GOOD AFTERNOON, FELLOW SIAMESE AND STAUNCH ALLIES. What I have chosen as the subject of my present discourse is not some bright, dazzling discovery but a puzzling matter which fills me with increasing concern and wonder. So I come before you with a think piece in the hope of sharing with you my grappling with a problem whose importance I leave up to you to confirm or dispute. Let me add that my career has pretty thoroughly pried me loose from any mathematico-centric view of the universe. I hope, therefore, to be in a position of presenting from a constructive viewpoint above the lines of faction which appear once more to be drawn across the face of mathematics, one of the particularly pivotal issues in this whole business-namely what has been, is, and should be the role of geometry in mathematics. The two major intellectual revolutions in mathematics which have affected Western thought far beyond the confines of our own chosen field clearly stem from geometry: the laying down of Euclid's elements and the conception, some two millenia later, of non-euclidean geometries. I shall use the second of these as my starting point for an appreciation of what geometry has meant to mathematics, for its out-runlers, I maintain, are even today significantly and indelibly shaping the current scene of mathematical research. This significance has been stated often and in various ways: It established once and for all that mathematics was not in any superficially direct and God-given way tied in its structure to this one particular world in which we happen to live-that, instead, logically consistent, intellectually meaningful and attractive systems could be found which, in content, were not only unrelated, but-as far as geometry was concerned, at variance with the then canonical system of natural philosophy. Once geometry had been made intrinsic to mathematics, rather than figuring as one of its most striking contributions to the scientific analysis of the world around us, the road was open for the confident development of the field along the lines of an internal, self-determined, essentially artistic freedom in the choice of its objects and in the examination of their interrelations. We shall later have to comment on the well-known fact that, in the course of time, the constructions which arose in this way constituted again one of the most significant contributions of mathematics to science. Suffice it to say here that, in the overwhelming majority of instances, this was not their motivation. What is of interest for the moment is that the enrichment of mathematics which immediately ensued from this newly gained freedom during the remainder of the 19th century affected with particular vitality the emergilng branches of
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