Going Beyond the Data as the Patching (Sheaving) of Local Knowledge - Phillips, Steven (Frontiers in Psychology, 2018)
Consistently predicting outcomes in novel situations is colloquially called ``going beyond the data,'' or ``generalization.'' Going beyond the data features in spatial and non-spatial cognition, raising the question of whether such features have a common basis---a kind of systematicity of generalization. Here, we conceptualize this ability as the patching of local knowledge to obtain non-local (global) information. Tracking the passage from local to global properties is the purview of sheaf theory, a branch of mathematics at the nexus of algebra and geometry/topology. Two cognitive domains are examined: (1) learning cue-target patterns that conform to an underlying algebraic rule, and (2) visual attention requiring the integration of space-based feature maps. In both cases, going beyond the data is obtained from a (universal) sheaf theory construction called ``sheaving'', i.e. the ``patching'' of local data attached to a topological space to obtain a representation considered as a globally coherent cognitive map. These results are discussed in the context of a previous (category theory) explanation for systematicity, vis-a-vis, categorical universal constructions, along with other cognitive domains where going beyond the data is apparent. Analogous to higher-order function (i.e. a function that takes/returns a function), going beyond the data as a higher-order systematicity property is explained by sheaving, a higher-order (categorical) universal construction.