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… nN). 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.