Why are so many giving their lives to mathematics today; why have the past 50 years been so productive mathematically? The mathematical groundwork laid by our predecessors, the universities, societies, foundations, libraries, etc., have furnished unusual opportunity, incentive, and cultural material. In addition, the processes of evolution and diffusion have greatly accelerated. Of the two, the latter seems to have played the greater role in the recent activity. For during the past 50 years there has been an exceptional amount of fusion of different branches of mathematics, as you well know. A most unusual cultural factor affecting the development of mathematics has been the emigration of eminent mathematicians from Germany, Poland, and other countries to the United States during the past 30 years. Men whose interests had been in different branches of mathematics were thrown together and discovered how to merge these branches to their mutual benefit, and frequently new branches grew out of such meetings. The cultural history of mathematics during the past 50 years, taken in conjunction with that of mathematics in ancient Greece, China, and Western Europe, furnishes convincing evidence that no branch of mathematics can pursue its course in isolation indefinitely, without ultimately reaching a static condition.
Of the instruments for diffusion in mathematics, none is more important, probably, than the journals. Without sufficient outlet for the results of research, and proper distribution of the same, the progress of mathematics will be severely hampered. And any move that retards international contacts through the medium of journals, such as restriction to languages not widely read, is distinctly an anti-mathematical act. For it has become a truism that today mathematics is international.
This brings us to a consideration of symbols. For the so-called "international character" of mathematics is due in large measure to the standardization of symbols that it has achieved, thereby stimulating diffusion. Without a symbolic apparatus to convey our ideas to one another, and to pass on our results to future generations, there wouldn't be any such thing as mathematics - indeed, there would be essentially no culture at all, since, with the possible exception of a few simple tools, culture is based on the use of symbols. A good case can be made for the thesis that man is to be distinguished from other animals by the way in which he uses symbols [
As an aspect of our culture that depends so exclusively on symbols, as well as the investigation of certain relationships between them, mathematics is probably the furthest from comprehension by the non-human animal. However, much of our mathematical behaviour that was originally of the symbolic initiative type drops to the symbolic reflex level. This is apparently a kind of labour-saving device set up by our neural systems. It is largely due to this, I believe, that a considerable amount of what passes for "good" teaching in mathematics is of the symbolic reflex type, involving no use of symbolic initiative. I refer of course to the drill type of teaching which may enable stupid John to get a required credit in mathematics but bores the creative minded William to the extent that he comes to loathe the subject! What essential difference is there between teaching a human animal to take the square root of 2 and teaching a pigeon to punch certain combinations of coloured buttons? Undoubtedly the symbolic reflex type of teaching is justified when the pupil is very young - closer to the so-called "animal" stage of his development, as we say. But as he approaches maturity, more emphasis should be placed on his symbolic initiative. I am reminded here of a certain mathematician who seems to have an uncanny skill for discovering mathematical talent among the undergraduates at his university. But there is nothing mysterious about this; he simply encourages them to use their symbolic initiative. Let me recall parenthetically here what I said about the perennial presence of potential "great men;" there is no reason to believe that this teacher's success is due to a preference for his university by the possessors of mathematical talent, for they usually have no intention of becoming mathematicians when they matriculate. It moves one to wonder how many potentially great mathematicians are being constantly lost to mathematics because of "symbolic reflex" types of teaching.
I want to come now to a consideration of the Foundations of Mathematics. We have witnessed, during the past 50 years, what we might call the most thorough soul-searching in the history of mathematics. By 1900, the Burali-Forti contradiction had been found and the Russell and other antinomies were soon to appear. The sequel is well known: Best known are the attempt of Russell and Whitehead in their monumental Principia Mathematica to show that mathematics can be founded, in a manner free of contradiction, on the symbolically expressed principles and methods of what were at the time considered universally valid logical concepts; the formulation, chiefly at the hands of Brouwer and his collaborators, of the tenet of Intuitionism, which although furnishing a theory evidently free of contradiction, introduces a highly complicated set theory and a mathematics radically restricted as compared with the mathematics developed during the 19th century; and the formalization of mathematics by Hilbert and his collaborators, together with the development of a metamathematical proof theory which it was hoped would lead to proofs of freedom from contradiction for a satisfactory portion, at least, of the classical mathematics. None of these "foundations" has met with complete success. Russell and Whitehead's theory of types had to be bolstered with an axiom which they had to admit, in the second edition of Principia Mathematica, has only pragmatic justification, and subsequent attempts by Chwistek, Wittgenstein, and Ramsey to eliminate or modify the use of this axiom generally led to new objections. The restricted mathematics known as Intuitionism has won only a small following, although some of its methods, such as those of a finite constructive character, seem to parallel the methods underlying the treatment of formal systems in symbolic logic, and some of its tenets, especially regarding constructive existence proofs, have found considerable favour. The possibility of carrying out the Hilbert program seems highly doubtful, in view of the investigations of Gödel and others.
Now the cultural point of view is not advanced as a substitute for such theories. In my title I have used the word "basis" instead of "foundations" in order to emphasize this point. But it seems probable that - the recognition of the cultural basis of mathematics would clear the air in Foundation theories of most of the mystical and vague philosophical arguments which are offered in their defence, as well as furnish a guide and motive for further research. The points of view underlying various attempts at Foundations of Mathematics are often hard to comprehend. In most cases it would seem that the proponents have decided in their own minds just what mathematics is, and that all they have to do is formulate it accordingly - overlooking entirely the fact that because of its cultural basis, mathematics as they know it may be not at all what it will be a century hence. If the thought underlying their endeavours is that they will succeed in trapping the elusive beast and confining it within bounds which it will never break, they are exceedingly optimistic. If the culture concept tells us anything, it should teach us that the first, rule for setting up any Foundation theory is that it should only attempt to encompass specific portions of the field as it is known in our culture. At most, a Foundation theory should be considered as a kind of constitution with provision for future amendments. And in view of the situation as regards such principles as the choice axiom, for instance, it looks at present as though no such constitution could be adopted by a unanimous vote!
I mentioned "mysticism and vague philosophical arguments" and their elimination on the cultural basis. Consider, for example, the insistence of Intuitionism that all mathematics should be founded on the natural numbers or the counting process, and that the latter are "intuitively given." There are plausible arguments to support the thesis that the natural numbers should form the starting point for mathematics, but it is hard to understand just what "intuitively given" means, or why the classical conception of the continuum, which the Intuitionist refuses to accept, should not be considered as "intuitively given." It makes one feel that the Intuitionist has taken Kronecker's much-quoted dictum that "The integers were made by God, but all else is the work of man"' and substituted "Intuition" for "God." However, if he would substitute for this vague psychological notion of "intuition" the viewpoint that inasmuch as the counting process is a cultural invariant, it follows that the natural numbers. form for every culture the most basic part of what has been universally called "mathematics," and should therefore serve as the starting point for every Foundations theory; then I think he would have a much sounder argument. I confess that I have not studied the question as to whether he can find further cultural support to meet all the objections of opponents of Intuitionism. It would seem, however, that he would have to drop his insistence that in construction of sets (to quote Brouwer [
Or consider the thesis that all mathematics is derivable from what some seem to regard as primitive or universal logical principles and methods. Whence comes this "primitive" or "universal" character? If by these terms it is meant to imply that these principles have a culturally invariant basis like that of the counting process, then it should be pointed out that cultures exist in which they do not have any validity, even in their qualitative non-symbolic form. For example, in cultures which contain magical elements (and such elements form an extremely important part of some primitive cultures), the law of contradiction usually fails. Moreover, the belief that our forms of thought are culturally invariant is no longer held. As eminent a philosopher as John Stuart Mill stated, [
It is probably fair to say that the Foundations of Mathematics as conceived and currently investigated by the mathematical logicians finds greatest support on the cultural basis. For inasmuch as there can exist, and have existed, different cultures, different forms of thought, and hence different mathematics, it seems impossible to consider mathematics, as I have already indicated, other than man-made and having no more of the character of necessity or truth than other cultural traits. Problems of mathematical existence, for example, can never be settled by appeal to any mathematical dogma. Indeed, they have no validity except as related to special foundations theories. The question as to, the existence of choice sets, for instance, is not the same for an Intuitionist as for a Formalist. The Intuitionist can justifiably assert that "there is no such problem as the continuum problem" provided he adds the words "for an Intuitionist" - otherwise he is talking nonsense. Because of its cultural basis, there is no such thing as the absolute in mathematics; there is only the relative.
But we must not be misled by these considerations and jump to the conclusion that what constitutes mathematics in our culture is purely arbitrary; that, for instance, it can be defined as the "science of p implies q", or the science of axiomatic systems. Although the individual person in the cultural group may have some degree of variability allowed him, he is at the same time subject to the dominance of his culture. The individual mathematician can play with postulational systems as he will, but unless and until they are related to the existing state of mathematics in his culture they will only be regarded as idiosyncrasies. Similar ties, not so obvious however, join mathematics to other cultural elements. And these bonds, together with those that tie each and every one of us to our separate mathematical interests, cannot be ignored even if we will to do so. They may exert their influence quite openly, as in the case of those mathematicians who have recently been devoting their time to high speed computers, or to developing other new and unforeseen mathematics induced by the recent wartime demands of our culture. Or their influence may be hidden, as in the case of certain mathematical habits which were culturally induced and have reached the symbolic reflex level in our reactions. Thus, although the postulational method may turn out to be the most generally accepted mode of founding a theory, it must be used with discretion; otherwise the theories produced will not be mathematics in the sense that they will be a part of the mathematical component of our culture.
But it is time that I closed these remarks. It would be interesting to study evidence in mathematics of styles and of cultural patterns; these would probably be interesting subjects of investigation for either the mathematician or the anthropologist, and could conceivably throw some light on the probable future course of the field. I shall have to pass on, however, to a brief conclusion:
In man's various cultures are found certain elements which are called mathematical. In the earlier days of civilization, they varied greatly from one culture to another so much so that what was called "mathematics" in one culture would hardly be recognized as such in certain others. With the increase in diffusion due, first, to exploration and invention, and, secondly, to the increase in the use of suitable symbols and their subsequent standardization and dissemination in journals, the mathematical elements of the most advanced cultures gradually merged until, except for minor cultural differences like the emphasis on geometry in Italy, or on function theory in France, there has resulted essentially one element, common to all civilized cultures, known as mathematics. This is not a fixed entity, however, but is subject to constant change. Not all of the change represents accretion of new material; some of it is a shedding of material no longer, due to influential cultural variations, considered mathematics. Some so-called "borderline" work, for example, it is difficult to place either in mathematics or outside mathematics.
From the extension of the notion of number to the transfinite, during the latter half of the 19th century, there evolved certain contradictions around the turn of the century, and as a consequence the study of Foundations questions, accompanied by a great development of mathematical logic, has increased during the last 50 years. Insofar as the search for satisfactory Foundation theories aims at any absolute criterion for truth in mathematics or fixation of mathematical method, it appears doomed to failure, since recognition of the cultural basis of mathematics compels the realization of its variable and growing character. Like other culture traits, however, mathematics is not a thoroughly arbitrary construction of the individual mathematician, since the latter is restricted in his seemingly free creations by the state of mathematics and its directions of growth during his lifetime, it being the latter that determines what is considered "important" at the given time.
In turn, the state and directions of growth of mathematics are determined by the general complex of cultural forces both within and without mathematics. Conspicuous among the forces operating from without during the past 50 years have been the crises through which the cultures chiefly concerned have been passing; these have brought about a large exodus of mathematicians from Western Europe to the United States, thereby setting up new contacts with resulting diffusion and interaction of mathematical ideas, as well as in the institution of new directions or acceleration of directions already under way, such as in certain branches of applied mathematics.
What the next 50 years will bring, I am not competent to predict. In his Decline of the West, Spengler concluded [
UNIVERSITY OF MICHIGAN,
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