered simultaneously and independently the explanation of radioactivity (one in Russia, the other in this country), came to spend the last ten years of their lives within a hundred yards of each other in Boulder. They had become friends even though Condon often felt that neither he nor his collaborator Gurney had received their due share of credit for the discovery.

Condon was a marvelous person. For me he typified the best in the native American character, earthy, super-honest, solid, simple, and at the same time very perspicacious. His political views often coincided with mine. He did not like Nixon, who had hounded him on the Un-American Activities Committee to the point that he resigned from the directorship of the Bureau of Standards. He joined the physics department in Boulder after developing heart trouble; the year before his death in 1973, he had been given an artificial heart valve, which rendered his last months more active and comfortable.

In the relative greater freedom of university life, longer vacations, no fixed schedule except for some teaching, I was returning to a more academic type of science in a milieu of mathematicians and physicists. The mathematics department was acquiring excellent researchers in the foundations of mathematics, set theory, logic, and number theory. Wolfgang Schmidt, an Austrian by birth, was one of them, powerful and original in the latter. Another is a younger, brilliant Pole, Jan Mycielski, a student of Steinhaus, whom I invited to accept a professorship when I was chairman of the department. We have since collaborated in problems of game theory, combinatorics, set theory and — during the last several years — on mathematical schemata connected with the study of the nervous system. Mycielski, with Rota and a Los Alamos mathematician, William Beyer, gathered and edited the first volume of my collected works, which has been published by the MIT Press under the title Sets, Numbers, Universes. The Boulder mathematics department also has a number of young people strong in analysis and topology.

In 1967 the mathematician Mark Kac and I were invited by the editors of the Encyclopaedia Britannica to write a long article which was to be part of a series of special appendices to a new edition of the Britannica. Since then it has appeared separately under the title Mathematics and Logic. It received very favorable reviews and has been translated into French, Spanish, Russian, Czech, and Japanese. It was rather difficult for us to find the right level of presentation. Designed not so much for the broad public but rather for scientists in other fields, we tried to make it a semi-popular presentation of modern ideas and perspectives of the great concepts of mathematics.

As the reader may have noticed, much of my work seems to have been done in collaboration with others (just as this book was assembled in collaboration with Françoise). One of the reasons for this is my leaning on conversation as a stimulant to thinking; the other is my well-known impatience with detail and a certain distaste for reading what I have written. When I see one of my papers in print, I have a childish complex, a tiny nagging doubt that it might be wrong or that it may not contain anything interesting, and I discard it after a quick glance.

Mark Kac had also studied in Lwów, but since he was several years younger than I (and I had left when only twenty-six myself), I knew him then only slightly. He told me that as a young student he had been present at my doctorate ceremony and had been impressed by it. He added that these first impressions usually stay, and that he still considers me “a very senior and advanced person,” even though the ratio of our ages is now very close to one. He came to America two or three years after I did. I remembered him in Poland as very slim and slight, but here he became rather rotund. I asked him, a couple of years after his arrival, how it had happened. With his characteristic good humor he replied: “Prosperity!” His ready wit and almost constant joviality make him extremely congenial.

After the war he visited Los Alamos, and we developed our scientific collaboration and friendship. After a number of years as a professor at Cornell he became a professor of mathematics at The Rockefeller Institute in New York (now The Rockefeller University.) He and the physicist George Uhlenbeck have established mathematics and physics groups at this Institute, where biological studies were the principal and almost exclusive subject before.

Mark is one of the very few mathematicians who possess a tremendous sense of what the real applications of pure mathematics are and can be; in this respect he is comparable to von Neumann. He was one of Steinhaus’s best students. As an undergraduate he collaborated with him on applications of Fourier series and transform techniques to probability theory. They published several joint papers on the ideas of “independent functions.” Along with Antoni Zygmund he is a great exponent and true master in this field. His work in the United States is prolific. It includes interesting results on probability methods in number theory. In a way, Kac, with his superior common sense, as a mathematician is comparable to Weisskopf and Gamow as physicists in their ability to select topics of scientific research which lie at the heart of the matter and are at the same time of conceptual simplicity. In addition — and this is perhaps related — they have the ability to present to a wider scientific audience the most recent and modern results and techniques in an understandable and often very exciting manner. Kac is a wonderful lecturer, clear, intelligent, full of sense and avoidance of trivia.

Among the mathematicians of my generation who influenced me the most in my youth were Mazur and Borsuk. Mazur I have described earlier. As for Borsuk, he represented for me the essence of geometrical intuition and truly meaningful topology. I gleaned from him, without being able to practice it myself, the workings of n-dimensional imagination. Today Borsuk is continuing his creative work in Warsaw. His recent theory of the “shape” in topology shows increasing power and applications. His general interests and mathematical outlook are very close to mine, and our old friendship was renewed after the war during his visits to the United States and my brief trip to Poland in 1973 when I saw him in his country house near Warsaw.

One could go on ad infinitum recollecting from memory, reflecting and writing down. If the reader is still with me, he may have derived from the preceding a sort of existentialist (the word is in vogue) picture of my life, these times, and the many scientists I have known. By way of a conclusion to this chapter I will add a self-portrait which I sent to Françoise before we were married. I am translating from the French which accounts in part for its awkwardness.

“Self-portrait of Mr. S. U.

“His expression is usually ironic and quizzical. In truth he is very much affected by all that is ridiculous. Perhaps he has some talent to recognize and feel it at once, so it is not surprising that this is reflected in his facial expression.

“His conversation is very uneven, sometimes serious, sometimes gay, but never tiring or pedantic. He only tries to amuse and distract the people he likes. With the exception of the exact sciences, there is nothing which appears so certain or obvious to him that he would not allow for differing opinions: on almost any subject one can say almost anything.

“He brought to the study of mathematics a certain talent and facility which allowed him to make a name for himself at an early age. Dedicated to work and solitude until he was twenty-five, he became more worldly rather late. Nevertheless he is never rude because he is neither coarse nor hard. If he sometimes offends it is through inattention or ignorance. In speech he is neither gallant nor graceful. When he says kind things it is because he means them. Therefore the essence of his character is a frankness and truthfulness which are sometimes a little strong but never really shocking.

“Impatient and choleric to the point of violence, everything that contradicts or wounds him affects him in an uncontrollable way, but this usually disappears when he has vented his feelings.

“He is easy to influence or govern provided he is unaware that this is intended.

“Some people think that he is malicious because he makes merciless fun of pretentious bores. His temperament is naturally sensitive and renders him subject to delicate moods. This makes him at once gay and melancholy.

“Mr. U. behaves according to this general rule: he says a lot of foolish things, seldom writes them and never does any.

When Françoise read this description she felt it agreed well with what she then knew of me but was very surprised at the quality of my French, until she came to the last paragraph:

“And now I shall change from my text which I came upon by chance yesterday. The above are verbatim extracts from a letter of d’Alembert to Mademoiselle de Lespinasse written some two hundred years ago!” (D’Alembert was a famous French mathematician and encyclopedist of the eighteenth century). Francoise was very amused.

Some thirty years have elapsed since I copied this little text. I will now add as a finishing touch that I don’t think I have changed much, but that there is one trait which d’Alembert did not mention that I possess — all this merely si parva magnis comparare licet — it is a certain impatience. I have been afflicted with this all my life. It may be increasing with advancing years. (If Einstein or Cantor came to lecture here today I would have the split reaction of a schoolboy — wanting to learn on the one hand and to skip class on the other.) While I still feel quite happy giving lectures, talks, or discussions I am becoming less and less able to sit through hours of such given by others. I am, I told some colleagues, “like an old boxer who can still dish it out but can’t take it any more.” This amused them no end.

Chapter 15. Random Reflections on Mathematics and Science

This chapter will be somewhat different in content from the preceding account of my ’’adventures” and of scientists I have known.

Here I have tried to gather, review, and sometimes amplify some of the general ideas I have touched upon so lightly throughout the book. I hope that in their randomness these reflections will give the reader an added glimpse into the manifold aspects of science and especially the relation of mathematics to other sciences. It is merely about the “gist of the gist.” For greater detail, I can only refer the reader to some of my more general scientific publications.What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers and figures, with patterns, relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes. They also know that it purports to form the foundations of all rational thought.

Some could say it is the external world which has molded our thinking — that is, the operation of the human brain — into what is now called logic. Others — philosophers and scientists alike — say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution “independently” of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents.

The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.

The aesthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is. Few non-mathematicians, even among other scientists, can fully appreciate the aesthetic value of mathematics, but for the practitioners it is undeniable. One can, however, look conversely at what might be called the homely side of mathematics. This homeliness has to do with having to be punctilious, of having to make sure of every step. In mathematics one cannot stop at drawing with a big, wide brush; all the details have to be filled in at some time.

“Mathematics is a language in which one cannot express unprecise or nebulous thoughts,” said Poincaré, I believe in a speech on world science which he gave at the St. Louis fair many years ago. And he gave as an example of the influence of language on thought a description of how differently he felt using English instead of French.

I tend to agree with him. It is a truism to say that there is a clarity to French which is not there in other tongues, and I suppose this makes a difference in the mathematical and scientific literature. Thoughts are steered in different ways. In French generalizations come to my mind and stimulate me toward conciseness and simplification. In English one sees the practical sense; German tends to make one go for a depth which is not always there.

In Polish and Russian, the language lends itself to a sort of brewing, a development of thought like tea growing stronger and stronger. Slavic languages tend to be pensive, soulful, expansive, more psychological than philosophical, but not nebulous or carried by words as much as German, where words and syllables concatenate. They concatenate thoughts which sometimes do not go very well together. Latin is something else again. It is orderly; clarity is always there; words are separated; they do not glue together as in German; it is like well-cooked rice compared to overcooked.

Generally speaking, my own impressions of languages are the following: When I speak German everything I say seems overstated, in English on the contrary it feels like an understatement. Only in French does it seem just right, and in Polish, too, since it is my native language and feels so natural.

Some French mathematicians used to manage to write in a more fluent style without stating too many definite theorems. This was more agreeable than the present style of the research papers or books which have so much symbolism and formulae on every page. I am turned off when I see only formulas and symbols, and little text. It is too laborious for me to look at such pages not knowing what to concentrate on. I wonder how many other mathematicians really read them in detail and enjoy them.

There do exist, though, important, laborious and inelegant theorems. For example some of the work connected with partial differential equations tends to be less “beautiful” in form and style, but it may have “depth,” and may be pregnant with consequences for interpretations in physics.

How does one arrive at a value judgment nowadays?

Mathematicians, whose job in a sense is to analyze the motivation and origin of their work, fool themselves and may be remiss when they think their main business is to prove theorems without at least indicating why they may be important. If left entirely to aesthetic criteria, doesn’t it compound the mystery?

I believe that in the decades to come there will be more understanding, even on a formal level, of the degree of beauty, though by that time the criteria may have shifted and there will be a super beauty in unanalyzable higher levels. So far when anyone has tried to analyze the aesthetic criteria of mathematics too precisely, whatever was proposed has seemed too narrow. It has to appeal to connections with other theories of the external world or to the history of the development of the human brain, or else it is purely aesthetic and very subjective in the sense that music is. I believe that even the quality of music will be analyzable — to an extent only, of course — at least by formal criteria, by mathematizing the idea of analogy.

Some of the old problems, unsolved for many years, are being settled. Some are solved with a bang, and others with a whimper, so to speak. This applies to problems seemingly equally important and a priori interesting, but some, even famous classical ones, are solved in such a specific way that there is nothing more to be asked or said. Some others, less famous, immediately upon solution become sources of curiosity and activity. They seem to open new vistas.

As for publications, mathematicians nowadays are almost forced to conceal the way they obtain their results. Evariste Galois, the young French genius who died at the age of twenty-one, in his last letter written before his fatal duel, stressed how the real process of discovery is different from what finally appears in print as the process of proof. It is important to repeat this again and again.

On the whole and in the large lines there does seem to exist a consensus among working mathematicians about the value of individual achievements and the value of new theories. There must therefore be something objective if not yet defined about the feeling of beauty which mathematics offers, dependent sometimes also on how useful it turns out to be in other branches of itself or of other sciences. Why mathematics is really so useful in the description of the physical world, for me at least remains philosophically a mystery. Eugene Wigner once wrote a fascinating article on this “implausible” usefulness of mathematics and titled it “The Unreasonable Effectiveness of Mathematics.”

It is, of course, one very concise way of formalizing all rational thought.

It also has manifestly, in elementary, secondary and advanced schools the value of training our brain, since practice, just as in any other game, sharpens the organ. I cannot say whether a mathematician’s brain is today sharper than it was in the time of the Greeks; nevertheless on the longer scale of evolution it must be so. I do believe mathematics may have a great genetic role, it may be one of the few means of perfecting the human brain. If true, nothing could then be more important for humanity, whether to arrive at some new destiny as a group or for individuals. Mathematics may be a way of developing physically, that is anatomically, new connections in the brain. It has a sharpening value even though the enormous proliferation of material shows a tendency to beat things to death.

Yet every formalism, every algorithm, has a certain magic in it. The Jewish Talmud, or even the Kabbalah, contains material which does not appear particularly enlightening intellectually, being a vast collection of grammatical or culinary recipes, some perhaps poetic, others mystical, all rather arbitrary. Over centuries thousands of minds have pored over, memorized, dissected, and classified these works. In so doing people may have sharpened their memories and deductive practice. As one sharpens a knife on a whetstone, the brain can be sharpened on dull objects of thought. Every form of assiduous thinking has its value.

There exist in mathematics propositions, such as the one called “Fermat’s Great Theorem,” which, standing by themselves, seem special and unrelated to the main body of number theory. They are very simple to state but have defied all the efforts of the greatest minds to prove them. Such statements have stimulated young minds (my own included) to more general wonder and curiosity. In the case of Fermat’s problem, special or irrelevant as it is by itself, it has stimulated through the last three centuries of mathematics, the creation of new living objects of mathematical thought, in particular the so-called theory of ideals in algebraic structures. The history of mathematics knows a number of such creations.

The invention of imaginary and complex numbers (which are pairs of real numbers with a special rule for their addition and multiplication) beyond the immediate purpose and the use to which they were put, opened new possibilities and led to the discovery of miraculous properties of the complex variables. These analytic functions (the examples of which are, to mention the simplest, z = √w, z = e^w, z = log w), possess unexpected, simple and a priori unforeseen properties deriving from the few general rules which govern them. They have convenient algorithms and rather deep connections with the properties of geometrical objects and also with the mysteries concerning the seemingly so familiar natural numbers, the ordinary integers. It is as if some invisible different universe governing our thought became dimly perceptible through it, a universe with some laws, and yes, facts, of which we become only vaguely aware.

The fact that some seemingly very special functions, like the Riemann Zeta function, have such deep connections with the behavior of integers, of prime numbers, is hard to explain a priori and in depth. This is really not well understood to this day. Somehow these entities, these special analytic functions defined by infinite series, have been generalized more recently to spaces other than the plane of all complex numbers, such as to algebraic surfaces. These entities show connections between seemingly diverse notions. They also seem to show the existence (to make a metaphor stimulated by the subject itself) of another surface of reality, another Riemann surface of thought (and connections of thought) of which we are not consciously aware.

Some of the properties of the analytic functions of the complex numbers turn out to be not merely convenient, but very fundamentally tied to physical properties of matter, in the theory of hydrodynamics, in the description of the motions of incompressible fluids such as water, in electrodynamics, and in the foundations of quantum theory itself.

The creation of a general idea of a space, abstracted to be sure, but not really completely dictated or uniquely indicated by the physical space of our senses, the generalization to the n-dimensional space where n is greater than three and even to infinitely many dimensions, and so useful at least as a language for the foundations of physics itself, are these marvels of the power of the human brain? Or is it the nature of the physical reality which reveals it to us? The very invention, or is it “discovery,” that there are different degrees or different kinds of infinity has had not only a philosophical, but beyond that, a striking psychological influence on receptive minds.

Speaking of the fascination of surprises, the mysterious attraction of mathematics and, of course, of other sciences — physics especially — it may be remarked how often it happens that in the game of chess one may observe weak players or even rank beginners getting into deep and fascinating positions. I have often watched amateurs or non-talented beginners, looked at their game after some fifteen moves, observed that their position arrived at perhaps by chance, certainly not by design, was full of marvelous possibilities for both sides. And I wonder how it is that the game itself produces these positions of great appeal and art without these simple fellows being even aware of it. I do not know whether an analogous experience is possible in the game of Go. I cannot myself judge, not knowing much about the intricacies of that beautiful game, but I wonder whether a master looking at a position can tell whether it was arrived at by chance or by a logically developed correct and thoughtful play.

In science, and in mathematics in particular, there seems to be a similar magical interest in certain algorithms. They appear to have a power to produce by themselves, as it were, solutions to problems or vistas of new perspectives. What seemed to be at first mere tools designed for special purposes can bring about some unforeseen and unexpected new uses.

By the way, a little philosophical conundrum occurred to me which I do not know how to resolve: Consider a game like a solitaire, or a game between two persons. Assume that the players may cheat once or twice during the course of the game. For instance, in a Canfield solitaire, if one changes the position of one or two cards once and once only, the game is not destroyed. It would still be a precisely, completely, mathematically meaningful, albeit different game. It would become simply a bit richer, more general. But if one takes a mathematical system, a system of axioms and allows the addition of one or two false statements, the result is immediate nonsense because once one has a false statement, one can deduce anything one wants to. Where does the difference lie? Perhaps it lies in the fact that only a certain class of motions is allowed in the game, whereas in mathematics once an incorrect statement is introduced one may immediately get the statement: zero equals one. There must then be a way to generalize the game of mathematics so that one could make a few mistakes and instead of getting complete nonsense, obtain merely a wider system.

Hawkins and I have speculated on the following related problem: a variation on the game of Twenty Questions. Someone thinks of a number between one and one million (which is just less than 2²⁰). Another person is allowed to ask up to twenty questions, to each of which the first person is supposed to answer only yes or no. Obviously the number can be guessed by asking first: Is the number in the first half million? then again reduce the reservoir of numbers in the next question by one-half, and so on. Finally the number is obtained in less than log₂(1,000,000). Now suppose one were allowed to lie once or twice, then how many questions would one need to get the right answer? One clearly needs more than n questions for guessing one of 2ⁿ objects because one does not know when the lie was told. This problem is not solved in general.

In my book on unsolved problems I claim that many mathematical theorems can be payzised (a Greek word which means to play). That is, that they can be formulated in game-theoretical language. For example, a rather general schema for playing a game can be set up as follows:

Suppose N is a given integer and two players are to build two permutations of N letters (n₁, n₂… n_N). They are constructed by the two players in turn, as follows. First permutation, the first player takes n₁, the second n₂, the first takes n₃ and so on. Finally the first permutation is accomplished. Then they play for the second permutation, and if the two permutations generate the group of all permutations, the first player wins, if not the second wins. Who has a winning strategy in this game? This is merely a small example of how, in any domain of mathematics — in this case in finite group theory — one can invent gamelike schemata which lead to purely mathematical problems and theorems.

One can also ask questions of a different type: If this is done at random what are the chances? This is a problem that combines measure theory, probability and combinatorics. One may proceed this way in many domains of mathematics.

Set theory revolutionized mathematics toward the end of the nineteenth century. What started this was that Georg Cantor proved (i.e., discovered) that the continuum is not countable. He did have predecessors in these speculations on the logic of infinity, Weierstrass and Bolzano, but the first precise study of degrees of infinity was certainly his. This arose from his discussion of trigonometric series, and very quickly transformed the shape and flavor of mathematics. Its spirit has increasingly pervaded mathematics; recently it has had a new and technically unexpected renewed development, not only in its most abstract form, but also in its immediate applications. The formulations of topology, of algebraical ideas in their most general form received impetus and direction from the activities of the Polish school, much of it from Lwów where the interests centered around what can be called roughly functional analysis in a geometrical and algebraical spirit.

To give an oversimplified description of the origin of much of these activities: After Cantor and the mathematicians of the French school, Borel, Lebesgue and others, this kind of investigation found a home in Poland. In her book Illustrious Immigrants, Laura Fermi expresses a surprised admiration at the large percentage of Polish mathematicians in the United States who contributed so much significant work to the flourishing of this field. Many came here to settle and continue such work. Simultaneously the studies of analysis of Hilbert and other German mathematicians brought about a simple general mathematical construction of infinitely dimensional functional spaces, also later further developed by the Polish school. Independently and at the same time, the work of Moore, Veblen, and others in America brought about a meeting of the geometrical and algebraical points of view, and a unification — only to some extent to be sure — of mathematical activities.

It appears that in spite of increasing diversity and even overspecialization, the choice of mathematical topics of research thus follows prevalent currents, threads, and trends which come together from independent sources.

Some few individuals with a few new definitions are apparently able to start a whole avalanche of work in special fields. This is partly due to fashion and self-perpetuation by the sheer force of the teachers’ influence. When I first came to this country I was amazed at what seemed to me an exaggerated concentration on topology. Now I feel there is perhaps too much work in the domain of algebraic geometry.

A second epochal landmark was Gödel’s work, recently made more specific by Paul Cohen’s results. Gödel, the mathematical logician at the Institute for Advanced Studies in Princeton, found that any finite system of axioms or even countably infinite systems of axioms in mathematics, allows one to formulate meaningful statements within the system which are undecidable — that is to say, within the system one will not be able to prove or disprove the truth of these statements. Cohen opened the door to a whole class of new axioms of infinities. There is now a plethora of results showing that our intuition of infinity is not complete. They open up mysterious areas in our intuitions to different concepts of infinity. This will, in turn, contribute indirectly to a change in the philosophy of foundations of mathematics, indicating that mathematics is not a finished object as was believed, based on fixed, uniquely given laws, but that it is genetically evolving. This point of view has not yet been accepted consciously, but it points a way to a different outlook. Mathematics really thrives on the infinite, and who can tell what will happen to our attitudes toward this notion during the next fifty years? Certainly, there will be something — if not axioms in the present sense of the word, at least new rules or agreements among mathematicians about the assumption of new postulates or rather let us call them formalized desiderata, expressing an absolute freedom of thought, freedom of construction, given an undecidable proposition, in preference to true or false assumption. Indeed some statements may be undecidably undecidable. This should have great philosophical interest.

The interest in the foundations of mathematics is to some extent also philosophical, though eventually it does pervade everything, like set theory. But the word ’’foundations” is a misnomer; for the time being, it is just one more mathematical specialty, fundamental to be sure.

The great at dichotomy in the origin and in the inspiration of mathematical thought — stimulated by the influence of external reality, the physical universe, on one hand, and by the developing processes of physiology, almost perhaps of the human brain, on the other — has in a small and special way a homomorphic image in the present and near-future use of electronic computers.

Even the most idealistic point of view about mathematics as a pure creation of the human mind must be reconciled with the fact that the choice of definitions and axioms of geometry — in fact of most mathematical concepts — is the result of impressions obtained through our senses from external stimuli and inherently from observations and experiments in the “external world.” The theory of probability, for example, came about as a development of a few questions concerning games of chance. Now, computing machines constructed to solve special problems of mathematics promise to enlarge very greatly the scope of the Gedanken Experimente, the idealization of experience, and our more abstract schemata of thought.

It appears that experimentation on models of games played by self-organizing living matter through chemical reactions in living organisms will lead to novel abstract mathematical schemata. The new study of the mathematics of growing patterns, and the possibility of studying experimentally on computing machines the course of competitions or contests between geometrical configurations imitating the fight for survival, these might give rise to new mathematical setups. One could again give names like “payzonomy” to the combinatorics of contesting reactions and “auxology” to a yet-to-be-developed theory of growth and organization, this latter ultimately including the growing tree of mathematics itself.

So far only the very simplest and crude mathematical schemata have been proposed to mirror the mathematical properties of geometrical growth. (An account of my own simple-minded models can be found in a recent book edited by Arthur Burke, A Theory of Cellular Automata, published by the University of Illinois Press.)

An especially ingenious set of rules was devised by the English mathematician John Conway, a number theorist. The Conway Game of Life is an example of a game or pastime which, perhaps much like the early problems involving dice and cards, has led ultimately to the present edifice of probability theory, and may lead to a vast new theory describing the “processes” which Alfred North Whitehead studied in his philosophy.

The use of computers seems thus not merely convenient, but absolutely essential for such experiments which involve following the games or contests through a very great number of moves or stages. I believe that the experience gained as a result of following the behavior of such processes will have a fundamental influence on whatever may ultimately generalize or perhaps even replace in mathematics our present exclusive immersion in the formal axiomatic method.

The already-mentioned recent results of Paul Cohen and others — Peter Novikoff, Hao Wang, Yuri Matiasevic — on the independence from the traditional system of axioms of some of the most fundamental mathematical statements, indicate a new role for pragmatic approaches. Work with automata will help indicate whether a problem can be solved by existing means.

To illustrate what we have in mind let us consider for example a “little” special problem in three dimensions: given a closed curve in space and a solid body of given shape, the problem is to push the body through the curve. There are no clear mathematical criteria to decide whether it can be done or not. One has to rotate, wiggle, squeeze, and “try,” to see whether it can be done. In a higher number of dimensions, like five, one can have an analogous problem. The idea is to set it up on a computer and try various possible motions. Perhaps, after very many tries, one would acquire a feeling for the freedom of maneuvering in this high dimensional space and a new type of an almost tactile intuition. Of course, this is a special, small and unimportant example, but I feel that one could develop new imaginations by suitable experimentation with these new tools, electronic computers especially, in setting up and observing the various growth processes and evolutionary developments.

It seems to me the impact and role of the electronic computer will significantly affect pure mathematics also, just as it has already done so in the mathematical sciences, principally in physics, astronomy and chemistry.

These conjectured excursions into aspects of the future of mathematics take us far from von Neumann and his contemporaries, and their role in the evolution of science a quarter-century ago. The rate of growth in the organized activities of the human mind, accelerated no doubt by the advent of computers, seems to increase in a way which forebodes qualitative changes in our way of thinking and living. As Niels Bohr said in one of his amusing remarks: “It is very hard to predict, especially the future.” But I think mathematics will greatly change its aspect. Something drastic may evolve, an entirely different point of view on the axiomatic method itself. Instead of detailed work on special theorems which now number in the millions, instead of thinking in terms of rules operating with symbols given once and for all, it may be that mathematics will consist more and more of problems, or desiderata, or programs for work of a general nature. No longer will there be additional multitudes of special spaces, definitions of special manifolds, of special mappings of this and that — though a few will survive: “apparent rari nantes in gurgite vasto,” no new collections of individual theorems, but instead general sketches or outlines of larger theories, of vaster enterprises, and the actual working out of proofs of theorems will be left to students or even to machines. It may become comparable to impressionistic painting in contrast to the painful, detailed drawing of earlier days. It could be a more living and changing scene, not only in the choice of definitions but in the very rules of the game, this great game whose rules until now have not changed since antiquity.

If the rules have not changed, very great changes have already taken place in the scope of mathematics in the space of my own lifetime. In the nineteenth century the applications of mathematics were all-inclusive in physics, astronomy, chemistry, in mechanics, engineering, and all the other facets of technology. More recently, mathematics serves to formulate the foundations of other sciences as well, so-called mathematical physics is really the theory of all physics, reaching into its most abstract parts like quantum theory, the very strange four-dimensional continuum of space-time. These belong specifically to the twentieth century. In the short span of sixty to one hundred years the proliferation of the use of mathematical ideas has been unbelievably varied. It was accompanied by, one could say, an explosive creation of new mathematical objects, large and small, and a tendency to “beat things to death” with proliferation and hairsplitting studies of minute details that is almost Talmudic.

At a talk which I gave at a celebration of the twenty-fifth anniversary of the construction of von Neumann’s computer in Princeton a few years ago, I suddenly started estimating silently in my mind how many theorems are published yearly in mathematical journals. (A theorem being defined as a statement which is just labeled “theorem,” and is published in a recognized mathematical journal.) I made a quick mental calculation, amazing myself that I could do this while talking about something entirely different and came to a number like one hundred thousand theorems per year. Quickly changing my topic I mentioned this and the audience gasped. It may interest the reader that the next day two of the younger mathematicians in the audience came to tell me that impressed by this enormous figure they undertook a more systematic and detailed search in the Institute library. By multiplying the number of journals by the number of yearly issues, by the number of papers per issue and the average number of theorems per paper, their estimate came to nearer two hundred thousand theorems a year. Such an enormous number should certainly give food for thought. If one believes that mathematics is more than games and puzzles, here is something to worry about. Clearly the danger is that mathematics itself will suffer the fate of splitting into different separate sciences, into many independent disciplines tenuously connected. My own hope is that this will not happen, for if the number of theorems is larger than one can possibly survey, who can be trusted to judge what is “important”? The problem becomes one of record keeping, of storage and retrieval of the results obtained. This problem now becomes paramount; one cannot have survival of the fittest if there is no interaction.

It is actually impossible to keep abreast of even the more outstanding and exciting results. How can one reconcile this with the view that mathematics will survive as a single science? Just as one cannot know all the beautiful women or all the beautiful works of art and one finally marries one beautiful person, one can say that in mathematics one becomes married to one’s own little field. Because of this, the judgment of value in mathematical research is becoming more and more difficult, and most of us are becoming mainly technicians. The variety of objects worked on by young scientists is growing exponentially. Perhaps one should not call it a pollution of thought; it is possibly a mirror of the prodigality of nature which produces a million species of different insects. Somehow one feels, though, that it goes against the grain of one’s ideals of science, which aims to understand, abbreviate, summarize, and, in particular, to develop, a notation system for the phenomena of the mind and of nature.

It is the unexpected in the development of science, the way really new ideas and concepts strike a young mind, that mold it irreversibly. Later, for the mature or older mind, the unexpected causes a wonder which induces new stimulation, even when one has become less impressionable or even jaded. To quote Einstein, “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”

Mathematics creates new objects of thought — one could call it a meta-reality — by engendering ideas which begin to live their own life in an independent development. Once born these cannot any more be controlled by a single person, only by a collection of brains which are the perpetuating set of mathematicians.

Talent or genius in mathematics is hard to quantify. I tend to feel that there is an almost continuous passage from mediocrity to the highest levels of people like Gauss, Poincaré and Hilbert. So much depends not on the brain alone. There are definitely what I have called, for want of a better word, “hormonal factors” or traits of character: stubbornness, physical ability, willingness to work, what some call “passion.” These depend a great deal on habits mostly acquired in childhood or early youth when accidents of early impressions play a great role. Undoubtedly, much of the quality called imagination or intuition comes from the physiological structural properties of the brain, which in turn may be partly developed through experiences leading to certain habits of thought and of the direction of the train of thought.

The willingness to plunge into the unknown and the unfamiliar varies with different individuals. There are distinctly different types of mathematicians — those who prefer to attack existing problems or to build on what is already there, and those who like to imagine new schemata and new possibilities. The first perhaps constitute a majority, maybe more than eighty percent. When a young man wants to establish a reputation he will mostly attack an unsolved problem that has already been worked on. In this way, if he is lucky or strong enough, it will be comparable to an athlete beating a record, jumping higher than anyone before. Although what is often of greater value is the conception of a new idea, a young person is often unwilling to try this, not knowing whether the new thought will be appreciated even when to him it is important and beautiful.

I am of the type that likes to start new things rather than improve or elaborate. The simpler and “lower” I can start the better I like it. I do not remember using complicated theorems to prove more complicated ones. (Of course, this is all relative, “there’s nothing new under the sun” — everything can be traced back to Archimedes or even earlier.)

I also believe that changing fields of work during one’s life is rejuvenating. If one stays too much with the same subfields or the same narrow class of problems a sort of self-poisoning prevents acquisition of new points of view and one may become stale. Unfortunately, this is not uncommon in mathematical creativity.

With all its grandiose vistas, appreciation of beauty, and vision of new realities, mathematics has an addictive property which is less obvious or healthy. It is perhaps akin to the action of some chemical drugs. The smallest puzzle, immediately recognizable as trivial or repetitive can exert such an addictive influence. One can get drawn in by starting to solve such puzzles. I remember when the Mathematical Monthly occasionally published problems sent in by a French geometer concerning banal arrangements of circles, lines and triangles on the plane. “Belanglos,” as the Germans say, but nevertheless these figures could draw you in once you started to think about how to solve them, even when realizing all the time that a solution could hardly lead to more exciting or more general topics. This is much in contrast to what I said about the history of Fermat’s theorem, which led to the creation of vast new algebraical concepts. The difference lies perhaps in that little problems can be solved with a moderate effort whereas Fermat’s is still unsolved and a continuing challenge. Nevertheless both types of mathematical curiosities have a strongly addictive quality for the would-be mathematician which exists on all levels from trivia to the most inspiring aspects.

In the past there always were a few mathematicians who either explicitly or by implication gave specific ideas and choice of direction to the work of others — men like Poincaré, Hilbert and Weyl. This is now becoming increasingly difficult if not impossible. There is probably not one mathematician now living who can even understand all of what is written today.

A volume written more than thirty years ago by Eric Temple Bell, The Development of Mathematics, contains an excellent abbreviated account of the history of mathematics. (Perhaps I like it, because, to use G.-C. Rota’s language, my work is mentioned there even though the book was written when I was only twenty-eight years old and it is a rather small volume. There is more satisfaction in being mentioned in a short history than in one which has ten thousand pages!) But when Weyl was asked by a publisher to write a history of mathematics in the twentieth century he turned it down because he felt that no one person could do it.

Von Neumann, who could have aspired to such a role, admitted to me some thirty-five years ago that he knew less than a third of the corpus of mathematics. At his suggestion once I concocted for him a doctoral-style examination in various fields trying to select questions which he would not be able to answer. I did find some, one each in differential geometry, in number theory, in algebra, which he could not answer satisfactorily. (This by the way may also tend to show that doctoral exams have little permanent meaning.)

As for myself, I cannot claim that I know much of the technical material of mathematics. What I may have is a feeling for the gist, or maybe only the gist of the gist, in a number of its fields. It is possible to have this knack for guessing or feeling what is likely to be new or already known, or else not known, in some branch of mathematics where one does not know the details. I think I have this ability to a degree and can often tell whether a theorem is known, i.e. already proved, or is a new conjecture. This is a sort of feeling that comes from the way the quantifiers are arranged, from the tone or the music of the statement, so to speak.

Speaking of this analogy: I can remember tunes and am able to whistle various melodies rather correctly. But when I try to invent or compose some new “catchy” tune, I find rather impotently that what I do is a trivial combination of what I have heard. This in complete contrast to mathematics where I believe with a mere “touch” I can always propose something new.

Collaboration in mathematics is a very interesting and new phenomenon which developed during the last several decades.

It is natural in experimental physics that investigators work together on the different phases of instrumentation. By now every experiment is really a class of technical projects, especially on the great machines which require hundreds of engineers and specialists for their construction and operation. In theoretical physics this is perhaps not as evident, but it exists, and strangely enough in mathematics also. We have seen that the creative effort in mathematics requires intense concentration and constant thinking in depth for hours on end, and that it is often shared by two individuals who just look at each other and occasionally make a few remarks when they collaborate. It is now definitely so that even in the most abstruse mathematical questions two or more persons work together on trying to find a proof. Many papers have now two, sometimes three or more authors. The exchange of conjectures, suggesting tentative approaches, helps to build up partial results along the way. It is easier to talk than to write down every thought. There is here an analogy to analyzing a game of chess.

It may be that in the future large groups of mathematicians working together will produce important, beautiful, and simple results. Some have already been produced this way in recent years. For example, the solution of one of Hilbert’s problems about the existence of algorithms to solve diophantine equations was really obtained (not in parallel to be sure but in sequence) by several scientists in this country, and at the end by a young Russian, Yuri Matiasevic, who took the last step. Several mathematicians working independently in the United States and in Poland but aware of each other’s results, solved an old problem of Banach’s about the homeomorphism of his spaces. They were able to climb on each other’s shoulders, so to speak.

It was after the publicity surrounding the construction of the atomic bomb in Los Alamos that the expression “critical mass” became current as a metaphoric description of the required minimal size of a group of scientists working together in order to obtain successful results. If large enough, the group produces results explosively. When the critical mass is reached, due to mutual stimulation the multiplication of results, like that of neutrons, becomes exponentially larger and more rapid. Before such a mass is attained, progress is gradual, slow and linear.

Other variations in the working habits of scientists have been slower. The mode of life in the ivory tower world of science now includes more scientific meetings, more involvement in governmental work.

A simple but important thing like letter-writing has also undergone a noticeable change. It used to be an art, not only in the world of literature. Mathematicians were voluminous letter writers. They wrote in longhand and communicated at length intimate and personal details as well as mathematical thoughts. Today the availability of secretarial help renders such personal exchanges more awkward, and as it is difficult to dictate technical material scientists in general and mathematicians in particular exchange fewer letters. In my file of letters from all the scientists I have known, a collection extending over more than forty years, one can see the gradual, and after the war accelerated, change from long, personal, handwritten letters to more official, dry, typewritten notes. In my correspondence of recent years, only two persons have continued to write in longhand: George Gamow and Paul Erdös.

Chen Ning Yang, the Nobel prize physicist, tells a story which illustrates an aspect of the intellectual relation between mathematicians and physicists at present:

One evening a group of men came to a town. They needed to have their laundry done so they walked around the city streets trying to find a laundry. They found a place with the sign in the window, ’’Laundry Taken in Here.” One of them asked: “May we leave our laundry with you?” The proprietor said: “No. We don’t do laundry here.” “How come?” the visitor asked. “There is such a sign in your window.” “Here we make signs,” was the reply. This is somewhat the case with mathematicians. They are the makers of signs which they hope will fit all contingencies. Yet physicists have created a lot of mathematics.

In some of the more concrete parts of mathematics — for example probability theory — physicists like Einstein and Smoluchowski have opened certain new areas even before mathematicians. The ideas of information theory, of entropy of information and its role in general continuum originated with physicists like Leo Szilard and an engineer, Claude Shannon, and not with “pure” mathematicians who could and ought to have done so long before. Entropy, a property of a distribution, was a notion originating in thermodynamics and was applied to physical objects. But Szilard (in very general terms) and Shannon defined this notion for general mathematical systems. True, Norbert Wiener had some part in the origin of it and wonderful mathematicians like Andrei Kolmogoroff later developed, generalized, and applied it to purely mathematical problems.

In the past some mathematicians, Poincaré for example, knew a lot of physics. Hilbert did not seem to have too much true physical instinct, but he wrote very important papers about the techniques and the logic of physics. Von Neumann knew a good deal of physics too, but I would say that he did not have the physicist’s natural feeling for and recourse to experiment. He was interested in the foundations of quantum mechanics as long as they could be mathematized. The axiomatic approach to physical theories is to physics what grammar is to literature. Such mathematical clarity need not be conceptually crucial for physics.

On the other hand, much of the apparatus of theoretical physics and occasionally some precursor ideas came from pure mathematics. The general non-Euclidean geometries prophetically envisaged by Riemann as having future importance for physics, came before general relativity, and the definition and study of operators in Hilbert space came before quantum mechanics. The word spectrum, for example, was used by mathematicians long before anybody would have dreamed of using the spectrum representation of Hilbert space operators to explain the actual spectrum of light emitted by atoms.

I have often wondered why mathematicians have not generalized the special theory of relativity into different types of “special relativities,” so to speak (not into the presently known general theory of relativity). I am sure there are other “relativities” possible in general spaces, yet hardly anything has been attempted by mathematicians. Endless papers exist on metric spaces generalizing the ordinary geometry without the dimension of time in it. Put in time and space together, and mathematicians stay out! Topologists stay with spatial spaces; they have not considered ideas which would generalize the four-dimensional time-space. This is very curious to me, epistemologically and psychologically. (I can think of one paper by van Danzig, which speculates philosophically around the notion of time topology; he says it might be a solenoidal variable. I like this, but clearly one should do much more imaginative work with time-like spaces.)

As is well known, the theory of special relativity postulates and is built entirely on the fact that light always has the same velocity regardless of the motion of the source or the observer. From this postulate alone everything follows, including the famous formula E = mc². Mathematically speaking, the invariance of the cones of light lead to the Lorentz group of transformations. Now a mathematician could, just for mathematical fun, postulate that the frequency, for example, remains the same, or that some other class of simple physical relation is invariant. By following logically one could see what the consequences would be in such a picture of a not “real” universe.

Mathematics is now so completely different from what it was in the nineteenth century, even if ninety-nine percent of mathematicians have no feeling for physics. There are so many ideas in physics begging for mathematical inspiration — new formulations, new mathematical ideas. I do not mean the use of mathematics in physics, but the other way round: physics as a stimulant for new mathematical concepts.

Contrary to mathematics, in physics one can, in principle, keep more abreast of what is going on in research. Every physicist can know the gist of most of physics. There are very few fundamental problems now such as the problem of the nature of elementary particles or what is the nature of the physical space and time.

In present-day research in theoretical physics, even though many of the young people are very clever, ingenious, and technically superb, their fundamental ideas tend to be orthodox, and on the whole only small variations on what has been done are produced, elaboration of details, and continuation along lines that already have been started.

Perhaps this has always been so and really new ideas are exceptional.

Sometimes, half in jest, to needle contemporary young physicist friends who spend all their time examining a few very strange particles, I tell them that it is not necessarily the best way to get new inspiration about the foundations of physics and the scheme of things in space and time.

Of course, it is not a precise problem or recognized as such, but what to my mind is a first question in physics is whether there exists a true infinity of structures going down into smaller and smaller dimensions. If so it would be worthwhile for mathematicians to speculate on whether space and time change, even in their topology, in smaller and smaller regions. We had in physics an atomistic or field-structure base. If the ultimate reality consists of a field, then its points are true mathematical points and indistinguishable. There is a possibility that in reality we have a strange structure of infinitely many stages, each stage different in nature. This is a fascinating picture which becomes more physical and not merely a philosophical conundrum. Recent experiments show definitely the increasing complication of structures. In a single nucleon we may have partons, as Feynman calls them. These partons may be the hypothetical quarks or other structures. The recent theoretical attempts no longer explain the experimental models by simple quarks, but one has to involve colored quarks of different types. Perhaps one has reached a point where it might be preferable to consider the succession of structures as going on ad infinitum.

Theoretical physics is possible because there are many identical or nearly identical copies of objects and situations. If one takes the universe by definition as only one (even though it is true that galaxies resemble each other) and the world as a whole being one, the questions asked about the cosmos as a whole have a different character. The stability with respect to adding a few more elements to an already large number is no longer guaranteed. We have no way to observe or experiment with a number of universes. Therefore problems of cosmology and cosmogony have a different character from those of even the most fundamental physics.

Science would not be possible, physics would not be conceivable, if there was not this similarity or identity of vast numbers of points or subsets or groups of points in this universe. All individual protons seem to resemble each other, all electrons seem to resemble each other, the attraction between any two celestial bodies seems to be similar, depending only on distance and mass. So the role of physics appears, inter alia, to divide the existing groupings into entities of which there are very many examples that are isomorphic or almost isomorphic to each other. The hope for physics lies in the fact that one can almost repeat situations, or if not exactly repeat, the addition of one or more small changes makes relatively little difference. Whether there are twenty or twenty-two bodies does not make their behavior change radically. A belief in some fundamental stability! Somehow the hope is to describe physics in terms of simpler entities and identity of parts by some kind of union or counting. For example, physicists believed, at least until recently, that if one had many points, the behavior of their mass could be explained by two-body interactions — this means adding up the potentials between any two bodies. Otherwise, if every time through adding a few bodies one changed the behavior of the whole system, there would be no science of physics. This point is not sufficiently brought out in physics textbooks.

One can relate the notion of entropy to the notion of complexity, if one defines the distance between two algebraical structures and the total work necessary to prove a statement or a theorem as energy. Results exist stating that in given systems in order to prove such and such formula one needs so many steps. The minimum sufficient number of steps can be defined as an analogue of work or energy. This is worth thinking about. To make a sensible theory of it requires erudition, imagination and common sense. There is no axiomatic system even for the presently established body of physics.

Just as in pure mathematics, in theoretical physics we can see a dichotomy between the great new “unexpected” ideas and the great syntheses of established theories. Such syntheses are in a sense complementary or opposed to the new concepts. They summarize previous theories in a non-obvious way. Let me illustrate this distinction: the special theory of relativity is a priori a very strange and mysterious concept. It involves an almost irrational insight and an a priori implausible axiom based on the experimental fact that light velocity seems to be the same for a moving observer from a fixed emission point or vice versa. When the emission point moves away or toward the observer the velocity of light relative to the observer is the same no matter what the relative velocity is. From this alone a great theoretical edifice was built, a physical theory of space and time with so many surprising — and as we now know — technologically shattering consequences.

Quantum theory involves similarly, in a way, an a priori non-intuitive or unexpected set of concepts.

Maxwell’s theory of electromagnetism would be an example of a great synthesis. It came after a great number of experimental facts were developed which were perhaps not so strange to their first discoverers. The theory that explains these observational facts in one set of mathematical equations constitutes one of the most impressive achievements of human thought. Epistemologically this theory is of a different nature, or so it seems to me, from relativity and quantum theory which were, one might say, more unexpected.

In astronomy, the recent observational discoveries show the continuing strangeness of the cosmos in the variety of different types of stars, of conglomerations of stars, of clusters, galaxies, and strange new objects. These include neutron stars, black holes, and other very peculiar and until recently unsuspected properties of assemblies of matter, and enormous clouds of molecules, some of them “pre-organic” in the interstellar space. Again this is an indication of the strangeness of the universe relative to our conceptions, which were acquired by previous sets of observations and under previous canons of learning and knowledge.

In physics one gets surprises, too, frequently now in the more technological or practical consequences of some physical discoveries. For example, the applications of the conception and development of holograms and their uses are very perplexing at first. Similarly, the new laser techniques are in general very impressive.

The recent discoveries in biology, revolutionary and promising as they are in their introduction of fantastic new vistas of the future change of the mode of life on our earth, have a different epistemological character. I am struck by the “reasonableness” of the arrangements on which life has been shown to be based. The discoveries of the way living matter replicates, everything that followed from Crick’s and Watson’s models, the nature of the biological code and “tout ce qui s’y rattache,” as the French say, show on the contrary a sort of very comprehensible and almost nineteenth-century type of mechanical arrangements which do not require basic physics for the understanding of how they operate. Quantum theory is important for the explanation of the phenomena of the basic molecular reaction, the basis of the arrangement, but the arrangements themselves seem to be quasi-mechanical or almost quasi-engineering in the way they involve the very fabric of the life processes.

One could ask why is this so? Why is it that our understanding of the physical world and perhaps the world of living matter or ourselves or the pattern of our thoughts, does not seem to proceed or accrue continuously? Instead of a logical development of’ steady growth we observe discrete quantum stages. Is it that the world is really simple in its ineffable structure, but that the apparatus of the nervous system which brings it to consciousness or renders its understanding communicable must, of necessity, be complicated?

Is it that the structure of our brain with all its neurons and connections, an admittedly very complicated arrangement, is not best suited to a direct description of the universe? Or perhaps the other way round, reality is on some very complicated objective scale which we do not yet even conceive and we, in our simpleminded way, try to glean it and describe it by simple steps in successive approximations as Descartes prescribed in his Discours de la Méthode?

(For a more detailed consideration of the possibilities of a future role of mathematics in biological research, I refer the reader to an article I wrote entitled “Some Ideas and Prospects in Biomathematics.” The technical aspect of the article is somewhat beyond the scope of these general remarks, but interested readers may like to look at it.)

In social sciences, a layman like myself feels that there is no theory or deeper knowledge at the present time. Perhaps this is due to my ignorance but I often have the feeling that by just observing the scene or reading, say, The New York Times, one can have as much foresight or knowledge in economics as the great experts. I don’t think that for the present they have the slightest idea what causes the major economic or socio-political phenomena except for the trivialities everyone should know.

A development whose effect we cannot even estimate and which would have an impact greater by far, I think, than any of the established religions, will be the discovery of the existence of other intelligences in the universe — perhaps thousands of light-years away from the solar system. It is entirely possible that there are waves which have traveled for a long time which we could suddenly decipher. An inkling or a proof of existence without the possibility of communicating two ways would have an overwhelming effect on humanity. This could happen very soon, and it might create panic or, on the contrary, a new type of religion.

We have all read about flying saucers and other unidentified flying objects. Edward U. Condon directed a very thorough study of the subject. Most cases were easily proved to be either illusions, optical or otherwise, or natural atmospheric phenomena, but there remain a few cases of authenticated UFO’s which are most puzzling. Take the group of Mount Wilson astronomers who were on a walk and saw a very strange meteoric object, and when they returned to the observatory saw indications of high peaks of radioactivity. There are also a few cases of objects which were simultaneously followed visually, by plane and by radar, which have never been explained.

Fermi used to ask: “Where is everybody? Where are the signs of other life?”

In my opinion, more than anything else it is the new biology which will change the way of life in our world during the next ten or fifteen years. Discoveries which at first seemed rather ordinary have already had more effect on the composition of the world than even big wars: new drugs like penicillin on the one hand, contraceptives on the other, have changed the balance of population.

To illustrate the fast pace of discoveries in biology, I recently heard of two important developments in cancer studies in the space of one week. One is that a Michigan scientist has discovered a virus in the human breast cancer cell. The other is that an experiment which was actually performed in Boulder, where there is a very good electron microscope, resulted in a surprising new technique. Keith Porter and his associates were able to produce cells from which the nuclei could be extracted. These undamaged nuclei can be transferred to other denucleized cells, so in effect it is an exchange of nuclei between cells. For example, a cancerous cell can have its nucleus removed and put into a healthy cell. Then the new cell may become normal. This is most remarkable and shows that some instructions may be coming not from the nucleus, as was believed, but from the cytoplasm.

In the future, new ways to produce or replace food will have much more influence on the shape and mode of human life here on earth than any politico-socio-economic developments in the present sense of the words. All this may be obvious, but sometimes obvious things need to be repeated over and over before they are realized. The world will be so different. I am reminded of a book which was published not so long ago under the title The Next Million Years. What lack of imagination it shows!

Gamow’s interests, von Neumann’s foresight, Banach’s and Fermi’s work, among others, have all contributed to enlarge the aspect of today’s science, and enormously widen the perspectives of physics and mathematics. It is a marvel that so many new vistas and achievements are due to such a fortuitous and fortunate confluence of all the divisions of science.