  We formulate five basic tenets of enactivist cognitive science that we have
carefully identified in the relevant literature as the main underlying
principles of that philosophy. We then develop a mathematical framework to talk
about cognitive systems (both artificial and natural) which complies with these
enactivist tenets. In particular we pay attention that our mathematical
modeling does not attribute contentful symbolic representations to the agents,
and that the agent's brain, body and environment are modeled in a way that
makes them an inseparable part of a greater totality. The purpose is to create
a mathematical foundation for cognition which is in line with enactivism. We
see two main benefits of doing so: (1) It enables enactivist ideas to be more
accessible for computer scientists, AI researchers, roboticists, cognitive
scientists, and psychologists, and (2) it gives the philosophers a mathematical
tool which can be used to clarify their notions and help with their debates.
Our main notion is that of a sensorimotor system which is a special case of a
well studied notion of a transition system. We also consider related notions
such as labeled transition systems and deterministic automata. We analyze a
notion called sufficiency and show that it is a very good candidate for a
foundational notion in the "mathematics of cognition from an enactivist
perspective". We demonstrate its importance by proving a uniqueness theorem
about the minimal sufficient refinements (which correspond in some sense to an
optimal attunement of an organism to its environment) and by showing that
sufficiency corresponds to known notions such as sufficient history information
spaces. We then develop other related notions such as degree of insufficiency,
universal covers, hierarchies, strategic sufficiency. In the end, we tie it all
back to the enactivist tenets.
