The difference between cognitive linguistics and hierarchical complexity

Sinha's chapter in The Handbook of Cognitive Linguistics (2007) explains this key difference. Granted he explains the difference between cognitive linguistics (CL) and classic cognitive science (CCS), but the premises of the latter are also taken by the model of hierarchical complexity (MHC) and hence to me applicable to the latter. This explains a lot toward understanding why CL does not accept the MHC.

CL also finds a basis for progressive "higher mental functioning" but from a different source.  For example:

"'Rule' vs. 'schema.' The single most important theoretical concept in traditional and
formal linguistics is the rule, adopted by CCS in the specific form of the algorithm.
CL is a usage-based, not a rule-based theory. The CL unit of analysis that most
readily corresponds to 'rule' is 'schema,' which is employed in a variety of different contexts (e.g. image schema, event schema, construction schema), and recurs throughout this Handbook" (5).

CL also has roots in Piaget:  

"Piaget’s account of sensori-motor development in infancy is one in which successive re-organizations and co-ordinations of action schemata, arising from bodily movement and interactions with the physical world, lead to increasingly abstract cognitive representations (or internalized operational structures). [...] For Piaget, all schemata originate in basic bodily actions." (12-13).  

A key difference with CL though:

"Piaget believed that perception was subordinate to action, and he downplayed the role of imagery: an assumption which is, of course, not shared by CL. [...]  He regarded what he called 'figurative thought' as non-progressive, and in some sense primitive. This was because he sought to formalize his stage theory of cognitive development in terms of the mathematical theory of groups" (13).

Hence the divergence from Piaget's coordination of action schema into progressive figurative thought instead of strictly abstract, algorithmic, mathematical thought. Granted CL can account for the latter, but from a different basis. Conclusion:

"The philosophical basis of CCS was Objectivist (Lakoff, 1987), based upon the idea of a correspondence mapping between external world and internal mental representation. CCS, even though it claimed to be realist, was in fact hopelessly enmeshed in the insoluble antinomies of Cartesian dualism. G2CS [2nd generation cognitive science] is realist, but not Objectivist. It seeks its grounding of the mind not in 'mental representation,' but in the activity, movement and engagement of the organism with its environment: a point of view which clearly resonates with the pragmatist tradition" (25). 

 Also see this Facecrook thread on the topic.