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INTRODUCTION THE ESSAYS of this collection are all concerned with the role of mathematics in the rise and unfolding of Western intellectuality, with the sources and manifestations of the clarity and the mystique of mathematics, and with its ubiquity, universality, and indispensability. We will frequently confront the mathematics of today with the mathematics of the Greeks; and in such a confrontation it is pertinent to take the entire mathematical development since A.D. 1600 as one unit. Therefore, "modern mathematics" will mean for us, invariably, mathematics since 1600, and not since some later date, even though, for good reasons, the mathematics of most of the 20th century is clamoring for an identity and definition of its own. For our retrospections, the sobering fact that Greek mathematics, the mighty one, eventually died out in its own phase is of much greater import than the glamorous fact that Greek mathematics had come into being at all, although historians rather exult in its birth than grieve at its death. T. L. Heath in his standard history of Greek mathematics1 finds it awe-inspiring to contemplate how much Greek mathematics achieved "in an almost incredibly short time"; and the context makes it clear that Heath means the time from 600 B.C., when the first Thales-like geometry began to stir, to 200 B.C., when Apollonius wrote his treatise on Conies. This makes 400 long years, and pronouncements like that of Heath are insultingly condescending to the Greeks, especially painfully so since immediately after the Conies, and thus 50 years after the floruit of Archimedes, Greek mathematics was at a loss where to turn next. In irreconcilable contrast to this, in the 400 years since A.D. 1565 mathematical intellectuality has turned the world topsy-turvy many times over, and shows no sign of abating. 1 T. L. Heath, A History of Greek Mathematics (1921), I, 1. [3] INTRODUCTION Any major intellectual growth has to have, from time to time, bursts of unconscionably fast developments and stages of dizzying precipitousness, unless the developments are in anthropologically early phases of pre-intellectuality only. Much more "incredible" than the development of Greek mathematics in the course of 400 years was the reorientation of 20th-century physics, by a handful of youngsters, in the course of the four years from 1925 to 1928. Or, if to stay within Greek antiquity, nothing in the development of Greek mathematics can match, in sheer speed, Aristotle's amassment and articulation of logical, metaphysical, physical, cosmological, biological, and sociological knowledge within at most 40 years of his life (he died at the age of 62). As regards the growth of intellectuality , the plea of G. B. Shaw that man's life span ought to be at least 300 years2 is of dubious merit. Greek mathematics, whatever its inspiredness and universality , was slow, awkward, clumsy, bungling, and somehow sterile; and the limitations of Greek mathematics have wide implications. By its nature and circumstances Greek mathematics was part of Greek philosophy, that is, of the philosophies of Parmenides, Plato, and Aristotle, and it thus was a faithful image of a large segment of Greek intellectuality as a whole. Therefore, weaknesses of the mathematics of the Greeks were weaknesses of their intellectuality as such; and just as the death of the Great Pan signified the end of Greece's chthonic vitality,3 so also the death of the mathematics of Archimedes signified, perforce, the end of Olympian 4 intellectuality. 2 In the Preface to Back to Methuselah. 3 Jane Harrison, Prolegomena to the Study of Greek Religion, 3rd ed. (1922), also reprinted as Meridian Book (1957), p. 651; Archer Taylor, "Northern Parallels to the Death of Pan," Washington University {St. Louis) Studies, Vol. 10, Part 2, No. 1 (October 1922), p. 3. 4 Our distinction between "chthoric" and "Olympian" is that of the work of Jane Harrison, n. 3. W INTRODUCTIOK Yet the Greeks had extraordinary anticipations of the role of mathematics. It is an attribute of our times that mathematics is growing differently from other disciplines. AU areas of knowledge are growing by expansion from within and accretion from without, by internal subdivision and external overlappings with neighboring areas. Mathematics , in addition to that, is 'also penetrating into other areas of knowledge one-sidedly, for their benefit, and more by invitations and urgings of the recipient fields than from an expansionist drive. Now, a glimmer of this peculiar and unique universality of mathematics was noticed, however dimly and inarticulately, by the "typically...

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