Mathematical Understanding and the role of Counterexamples and Pathologies: a case study in Mathematical Analysis.

Pathological objects and counterexamples 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 try to give a partial response to these questions from the standpoint of mathematical analysis in the nineteenth and twentieth centuries. We will describe briefly how the notion of function changed dramatically in the nineteenth century and we will study how this change brought on important philosophical consequences for the subject implying that the notion of pathology relies upon certain properties occurring only in a few instances.