A computational account of conceptual blending in basic mathematics

  • Authors:

  • Markus Guhe;Alison Pease;Alan Smaill;Maricarmen Martinez;Martin Schmidt;Helmar Gust;Kai-Uwe Kühnberger;Ulf Krumnack

  • Affiliations:

  • University of Edinburgh, School of Informatics, Informatics Forum, 10 Crichton Street, Edinburgh EH8 9AB, Scotland, United Kingdom;University of Edinburgh, School of Informatics, Informatics Forum, 10 Crichton Street, Edinburgh EH8 9AB, Scotland, United Kingdom;University of Edinburgh, School of Informatics, Informatics Forum, 10 Crichton Street, Edinburgh EH8 9AB, Scotland, United Kingdom;University of Osnabrück, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabrück, Germany and Universidad de los Andes, Department of Mathematics, Cra 1a 18A-10, Bogotá, Co ...;University of Osnabrück, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabrück, Germany;University of Osnabrück, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabrück, Germany;University of Osnabrück, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabrück, Germany;University of Osnabrück, Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabrück, Germany

  • Venue:
  • Cognitive Systems Research
  • Year:
  • 2011

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Abstract

We present an account of a process by which different conceptualisations of number can be blended together to form new conceptualisations via recognition of common features, and judicious combination of their distinctive features. The accounts of number are based on Lakoff and Nunez's cognitively-based grounding metaphors for arithmetic. The approach incorporates elements of analogical inference into a generalised framework of conceptual blending, using some ideas from the work of Goguen. The ideas are worked out using Heuristic-Driven Theory Projection (HDTP, a method based on higher-order anti-unification). HDTP provides generalisations between domains, giving a crucial step in the process of finding commonalities between theories. In addition to generalisations, HDTP can also transfer concepts from one domain to another, allowing the construction of new conceptual blends. Alongside the methods by which conceptual blends may be constructed, we provide heuristics to guide this process.