R L Wilder: Cultural Basis of Mathematics II
Here is a link to the First part of Wilder's address
Here is a link to the First part of Wilder's address
THE CULTURAL BASIS OF MATHEMATICS Part II
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 [
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 [
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 [
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) [
The Chinese-Japanese mathematics is of interest here. Evidently, as pointed out by Mikami [
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 [
Spengler states it this way [
- 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 [
- L A White, The science of culture (Farrar, Straus, New York, 1949).
Here is a link to the Third part of Wilder's address
JOC/EFR March 2006
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