Abstract
This paper describes the use of diagrams as pictorial, non analogical representations in mathematics. Three interpretations of diagrams, of increasing complexity, are discussed: commutative diagrams, exact sequences, universal diagrams. Typically, reasoning with diagrams (in the case under scrutiny, geometrical structures of arrows) involves three steps: representation, construction, inspection and interpretation. We show how reasoning with diagrams makes a metaphoric use of the properties of the representations and suggest how extensions of the existing paradigms can enrich the emergent domain of hybrid reasoning.
We think that much insight on how to use picture-based reasoning can be gained by analyzing the way diagrammatic reasoning is used by humans in existing fields of research. This study complements the more familiar topic of common-sense reasoning with pictures and diagrams: here, expert knowledge, rather than common-sense knowledge, is involved.
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References
Barwise, J., Etchemendy, J.: Heterogeneous logic. In: Glasgow, J., Hari Narayanan, N., Chandrasekaran, B. (eds.) Diagrammatic Reasoning: Cognitive and Computational Perspectives, pp. 211–234. AAAI Press/ The MIT Press (1995)
Bredon, G.: Topology and Geometry. Springer, Heidelberg (1993)
Eilenberg, S., Steenrod, N.: Foundations of algebraic topology. Princeton University Press, Princeton (1952)
Faltings, B.: Qualitative Spatial Reasoning Using Algebraic Topology. In: Kuhn, W., Frank, A.U. (eds.) COSIT 1995. LNCS, vol. 988, pp. 17–30. Springer, Heidelberg (1995)
Ligozat, G.: Towards a general characterization of conceptual neighborhoods in temporal and spatial reasoning. In: AAAI-1994 Workshop on Spatial and Temporal Reasoning, Seattle, WA (1994)
MacLane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)
Massey, W.S.: Algebraic topology: an introduction. Springer, Heidelberg (1967)
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
Spanier, E.E.: Algebraic Topology. Mc Graw-Hill, New York (1966)
Tennant, N.: The Withering Away of Formal Semantics. Mind and Language 1 (1986)
Wang, D., Lee, J., Zeevat, H.: Reasoning with diagrammatic representations. In: Glasgow, J., Hari Narayanan, N., Chandrasekaran, B. (eds.) Diagrammatic Reasoning: Cognitive and Computational Perspectives, pp. 339–393. AAAI Press/ The MIT Press (1995)
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© 1999 Springer-Verlag Berlin Heidelberg
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Ligozat, G. (1999). Reasoning with Diagrams: The Semantics of Arrows. In: Imam, I., Kodratoff, Y., El-Dessouki, A., Ali, M. (eds) Multiple Approaches to Intelligent Systems. IEA/AIE 1999. Lecture Notes in Computer Science(), vol 1611. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48765-4_27
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DOI: https://doi.org/10.1007/978-3-540-48765-4_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66076-7
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