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A Theory of Concepts and Their Combinations II: A Hilbert Space Representation
 

Summary: A Theory of Concepts and Their Combinations II:
A Hilbert Space Representation
Diederik Aerts
Center Leo Apostel for Interdisciplinary Studies
Department of Mathematics and Department of Psychology
Vrije Universiteit Brussel, 1160 Brussels, Belgium
E-Mail: diraerts@vub.ac.be
Liane Gabora
Center Leo Apostel for Interdisciplinary Studies
Vrije Universiteit Brussel and Department of Psychology
University of California at Berkeley, Berkeley, CA 94720-1650, USA
E-Mail: liane@berkeley.edu
Abstract
The sets of contexts and properties of a concept are embedded in the complex Hilbert space of quantum
mechanics. States are unit vectors or density operators, and contexts and properties are orthogonal pro-
jections. The way calculations are done in Hilbert space makes it possible to model how context influences
the state of a concept. Moreover, a solution to the combination of concepts is proposed. Using the tensor
product, a procedure for describing combined concepts is elaborated, providing a natural solution to the
`pet fish' problem. This procedure allows the modeling of an arbitrary number of combined concepts. By
way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented.

  

Source: Aerts, Diederik - Leo Apostel Centre, Vrije Universiteit Brussel

 

Collections: Multidisciplinary Databases and Resources; Physics