The interest in the foundations of mathematics is to some extent also philosophical, though eventually it does pervade everything, like set theory. But the word ''foundations" is a misnomer; for the time being, it is just one more mathematical specialty, fundamental to be sure.
The great at dichotomy in the origin and in the inspiration of mathematical thought — stimulated by the influence of external reality, the physical universe, on one hand, and by the developing processes of physiology, almost perhaps of the human brain, on the other — has in a small and special way a homomorphic image in the present and near-future use of electronic computers.
Even the most idealistic point of view about mathematics as a pure creation of the human mind must be reconciled with the fact that the choice of definitions and axioms of geometry — in fact of most mathematical concepts — is the result of impressions obtained through our senses from external stimuli and inherently from observations and experiments in the "external world." The theory of probability, for example, came about as a development of a few questions concerning games of chance. Now, computing machines constructed to solve special problems of mathematics promise to enlarge very greatly the scope of the Gedanken Experimente, the idealization of experience, and our more abstract schemata of thought.
It appears that experimentation on models of games played by self-organizing living matter through chemical reactions in living organisms will lead to novel abstract mathematical schemata. The new study of the mathematics of growing patterns, and the possibility of studying experimentally on computing machines the course of competitions or contests between geometrical configurations imitating the fight for survival, these might give rise to new mathematical setups. One could again give names like "payzonomy" to the combinatorics of contesting reactions and "auxology" to a yet-to-be-developed theory of growth and organization, this latter ultimately including the growing tree of mathematics itself.
So far only the very simplest and crude mathematical schemata have been proposed to mirror the mathematical properties of geometrical growth. (An account of my own simple-minded models can be found in a recent book edited by Arthur Burke, A Theory of Cellular Automata, published by the University of Illinois Press.)
An especially ingenious set of rules was devised by the English mathematician John Conway, a number theorist. The Conway Game of Life is an example of a game or pastime which, perhaps much like the early problems involving dice and cards, has led ultimately to the present edifice of probability theory, and may lead to a vast new theory describing the "processes" which Alfred North Whitehead studied in his philosophy.
The use of computers seems thus not merely convenient, but absolutely essential for such experiments which involve following the games or contests through a very great number of moves or stages. I believe that the experience gained as a result of following the behavior of such processes will have a fundamental influence on whatever may ultimately generalize or perhaps even replace in mathematics our present exclusive immersion in the formal axiomatic method.
The already-mentioned recent results of Paul Cohen and others — Peter Novikoff, Hao Wang, Yuri Matiasevic — on the independence from the traditional system of axioms of some of the most fundamental mathematical statements, indicate a new role for pragmatic approaches. Work with automata will help indicate whether a problem can be solved by existing means.
To illustrate what we have in mind let us consider for example a "little" special problem in three dimensions: given a closed curve in space and a solid body of given shape, the problem is to push the body through the curve. There are no clear mathematical criteria to decide whether it can be done or not. One has to rotate, wiggle, squeeze, and "try," to see whether it can be done. In a higher number of dimensions, like five, one can have an analogous problem. The idea is to set it up on a computer and try various possible motions. Perhaps, after very many tries, one would acquire a feeling for the freedom of maneuvering in this high dimensional space and a new type of an almost tactile intuition. Of course, this is a special, small and unimportant example, but I feel that one could develop new imaginations by suitable experimentation with these new tools, electronic computers especially, in setting up and observing the various growth processes and evolutionary developments.
It seems to me the impact and role of the electronic computer will significantly affect pure mathematics also, just as it has already done so in the mathematical sciences, principally in physics, astronomy and chemistry.
These conjectured excursions into aspects of the future of mathematics take us far from von Neumann and his contemporaries, and their role in the evolution of science a quarter-century ago. The rate of growth in the organized activities of the human mind, accelerated no doubt by the advent of computers, seems to increase in a way which forebodes qualitative changes in our way of thinking and living. As Niels Bohr said in one of his amusing remarks: "It is very hard to predict, especially the future." But I think mathematics will greatly change its aspect. Something drastic may evolve, an entirely different point of view on the axiomatic method itself. Instead of detailed work on special theorems which now number in the millions, instead of thinking in terms of rules operating with symbols given once and for all, it may be that mathematics will consist more and more of problems, or desiderata, or programs for work of a general nature. No longer will there be additional multitudes of special spaces, definitions of special manifolds, of special mappings of this and that — though a few will survive: "apparent rari nantes in gurgite vasto," no new collections of individual theorems, but instead general sketches or outlines of larger theories, of vaster enterprises, and the actual working out of proofs of theorems will be left to students or even to machines. It may become comparable to impressionistic painting in contrast to the painful, detailed drawing of earlier days. It could be a more living and changing scene, not only in the choice of definitions but in the very rules of the game, this great game whose rules until now have not changed since antiquity.
If the rules have not changed, very great changes have already taken place in the scope of mathematics in the space of my own lifetime. In the nineteenth century the applications of mathematics were all-inclusive in physics, astronomy, chemistry, in mechanics, engineering, and all the other facets of technology. More recently, mathematics serves to formulate the foundations of other sciences as well, so-called mathematical physics is really the theory of all physics, reaching into its most abstract parts like quantum theory, the very strange four-dimensional continuum of space-time. These belong specifically to the twentieth century. In the short span of sixty to one hundred years the proliferation of the use of mathematical ideas has been unbelievably varied. It was accompanied by, one could say, an explosive creation of new mathematical objects, large and small, and a tendency to "beat things to death" with proliferation and hairsplitting studies of minute details that is almost Talmudic.