ments. None are comparable to Johnny.

By the way, you were supposed to ask about the foods von Neumann liked.

ROTA: List the foods von Neumann liked and those he did not like!

ULAM: He was not a gourmet, but he liked to eat. He liked to go to restaurants, mainly, I think, to escape from the usual scene or routine. It was an excuse for not working, because he was a very hard worker. At home he worked at a desk, writing, a thing which irritated me a bit. When I stayed at his house and saw him suddenly leave to go upstairs and write, I, cruelly and foolishly I must say, would make fun of it. So for relaxation he liked to drive out for dinner. In Princeton we often went to a restaurant called Marot, on the highway to Trenton.

He never smoked, but he ate voluminously, which accounted for his increasing rotundity and portliness as the years went by. Sometimes when Klari, his second wife, could not finish what was on her plate, she would give it to Johnny or to me and say, “Both of you are human garbage cans!" Klari, by the way, was a very intelligent, very nervous woman who had a deep complex that people paid attention to her only because she was the wife of the great von Neumann, which was not true of course.

Johnny liked Mexican food, hot peppery stuff. I suspect it was because if he had a stomachache later, he would know what to blame it on! I always have such Machiavellian suspicions. It is probably just that he was used to Hungarian goulashes and hot paprika. He liked sweets too, but on the whole what he wanted was volume, like me, like you too. You like the volume of pasta.

He had this nervous trait, an almost automatic response. For example, whenever he saw the words chicken mole on a menu, he would automatically intone Moles Hadriani, and I would respond Jacques de Molay-you know, the Grand Master of the Knights Templar. It was a

game of association, just like you always add Pal [Hungarian for Paul] when you hear the word Erdös!

He also had occasionally an infrequent but noticeable stutter. He would say a word and repeat it two or three times in quick succession. I wonder whether it could have been an incipient physical lesion, for he died of things affecting his brain. Actually, on second thought it could not, because his cancer started somewhere else. Sometimes I suspect that his stutter was in order to gain time while thinking over a riposte or considering quickly some other angle for a statement, like a person lighting a pipe to gain time.

ROTA: How long did you know von Neumann?

ULAM: I first met him in Warsaw in 1935, but I had already started corresponding with him the year before, and that is when he invited me to visit him at the Institute in Princeton.

ROTA: What was he working on at Los Alamos during the war?

ULAM: On everything. He was one of the originators, one of the "influencers" of implosion. By the way, you are my most eminent "influencee"; it is a relationship different from teacher-student.

He worked on the whole project, scientifically and politically, especially with the hydrogen work.

ROTA: But actual work?

ULAM: Of course, mostly hydrodynamics. ROTA: Did he know much physics? ULAM: To some extent, but he did not have the physicist's feeling for experiment. His interest was more modern than Hilbert's. His interest was in the foundations of quantum mechanics, which were mathematical. And that could be taken as an example of mathematics not really useful for real physics.

But there was no bullshit in him. That

is an expression he used about certain people. He would say, "It is very rare, but there is no bullshit in so-and-so."

Of course he worked, in answer to your question. In fact he was unable to play the role of senior scientist or advisor without being actively engaged, like with computing. Even towards the end of his life, when he was chairman of the ICBM Committee, a committee established by the President after Sputnik.

ROTA: I still don't have a picture of von Neumann's personality.

ULAM: He loved jokes, though I don't think he invented many, but he remembered and repeated them, and occasionally he made original and very witty remarks or saw comparisons which were comical. Most are unprintable.

A propos of the church knowing about the atom bomb, he said, "Priests will bless the active cores." And when he noticed all the churches of Los Alamos, he was much amused when I pointed to one church and called it "San Giovanni delle Bombe"! One of the first solid non-wood buildings in Los Alamos was built for the offices of the AEC. He called it “El Palacio de la Seguridad”!

Oh! One thing about Johnny, he tended to tell people what they wanted to hear. He also used to tell me his little tactical discoveries. Once he said, "In Los Alamos it is very difficult to introduce novelty, but once introduced, it is impossible to get rid of it!"

After the war he was for a Pax Americana, and one could probably have established it, but the historical perspective, the desire to do it were not present in the country. The general population was not thinking in those terms. Although, when World War II ended, Americans were like Roman citizens during the Roman Empire. By commuting through the American bases one could go anywhere in Europe without encountering the native populations. This was really a beginning of that sort of thing, but for good or for

bad-who knows-it quickly dissipated.

What else would you like to know about von Neumann?

ROTA: Always well dressed, wasn't he? ULAM: Not really well dressed, but simple, decent, well-cut, classic city dress.

ROTA: I still don't have a picture of the

man.

ULAM: He became an important government figure and very influential in ballistic missile development.

ROTA: It is strange how you like everything about him except his work in mathematics.

ULAM: Really? No, not quite so. he was not a mathematician's mathematician. He did little in number theory, some in continuous geometry and operators and Hilbert space, and some in measure theory and group theory.

To my mind and to my taste, the most important work he did is what he did when he was getting older, which mathematicians don't appreciate, namely his speculations on automata, on the brain, and his contributions to computing and to problems in hydrodynamics.

He knew about quantum theory and some parts of theoretical physics, which few mathematicians did. He contributed to the grammar of physics, so to say. One must also mention the theory of games. What interested me less was his work in the almost-periodic functions of groups.

ROTA: Can you tell me something about how his mind worked?

ULAM: It is curious to me that in our many mathematical conversations on topics belonging to set theory and allied fields, he always seemed to think formally. Most mathematicians, when discussing problems in these fields, seem to have an intuitive framework based on geometrical or almost tactile pictures of abstract sets, transformations, and such. Johnny

gave the impression of operating sequentially by formal deductions. His intuitions seemed very abstract; they involved a complementarity between the formal appearance of a collection of symbols, the games played with them, and the interpretation of their meanings. Something like the distinction between a mental picture of the physical chess board and a mental picture of a sequence of moves on it written down in algebraic notation!

The quickness of his thinking was quite remarkable. He saw immediately the possibilities of Monte Carlo. To my mind this was much more important than one hundred papers in partial differential equations! It is at least a general procedure I would not quite call it a method-and he invented many tricks for it and specific ways to get random distributions. It was very pleasant to discuss it with him.

Too bad he did not live to see how computers have revolutionized everything and what influence they will have on science in general and even on pure mathematics. His role in their development was tremendous, and if I may say so I would say I too played a modest role in showing how to use computers!

ROTA: How would you characterize his influence?

ULAM: There used to be a time when there were mathematicians who gave specific ideas and choice of topics and directions either explicitly or by implication to the work of other mathematicians. Not to go back centuries but less than a hundred years, let us say Poincaré, Hilbert, in more recent times Herman Weyl. Hilbert had laid what was hoped would be a foundation for the final axiomatization of mathematics and beyond, of all science. Little did he know that in the thirties the unavoidable limitations of this approach would be revealed.

Von Neumann was one of these giants too in the breadth of his knowledge, especially when one remembers that now the

diversity and complexity of contemporary problems enormously surpass the situation confronting Poincaré and Hilbert. Yet, he admitted to me that he felt he did not know even a third of mathematics, that he did not think it was possible nowadays for any one brain to have more than a passing knowledge of more than one-third of pure mathematics.

So, at his suggestion and for his amusement I concocted an oral doctoral examination in various fields in such a way that he would not be able to pass it. And indeed, when I thought about what problems to give him in each domain, I found one in differential geometry, one in number theory, one in algebra and a couple of others. And he agreed that he could not have answered any of the questions and the exam would have been a complete failure. Which goes to show that doctoral exams are to some extent meaningless. Of course, if one prepares for some specific topics, that is something else. ROTA: Who was von Neumann a student of?

ULAM: He considered himself a student of Ehrardt Schmidt. It was not easy for me to get to the bottom of this. One reason, I suspect, is that Schmidt did some work in combinatorics which always interested Johnny very much.

ROTA: It was the Hilbert space. Schmidt was the only person at the time who studied nonlinear operators.

ULAM: But Johnny did not.

ROTA: That is why he admired Schmidt! ULAM: Also I remember that he told me that Schmidt did not like to write. That surprised Johnny. I also think he secretly admired it. He said that Schmidt had told him that he felt faint whenever he saw a blank sheet of paper. Johnny was not at all like that. On the contrary, whenever he had a mathematical thought, he immediately wanted to write it down and elaborate.

ROTA: Did he have any students?

ULAM: Not really, even though at the Institute he gave several courses every year. Murray and Halperin may be considered his students.

ROTA: What about Gödel and von Neumann?

ULAM: One summer before the war when I was returning to the States, Johnny was waiting for me at the pier. His first words were that Gödel had shown that the continuum hypothesis was undecidable. This was how I heard for the first time about the existence of undecidable propositions in any formal system. So I said to him, "Oh! That is because he defines what is meant by a set." Johnny opened his eyes wide and expressed surprise that I had seen right away what was indeed the essential point. He thought I had some supernatural intuition.

I asked him whether Gödel was not a little afraid that his result was nothing but a sort of super paradox of the existing set theory, merely a diagonal method. In a sense it is a diagonalization. He agreed that this was probably right and that Gödel did not quite realize the importance of his discovery because of the fear that it would turn out to be merely another version of the whole series of settheoretical paradoxes. Of course it was much more than that because he had made it all formal. The other paradoxes were special and dependent on metamathematical considerations that were not truly part of mathematics, whereas his results were. Curious how nervous people can be about their own work when it is the work!

ROTA: You have a higher opinion of Gödel than I have.

ULAM: Yes, I know. It was so unexpected at the time, and poor Hilbert was ... ROTA: Not to speak of poor von Neu

mann.

ULAM: Johnny told me that Gödel's re

sults made him very downcast, not quite despairing but disappointed. You must remember that his work on the axiomatization of set theory, which was way back in the twenties, constitutes to this day one of the best foundations for set-theoretical mathematics. Basically he believed in Hilbert's goal of a final and conclusive axiomatization of mathematics, and yet,

in a 1925 paper, in a mysterious flash of intuition, he pointed out the limits of any axiomatic formulation of set theory. That was perhaps a vague forecast of Gödel's result. But it was left to Gödel to follow it through, and it has changed the direction of all science.

ROTA: What is the difference between the Slavic mind and the German mind? ULAM: The German mind is systematic; the Slavic is not. Slavs tend to be soulful, expansive, pensive, but they are not as nebulous or as much carried away by the sound of words as Germans are. In the German language syllables and words concatenate, and they concatenate thoughts which sometimes don't go very well together.

ROTA: Whereas the Slavic?

ULAM: Slavs tend, I think, to be selfanalytic, more psychological than philosophical, full of regrets, feelings of guilt, but more fundamentally optimistic than the German, and with humor, which if it is not showing, is at least not far away. German humor is based on ridicule, I don't know why. Latins are something else.

ROTA: Describe the Latin mind.

ULAM: Order. Clarity is always there. Words are separated, they don't stick to

Gainesville January 1974

gether. It is like well-cooked rice compared to the sticky overcooked stuff that comes out of a German mess.

What would you say about the Jews? Would you say there is a Jewish mind? ROTA: I don't think so. Italian Jews are Italian, German Jews are German, and so

on.

ULAM: Don't you think that the Jewish mind is a little truculent, that there is a bent for contradiction? I feel I have myself this Jewish characteristic of always wanting to change what exists. It is a sort of rebelliousness, an inability to kowtow to authority. Think of the great revolutionaries-Jesus, Marx, Freud, Einstein, Cantor. Cantor by the way was only half Jewish. Most Jews are only part Jewish, you know, but the Jewishness comes through all the same.

This rebellious spirit of the Jews does not show in music, where the Jews are much less creators than performers, interpreters.

Gainesville January 1974

ULAM: It is Gamow who brought Teller to George Washington University originally. ROTA: From zero?

ULAM: From Europe, in Hitler time, when he had no job.

ROTA: How did Gamow and Teller get along?

ULAM: Gamow ruined Teller a bit. Gamow had this fantastic talent-an intu

ition, a lightness of touch for what is important, without doing too much work, without much mathematics, without any laborious Gründlichkeit. Teller wanted to be like that. He had other talents, complementary perhaps. Comedians always want to be tragedians and vice versa. Under Gamow's influence Teller wanted to have "ideas" at any cost.

ULAM: Plutarch compared lives, and it may have a certain sense, a certain value, to compare pairs of mathematicians. Take Erdös. Erdös and I have something in common, a tremendous facility for finding difficult combinatorial problems out of thin air.

I'd like to take Erdös, Rota, and Everett and, like in the theory of colors, see whether by mixing them one could produce all other colors!

ROTA: Your style is completely different from Erdös's. He is interested in proofs; you are not interested in proofs. You are interested in problems interesting to state and don't care very much how they are solved. Erdös cares about techniques that he uses all the time.

ULAM: Really? He likes to think from the beginning; he does not quote somebody's theorem to prove something else.

ROTA: Your typical problem can almost always be restated as follows: Develop a theory of... along the following lines. That is what your problems are about, whereas Erdös's are never this way.

Gainesville January 1974

ULAM: Maybe he exaggerates by trying to put everything on paper immediately. ROTA: There is a primitivism to Erdös. ULAM: Yes. I have that feeling too, very much. Once you said something which if true is very flattering, namely that things I mention are germs of whole theories, whereas his on the whole are more special.

Erdös is not really narrow, but it is hard to get things out of him. I think he knows a great deal, though I don't think he has read much belles-lettres.

ROTA: He has no outside interests.

ULAM: I think he reads quickly and efficiently and gets the gist of things. I don't know how much he knows, say, of French literature, the classics, history. He does know some history because he is interested in politics. He reads about current things, progress in medicine, a little about physics. He forms impressions.

He is really very nice, never diminishes people, does not make fun of anybody, and is very much interested in young peo

ple in the sense that he is always searching for young geniuses. Wouldn't you say that in a sense he is more human than von Neumann or Fermi? Fermi was enormously aware of but not warmly interested in others.

ROTA: I really don't understand Erdös as a person. I understand him mathematically. ULAM: He wants to be famous. He is very well known. Every mathematician knows him. He has written over 800 papers. You know the "Erdös number”— who wrote papers with him. People have a weakness for him. He has some sense of humor. Politically he is not naive at all. He is very well wishing, and really I have never heard him speak badly of anyone. Very few people are like that. You speak badly of people. I speak badly of people.

The death of his mother was a terrible blow; he still has not recovered. She was ninety-one, and he says she still could have lived another three or four years.

Erdös is interested in human destiny, in sickness, in death. He has no home. Now he refuses to go to Hungary because of their attitude towards two Israelis. Last summer, at the time of a meeting in his honor, Hungary did not let two Israeli mathematicians in. This infuriated him, and he said he would not return for several years. He is a true man of principles and in a way very courageous.

Erdös

Gainesville March 1974

Paul Erdös is the most prolific mathematician of modern times and is second only to Euler in the volume of work produced. He was a long-time friend and collaborator of Ulam. In Ulam's files are 191 letters from Erdös, mostly in longhand. Erdös collaborates the world over and has done more for collaboration in mathematics than anyone else.

ROTA: What are your views on classical physics versus quantum mechanics?

ULAM: Quantum mechanics uses variables of higher type. Instead of idealized points, or groups of points or little spheres or atoms or bodies, the primitive notion is a probability measure. Quite a logical leap from the classical point of view.

Nevertheless you find in quantum mechanics the strange phenomenon that a theory dealing with variables of higher type has to be imaged on variables of lower type. It is the complementarity between electron and wave.

In our minds, because of habit or historical conditions, an electron is a localized small object, whereas a wave is something diffuse. But some phenomena show a dual nature; they share properties of one and the other. I don't think there is yet a satisfactory logical or mathematical discussion of this duality. In my opinion it does not do any good to write down axioms which sanctify the usual dicta. People accept what works. Quantum theory is very successful at describing atomic phenomena, and some of its general features seem to be valid even in the subatomic nuclear and elementaryparticle phenomena. But the overall success is not too striking, except perhaps in quantum electrodynamics.

To me the situation in theoretical physics seems to be the following. There are about one hundred bright young physicists in the country, all mathematically very skillful and learned-too much so for my taste! To predict or explain some of their observations, they fudge a little, which is only natural. However the next experiments at CERN or Fermilab always seem to invalidate their calculations. You would think that among so many guys making so many different predictions, at least a few would get some

correct answers, but no! Whatever the prevailing beliefs or attempts, the new experiments show something else. How can this be? Nature is not that malicious. Maybe today's physicists are technically very skilled but not really imaginative or innovative enough.

ROTA: What is to your mind problem number one in physics?

ULAM: Is there a true infinity of structures going down into smaller and smaller dimensions? Is it not a precise problem, or recognized as such.

In physics there has always been an atomistic or a field point of view. If there is a field, then points are mathematical points and they are all the same. But another possibility is a very strange structure of successive stages, each stage different. The topology or the scene on which they exist, that is, space and time themselves, need not be the uniform, smooth Euclidean topology. The miracle is that physics would not be possible if protons and electrons were not very much the same. If this similarity or identity of subsets of the universe did not exist, there would be no physics. The role of physics to some extent is to divide the existing groupings call them particles—into entities isomorphic or almost isomorphic to each other.

The great hope of physics lies in the fact that one can almost repeat the same situations. Having twenty or twenty-two bodies does not radically change a physical law. In mathematics too there are similar analogies. In physics such analogies are essential.

It may be that in reality for phenomena in the small and involving high energy, there may be an underlying true infinity that does not allow for similarities. It may be that at the present stage of evolution of the universe a sufficient number of identi

cal situations has not yet been produced. If this is so, then physics will become fundamentally more complicated.

Who knows whether there are not fundamental complications in the nature of subparticles? Are the billions of protons that compose our bodies or this table really the same? This stability is far from guaranteed. There might be critical numbers, critical crises not only in technology but in fundamental physics itself.

Since Gödel, even in mathematics it is not simple anymore. Have I told you that van Hove asked me to give a talk on infinities in physics at CERN? ROTA: What did you say?

ULAM: I intend to write it up in my future Physics for Mathematicians.

In recent years you seem to have lost your feeling of horror towards physics! ROTA: I did not understand. I like to understand.

ULAM: Do you understand mathematics? It is easier to get accustomed quickly to a fixed symbolism, like that of mathematics. But this is largely an illusion. Mathematics has a restricted range; it has not changed since Archimedes. There are axioms, proofs, lemmas, theorems. In physics it is not clear what one really does and at what point one becomes satisfied that the formulation is correct.

Santa Fe July and August 1974