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Cognitive Morphodynamics

Dynamical Morphological Models of Constituency in Perception and Syntax

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Jean Petitot

This book – written in collaboration with René Doursat, director of the Complex Systems Institute, Paris – adds a new dimension to Cognitive Grammars. It provides a rigorous, operational mathematical foundation, which draws from topology, geometry and dynamical systems to model iconic «image-schemas» and «conceptual archetypes». It defends the thesis that René Thom’s morphodynamics is especially well suited to the task and allows to transform the morphological structures of perception into Gestalt-like, abstract, proto-linguistic schemas that can act as inputs into higher-level specific linguistic routines.
Cognitive Grammars have drawn upon the view that the deep syntactic and semantic structures of language, such as prepositions and case roles, are grounded in perception and action. This study raises difficult problems, which thus far have not been addressed as a mathematical challenge. Cognitive Morphodynamics shows how this gap can be filled.

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Conclusion In his paper A Suggestion for a Linguistics with Connectionist Foundation [199], George Lakoff generalized Regier’s idea (see Section 5.3 of Chapter 3) to the conjecture that Ullman-style visual routines (...) are sufficient to characterize all known structures in cognitive topology. We think that we have demonstrated Lakoff-Regier’s conjecture for actantial relations and interactions. Starting from the Smolensky/Fodor-Pylyshyn debate concerning connec- tionist modeling of constituency, we first stressed that the main problem was to achieve a connectionist configurational definition of semantic (actantial) roles, that is, according to Jerry Fodor and Zenon Pylyshyn, of that “geometrical whole, where the geometrical relations are themselves semantically significant”, and which constitutes the geometrical basis of constituent-structures. We have then emphasized that, in order to solve this non-trivial problem, we first need a “good” linguistic theory. We selected cognitive grammars in Langacker’s, Talmy’s, Jackendoff’s and Lakoff’s sense and we stressed the cen- tral role of the localist hypothesis. Using this general perceptual, iconic and schematic grounding of basic elementary syntactic structures, we reduced the main problem to “perceptual” constituency. We then introduced contour diffusion/propagation routines that generalize, to higher-order representational levels, well-known routines of computational vision. We treated two cases of spreading activation triggered by boundaries: contour diffusion (heat equation) and contour propagation (wave equation). In these two cases we showed, according to deep theorems such as Morse’s theo- rem, that the singularities of the diffusion/propagation processes are singular structures that can be locally and...

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