Mathematics for Cognitive Science
Creators: name: Posina, Venkata Rayudu
Publisher: name: Zenodo
Year: 2020
Subjects: Category theory
Cognitive science
Functorial semantics
Representation

description: The unsatisfactory contemporary state-of-affairs of cognitive science was highlighted in the recent study: Núñez, R. et al. What happened to cognitive science? Nat. Hum. Behav. 3, 782-791 (2019). Based on the kinship between mathematics [in particular and science in general] and cognition (cf. cognition is science writ small; Daedalus 135: 86, 2006), I make a case for mathematical abstraction as a pathway for the advancement of cognitive science. It may be noted here that the parallels between mathematical knowing and knowing in general, which I bring into focus, provide a means "to connect cognitive science theories to computational foundations" (Nat. Hum. Behav. 3: 782, 2019). I outline the mathematics of calculating representation(s)—a fundamental notion in cognitive science. I felt that it is important to explicitly state that the notion of 'representation' figuring in the foundational tenet—"cognition is computation of representations" (Nat. Hum. Behav. 3, 782, 2019)—of cognitive science is indispensable in theorizing about cognition. However, we may need to replace computation with calculation [of representations] so as to move past the computer metaphor and bring the insights of functorial semantics to bear on cognitive science. Representation (or model), according to functorial semantics, is an interpretation of a theory into a background category (Reprints in Theory and Applications of Categories 5, 8-11, 2004). The category of all models of a theory T in a background B is a functor category B^T of all functorial interpretations T --&gt; B of the theory T in the background B. Changing theories (T1 --&gt; T2) induces contravariant changes in representations (B^T2 --&gt; B^T1), while changing backgrounds (B1 --&gt; B2) induces covariant changes in representations (B1^T --&gt; B2^T). I discuss these mathematical insights into representation in a manner readily accessible to multidisciplinary audience. This discussion can help insure against: throwing the baby (representation) with bathwater (hexagon; Fig 1a in Nat. Hum. Behav. 3, 782, 2019). In closing, my manuscript brings out the reach of functorial semantics [of calculating representations] into figural salience--for all to see--so as to facilitate ready recognition of the relevance of functorial semantics for the development of cognitive science.
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