As we passed the battlefields of the Civil War, Johnny recounted the smallest details of the battles. His knowledge of history was really encyclopedic, but what he liked and knew best was ancient history. He was a great admirer of the concise and wonderful way the Greek historians wrote. His knowledge of Greek enabled him to read Thucydides, Herodotus, and others in the original; his knowledge of Latin was even better.
The story of the Athenian expedition to the island of Melos, the atrocities and killings that followed, and the lengthy debates between the opposing parties fascinated him for reasons which I never quite understood. He seemed to take a perverse pleasure in the brutality of a civilized people like the ancient Greeks. For him, I think it threw a certain not-too-complimentary light on human nature in general. Perhaps he thought it illustrated the fact that once embarked on a certain course, it is fated that ambition and pride will prevent a people from swerving from a chosen course and inexorably it may lead to awful ends, as in the Greek tragedies. Needless to say this prophetically anticipated the vaster and more terrible madness of the Nazis. Johnny was very much aware of the worsening political situation. In a Pythian manner, he foresaw the coming catastrophe.
It was during this trip also that for the first time I sensed that he was having problems at home. He exhibited a certain restlessness and nervousness and would frequently stop to telephone to Princeton. Once he came back to the car very pale and obviously unhappy. I learned later that he had just found out that his marriage to Marietta was definitely breaking up. She would leave him shortly thereafter to marry a younger physicist, one of the frequent guests at the numerous parties which the von Neumanns gave in Princeton.
On the way back from the meeting I posed a mathematical problem about the relation between the topology and the purely algebraic properties of a structure like an abstract group: when is it possible to introduce in an abstract group a topology such that the group will become a continuous topological group and be separable? "Separable" means that there exists a countable number of elements dense in the whole group. (Namely, every element of the group can be approximated by elements of this countable set.) The group, of course, has to be of power continuum at most — obviously a necessary condition. It was one of the first questions which concern the relation between purely algebraic and purely geometric or topological notions, to see how they can influence or determine each other.
We both thought about ways to do it. Suddenly, while we were in a motel I found a combinatorial trick showing that it could not be done. It was, if I say so myself, rather ingenious. I explained it to Johnny. As we drove Johnny later simplified this proof in the sense that he found an example of a continuum group which is even Abelian (commutative) and yet unable to assume a separable topology. In other words, there exist abstract groups of power continuum in which there is no possible continuous separable topology. What is more, there exist such groups that are Abelian. Johnny, who liked verbal games and to play on words, asked me what to call such a group. I said, "nonseparabilizable." It is a difficult word to pronounce; on and off during the car ride we played at repeating it.
Mathematicians have their own brand of "in" humor like this. Generally speaking, they are amused by stories involving triviality of identity of two definitions or "tautologies." They also like jokes involving vacuous sets. If you say something which is true "in vacuo," that is to say, the conditions of the statement are never satisfied, it will strike them as humorous. They appreciate a certain type of logical non sequitur or logical puzzle. For instance, the story of the Jewish mother who gives a present of two ties to her son-in-law.
The next time she sees him, he is wearing one of them, and she asks, "You don't like the other one?"
Some of von Neumann's remarks could be devastating, even though the sarcasm was of an abstract nature. Ed Condon told me in Boulder of a time he was sitting next to Johnny at a physics lecture in Princeton. The lecturer produced a slide with many experimental points and, although they were badly scattered, he showed how they lay on a curve. According to Condon, von Neumann murmured, "At least they lie on a plane."
Some people exhibit an ability to recall stories and tell them to others on appropriate occasions. Others have the ability to invent them by recognizing analogies of situations or ideas. A third group has the ability to laugh and enjoy other people's jokes. I sometimes wonder if types of humor could be classified according to personality. My friends and collaborators, C. J. Everett in the United States and Stanislaw Mazur in Poland, each had a wry sense of humor, and physically and in their handwriting they also resemble each other.
Generally von Neumann preferred to tell stories he had heard; I like to invent them. "I have some wit; it is a tremendous quality," my wife says I once told her. When she pointed out that I was bragging, I promptly added, "True. My faults are infinite, but modesty prevents me from mentioning them all."
In addition to "in" jokes, mathematicians also practice a form of "in" language. For example, they use the word "trivial." It is an expression they are very fond of, but what does it really mean? Easy? Simple? Banal? A colleague of my friend Gian-Carlo Rota once told him that he did not like teaching calculus because it was so trivial. Yet, is it? Simple as it is, calculus is one of the great creations of the human mind, with beginnings dating back to Archimedes. It was "invented'' by Newton and Leibnitz, and amplified by Euler, Lagrange, and others. It has a beauty and an importance going far beyond most of the mathematics of our present culture. So what is "trivial"? Certainly not Cantor's great set theory, technically very simple, but deep and wonderful conceptually without being difficult or complicated.
I have heard mathematicians sneer at the special theory of relativity, calling it nothing but a technically trivial quadratic equation and a few consequences. Yet it is one of the monuments of human thought. So what is "trivial"? Simple arithmetic? It may be trivial to us, but is it to the third-grade child?
Let us consider some other words mathematicians use: what about the adjective "continuous"? Out of this one word came all of topology. Topology may be considered as a big essay on the word "continuous" in all its ramifications, generalizations, and applications. Try to define logically or combinatorially an adverb like "even" or "nevertheless." Or take an ordinary word like ''key," a simple object. Yet it is an object far from easy to define quasi-mathematically. "Billowing" is a motion of smoke, for example, in which puffs are emitted from puffs. It is almost as common in nature as wave motion. Such a word may give rise to a whole theory of transformations and hydrodynamics. I once tried to write an essay on the mathematics of three-dimensional space that would imitate it.
Were I thirty years younger I might try to write a mathematical dictionary about the origins of mathematical expressions and concepts from commonly used words, imitating the manner of Voltaire's Dictionnaire Philosophique.