David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
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Foundations of Science 14 (1-2):75-96 (2009)
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.
|Keywords||Diagrammatics Diagrammatic reasoning Rigorous reasoning Reasoning errors Generalization Diagram particularity Divergence|
|Categories||categorize this paper)|
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References found in this work BETA
M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.
Gottlob Frege (1879). Begriffsschrift: Eine Der Arithmetische Nachgebildete Formelsprache des Reinen Denkens. L. Nebert.
Sun-Joo Shin & Giovanna Corsi (1997). The Logical Status of Diagrams. British Journal for the Philosophy of Science 48 (2):290-291.
Rudolf Arnheim (1970). Visual Thinking. London,Faber.
Ian Dove (2002). Can Pictures Prove? Logique Et Analyse 45.
Citations of this work BETA
Nicholaos Jones (2014). Bowtie Structures, Pathway Diagrams, and Topological Explanation. Erkenntnis 79 (5):1135-1155.
Andrew Aberdein (2013). Mathematical Wit and Mathematical Cognition. Topics in Cognitive Science 5 (2):231-250.
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