Mathematical Work of Francisco Varela

Kauffman, Louis H.

Volume 13 · Number 1 · Pages 11–17

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Mathematical Work of Francisco Varela

Louis H. Kauffman

Abstract

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.

Key words: Autonomy, autopoiesis, eigenform, reflexivity, reflexive domain, observer, self-reference, category, functor, adjoint functor, distinction.

Citation

Kauffman L. H. (2017) Mathematical work of Francisco Varela. Constructivist Foundations 13(1): 11–17. http://constructivist.info/13/1/011

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References

Barendregt H. P. (1981) The lambda calculus its syntax and semantics. North Holland, Amsterdam. ▸︎ Google︎ Scholar

Foerster H. von (1981) Notes on an epistemology for living things. In: Foerster H. von, Observing systems. Intersystems, Salinas CA: 258–271. Originally published in 1972. http://cepa.info/2258

Foerster H. von (1981) Objects: Tokens for (eigen-)behaviors. In: Foerster H. von, Observing systems. Intersystems, Salinas CA: 274–285. Originally published in 1976. http://cepa.info/1270

Goguen J. A. & Varela F. J. (1979) Systems and distinctions: Duality and complementarity. International Journal of General Systems 5(1): 31–43. http://cepa.info/2060

Kauffman L. H. & Varela F. J. (1980) Form dynamics. Journal for Social and Biological Structures 3: 171–206. http://cepa.info/1841

Kauffman L. H. (1978) DeMorgan algebras: Completeness and recursion. In: Proceedings of the eighth international conference on multiple valued logic. IEEE Computer Society Press, Los Alamitos CA: 82–86. ▸︎ Google︎ Scholar

Kauffman L. H. (1978) Network synthesis and Varela’s calculus. International Journal of General Systems 4: 179–187. http://cepa.info/1822

Kauffman L. H. (2005) Reformulating the map color theorem. Discrete Mathematics 302(1–3): 145–172. ▸︎ Google︎ Scholar

Linde C. & Goguen J. (1978) Structure of planning discourse. Journal of Social and Biological Structures 1: 219–251. ▸︎ Google︎ Scholar

Scott D. S. (1980) Relating theories of the lambda calculus. In: Seldin P. & Hindley R. (eds.) To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism. Academic Press, New York: 403–450. ▸︎ Google︎ Scholar

Spencer Brown G. (1969) Laws of form. George Allen and Unwin, London. ▸︎ Google︎ Scholar

Stcherbatsky T. (1968) Buddhist logic. Mouton de Gruyter, Berlin. ▸︎ Google︎ Scholar

Varela F. J. & Goguen J. A. (1978) The arithmetic of closure. Cybernetics and System 8(3–4): 291–324. ▸︎ Google︎ Scholar

Varela F. J. (1975) A calculus for self-reference. International Journal of General Systems 2: 5–24. Available at http://cepa.info/1840

Varela F. J. (1979) Principles of biological autonomy. North Holland, New York. ▸︎ Google︎ Scholar

Varela F. J., Maturana H. R. & Uribe R. (1974) Autopoiesis: The organization of living systems, its characterization and a model. Biosystems 5(4): 187–196. http://cepa.info/546

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