Abstract Geuder & Weisgerber

On the Geometrical Representation of Concepts.

It is fairly uncontroversial, in model-theoretic as well as in cognitive semantics, that a theory of word meaning requires a qualitative analysis of concepts (not merely sets of potential referents). In Gärdenfors (2000), a framework of "conceptual spaces", i.e. a geometrical representation of the internal structure of concepts, is proposed. We offer a detailed investigation of this proposal; we point out that this framework provides a number of important insights, but reject the claim that concepts should receive a geometrical representation in a unified metrical space. Instead, we argue, a modular representation is required for the internal structure of concepts.

The idea that underlies the framework of Conceptual Spaces (CS) is that concepts can be decomposed into quality dimensions. Each quality dimension consists of a range of feature values, ideally forming a scale. Objects are represented via their coordinates (feature values) in different conceptual dimensions, and sets of objects (categories) therefore appear as regions in that space. Moreover, the ordering of property values reflects similarity: distances between two points in the space are inversely related to the similarity of the corresponding objects. If conceptual categories are clusters of similar objects, then the representation of a category appears as a connected, possibly convex, region in conceptual space.

In the first part of the talk, we illustrate the workings of the model via the example of the colour space, which describes colour properties via the three dimensions hue, saturation and brightness. The geometrical representation proves to be an interesting tool because the interdependencies that exist between admissible values of hue, saturation and brightness can be succinctly encoded via the curvature of the space. At the same time, this means that the representation will always be quite specifically tailored to the demands of one particular conceptual domain. Moreover, we argue that there is a latent ambiguity in the notion of a dimension, because either the similarity metric or the qualitative scale could be taken to put up the space.

This ambiguity has consequences for the representation of concepts. In the second part, we apply the model to cases of concept formation that are not hard-wired but constitute an open-ended learning process. This requires a method of building up complex conceptual representations by assembling complex units from correlations among simpler properties. It is an advantage of a geometrical model that a distinction between simple properties and complex concepts can be formulated via the dimensionality of the space and the interrelations among values; the distinction is not captured in standard predicate logical calculus. However, if concepts are formed via free couplings of qualitative feature values, there is a clash with the fact that the similarity metric may dictate different curvatures of the spaces that represent the individual domains. There is no straightforward graphic / algebraic solution for the representation of a complex concept in such a case (as we are going to demonstrate with the example of colour and taste spaces, which would have to cooccur e.g. in the description of fruit). We conclude that a literally geometrical approach to concept representation is not possible. Instead of representations with a unified metrical space, we propose that concept representation should be executed as a network of correlated domains in terms of modules. On the modular view, only the similarity measure but not the distance metrics applies across conceptual domains, in contrast to the geometrical model, in which similarity is linearly correlated to distance in inner-domain as well as cross-domain applications. We argue, finally, that the modular view has further advantages over the geometrical model, e.g. in that the former can handle the problem of incommensurability in similarity judgements.

Reference: Gärdenfors, P. (2000): Conceptual Spaces. MIT Press.