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C H A P T E R 1 FROM MYTH TO MATHEMATICS TO KNOWLEDGE * 1. WHAT IS MATHEMATICS? WHAT INDEED is mathematics? This question, if asked in earnest, has no answer, not a satisfactory one; only part answers and observations can be attempted. A neat little answer, a citizen's description of mathematics in capsule form, is preserved in the writings of a Chureh Father of the 3rd century A.D.; Anatolius of Alexandria , bishop of Laodicea, reports that a certain (unnamed ) "jokester," using words of Homer which had been intended for something entirely different, put it thus: * Small at her birth, but rising every hour, While scarce the skies her horrid [mighty] head can bound, She stalks on earth and shakes the world around. (Iliad, IV, 442-443, Pope's translation.) For, explains Anatolius, mathematics begins with a point and a line, and forthwith it takes in heaven itself and all things within its compass. If the bishop were among us today, he might have worded the same explanation thus: * Originally in Journal of the History of Ideas, Vol. 26 (1965), pp. 3-24, under the title "Why Mathematics Grows." Enlarged. 1 The Ante-Nicene Fathers, edited by Alexander Roberts and James Donaldson, VI, 152. The leading edition of the Greek text of Anatolius is due to F. Hultsch. This text, which is only a surviving fragment, is included in the collection Anonymi variae collectiones which Hultsch appended to his edition of the mathematical works which go under the name of Heron (Berlin, 1864). For bibliography about Anatolius see George Sarton, Introduction to the History of Science, 1(1927),337. [13] MYTH, MATHEMATICS, KNOWLEDGE For mathematics, as a means of articulation and theoretization of physics, spans the universe, all the way from the smallest elementary particle to the largest galaxy at the rim of the cosmos. The gist of this and similar explanations is a declaratory statement or assertion, with some illustrations, that mathematics is distinctive and effective, and thus important. But how have these traits of distinctiveness and effectiveness come about, and how do they sustain themselves so powerfully ? And whence comes the urge, the intellectual one, to spread mathematics into ever more areas of knowledge, old and new, especially new ones? Perhaps such questions cannot be answered at all. But we will try, and our approach will be analytical and evolutional, both. 2. MATHEMATICS AND MYTHS We will proceed from a quotation: Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfillment in the act but must proclaim and elaborate a poetic form of truth. This appealing sentence is not my own; I wish it were. It is taken from a book on the awakening of intellectuality in Egypt and Mesopotamia, the two near-Mediterranean areas in which, by Chance or Providence, inexplicably but unmistakably, our present-day "Western" civilization germinated first.2 However, in the book from which it is 2 See the book of Henri Frankfort and others, The Intellectual Adventures of Ancient Man (Chicago, 1946), p. 8; also in a Penguin edition, under the title, Before Philosophy, p. 16. Another statement from this book which is meaningful for mathematics instead of myth is the following one from page 12 (or page 21 in the Penguin edition): ". . . there is coalescence of the symbol [14] MYTH, MATHEMATICS, KNOWLEDGE taken, the sentence is a pronouncement not on mathematics but on myth. That is, in the original the sentence runs as follows: Myth is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning. . . . Also, the book from which we quote does not concern itself with mathematics at all; but it might have done so, because in Egypt and Mesopotamia, severally, mathematics started very early, surprisingly so. For instance, it started there about 2,000 years before the famed "classical" mathematics of the Greeks, which was a summit achievement of their natural philosophy, began to shape itself, in its separateness, in the 6th and 5th centuries B.C. It is true that in aspects of intellectuality Greek mathematics was greatly superior to whatever earlier mathematics it built on. But it is also a fact that, by evolutionary precedence, the two mathematics of Egypt and Mesopotamia were firsts, unqualifiedly so, not as anthropological phenomena...


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