CSCTR – Session 6 Dana Retová Conceptual spaces CSCTR – Session 6 Dana Retová

Conceptual spaces (Gärdenfors) Consist of a number of quality dimensions Building blocks of representations Weight, temperature, brightness, pitch, height, width, depth Abstract non-sensory dimensions Represent various qualities of objects Independent of symbolic representations (language) Abstract representation for modeling Do not claim to have any immediate physical realization

Dimensions Innate – hardwired in nervous system Learned Learning involves expanding conc. Space with new quality dimensions Culturally dependent Time Scientific Weight vs. mass “Evidence suggests that dimensions that are easily separated by adults, such as the brightness and size of a square, are treated as fused together for children .... For example, children have difficulty identifying whether two objects differ on their brightness or size even though they can easily see that they differ in some way. Both differentiation and dimensionalization occur throughout one's lifetime.” (Goldstone & Bartalou 1998)

Dimension of ‘time’ In our culture and in science In other cultures One-dimensional structure isomorphic to the line of real numbers In other cultures Circular structure

Dimension of ‘pitch’ 1-D structure from low tones to high Logarithmic scale Acoustic frequency is spatially coded in chochlea

Color space Hue Brightness Color

Contrast classes Skin color Possible colors are the subset of the full color space Can be irregular Subset “stretched” to form a space with the same topology Color terms can be used even if they do not correspond to the original hues “Metaphor” In case of contrast classes, one set of dimensions is not really mapped onto another set but the conceptual space is mapped onto a subspace of itself retaining the same topological structure.

Conceptual spaces Similarity - defined via distance between representing points Object – point in a conceptual space Property /Concept – region of a conceptual space

Metaphors in conceptual space A metaphor expresses a similarity in topological or metrical structure between different quality dimensions A word that represent a particular structure in one quality dimension can be used as a metaphor to express a similar structure about another dimension Metaphors transfer knowledge about one conceptual dimension to another E.g. space mapped to time In case of contrast classes, one set of dimensions is not really mapped onto another set but the conceptual space is mapped onto a subspace of itself retaining the same topological structure.

Primary and secondary properties Predicates are assigned regions of space (red) Secondary properties (tall) “Parasitical” on other properties “Big chihuahua”

Marr (1982) Cylinders Prototypical vector for an object – image schema Length Width Angle between the dominating and the other one Position of the added cylinder Prototypical vector for an object – image schema Subordinate cat. – subregions of the convex region on each level of the hierarchy an object is described by a comparatively small number of coordinates based on lengths and angles. Thus the object can be identified as a hierarchially structured vector in a (higher order) conceptual space. Furthermore, a ’prototypical’ vector for an object category like ’bird’ identifies a spatial structure that can serve as an image schema for that category. Such an image schema represents a basic level category (cf. Section 3), while subordinate categories like ’ostrich’ are represented by subregions of the convex region associated with the prototypical object. Superordinate categories like ’animal’ do not have any associated image schemas.

Action space Spatio-temporal patterns of forces that generate the movement

Functional concepts Function of an object can be analysed Actions it affords Functional concept = convex region in action space

Conceptual spaces Ideal to represent Concepts on basic level of conceptualization Spatial-relations concepts Rules follow from the topological structure For example: A point in a conceptual space will always have an internally consistent set of properties Something cannot be blue and yellow at the same time Everything that is green is also colored Nothing is in the same place in the same time Transitivity – as in “earlier than”

Questions?