Date: 17th March 2021

Speaker of the day: Louis Kauffman

Articles: Varela, “Calculus of Self-Reference” (1975) and Varela & Gougen, “The Arithmetic of Closure” (1978)

Abstract: In the lecture Louis Kauffman explains Varela’s Calculus of Self-Reference and The Arithmetic of Closure, in which – through the expansion of Spencer-Brown’s Theory of Form – Varela tries to develop a consistent formal account of self-referential objects.

Keywords: system-wholes, operational closure, distinction, circularity, autopoiesis, self-indication, autonomous form, Flagg resolution



In Arithmetic of Closure VARELA tries to develop a formal system which would make it possible to think about SYSTEM-WHOLES. These are characterised by their stability and coherence, both of which arise out of their OPERATIONAL CLOSURE; as the principle bearing its name states: every system-whole is operationally closed.

By considering system-wholes, one has to specify the way in which these are distinct from their surroundings. “The basic act of splitting the world into what we consider separable and significant entities” is taken to be that of DISTINCTION (its formal instantiation originating from Spencer-Brown’s Laws of Form). By investigating its formal character – which seems to encapsulate yet not be reducible to possible (empirical) kinds of distinction – we stumble upon the problem of CIRCULARITY: since the act of defining is in a certain sense an act of making a distinction, defining it would amount to distincting distinction.

However, circularity can be considered as a positive dynamic of the formal system. In knot theory, for example, one finds that a string can obtain topological invariant features by binding its outer ends together, thus transforming the string itself into a kind of self-referential ouroboros. Moreover, this kind of invariance seems to be central to the theory of AUTOPOIESIS.

In A Calculus of Self-reference, Varela constructs a set of axioms which would support the formal treatment of system-wholes. Insofar as these maintain and enable their own distinction from the surrounding environment, they are said to be SELF-INDICATING. With the concept of SELF-INDICATION one can formally express integrity of system-wholes by which they resist perturbations.

Formulated symbolically: provided J = ( ( ( ( … ) ) ) ), which includes infinite amount of nested distinctions, J becomes indistinguishable from ( J ). With the operation J = ( J ) = ( ( J ) ) = … , J comes to represent an AUTONOMOUS FORM, generating its own distinction – self-indicating. In this way one could symbolically express von Foerster’s statement “I am the observed relation between myself and observing myself” thusly I = ( I ( I ) ), where I stands for the self-indicating first-person pronoun “I”. VARELA’s unique contribution is the formalisation of rules used when operating with self-indication and its addition as the third object besides the marked and unmarked states introduced by Spencer-Brown.


1) GAMBARROTO: What legacy did/do these approaches have in AI modelling?

KAUFFMAN: These two articles were influential in the school of second order cybernetics. They should be considered central to the task of relating autopoiesis to AI and AL.

 2) VÖRÖS: Why did Varela stop developing a formal system of circularity/autopoiesis?

THOMPSON: Varela said that he was immersed in the thought of Spencer-Brown at the time, but at a certain point he realised that “he had done what he had to do”. Impulsiveness in adopting novel research directions was characteristic of his work.

VIJVER: Second order cybernetics requires the observer to enter the world of the observer; this approach is of course foreign to prevailing epistemologies.

3) VIJVER: There is the mark, and the return upon the mark. Does it involve the same mark? In other words, how does the mark return upon itself and become a self-indication?

KAUFFMAN: It is an empirical fact that distinctions arise in our everyday life. Moreover, “in positing an axiomatic, mathematical system one always has false foundations for it.” Since these assumed false foundations are not considered to be dogmatic, we are allowed to question the assumptions. Nevertheless, mathematics is a game with assumed objects and rules which hold between them; “after a certain point we [mathematicians] want to be following out all the consequences of the rules.” Thus investigating the birth of distinction would thus be somewhat ‘un-mathematical’.

 4) THOMPSON: How can one avoid the explosion of free deduction following the contradictions of self-referential inconsistencies?

KAUFFMAN: To mitigate the explosion of free deduction resulting from the inconsistent system one may (a) ban such an outcome, (b) exclude the Law of excluded middle or (c) adopt the so-called FLAGG RESOLUTION and treat the paradox as a new kind of object that possess its own rule of substitution: An operation with J has to be performed on all or none of the appearances of J.

5) HEPPNER: How does change come into this infinite circle of repetition (the J circle)? For Hegel this would be what he calls ‘bad infinity’ – infinity represented as a non-creative non-evolving formula.

KAUFFMAN: Firstly, the seemingly static symbol of the self-referential structure has an oscillatory dynamic when considered with respect to time. Secondly, the sheer amount of steps might produce unpredictable and thus creative outcomes; if one designs a simple set of rules, this by itself does not mean that the dynamics of the created structure are simple.

6) VÖRÖS: Time seems to play a crucial role in these explanations, yet it is never treated explicitly. Might this pose a problem to the formal way of defining the rule-following process?

KAUFFMAN: The concept of time is formally incorporated using the concept of step. In chess, it is impossible to compute all the possible outcomes, even if we know the exact transformations a particular step can produce. Accordingly, to know time would mean to be able to understand all the possible higher-order dynamics of a game and be able to compute them in advance. Therefore, even though the rules are all there, we still know very little. This exact situation seems to arise even in normal infinite mathematics, where one lacks proofs of complex properties which appear when calculating with very large numbers.

7) LÜBBERT: Cells maintain themselves in a substrate and the stability of the substrate seems to be crucial for the stability of the cellular organisation. How does the distinction (and thus operational/organisational closure) come up in the first place, provided we first need the substrate? 

KAUFFMAN: Since organisational closure is a category of perceptive order, the question should be, how complicated the structure would have to be in order for us to perceive it as a self-indication. “If we weren’t able to find them, the proto-cells wouldn’t really be there.”

 8) LIPIČ: How can one express the sense of oneness, when in meditation the perception of boundaries dissipates? What does the work of Spencer-Brown and Varela contribute to an understanding of such experiences?

KAUFFMAN: By meditating on distinctions one can realise their transitive nature and especially the transitive nature of particular distinctions, which are revealed not as objects, but specific states of organisation.

VÖRÖS: One cannot but make distinctions, however one need not be caught in them – they are empty, fluid and dynamic. This is especially present in the Mahayana dictum: “Form is emptiness and emptiness is form.”

KAUFFMAN: The plane preceding the drawn circle is undoubtedly whole; it is we, who have drawn a distinction, that separate the originally unified plane into an outside and an inside.