Factors Inside and Outside Mathematics which are Important in the Generation and Development of New Mathematic Disciplines

Abstract

In the discussion I want to introduce explicitly a new element which does not seem to be included in the interesting catalogue of questions prepared for the first Session. That is the social economical background. I think that even in mathematics this background is of greatest importance, not only for the early development of mathematics but also for the tremendous achievements and new mathematical ideas in our century. I claim that basically the emergence of a new field in mathematics always follows the same pattern. When the economical development of a human society has reached a certain level, problems of an empirical origin occur. The human mind is then able to develop methods belonging to a branch of applied mathematics by means of which these problems can be mastered. When this branch has reached a certain maturity, then mathematics as a discipline in its own right emerges from here. The beginning of this transition is due to outstanding personalities. When mathematics itself has reached a high level, this transition from applied mathematics to a new mathematical field may occur on a very high level of abstraction. Moreover the empirical background of this field may sometimes be rediscovered only by tracing back a long chain of development.