A typology of mathematical diagrams

Abstract

In this paper, we develop and discuss a classification scheme that allows us to distinguish between the types of diagrams used in mathematical research based on the cognitive support offered by diagrams. By cognitive support, we refer to the gain that research mathematicians get from using diagrams. This support transcends the specific mathematical topic and diagram type involved and arises from the cognitive strategies mathematicians tend to use. The overall goal of this classification scheme is to facilitate a large-scale quantitative investigation of the norms and values governing the publication style of mathematical research, as well as trends in the kinds of cognitive support used in mathematics. This paper, however, focuses only on the development of the classification scheme. The classification scheme takes its point of departure from case studies known from the literature, but in this paper, we validate the scheme using examples from a preliminary investigation of developments in the use of diagrams. Building on these results, we discuss the potential and pitfalls in using one generic classification scheme, as done in this analysis. This approach is contrasted with attempts that respect and build on individual diagram types, and as part of this discussion, we report the problems we experienced when using that strategy. The paper ends with a description of possible next steps in using text corpora as an empirical approach to understanding the nature of mathematical diagrams and their relation to mathematical culture.