**R L Wilder**to the International Congress of Mathematicians was held in Cambridge, Massachusetts, USA.

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

### THE CULTURAL BASIS OF MATHEMATICS Part II

**R L Wilder**

Let us look for a few minutes at the history of mathematics. I confess I know very little about it, since I am not a historian. I should think, however, that in writing a history of mathematics the historian would be constantly faced with the question of what sort of material to include. In order to make a clearer case, let us suppose that a hypothetical person, A, sets out to write a complete history, desiring to include all available material on the "history of mathematics." Obviously, he will have to accept some material and reject other material. It seems clear that his criterion for choice must be based on knowledge of what constitutes mathematics! If by this we mean a definition of mathematics, of course his task is hopeless. Many definitions have been given, but none has been chosen; judging by their number, it used to be expected of every self-respecting mathematician that he would leave a definition of mathematics to posterity! Consequently our hypothetical mathematician A will be guided, I imagine, by what is called "mathematics" in his culture, both in existing (previously written) histories and in works called "mathematical," as well as by what sort of thing people who are called "mathematicians" publish. He will, then, recognize what we have already stated, that mathematics is a certain part of his culture, and will be guided thereby.

For example, suppose A were a Chinese historian living about the year 1200 (500 or 1500 would do as well). He would include a great deal about computing with numbers and solving equations; but there wouldn't be any geometry as the Greek understood it in his history, simply because it had never been integrated with the mathematics of his culture. On the other hand, if A were a Greek of 200 A.D., his history of mathematics would be replete with geometry, but there would be little of algebra or even of computing with numbers as the Chinese practiced it. But if A were one of our contemporaries, he would include both geometry and algebra because both are part of what we call mathematics. I wonder what he would do about logic, however?

Here is a subject which, despite the dependence of the Greeks on logical deduction, and despite the fact that mathematicians, such as Leibniz and Pascal, have devoted considerable time to it on its own merits, has been given very little space in histories of mathematics. As an experiment, I looked in two histories that have been popular in this country; Ball's [1] and Cajori's [5], both written shortly before 1900. In the index of Ball's first edition (1888) there is no mention of "logic;" but in the fourth edition (1908) "symbolic and mathematical logic" is mentioned with a single citation, which proved to be a reference to an incidental remark about George Boole to the effect that he "was one of the creators of symbolic or mathematical logic." Thus symbolic logic barely squeezed under the line because Boole was a mathematician! The index of Cajori's first edition (1893) contains four citations under "logic," all referring to incidental remarks in the text. None of these citations is repeated in the second edition (1919), whose index has only three citations under "logic" (two of which also constitute the sole citations under "symbolic logic"), again referring only to brief remarks in the text. Inspection of the text, however, reveals nearly four pages (407-410) of material under the title "Mathematical logic," although there is no citation to this subject in the index nor is it cited under "logic" or "symbolic logic." (It is as though the subject had, by 1919, achieved enough importance for inclusion as textual material in a history of mathematics although not for citation in the index!)

I doubt if a like situation could prevail in a history of mathematics which covers the past 50 years! The only such history that covers this period, that I am acquainted with, is Bell's *Development of Mathematics* [2]. Turning to the index of this book, I found so many citations to "logic" that I did not care to count them. In particular, Bell devotes at least 25 pages to the development of what he calls "mathematical logic." Can there be any possible doubt that this subject, not considered part of mathematics in our culture in 1900, despite the pioneering work of Peano and his colleagues, is now in such "good standing" that any impartial definition of mathematics must be broad enough to include it?

Despite the tendency to approach the history of mathematics from the biographical standpoint, there has usually existed some awareness of the impact of cultural forces. For example, in commencing his chapter on Renaissance mathematics, Ball points out the influence of the introduction of the printing press. In the latest histories, namely the work of Bell already cited, and Struik's excellent little two volume work [17], the evidence is especially strong. For example, in his introduction, Struik expresses regret that space limitations prevented sufficient "reference to the general cultural and sociological atmosphere in which the mathematics of a period matured - or was stifled." And he goes on to say "Mathematics has been influenced by agriculture, commerce and manufacture, by warfare, engineering and philosophy, by physics and by astronomy. The influence of hydrodynamics on function theory, of Kantianism and of surveying on geometry, of electromagnetism on differential equations, of Cartesianism on mechanics, and of scholasticism on the calculus could only be indicated [in his book]; - yet an understanding of the course and content of mathematics can be reached only if all these determining factors are taken into consideration." In his third chapter Struik gives a revealing account of the rise of Hellenistic mathematics, relating it to the cultural conditions then prevailing. I hope that future histories of mathematics will similarly give more attention to mathematics as a cultural element, placing greater emphasis on its relations to the cultures in which it is imbedded.

In discussing the general culture concept, I did not mention the two major processes of cultural change, evolution and diffusion. By diffusion is meant the transmission of a cultural trait from one culture to another, as a result of some kind of contact between groups of people; for example, the diffusion of French language and customs into the Anglo-Saxon culture following the Norman conquest. As to how much of what we call cultural progress is due to evolution and how much to diffusion, or to a combination of both, is usually difficult to determine, since the two processes tend so much to merge. Consider, for example, the counting process. This is what the anthropologist calls a universal trait - what I would prefer to call, in talking to mathematicians, a cultural invariant - it is found in every culture in at least a rudimentary form. The "base" may be 10, 12, 20, 25, 60 - all of these are common, and are evidently determined by other (variable) culture elements - but the counting process in its essence, as the Intuitionist speaks of it, is invariant. If we consider more advanced cultures, the notion of a zero element sometimes appears. As pointed out by the anthropologist A L Kroeber, who in his Anthropology calls it a "milestone of civilization," a symbol for zero evolved in the cultures of at least three peoples; the Neo-Babylonian (who used a sexagesimal system), the Mayan (who used a vigesimal system), and the Hindu (from whom our decimal system is derived) [10] (pp. 468-472). Attempts by the extreme "diffusionists" to relate these have not yet been successful, and until they are, we can surmise that the concept of zero might ultimately evolve in any culture.

The Chinese-Japanese mathematics is of interest here. Evidently, as pointed out by Mikami [13] and others, the Chinese borrowed the zero concept from the Hindus, with whom they established contact at least as early as the first century, A.D. Here we have an example of its introduction by diffusion, but without such contacts, the zero would probably have evolved in Chinese mathematics, especially since calculators of the rod type were employed. The Chinese mathematics is also interesting from another standpoint in that its development seems to have been so much due to evolution within its own culture and so little affected by diffusion. Through the centuries it developed along slender arithmetic and algebraic lines, with no hint of geometry as the Greeks developed it. Those who feel that without the benefit of diffusion a culture will eventually stagnate find some evidence perhaps in the delight with which Japanese mathematicians of the 17th and 18th centuries, to whom the Chinese mathematics had come by the diffusion process, solved equations of degrees as high as 3000 or 4000. One is tempted to speculate what might have happened if the Babylonian zero and method of position had been integrated with the Greek mathematics - would it have meant that Greek mathematics might have taken an algebraic turn? Its introduction into the Chinese mathematics certainly was not productive, other than in the slight impetus it gave an already computational tendency.

That the Greek mathematics was a natural concomitant of the other elements in Greek culture, as well as a natural result of the evolution and diffusion processes that had produced this culture in the Asia Minor area, has been generally recognized. Not only was the Greek culture conducive to the type of mathematics that evolved in Greece, but it is probable that it resisted integration with the Babylonian method of enumeration. For if the latter became known to certain Greek scholars, as some seem to think, its value could not have been apparent to the Greeks.

We are familiar with the manner in which the Hindu-Arabic mathematical cultures diffused via Africa to Spain and then into the Western European cultures. What had become stagnant came to life-analytic geometry appeared, calculus-and the flood was on. The mathematical cultural development of these times would be a fascinating study, and awaits the cultural historian who will undertake it. The easy explanation that a number of "supermen" suddenly appeared on the scene has been abandoned by virtually all anthropologists. A necessary condition for the emergence of the "great man" is the presence of suitable cultural environment, including opportunity, incentive, and materials. Who can doubt that potentially great algebraists lived in Greece? But in Greece, although the opportunity and incentive may have been present, the cultural materials did not contain the proper symbolic apparatus. The anthropologist Ralph Linton remarked [12] (p. 319) "The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, life-long application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes." Furthermore, the evidence points strongly to the sufficiency of the conditions stated: That is, suitable cultural environment is sufficient for the emergence of the great man. If your philosophy depends on the assumption of free will, you can probably adjust to this. For certainly your will is no freer than the opportunity to express it; you may will a trip to the moon this evening, but you won't make it. There may be potentially great blancophrenologists sitting right in this room; but if so they are destined to go unnoticed and undeveloped because blancophrenology is not yet one of our cultural elements.

Spengler states it this way [16] (vol. II, p. 507): "We have not the freedom to reach to this or to that, but the freedom to do the necessary or to do nothing. And a task that historic necessity has set will be accomplished with the individual or against him." As a matter of fact, when a culture or cultural element has developed to the point where it is ready for an important innovation, the latter is likely to emerge in more than one spot. A classical example is that of the theory of biological evolution, which had been anticipated by Spencer and, had it not been announced by Darwin, was ready to be announced by Wallace and soon thereafter by others. And as in this case, so in most other cases, - and you can recall many such in mathematics - one can after the fact usually go back and map out the evolution of the theory by its traces in the writings of men in the field.

**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 Third part of Wilder's address