
The following is a work of historical investigation that focuses on specific mathematical mo-ments where methods, concepts, and definitions underwent profound changes. There are two main goals. First to sketch the history of the development of a continuous concept of expo-nents, and second to examine carefully he epistemological setting in which these develop-ments took place. Put more simply, what was it that convinced certain mathematicians that their concepts were viable. It is not intended to be a complete historical discussion, but rather a series of illuminating snapshots. The mathematical details of these moments are provided to the extent necessary for an understanding of the epistemology.