Crítica, Revista Hispanoamericana de Filosofía, Volume 22, number 64, abril 1990
Estructuras y representaciones
Adolfo García de la Sienra
Instituto de Investigaciones Filosóficas
Universidad Nacional Autónoma de México

Abstract: The aim of the present paper is to set a philosophical basis in order to discuss the type of representation that holds between mathematical structures and those aspects of the real world which they represent. It is maintained that an actualized version of Aristotelian metaphysics is suited for this purpose. The connection between the abstract, rigid concepts of mathematics, and the concepts of metaphysics is attempted through the concept of a fundamental measurement. The existence and degree of uniqueness of a fundamental measurement is established as a representation theorem asserting the existence of a homomorphism from what I call an ontological structure into a numerical one. An ontological structure contains as elements real beings, and its relations represent —in a sense made precise thereof— real relations among these beings. The role of metaphysics in the establishment of a representation theorem is to provide the conceptual apparatus required to discuss and formulate the ontological axioms required to derive the theorem. The paper contains a very complete example of a fundamental measurement in the sense described, namely, the measurement of the height of a physical parallelepiped and that of its potential parts.

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