A Category Theory Explanation for Systematicity

Abstract

Classical and Connectionist theories of cognitive architec- ture explain systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific architecturesgrammars, or Connectionist networksthat do not account for systematicity. By analogy with the Ptole- maic (i.e., geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to explain it (Aizawa, 2003b). Category theory provides an alternative explanation based on the for- mal concept of adjunction, which consists of a pair of struc- ture preserving maps, called functors. A functor generalizes the notion of a map between representational states to in- clude a map between state transformations (processes). In a formal sense, systematicity is a necessary consequence of a higher-order theory of cognitive architecture, in contrast to the first-order theories derived from Classicism or Con- nectionism. Category theory offers a re-conceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universereplaced by the relationships between the maps that transform them.