Thinking with Diagrams: The Case of Mathematics

Silvia de Toffoli

Visual representations of various kinds are ubiquitous in pure and applied mathematics, in the natural and social sciences, as well as in many other human activities. By investigating these visualizations, many questions arise: What are they? How do they function? What are the conditions of their correct use? Why are they, at times, such effective aids to cognition? What type of knowledge can they promote? In this project, I focus on diagrams in pure mathematics, not exclusively in geometry where diagrams are common, but in different mathematical domains. Despite the extreme variety of representations in these domains, it is possible to distill fundamental properties of the nature and use of mathematical diagrams. Diagrams are not static illustrations simply recording information, but dynamic displays for advancing thought. An effective diagram, or a sequence of diagrams, sets the relevant reasoning into material, visual form. By manipulating these concrete external representations in prescribed ways, information about abstract mathematical structures can be obtained without going through a process of formal calculation. In this way cognitive abilities that no doubt evolved in order to manipulate concrete objects can be re-deployed in the abstract realm of mathematics.


 

Academic Year
2016-2017
Area of Study