Operational Constraints in Diagrammatic Reasoning

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Diagrammatic reasoning is reasoning whose task is partially taken over by operations on diagrams. It consists of two kinds of activities: (i) physical operations, such as drawing and erasing lines, curves, figures, patterns, symbols, through which diagrams come to encode new information (or discard old information), and (ii) extractions of information diagrams, such as interpreting Venn diagrams, statistical graphs, and geographical maps. Given particular tasks of reasoning, different types of diagrams show different degrees of suitedness. For example, Euler diagrams are superior in handling certain problems concerning inclusion and membership among classes and individuals, but they cannot be generally applied to such problems without special provisos. Diagrams make many proofs in geometry shorter and more intuitive, while they take certain precautions of the reasoner’s to be used validly. Tables with particular configurations are better suited than other tables to reason about the train schedule of a station. Different types of geographical maps support different tasks of reasoning about a single mountain area. Mathematicians experience that coming up with the “right” sorts of diagrams is more than half-way to the solution of most complicated problems. Perhaps many of these phenomena are explained with reference to aspect (ii) of diagrammatic reasoning because some types of diagrams lets a reasoner retrieve a kind of information that others do not. or lets the reasoner retrieve it more “easily” than others. In fact, this is the approach that psychologists have traditionally taken. In this chapter, we take a different path and focus on aspect (i) of diagrammatic reasoning. Namely, we look closely at the process in which a reasoner applies operations to diagrams and in which diagrams come to encode new information through these operations. It seems that this process is different in some crucial points from one type of diagrams to another, and that these differences partially explain why some types of diagrams are better suited than others to particular tasks of reasoning.