What’s in a Diagram? On the Classification of Symbols, Figures and Diagrams

In this paper I analyze the cognitive function of symbols, figures and diagrams. The analysis shows that although all three representational forms serve to externalize mental content, they do so in radically different ways, and consequently they have qualitatively different functions in mathematical cognition. Symbols represent by convention and allow mental computations to be replaced by epistemic actions. Figures and diagrams both serve as material anchors for conceptual structures. However, figures do so by having a direct likeness to the objects they represent, whereas diagrams have a metaphorical likeness. Thus, I claim that diagrams can be seen as material anchors for conceptual mappings. This classification of diagrams is of theoretical importance as it sheds light on the functional role played by conceptual mappings in the production of new mathematical knowledge.