On the Isomorphism, or Lack of It, of Representations

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.