The Organising Committee invited the following mathematicians to address the Congress:

**Wednesday 30 August:** A Beurling, H Hopf, Henri Cartan, R L Wilder.

**Thursday 31 August:** S Bochner, K Gödel, O Zariski, A Weil.

**Friday 1 September:** M Morse, A Rome.

**Saturday 2 September:** Abraham Albert.

**Monday 4 September:** A Tarski, J von Neumann.

**Tuesday 5 September:** A Wald, H Whitney, W V D Hodge, J F Ritt, H Davenport, L Schwartz, S Kakutani, S S Chern,

**Wednesday 6 September:** Norbert Wiener.

**R L Wilder** gave his invited address to the Congress on the afternoon of Wednesday 30 August in Fogg Large Lecture Room on *The Cultural Basis of Mathematics.*

### THE CULTURAL BASIS OF MATHEMATICS Part I

**R L Wilder**

I presume that it is not inappropriate, on the occasion of an International Mathematical Congress which comes at the half-century mark, to devote a, little time to a consideration of mathematics as a whole. The addresses and papers to be given in the various conferences and sectional meetings will in general be concerned with special fields or branches of mathematics. It is the aim of the present remarks to get outside mathematics, as it were, in the hope of attaining a new perspective. Mathematics has been studied extensively from the abstract philosophical viewpoint, and some benefits have accrued to mathematics from such studies-although generally the working mathematician is inclined to look upon philosophical speculation with suspicion. A growing number of mathematicians have been devoting thought to the Foundations of Mathematics, many of them men whose contributions to mathematics have won them respect. The varying degrees of dogmatism with which some of these have come to regard their theories, as well as the sometimes acrimonious debates which have occurred between holders of conflicting theories, makes one wonder if there is not some vantage point from which one can view such matters more dispassionately.

It has become commonplace today to say that mankind is in its present "deplorable" state because it has devoted so much of its energy to technical skills and so little to the study of man itself. Early in his civilized career, man studied astronomy and the other physical sciences, along with the mathematics these subjects suggested; but in regard to such subjects as anatomy, for example, it was not easy for him to be objective. Man himself, it seemed, should be considered untouchable so far as his private person was concerned. It is virtually only within our own era that the study of the even more personal subjects, such as psychology, has become moderately respectable! But in the study of the behaviour of man en masse, we have made little progress. This is evidently due to a variety of reasons such as (1) inability to distinguish between group behaviour and individual behaviour, and (2) the fact that although the average person may grudgingly give in to being cut open by a surgeon, or analysed by a psychiatrist, those group institutions which determine his system of values, such as nation, church, clubs, etc., are still considered untouchable.

Fortunately, just as the body of the executed criminal ultimately became available to the anatomist, so the "primitive" tribes of Australia, the Pacific Islands, Africa, and the United States, became available to the anthropologist. Using methods that have now become so impersonal and objective as to merit its being classed among the natural sciences rather than with such social studies as history, anthropology has made great advances within the past 50 years in the study of the group behaviour of mankind. Its development of the culture concept and investigation of cultural forces will, perhaps, rank among the greatest achievements of the human mind, and despite opposition, application of the concept has made strides in recent years. Not only are psychologists, psychiatrists, and sociologists applying it, but governments that seek to extend their control over alien peoples have recognized it. Manifold human suffering has resulted from ignorance of the concept, both in the treatment of colonial peoples, and in the handling of the American Indian, for example.

Now I am not going to offer the culture concept as an antidote for all the ills that beset mathematics. But I do believe that only by recognition of the cultural basis of mathematics will a better understanding of its nature be achieved; moreover, light can be thrown on various problems, particularly those of the Foundations of Mathematics. I don't mean that it can solve these problems, but that it can point the way to solutions as well as show the kinds of solutions that may be expected. In addition, many things that we have believed, and attributed to some kind of vague "intuition," acquire a real validity on the cultural basis.

For the sake of completeness, I shall begin with a rough explanation of the concept. (For a more adequate exposition, see [10] (Chap. 7) and [19].) Obviously it has nothing to do with culture spelled with a "K", or with degrees from the best universities or inclusion in the "best" social circles. A culture is the collection of customs, rituals, beliefs, tools, mores, etc., which we may call cultural elements, possessed by a group of people, such as a primitive tribe or the people of North America. Generally it is not a fixed thing but changing with the course of time, forming what can be called a "culture stream." It is handed down from one generation to another, constituting a seemingly living body of tradition often more dictatorial in its hold than Hitler was over Nazi Germany; in some primitive tribes virtually every act, even such ordinary ones as eating and dressing, are governed by ritual. Many anthropologists have thought of a culture as a super-organic entity, having laws of development all its own, and most anthropologists seem in practice to treat a culture as a thing in itself, without necessarily referring (except for certain purposes) to the group or individuals possessing it.

We "civilized" people rarely think of how much we are dominated by our cultures - we take so much of our behaviour as "natural." But if you were to propose to the average American male that he should wear earrings, you might, as you picked yourself off the ground, reflect on the reason for the blow that you have just sustained. Was it because he decided at some previous date that every time someone suggested wearing earrings to him he would respond with a punch to the nose? Of course not. It was decided for him and imposed on him by the American culture, so that what he did was, he would say, the "natural thing to do." However, there are societies such as Navajo, Pueblo, and certain Amazon tribes, for instance, in which the wearing of earrings by the males is the "natural thing to do." What we call "human nature" is virtually nothing but a collection of such culture traits. What is "human nature" for a Navajo is distinctly different from what is "human nature" for a Hottentot.

As mathematicians, we share a certain portion of our cultures which is called "mathematical." We are influenced by it, and in turn we influence it. As individuals we assimilate parts of it, our contacts with it being through teachers, journals, books, meetings such as this, and our colleagues. We contribute to its growth the results of our individual syntheses of the portions that we have assimilated.

Now to look at mathematics as a cultural element is not new. Anthropologists have done so, but as their knowledge of mathematics is generally very limited, their reactions have ordinarily consisted of scattered remarks concerning the types of arithmetic found in primitive cultures. An exception is an article [18] which appeared about three years ago, by the anthropologist L A White, entitled The locus of mathematical reality, which was inspired by the seemingly conflicting notions of the nature of mathematics as expressed by various mathematicians and philosophers. Thus, there is the belief expressed by G H Hardy [8] (pp. 63-64) that "mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations." On the other hand there is the point of view expressed by P W Bridgman [3] (p. 60) that "it is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention." Although these statements seem irreconcilable, such is not the case when they are suitably interpreted. For insofar as our mathematics is a part of our culture, it is, as Hardy says, "outside us." And insofar as a culture cannot exist except as the product of human minds, mathematics is, as Bridgman states, a "human invention."

As a body of knowledge, mathematics is not something I know, you know, or any individual knows: It is a part of our culture, our collective possession. We may even forget, with the passing of time, some of our own individual contributions to it, but these may remain, despite our forgetfulness, in the culture stream. As in the case of many other cultural elements, we are taught mathematics from the time when we are able to speak, and from the first we are impressed with what we call its "absolute truth." It comes to have the same significance and type of reality, perhaps, as their system of gods and rituals has for a primitive people. Such would seem to be the case of Hermite, for example, who according to Hadamard [7] (p. xii) said, "We are rather servants than masters in Mathematics;" and who said [6] (p. 449) in a letter to Königsberger, " - these notions of analysis have their existence apart from us, - they constitute a whole of which only a part is revealed to us, incontestably although mysteriously associated with that other totality of things which we perceive by way of the senses." Evidently Hermite sensed the impelling influence of the culture stream to which he contributed so much!

In his famous work *Der Untergang des Abendlandes* [15], 0 Spengler discussed at considerable length the nature of mathematics and its importance in his organic theory of cultures. And under the influence of this work, C J Keyser published [9] some views concerning Mathematics as a Culture Clue, constituting an exposition and defence of the thesis that "The type of mathematics found in any major Culture is a clue, or key, to the distinctive character of the Culture taken as a whole." Insofar as mathematics is a part of and is influenced by the culture in which it is found, one may expect to find some sort of relationship between the two. As to how good a "key" it furnishes to a culture, however, I shall express no opinion; this is really a question for an anthropologist to answer. Since the culture dominates its elements, and in particular its mathematics, it would appear that for mathematicians it would be more fruitful to study the relationship from this point of view.

**References:**

- W W R Ball,
*A short account of the history of mathematics*(Macmillan, London, 1888; 4th ed.,1908). - E T Bell,
*The development of mathematics*(McGraw-Hill, New York, 2nd ed., 1945). - P W Bridgman,
*The logic of modern physics*(Macmillan, New York, 1927). - L E J Brouwer, Intuitionism and formalism (tr. by A Dresden),
*Bull. Amer. Math. Soc.***20**(1913-1914), 81-96. - F Cajori,
*A history of mathematics*(Macmillan, New York, 1893; 2nd ed., 1919). - A Dresden, Some philosophical aspects of mathematics,
*Bull. Amer. Math. Soc.***34**(1928), 438-452. - J Hadamard,
*The psychology of invention in the mathematical field*(Princeton University Press, Princeton, 1945). - G H Hardy,
*A mathematician's apology*(Cambridge University Press, Cambridge, 1941). - C J Keyser, Mathematics as a culture clue,
*Scripta Mathematica***1**(1932-1933), 185-203; reprinted in a volume of essays having same title*Scripta Mathematica*(New York, 1947). - A L Kroeber,
*Anthropology*(Harcourt, Brace, New York, rev. ed., 1948). - D D Lee, A primitive system of values,
*Philosophy of Science***7**(1940), 355-378. - R Linton,
*The study of man*(Appleton-Century, New York, 1936). - Y Mikami,
*The development of mathematics in China and Japan*(Drugulin, New York, 1913). - J S Mill, Inaugural address, delivered to the University of St Andrews, 1 Feb. 1867 (Littell and Gay, Boston, 1867).
- 0 Spengler, Der Untergang des Abendlandes, München, C H Beek, vol. I, 1918, (2d ed., 1923), vol. II, 1922.
- 0 Spengler (tr. of [15] by C F Atkinson),
*The decline of the West*(Knopf, New York, vol. I, 1926, vol. II, 1928). - D J Struik,
*A concise history of mathematics,*2 vols. (Dover New York,1948). - L A White, The locus of mathematical reality,
*Philosophy of Science***14**(1947), 289-303; republished in somewhat altered form as Chapter 10 of [19]. - L A White, The science of culture (Farrar, Straus, New York, 1949).

Here is a link to the Second part of Wilder's address