From Triangles to Manifolds

Abstract

1. Geometry. I believe I am expected to tell you all about geometry; what it is, its developments through the centuries, its current issues and problems, and, if possible, a peep into the future. first question daes not have a clear-cut answer. meaning of the word geometry changes with time and with the speaker. With Euclid, geometry consists of the logical conclusions drawn from a set of axioms. This is clearly not sufficient with the horizons of geometry ever widening. Thus in 1932 the great geometers 0. Veblen and J. H. C. Whitehead said, A branch of mathematics is called geometry, because the name seems good on emotional and traditional grounds to a sufficiently large number of competent people [1]. This opinion was enthusiastically seconded by the great French geometer Elie Cartan [2]. Being an analyst himself, the great American mathematician George Birkhoff mentioned a disturbing secret fear that geometry may ultimately turn out to be no more than the glittering intuitional trappings of analysis [3]. Recently my friend Andre Weil said: The psychological aspects of true geometric intuition will perhaps never be cleared up. At one time it implied primarily the power of visualization in three-dimensional space. Now that higher-dimensional spaces have mostly driven out the more elementary problems, visualization can at best be partial or symbolic. Some degree of tactile imagination seems also to be involved [4]. At this point it is perhaps better to let things stand and turn to some concrete topics.