MATHEMATICAL UNDERSTANDING AND THE ROLE OF COUNTEREXAMPLES AND PATHOLOGIES: A CASE STUDY IN MATHEMATICAL ANALYSIS

Pathological objects play an important role in mathematical understanding even though there is no precise definition of them. What is a pathological object? What makes a mathematical object pathological? The aim of this paper is to give a partial response to these questions from the standpoint of mathematical analysis in the nineteenth century and the first quarter of the twentieth century. It will be briefly described how the notion of function changed dramatically in the nineteenth century and it will be studied how this change brought on important philosophical consequences for the subject, which lead to the conclusion that the notion of pathology relies upon certain properties that occur only in a few instances.