A Diagrammatic Subsystem of Hilbert's Geometry
Isabel Luengo · Oxford University Press eBooks · 1996
In the last few years there has been an increasing interest in the visual representation of mathematical concepts. The fact that computers can help us perform graphical tasks very easily has been translated into an increasing interest in diagrammatic representations in general. Several experiments have shown that diagrammatic reasoning plays a main role in the way in which experts in several areas solve problems (Gobert and Freferiksen [1992] and Kindfield [1992]). Two kinds of explanations have been given for the advantages of visual representations over linguistic ones. The first kind of explanation is psychological. It has been argued that visual representations are easier to use because they resemble the mental models hurnans build to solve problems Stenning and Oberlander [1991], Johnson-Laird and Byrne [1991], arid Tverski [1991]. The second kind of explanation is related to computational efficiency. Larkin and Simon [1987] have argued that diagrammatic representations are computationally more efficient than sentential representations because the location of each element in the diagram corresponds to the spatial or topological properties of the objects they represent. However, the efficiency of the use of diagrams is not enough justification for their use in analytical areas of knowledge. Mathematical discoveries often have been made using visual reasoning, but those very same discoveries were not justified by the visual reasoning. Diagrams are associated with intuitions and illustrations, not with rigorous proofs. Visual representations are allowed in the context of discovery, not in the context of justification. Many authors have considered diagrams in opposition to deductive systems. Lindsay [1988], for instance, has claimed that the main feature of visual representations is that they correspond to a non-deductive kind of inference system. Koedinger and Anderson [1991] have related diagrammatic reasoning in geometry to informal, inductive strategies to solve problems. Thus, though we have an empirical justification for the use of diagrams in mathematics (people use them and they work!) we do not usually have an analytical justification. In fact, the history of mathematics, and especially the history of geometry, is full of mistakes related to the use of diagrams.