Some Concluding Remarks on Mathematical Change

One of our major theses is that Leibniz's calculus combined with Descartes analytic geometry represented the beginnings of a mathematical analog of a Kuhnian "paradigm shift," a phrase which has become a cliché not only in the history of science but in journalism, corporate management theory, or automobile advertisements (one wonders what its inventor, a rigorously trained Harvard physicist, would have made of it all). But as we have argued in Chapters 1 and 2 the analogy with scientific change goes only so far. Rejected scientific theory is viewed as false. Its claims are no longer seen to correspond to the way the world is. Despite the assertions by sociological proponents of the "Strong Programme" or various "postmodern" theorists of science, this situation has been rarely the case in mathematics. The conclusions of Viète or Apollonius are as valid today as they were when they were made. On the other hand, mathematics can certainly abandon a previously dominant "thought style" or subject matter and go off in new directions in such a profound way that the previous work is essentially forgotten or even made inaccessible, and this was the fate of seventeenth century geometry…