ABSTRACT
Lawrence Sklar charges that Poincare wavered between reductionism and conventionalism. Kant's theory of geometry as a priori theory of space, which is given in intuition, is totally overturned in Poincare's geometric conventionalism. This chapter presents his argument for geometric conventionalism in two parts, first an argument against the a priori determination of metric and second an argument against the empirical determination of metric. Poincare argues that some elements of empirical science can be erected into principles they can be taken to be definitely true and never questioned. Poincare changes the traditional view of geometry as a priori science of space so profoundly, Ernest Nagel gives him a leading role in the development of the formal conception of mathematics in his widely cited paper "The Formation of Modern Conceptions of Formal Logic in the Development of Geometry". In pure geometry and physical science, Poincare expresses a formal conception of theories, which he sees as axiomatic systems with arbitrary starting points.
