Cognitive artifacts and their affordances in mathematical practice:Cluster algebras as a case study

This paper investigates the role that various kinds of signs, taken to be cognitive<br/>artifacts, play in contemporary mathematical practice. We examine two informationally equivalent ways of presenting mathematical content in light of some of their uses in the recently developed theory of cluster algebras. Following a short overview of the concept of cognitive artifacts as it pertains to mathematical practice, we present the conceptual frameworks of affordances and responsiveness types which are used in the analysis of the case study. These frameworks are employed to describe the sorts of epistemic operations that are available for a given cognitive artifact. We find that in the case of cluster algebras, the cognitive artifacts in question offer different opportunities for computational offloading, which effectively reduces cognitive load by the aid of external media. In addition, we provide a brief introduction to the theory of cluster algebras, albeit at a restricted level of generality, including some of the initial motivations which resulted in the formulation of this theory.