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.