Conservation Principles and Actio Schemes in the Synthesis of Geometric Concepts

  • 2009-09-24
  • Research

The theory is focused on a constructive process that synthesizes a function in the geometric domain representing a geometric concept.


In this talk a theory for the synthesis of geometric concepts is presented. The theory is focused on a constructive process that synthesizes a function in the geometric domain representing a geometric concept. Geometric theorems are instances of this kind of concepts. The theory involves four main conceptual components: conservation principles, action schemes, descriptions of geometric abstractions and reinterpretations of diagrams emerging during the generative process. A notion of diagrammatic derivation in which the external representation and its interpretation are synthesized in tandem is also introduced in this paper. The theory is exeplified with a diagrammatic proof of the Theorem of Pythagoras. The theory also illustrates how the arithmetic interpretation of this theorem is produced in tandem with its diagrammatic derivation under an appropriate representational mapping. The paper is concluded with a feflection on the expressive power of diagrams, their effectiveness in representation and inference, and the relation between synthetic and analytic knowledge in the realization of theorems and their proofs.


Luis Peneda has a PhD from the University of Edinburgh where he worked under the supervision of Prof. Ewan Klein (1986 - 1989); since then he has collaborated with the HCRC at Edinburgh (1989 - 1992), the Institute for Electrical Research in Cuernavaca, Mexico (1992 - 1998), and the Institute for Applied Mathematics and Systems (IIMAS) at the National Autonomous University of Mexico, where he headed the Computer Science Department from 1998 to 2002 and again from 2005 to the present date. He has worked on the semantics of graphics, diagrammatic reasoning, multimodal dialogue systems, speech acts, Spanish grammar, and speech computational linguistics and artificial itelligence. He was coordinator of the technical committee of the congress 50 Years of Computing in Mexico. He is a National Investigator since 1993 (SNI) and regular member of the Mexican Academy of Science since 2008.


Mit Unterstützung des Wolfgang-Pauli-Instituts und des Zentrums für Informatikforschung.


  • Luis A. Pineda, Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, Universidad Nacional Autonoma de Mexico

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