Reflections on Progress in Mathematics

Abstract

The vitality of mathematics, however, “is conditioned upon the connection of its parts.” What, however, are the “parts” and “connections”? Is there, perhaps, some general pattern to this ongoing enterprise? In other words, is there some recognisable order to the mathematical project, not as in something to be imposed, but an order that can be verified in actual works and collaborations? A main purpose of this paper is to offer an answer to this question in the affirmative. For there is accumulating evidence for the existence of an eight-fold periodic sequence of functionally related zones of enquiry H1, … , H8 – where for the rest of this paper these zones will be called functional specialties. In particular, each functional specialty would seem to have its own main objective and to involve its own differentiated type of enquiry.