Diagrammatic Proofs

Abstract

Diagrammatic reasoning comprises phenomena that range from the so-called "free-rides" (e.g. almost immediate understanding of visually perceived relationships) to conventions about tokens. Such reasoning must involve cognitive processes that are highly perceptual in content. In the domain of mathematical proofs where diagrams have had a long history, we have an opportunity to investigate in detail and in a controlled setting the various perceptual devices and cognitive processes that facilitate diagrammatically based arguments. This paper continues recent work by examining two kinds of diae;rammatic proofs, called Categories 1 and 3 by [Jamnik, et. al. 97], the first being one in which generalization of a diagram instance is implied, and the second being one in which an infinite completion is represented by an ellipsis. We provide explanations of why these proofs work, a semantics for ellipses, and conjectures about the underlying cognitive processes that seem to resonate with such proofs.