Conceptual revolutions and the history of mathematics: two studies in the growth of knowledge (1984)*

Abstract

Abstract It has often been argued that revolutions do not occur in the history of mathematics and that, unlike the other sciences, mathematics accumulates positive knowledge without revolutionizing or rejecting its past.1 But there are certain critical moments, even in mathematics, that suggest that revolutions do occur—that new orders are brought about and eventually serve to supplant an older mathematics. Although there are many important examples of such innovation in the history of mathematics, two are particularly instructive: the discovery by the ancient Greeks of incommensurable magnitudes, and the creation of transfinite set theory by Georg Cantor in the nineteenth century. Both examples are as different in character as they are separated in time, and yet each provides a clear instance of a major transformation in mathematical thought.