Abstract
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.
Highlights
A ubiquitous cognitive ability is the capacity to “go beyond the data.” That is, to put it broadly, an ability to successfully respond to stimuli not previously encountered
We present the basic sheaf theory and the sheaving construction considered as a “universal” basis for generalization
The first domain is a special case of the second domain. These results are discussed in the context of a previous explanation for systematicity, vis-a-vis, categorical universal constructions, along with other cognitive domains where going beyond the data is apparent
Summary
A ubiquitous cognitive ability is the capacity to “go beyond the data.” That is, to put it broadly, an ability to successfully respond to stimuli not previously encountered. What matters to the classical theory is not the particular syntactic relationship, but that the relationship employed is used consistently in all such situations In this way, a classical cognitive system with the capacity to juxtapose all such relevant combinations of symbols is supposed to explain the productivity and systematicity properties of language (Chomsky, 1980) and thought (Fodor and Pylyshyn, 1988), more generally. Such assumptions are characteristically ad hoc in being unconnected to the core principles of the theory, motivated solely to fit the data, and cannot be confirmed independently of confirming the theory—classical and connectionist theories fail to fully explain such properties (Aizawa, 2003) To address this challenge, a category theory (Eilenberg and Mac Lane, 1945; Mac Lane, 1998) approach was proposed whereby systematicity properties derive from universal (categorical) constructions (Phillips and Wilson, 2010). A preview of the sheaf theory approach is provided in the remainder of this introduction before delving deeper into the conceptual details and cognitive applications (subsequent main text), and supporting formal theory (Appendix in Supplementary Material)
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