E Pluribus Unum - Computational Creativity Research: Towards Creative Machines - page 178

Blend of two Blends

C

Blend 1 Blend 2

B1 B2

Upper Base

Input 1

colimit morphisms

Input 1

colimit morphisms

Input 2 Input 2

O1 O2 O3 O4

base morphisms base morphisms

Base Ontology 1 Base Ontology 2

Fig. 9.6 Blending two basic blends into a third

in the Goguen tradition, here based on the DOL language. However, the basic blend-
ing diagram only covers the most basic situation, that of an 'atomic blend' using
basic concepts and one base space. The real power of blending, however, is only
unleashed when blends are iterated and when partiality is allowed.
Lakoff and Núñez [ 46 ] give a detailed and powerful analysis of this in the field
of conceptual mathematics. A basic claim they make is that the most sophisticated
mathematical concepts have been created, over time, through a tower of blended
concepts, generating more and more abstract notions. A basic case is that of arith-
metic, where several metaphors, image schemas, and analogies are successively
blended into modern number systems such as rationals, reals, or complex numbers,
including 'arithmetic as object collection', 'object construction', the 'measuring stick
metaphor' and 'arithmetic as motion along a path' (see [ 29 , 46 ] for further details
and [ 15 ] for a conceptual blend of the complex numbers along these lines). A detailed
formal re-construction of such iterated blends is a challenging task, both conceptu-
ally and on a technical level. Figure 9.6 shows the basic diagrammatical structure of
such iterated blends.
Iteration of blends, however, is not the only variation of the basic blendoid struc-
ture. Figure 9.7 shows two triple blends; both have three input spaces, but the one
on the left has one base, the one on the right has two base spaces. For instance, we
might have 3 inputs that are simultaneously aligned with a basic image schema in the
base (left), or we have three ontologies that pairwise interpret different metaphors,
e.g. 'arithmetic as object collection' and 'arithmetic as motion along a path'.
Note that on a purely technical level, such complex diagrams can always be
reduced to a succession of squares, possibly by duplicating some nodes or adding