Abstract
Representations are invariably described as being somehow similar in structure to that which they represent. On occasion representations have been written of as being “morphisms,” “homomorphisms,” or “isomorphisms.” These terms suggest that the closeness of the similarity may vary from representation to representation, although the nature and implications of this variance have neither been studied in great detail nor with any degree of precision. The terms isomorphism and homomorphism have definite meanings in algebra, but their usage in describing representations has seldom been given a formal definition. We provide here an algebraic definition of representation which permits the formal definition of homomorphism, isomorphism and a range of further significant properties of the relation between representation and represented. We propose that most tractable representations (whether diagrammatic, textual, or otherwise) are homomorphisms, but that few are inherently isomorphic. Concentrating on diagrams, we illustrate how constraints imposed upon the construction and interpretation of representations can achieve isomorphism, and how the constraints necessary will vary depending upon the modality of the representational system.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Barwise and J. Etchemendy. Heterogeneous logic. In J. Glasgow, N.H. Narayan, and B. Chandrasekaran, editorsDiagrammatic Reasoning: Cognitive and Computational Perspectivespages 211–234. MIT Press, Cambridge, 1995.
D.G. Bobrow. Dimensions of representation. In D.G. Bobrow and A. Collins, editorsLanguage Thought and Culture: Advances in the Study of Cognitionpages 1–34. Academic Press, New York, 1975.
N. Goodman. Languages of Art. Hackett, Indianapolis, second edition, 1984.
P.J. Hayes. Some problems and non-problems in representation theory. In R.J. Brachman and H.J. Levesque, editorsReadings in Knowledge Representation.Morgan Kaufmann, Los Altos, 1985.
S.M. Kosslyn. Image and Mind, chapter 2. Harvard University Press, Cambridge, 1980.
J.H. Larkin and H.A. Simon. Why a diagram is (sometimes) worth ten thousand words.Cognitive Science11:65–99, 1987.
K. Myers and K. Konolige. Reasoning with analogical representations. In J. Glasgow, N.H. Narayan, and B. Chandrasekaran, editorsDiagrammatic Reasoning: Cognitive and Computational Perspectivespages 273–302. MIT Press, Cambridge, 1995.
S.E. Palmer. Fundamental aspects of cognitive representation. In E. Rosch and B.B. Lloyd, editorsCognition and Categorisationpages 259–303. Lawrence Erlbaum Associates, Hillsdale, 1978.
M. Petre and T.R.G. Green. Requirements of graphical notations for professional users: electronics CAD systems as a case study.Le Travail Humain55:47–70, 1992.
Z.W. Pylyshyn. What the mind’s eye tells the mind’s brain: a critique of mental imagery.Psychological Bulletin80:1–24, 1973.
J. Seligman. An algebraic appreciation of diagrams. Technical Report LP-94–03, Institute for Logic Language and Computation, University of Amsterdam, March 1994.
R.N. Shepard. Form, formation, and transformation of internal representations. In R.L. Solso, editorInformation processing and cognition: The Loyola Symposium.Lawrence Erlbaum Associates, Hillsdale, 1975.
R.N. Shepard and S. Chipman. Second-order isomorphism of internal representations: Shapes of states.Cognitive Psychology1:1–17, 1970.
A. Sloman. Interactions between philosophy and AI: The role of intuition and non-logical reasoning in intelligence. InProceedings Second IJCAI London.Morgan Kaufman, San Francisco, 1971.
A. Sloman. Afterthoughts on analogical representations. In R.J. Brachman and H.J. Levesque, editorsReadings in Knowledge Representation.Morgan Kaufmann, Los Altos, 1985.
K. Stenning and R. Inder. Applying semantic concepts to analysing media and modalities. In J. Glasgow, N.H. Narayan, and B. Chandrasekaran, editorsDiagrammatic Reasoning: Cognitive and Computational Perspectivespages 303–338. MIT Press, Cambridge, 1995.
D. Wang and J. Lee. Visual reasoning: Its formal semantics and applications.Journal of Visual Languages and Computing4:327–356, 1993.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gurr, C.A. (1998). On the Isomorphism, or Lack of It, of Representations. In: Marriott, K., Meyer, B. (eds) Visual Language Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1676-6_10
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1676-6_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7240-3
Online ISBN: 978-1-4612-1676-6
eBook Packages: Springer Book Archive