Mathematical Work of Francisco Varela

Purpose: This target article explicates mathematical themes in the work of Varela that remain of current interest in present-day second-order cybernetics. Problem: Varela’s approach extended biological autonomy to mathematical models of autonomy using reflexivity, category theory and eigenform. I will show specific ways that this mathematical modeling can contribute further to both biology and cybernetics. Method: The method of this article is to use elementary mathematics based in distinctions (and some excursions into category theory and other constructions that are also based in distinctions) to consistently make all constructions and thereby show how the observer is involved in the models that are so produced. Results: By following the line of mathematics constructed through the imagination of distinctions, we find direct access and construction for the autonomy postulated by Varela in his book Principles of Biological Autonomy. We do not need to impose autonomy at the base of the structure, but rather can construct it in the context of a reflexive domain. This sheds new light on the original approach to autonomy by Varela, who also constructed autonomous states but took them as axiomatic in his calculus for self-reference. Implications: The subject of the relationship of mathematical models, eigenforms and reflexivity should be reexamined in relation to biology, biology of cognition and cybernetics. The approach of Maturana to use only linguistic and philosophical approaches should now be reexamined and combined with Varela’s more mathematical approach and its present-day extensions.


Mathematical Work of Francisco Varela
Louis H. Kauffman