Main Problems of Diagrammatic Reasoning. Part I: The generalization problem

The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations of mathematicians expended on refining the methods of reasoning with formulae and predicate calculus. Next, two main groups of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained, methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several other problems constitute new, specifically diagrammatic ways of reliable reasoning.