The aim of this thesis is to shed light on how concepts and theories in mathematics develop. More specifically, it deals with two main themes in the philosophy of mathematical practice which concern the formation and evolution of mathematical concepts, on the one hand, and the role of signs in mathematics, on the other hand. These epistemological themes are investigated in light of a case study from contemporary mathematics which is situated at the intersection of algebra, combinatorics, and geometry. Two papers are included in this thesis:
Paper A considers the relationship between a name and a concept in mathematics. In particular, it addresses why a new name is sometimes given to denote a familiar mathematical concept as well as what it might mean for a concept to have a certain “attitude”. Paper B aims to articulate why there is a need for different mathematical signs used for the same purposes in the recently developed theory of cluster algebras.
This work as a whole contributes to the philosophy of mathematical practice in a number of ways. First, it introduces a novel case study of a contemporary mathematical theory into the philosophical literature. Second, it investigates roles of concepts and signs in mathematics which have not previously received the attention that they deserve. Third, it provides insights into how mathematics, understood both as a field of research and as a stable form of knowledge, progresses.
