ABSTRACT
This chapter introduces the strong affirmation of the essential autonomy of mathematics into a paper devoted to the central theme of the interaction of mathematics and the sciences. There was a mathematical counterpart of the new physical science, which served both as its precursor and principal tool. This was the mathematics of the new algebra and of the analytic movement of Vieta and Descartes, a mathematics which substituted calculation and manipulation of symbolic expressions for the deductive sophistication of the Greeks. Mathematical research, as a whole, balances the radical process of generation of new concepts and theories with the conservative tendency to maintain in existence all those domains, problems, and conceptual themes that once become established as foci of significant mathematical research. The greatest danger to the continued thrust of scientific and mathematical discovery is the possibility that the institutions which house that thrust will become routinized and bureaucratized.
