Mathematical concepts and investigative practice

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices.