1 ABSTRACTION, VISUALISATION AND GRAPHICAL PROOF Luis Pineda 1 , John Lee 2 and Gabriela Garza Abstract In this paper we investigate the process of learning and verifying graphical theorems through abstraction and visualisation. First, the notion of effectiveness of a representation is discussed from both a computational and a cognitive perspective; the nature of the relation between external representations and abstraction in these two views, and its implications for diagrammatic reasoning in AI, is also explored. Then we present a discussion of pragmatic aspects of reasoning with a system and reasoning from a system, and the relations of these views to recognising and learning graphical proofs. To understand more clearly the nature of diagrammatic proofs, a case study is presented from two different symbolic perspectives. In the first, the goal is that a system learns a proof from a sequence of graphical patterns by inductive learning; in the second, the emphasis is on the syntax, semantics and proof procedure of the inductive mathematical proof of the same problem. As both of these approaches lies within a logicist view of diagrammatic reasoning, the question is addressed of whether a diagrammatic argument can be visualised, and to what extent this visualisation constitutes a proof. To this end, a third approach to verifying and learning a diagrammatic proof of the case study through a “visualisation” with a “retina” is presented. The discussion results in a diagrammatic reasoning system with a declarative syntax and a compositional semantics but implemented with a distributed computing architecture. The paper is concluded with a discussion on the relation between abstraction, visualisation, interpretation change and learning, applied to understand a purely diagrammatic proof of the Theorem of Pythagoras. 1. INTRODUCTION Diagrammatic proofs are for many people usually easier to learn and understand than the corresponding proofs expressed in mathematical or logical notation. In diagrammatic proofs, proof procedures involve a limited number of operations which transform diagrams representing the premises of a theorem into a diagram representing its conclusion; proofs of geometric theorems, like the proof of the Theorem of Pythagoras in Figure 1.1, are probably the most typical examples of this kind. In an informal analogy between geometrical and logical proofs the different diagrams of the graphical proof would correspond to the premises of a logical argument and the final diagram, where the truth of the theorem can be appreciated, would correspond to the conclusion. Figure 1.1. Proof of the Theorem of Pythagoras There are also examples in which both theorem and proof are supposed to be read off from a concrete diagram, without external signs of the reasoning process involved, like the proof of the Pythagorean theorem shown in Figure 1.2. However, the absence of graphical transformations of the diagrams does not mean that the proof is grasped by a single, holistic, inference. In this latter case, the actual proof is a geometrical 1 Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas (IIMAS), UNAM, Mex., luis@leibniz.iimas.unam.mx 2 Human Communication Research Centre (HCRC), University of Edinburgh, UK, john@cogsci.ed.ac.uk