The Universal Domination of Geometry

Abstract

Contrary to popular belief, the idea of pure geometry, divorced from algebra, is quite recent (early nineteenth century) and was completely foreign to the Greeks. They had no algebra in the modern sense, but their geometry was really an algebro-geometry, a complex mixture of purely geometric arguments, and computations with ratios of segments. For instance, most properties of conics are proved by Apollonius by using what we now would call the cartesian equation of the curve with respect to two axes consisting of a tangent to the conic and the corresponding diameter. On the other hand, the solution of quadratic equations is presented as a geometric construction of areas. Finally the method of solving an algebraic problem by finding the intersection of two curves is one of the oldest in Greek mathematics: the solution given by Maenechmus to the problem of duplicating the cube (i.e., finding a ratio x/y such that (x/y)3 = a/b, a given ratio) was to intersect the two curves xy = ab, ay = x2, and this introduced the conics in mathematics. As soon as the invention of coordinates enormously enlarged the scope of geometry, both Descartes and Newton emphasized the possibility of using this new tool to solve far more complicated equations by intersection of curves. Thus, again contrary to what most people think, the method of coordinates worked both ways, linking algebra and geometry rather than replacing one by the other. This possibility of interpreting algebraic problems in geometric terms had a great appeal for mathematicians ever since the end of the seventeenth century. Unfortunately it was limited to problems dealing with 2 or 3 independent variables, but with the development of Mechanics and Astronomy, problems with an arbitrary number of variables (the degrees of freedom) became more and more common; furthermore, the passage from n variables to n + 1 variables usually did not modify the algebraic treatment in an appreciable way. The temptation to use in such problems a language inspired by geometry thus became irresistible by the middle of the nineteenth century; after 1870 it was generally agreed that one could use in mathematics a conventional language, derived from ordinary geometry, without of course claiming any more that it corresponded to an underlying physical reality. Instead of speaking of a system of n numbers, one would say it was a point in n-dimensional space; the set of such systems satisfying a linear equation would be called a hyperplane, etc. The success of this idea has been amazing, and has proliferated during the last century in an unexpected variety of ways. In the remainder of this paper I would like to emphasize some of the highlights of this evolution.