Abstract
Why does the capacity to think certain thoughts imply the capacity to think certain other, structurally related, thoughts? Despite decades of intensive debate, cognitive scientists have yet to reach a consensus on an explanation for this property of cognitive architecture—the basic processes and modes of composition that together afford cognitive capacity—called systematicity. Systematicity is generally considered to involve a capacity to represent/process common structural relations among the equivalently cognizable entities. However, the predominant theoretical approaches to the systematicity problem, i.e., classical (symbolic) and connectionist (subsymbolic), require arbitrary (ad hoc) assumptions to derive systematicity. That is, their core principles and assumptions do not provide the necessary and sufficient conditions from which systematicity follows, as required of a causal theory. Hence, these approaches fail to fully explain why systematicity is a (near) universal property of human cognition, albeit in restricted contexts. We review an alternative, category theory approach to the systematicity problem. As a mathematical theory of structure, category theory provides necessary and sufficient conditions for systematicity in the form of universal construction: each systematically related cognitive capacity is composed of a common component and a unique component. Moreover, every universal construction can be viewed as the optimal construction in the given context (category). From this view, universal constructions are derived from learning, as an optimization. The ultimate challenge, then, is to explain the determination of context. If context is a category, then a natural extension toward addressing this question is higher-order category theory, where categories themselves are the objects of construction.
Highlights
THE SYSTEMATICITY CHALLENGESystematicity is a property of cognitive architecture—the organization of basic processes affording cognition—where the capacity for certain cognitive abilities implies having the capacity for certain related cognitive abilities (Fodor and Pylyshyn, 1988)
Category theory approach to the systematicity problem
An often used example of systematicity is where having the capacity to understand the statement John loves Mary implies having the capacity to understand the statement Mary loves John. This property need not be restricted to language, Systematicity and Category Theory nor humans
Summary
Systematicity is a property of cognitive architecture—the organization of basic processes affording cognition—where the capacity for certain cognitive abilities implies having the capacity for certain related cognitive abilities (Fodor and Pylyshyn, 1988). The classical explanation appears to suffer the same kind of problem that was raised against connectionism: i.e., the ad hoc way in which grammatical (cf neural network) structures can be configured with and without support for systematicity (Aizawa, 2003). The challenge facing the classical approach to theories of cognition echoes the one facing the connectionist and other approaches: develop a theory of cognitive architecture whose core principles and assumptions provide the necessary and sufficient conditions from which systematicity follows, as required of a causal theory (Fodor and Pylyshyn, 1988; Fodor and McLaughlin, 1990; Aizawa, 2003). A challenge for the learning approach is to explain why a network is configured in just the right way to afford the desired generalization property, which echoes the original systematicity problem (Phillips and Wilson, 2010). We look at the implications of this (unique) category theory perspective on systematicity, and a new challenge that follows for cognitive science (Section 5)
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