ROTA: Logicians claim
*
but
*
is the same as
*
and
*
.

ULAM: No! Its meaning is entirely different. How would you describe but logically? Something that leads us to a conclusion but does not? A disappointment in probability’? A whole essay could be written about it. Someday there will be a tremendous theory devoted to its ramifications, It could be a germ like the word
*
continuous
*
. The study of topology is nothing else but the study of the word
*
continuous
*
.

ROTA: When I was at Princeton, Alonzo Church gave a two-hour lecture on the meaning of
*
but
*
and
*
and
*
. It is now written up in his great
*
Introduction to Mathematical Logic
*
.

ULAM: So you see! And what does he say? I never read it. I knew he was a logician but did not know he did things like that. Now let’s discuss things intelligently, professor.

ROTA: O.K. Let us begin with the word but, Stan?

ULAM: I would say that the word but suggests to me the following (we’ll be more precise later): an element of an algebra whose elements are uttered sentences. I can imagine it as a point in a universe of points interpreted as sentences—physical facts. I see that it won’t be easy to avoid circular definitions; we must not use the word but in developing a theory of but, right? The word but means that an element does not belong to a given set of points that was defined before. But—I am just saying this on purpose now—but expresses that an element belongs to a set which is similar or slightly larger than the already given set. Of course, 1 did not really need to use the word but in my explanation. However—Oh! I just used the word however: you see how hard it is to avoid these words?

[...]

ROTA: Can you tell me something about how his [von Neumann's] mind worked?

ULAM: It is curious tome 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 complementarily 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!

[...]

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 complementarily 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 elementary-

particle 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 - all 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 identical 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.

[...]

ULAM: Combinatorics is devoid of general methods, curiosities, it is Erdosian. I have nothing against it, no light on anything else.

ROTA: You are not being fair.

ULAM: Complex functions, the idea of entropy are broader, Ramsey’s theorem, interesting as it is, is like progress in zoology when a new species of insects with one red eye and one green eye has been discovered !

ROTA: Ramsey’s theorem tells more about the nature of sets than all the axioms of set theory!

ULAM: It is one of numerous properties of infinity. Why take two sets of pairs and divide them into two classes? My master’s thesis already contains that sort of thing. Some problems, big or small, are solved with a bang; they open new vistas. Others are solved with a whimper, in a way which is very specific and leaves nothing to be said or asked, regardless of whether the problems are important or interesting.