The present issue is a memorial issue for Francisco Varela both as a scholar and as a colleague. Varela passed away in his home in Paris on May 28 2001. He was part of the editorial board of this journal and thus in this memorial issue we would like to look into his heritage. Most of the papers we present have authors that have known and worked with Varela in some period of their and his life: Ranulph Glanville, Louis Kauffman, Andreas Weber. Weber makes the case that Varela’s thinking can provide a foundation for biosemiotics and as such it provides a further foundation for the cybersemiotic project. Most interesting and promising is his comparison with Varela’s concept of the organism and Bruno Latour’s concept of quasi-objects. The other articles all have some relationship to Varela’s elaboration on the work of Spencer-Brown. Using the metaphor of the Uroboros, Marks-Tarlow, Robertson, and Combs explore the notion of re-entry in Varela’s ‘A Calculus for Self-Reference ’ and his contribution to a theory of consciousness. In their articles, Glanville and Kauffman reflect upon their experience working with Varela on joint papers.

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

Network models are given for self-referential expressions in the calculus of indications (of G. Spencer Brown). A precise model is presented for the behavior of such expressions in time. The extension of Brown’s calculus by F. Varela is then shown to describe behavior invariant properties of these networks. Network design is discussed from this viewpoint.

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.

Reichel A. (2011) Snakes all the Way Down: Varela’s Calculus for Self-Reference and the Praxis of Paradise. Systems Research and Behavioral Science 28(6): 646–662. https://cepa.info/2321

This contribution seeks to commemorate Francisco Varela’s formal conceptions of self-reference, providing an overview of his writings while shedding some light in the praxis of self-reference, from where a future research agenda can be derived. The architecture of Varela’s thinking, determined by the interrelated notions of autopoiesis, autonomy, closure and self-reference, was examined. The emphasis was on the development and expansions of his calculus for self-reference from George Spencer Brown’s Laws of Form. After dealing with some of the criticism launched at both works, an appraisal of the praxis of self-reference of Varela’s thinking in action was given. The outlook rounds up this contribution, shedding some light on a possible future research agenda for the formalization of theory and praxis of self-reference.

Reichel A. (2017) From hardware to hardcore: Formalizing systems with form theory. International Journal of Systems and Society 4(1): 37–48. https://cepa.info/4258

The state and relevance of Systems as a field of research and a specific form of scientific inquiry into complex real-world problem situations, can be enhanced significantly by developing and applying more formalized and coherent tools: a new ‘hardware’ enabling to build a new ‘hardcore’ for systems science. The basis of this new hardware stems from a line of thought emanating from George Spencer-Brown and the ‘Laws of Form’, running through the work of Francisco Varela and his calculus for self-reference, being radicalized by Niklas Luhmann and his views on ‘Social Systems’, and continued by Dirk Baecker with the application of form theory to management and organizations. In this contribution, the author develops an understanding and appreciation of the potentials of a form-theoretical approach to formalizing systems (real-world phenomena) as well as Systems (field of research). Central aspects will be the power of the form-theoretical hardware as regards systems storytelling, systems diagnostics and abductive reasoning.

Schwartz D. G. (1981) Isomorphisms of Spencer-Brown’s law of forms and Varela’s calculus for self-reference. International Journal of General Systems 6(4): 239–255.

Exact isomorphisms with formal systems which employ the standard linguistic notations are established for Spencer-Brown’s primary algebra and F. J. Varela’s calculus for self-reference. The primary algebra is “essentially isomorphic” with classical propositional calculus, and the calculus for self-reference translates isomorphically into an axiomatization of S. C. Kleene’s three-valued logic of partial recursion. This correlates Varela’s “autonomy” with “total recursive undecidabilily,” and it suggests the use of Kleene-Varela type systems for discussing “mechanically unknowable” or empirically untestable system properties.

Schwartz D. G. (1981) Isomorphisms of Spencer-Brown’s laws of form and Varela’s calculus for self-reference. International Journal of General Systems 6(4): 239–255.

Exact isomorphisms with formal systems which employ the standard linguistic notations are established for Spencer- Brown’s primary algebra and F. J. Varela’s calculus for self-reference. The primary algebra is “essentially isomorphic” with classical propositional calculus, and the calculus for self-reference translates isomorphically into an axiornat- rzauon or S. C. Kleene’s three-valued logic of partial recursion. This correlates Varela’s “autonomy” with “total recursive undecidability,” and it suggests the use of Kleene-Varela type systems for discussing “mechanically unknowable” or empirically untestable system properties.