Mathematical Work of Francisco Varela · Mathematical Work of Francisco Varela Louis H. Kauffman...

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Mathematical Work of Francisco VarelaLouis H. Kauffman • University of Illinois at Chicago, USA • kauffman/at/uic.edu

> 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.

Introduction

« 1 » This is an article about the possi-bilities in the mathematical work of Fran-cisco Varela. The themes of this article will be related to my personal history of work with Varela since that work is directly in-volved in the mathematics he used and the possibilities inherent in that mathematics. In this introduction I shall summarize the mathematical ideas that Varela and I worked on together and I shall give some facts and opinions about their development. Many of these ideas were recorded by Varela in his seminal book Principles of Biological Autonomy (Varela 1979). See also Varela & Goguen (1978), Goguen & Varela (1979), Varela (1975), Varela, Maturana & Uribe (1974) and my publications as indicated in the list of references.

« 2 » Varela was one of the early people to recognize the significance of the work Laws of Form (Spencer Brown 1969). He saw that Laws of Form, based on the funda-mental idea of distinction, articulated a cru-cial concept that is foundational for biology and the biology of language. An organism is seen, by an observer, to make a distinction. By starting with a distinction we understand how (for an observer) the organism exhib-

its structural stability and autonomy, and becomes an exemplar of the living. This no-tion of distinction is crucial to our under-standing of the nature of an organism and the nature of life itself. The distinction is a joint creation of the organism in its environ-ment and the observer. Together they give life to the organism. The distinction does not appear without the observer, and the distinction that is the organism does not appear without the actions of the organism, producing itself from itself through com-ponents taken from and given back to the environment. A crucial model of this epis-temology is given by Humberto Maturana, Ricardo Uribe and Varela in their 1974 pa-per “Autopoiesis – the organization of living systems – its characterization and a model.”

« 3 » The notion that an organism must have autonomy of structure, and yet that structure is intimately related to the inter-action with the environment and with an observer, is the theme of Varela’s 1979 book Principles of Biological Autonomy. Varela took a very daring intellectual step in this book by creating a new extension of Laws of Form that included at its base a symbol for autonomy. This is the symbol of the reenter-ing mark:

« 4 » Let the letter K denote this reen-tering mark. The reader can take this sym-bol at the allegorical level as an ouroboros, a world-snake, the Snake that Eats its Tail, the mythical indication of the self-devouring, self-creating Universe.

« 5 » Just alongside the mythological level, we have the notion of the form that re-enters its own indicational space. The form sits inside itself, just as the ouroboros sits inside itself. We can indicate this reentrance by an equation

=

where the reentering mark is understood to have an indicational line that reenters its interior space, just as the snake’s tail enters its mouth. The mythology and the concept of the ouroboros demands more than this. At the mythological and conceptual level, the snake, as a whole, is within itself. The form does not just enter into its interior, it is wholly contained within its own interior. One can imagine a passageway that leads from the outside to the inside, but the ou-roboros demands more. We can indicate the demand by an equation of the form K = K so that the form K is actually situated within the mark of its own boundary. The reenter-ing mark is often taken to mean this autono-

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COnSTRUCTIVIST FOUndATIOns vol. 13, n°1

mous reentry as expressed by the equation above. Consequently, in this target article, I shall use the same symbol K, for both the reentering mark and for the entity that is identical to its own crossing.

« 6 » Using the original reentrant mark we write the equation below with the caveat that this is an interpretation of the reen-trance of K = K . That is, given the infinite nest as indicated below, then it follows that K does sit inside itself. It is not necessary to interpret K in this way. nevertheless, I shall use the same symbol for the infinite nest, the reentrant form and the self-pointing sym-bol.

K= …

« 7 » In this way, we indicate that K can appear as an infinite nest, and so appear identically within itself. I will say more about this theme later in the essay, but we should remark here that the infinite-nesting solution requires discussion of the nature of infinity and it certainly is a solution that goes outside the finite forms that a person might search for in looking for a solution to the equation K = K . I say a few words about the matter of infinity. The infinite nest is usually interpret-ed as a completed infinity. That is, one takes as given that there are a non-finite number of marks in the nest and that K corresponds to shifting each mark to the next mark, produc-ing a 1–1 correspondence of the marks in K with the marks in K . It is this 1–1 correspon-dence that is indicated by the equation K = K . Another interpretation of the nest can be that the number of marks in it is sufficiently large so that the observer cannot make the distinction between K and K . This confusion of K with K can be taken to be the meaning of the equation. The reader should appreciate the richness in these multiple interpretations and not be confused by them!

« 8 » The point about the “autonomous form” K is that it can be seen to cross the boundary between object and process. It is at once both an object and a process, just as a biological organism is both an object and a process for an observer. Varela took the step of identifying such an autonomous form as the basis for his “calculus for self-reference” and he placed it in alignment with the au-tonomous nature of a living organism. Self-reference is often associated with autonomy since it also involves an interaction of an en-

tity with itself. We here distinguish self-ref-erence and autonomy, but point out that the reentering mark can be seen as a mark that points to itself (in the act of pointing to its interior). Thus, for an observer, a reentering mark can be seen as self-referential. I point out that the mark itself can be regarded as self-referential, since it makes a distinction in the plane (for the observer of the plane) and the mark refers to that distinction. Any sign can be seen to stand for itself, but the mark not only stands for a distinction, it makes a distinction in the plane of the writ-ing. Of course, the distinction made by the mark requires an observer. Thus, in this sense, the mark itself is not autonomous in the sense of making its own structure. It is significant that Varela chose to interpret the reentering mark as a sign for autonomy.

Meeting and work with Varela« 9 » I discovered Laws of Form in 1974,

some time after the book had been pub-lished, and two years after I had completed a Phd in mathematics from Princeton Uni-versity. I was teaching at the University of Illinois at Chicago Circle (as it was called at that time). I encountered a special expe-rience in the foundations of thought and mathematics almost as soon as I picked up that book by George Spencer Brown. His book was a turning point in my intellectual life. Laws of Form is a lucid exposition of the foundations of mathematics. It embodies a movement from creativity, to creation, to symbol, to system and language and thought and self. Expressing that creation took away the apparent ground of my previous concep-tion. There was no longer any distinction be-tween the certainty or uncertainty of math-ematics and the certainty or uncertainty of present experience. There was no longer any distinction between geometry/topology and logic. There was no longer any possibility that logic could be the foundation of math-ematics, or that mathematics could have any foundation other than itself. There are many roads to this place. For me, Laws of Form came along at the right time. Later, I enjoyed reading accounts of similar experiences with Laws of Form by non-mathematicians such as Alan Watts and John Lilly.

« 10 » A year after my encounter with Laws of Form a seminar arose at the Circle Campus (now called the University of Illi-nois at Chicago). This seminar, devoted to Laws of Form, met every Wednesday at the apartment of Kelvin Rodolfo, a professor of geology at the Circle. The seminar met cor-dially for the whole evening in a comfortable living room with food and drink and ample time for discussion. We were an eclectic group: geographer david Solzman from geography, sociologists Rachel MacKenzie, Gerry Swatez, Joy Swatez, and Kathy Crit-tenden, anthropologist Mike Lieber, com-puter scientist Paul Uscinski, mathematician Brayton Gray, myself and others. We read the book, argued over it, free-associated to it, performed it and generally wandered in the space opened from the possibility of a single distinction. This seminar had a life of more than three years and it deeply influ-enced the lives of all its members.

« 11 » Uscinski and I became fascinated by the recursive and circuitous world of Chapter 11 in Laws of Form. We invented for ourselves an interpretation of the workings of those circuits, and we found ourselves writing the reentering mark to express these ideas.

« 12 » I discovered, around this time, an article in the Whole Earth Magazine about a young biologist, Varela, who had just writ-ten a paper (Varela 1975) about Laws of Form and had worked out an algebra that included the reentering mark! We found Va-rela’s paper and added it to the discussion in the seminar. I resolved to get in touch with him.

« 13 » A little research turned up Varela’s connection with Heinz von Foerster and the Biological Computer Laboratory at the University of Illinois at Urbana-Champaign, about 150 miles from Chicago. We had ear-lier in the seminar called up von Foerster to tell him that we were studying Laws of Form. Von Foerster had written the brilliant review of Laws of Form that appears in the Whole Earth Catalog, where he characterizes it as “Spencer Brown’s transistorized 20th cen-tury version of Occam’s razor.” When we called von Foerster and told him of our en-deavor he laughed and we laughed over the telephone.

« 14 » I was fascinated by the notion of imaginary Boolean values and the idea



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that the reentering mark and its relatives, the complex numbers, could be regarded as such values. The idea is that there are “logi-cal values” beyond true and false, and that these values can be used to prove theorems in domains that ordinary logic cannot reach. At that time, I was fascinated by the reentering mark, and I wanted to think about it, in and out of the temporal domain.

« 15 » The reentering mark has a value that is either marked or unmarked at any given time. But as soon as it is marked, the markedness acts upon itself and becomes unmarked. “It” disappears itself! However, as soon as the value is unmarked, then the unmarkedness “acts” to produce a mark! Having said this, let us look at the formal-ism. If K = <K> and K = <>, then K = <<>>, which is un-marked. If K = <<>>, then K = <K> = <<<>>> = <>, which is marked. It is in this formal sense that markedness and unmarkedmess alternate with the reentering mark. note that the structure of the reen-tering mark is given to persist throughout this recursion. We continue to come back to K = <K> and thus the reentrance persists beyond the temporal oscillation.

« 16 » You might well ask how un-markedness can “act” to produce marked-ness. How can we get something from noth-ing? The answer in Laws of Form is subtle. It is an answer that undoes itself. The answer is that any given “thing” is identical with what it is not. That is, that thing, taken together with what it is not, determines the entire dis-tinction, and when you fit them together, the distinction vanishes. What is marked is fully determined by what is not marked. Thus, markedness codetermines a corresponding unmarkedness. Light is determined by dark-ness. Everything is determined by the delin-eation of nothing. Comprehension and in-comprehension share a common boundary. Any duality is identical to its fitting together into union. In Tibetan Buddhist logic there is existence, nonexistence and that which nei-ther exists nor does not exist (Stcherbatsky 1968). Here is the realm of imaginary value.

« 17 » The condition of reentry, carried into time, reveals an alternating series of states that are marked or unmarked. This primordial waveform can be seen as

Marked, Unmarked, Marked, Unmarked, … Unmarked, Marked, Unmarked, Marked, …

« 18 » I decided to examine these two phases of the oscillation of the reentering mark, and I called them I and J respectively (Kauffman 1978a). These two imaginary val-ues fill out a world of possibility perpendicu-lar to the world of true and false (Figure 1).

« 19 » I wrote a paper about I and J, showing how they could be used to prove a completeness theorem for a four-valued logic based on True, False, I and J. I called this the “waveform arithmetic” associated with Laws of Form. In this theory, the imagi-nary values I and J participate in the proof that their own algebra is incomplete. This is a use of the imaginary value in a process of reasoning that would be much more dif-ficult (if not impossible) without it. Prior to that I had written a paper using Varela’s “Calculus for Self-Reference” to analyze the temporal behavior of self-referential circuits (Kauffman 1978b). My papers were inspired by Varela’s use of the reentering mark in his analysis of the completeness of the calculus for self-reference that he associated with that symbol.

« 20 » I also started corresponding with Varela, telling him all sorts of ideas and rec-reations related to self-reference. We agreed to meet, and I visited him in Boulder, Colo-rado in 1977. There we made a plan for a paper using the waveform arithmetic. This became the paper “Form dynamics,” even-tually published in the Journal for Social and Biological Structures (Kauffman & Varela 1980). An earlier attempt to publish it in the International Journal of General Systems was met by the criticism that we had failed to acknowledge the entire(!) Spanish School of Polish Logic. I still have the letter from that

referee. Later I learned to appreciate Span-ish Polish logic (a group of logicians in Spain working on deMorgan algebras and related matters in multiple-valued logic). Varela based a chapter of his book Principles of Bio-logical Autonomy on form dynamics. I re-member being surprised to find some of my words and phrases in the pages of his book.

« 21 » The point about form dynamics was to extend the notion of autonomy inher-ent in a timeless representation of the reen-tering mark to a larger context that includes temporality and the way that time can be implicit in a spatial or symbolic form. Thus, the reentering mark itself is beyond duality, but implicit within it are all sorts and forms of duality from the duality of space and time to the duality of temporal forms shifted in time from one another, to the duality of form and nothingness “itself.” I believe that both Varela and I felt that in developing Form dynamics we had reached a balance in relation to these dualities that was quite fruitful, creative and meditative. It was a wonderful aesthetic ex-cursion into basic science.

« 22 » This work relates at an abstract level to the notions of autonomy and auto-poiesis inherent in the earlier work of Mat-urana, Uribe and Varela (1974). There they gave a generalized definition of life (auto-poiesis) and showed how a self-distinguish-ing system could arise from a substrate of “chemical” interaction rules. I am sure that the relationship between the concept of the reentering mark and the details of this ear-lier model was instrumental in getting Va-rela to think deeply about Laws of Form and to focus on the calculus for self-reference. Later developments in fractal explorations,

I

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Figure 1 • Axes of Necessity (T, F) and Possibility (I, J) in relation to the reentering mark.



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artificial life and autopoiesis enrich the con-text of form dynamics.

« 23 » At the time (around 1980) that Varela and I discussed form dynamics we were concerned with providing a flexible framework within which one could have the “eigenforms” of von Foerster (1981b) and also the dynamical evolution of these forms as demanded by biology and by mathemat-ics. It was clear to me that Varela had a deep intuition about the role of these eigenforms in the organizational structure of biology. This is an intuition that comes forth in his work.

« 24 » There is a more general theme that has been around since that time. It is the theme of “unfolding from a singularity” as in catastrophe theory. In the metaphor of this theme the role of the fixed point is like the role of the singularity. The fixed point is an organizing center, but it is imaginary in rela-tion to the actual behavior of the organism, just as the “I” of an individual is imaginary in relation to the social/biological context. Of course, this remark is meant metaphorically, but it is also the case that in linguistic use the notion of imaginary is referent to constructs in the human imagination such as the refer-ence to the self as “I”. The Buddhists say that the “I” is a “fill-in.” The linguists point out that “I am the one who says ‘I’.” The process that is living never goes to the fixed point, is never fully stable. The process of approxi-mation that is the experiential and experi-enced I is a process lived in, and existing in the social/biological context. Mind becomes conversational domain and “mind” be-comes the imaginary value generated in that domain. Heinz von Foerster wrote, “I am the observed relation between myself and observing myself ” (Foerster 1981a: 8.45). Other examples of fixed points can occur within the domains in which the transfor-mation began. For example, T(x) = x2 has the fixed point 0 in the real numbers and so if T is regarded as a transformation of the real numbers then 0 is the fixed point that occurs within the original context for T. If S(x) = x2 + 1 and S is regarded as a mapping defined on the real numbers then there is no real fixed point for S. The complex number i with i2 = –1 is a fixed point for the transfor-mation F(x) = –1 / x. If we take the original domain for this transformation to be the real

numbers, then i is a fixed point that occurs outside the original domain. In §51 below I will discuss the lambda calculus method for finding fixed points in generalized domains.

« 25 » The biological context is a domain where structural coupling and coordination give rise to mind and language. The fixed point is fundamental to what the organism is not. In the imaginary sense, the organism becomes what it is not.

« 26 » Varela invited me to participate in summer science seminars held at the naropa Institute in Boulder, Colorado in the early 1980s. We had a group of scientists and courses of lectures: linguistics (Alton Becker, Kyoko Inoue), poetry (Haj Ross), poetry and linguistics (Haj Ross), geography (david Solzman), biology (Varela and Mat-urana), psychology (Eleanor Rosch), Laws of Form (Kauffman), constructive mathemat-ics (newcomb Greenleaf) and more. We talked and talked. I do not know how many of us also meditated, but the atmosphere of the Buddhist Institute provided a wonder-ful place for the gestation and exchange of ideas.

« 27 » After those naropa years we saw each other a few more times. Once we drove together from a cybernetics meeting (a Gor-don conference) to a weekend retreat at the Buddhist center, Karme Choling, in north-ern Vermont. I saw him again in Paris in 1989 and once at the conference “Einstein Meets Magritte” in Brussels in 1995.

« 28 » In those same years, from 1978 until the middle 1990s I had a long and com-plex correspondence with Spencer Brown that culminated in my paper “Reformulat-ing the Map Color Theorem” (Kauffman 2005) about his approach to the four-color map theorem. These conversations also re-volved around the nature of mathematics and the nature of the circuit structures in Chapter 11 of Laws of Form.

« 29 » Much of my work on cybernetics and Laws of Form grows out of this inter-action with Varela. The reader can consult more recent papers of mine to see how this work expands on eigenform and recursion and how a subject I call “iterants” grows out of form dynamics. There is much more to be done in understanding these ideas that come from a cybernetic and biological stance.

From categories and functors to laws of Form and eigenforms« 30 » Another important theme in

Varela’s book is the use of categories and functors to study biological autonomy. A category is a very general mathematical no-tion that is based on the philosophy that things, objects arise and acquire meaning through their relations and relationships. Just so, in a category there are objects, but we are not necessarily told anything about their “internal structure.” Indeed, the ob-jects may not have any internal structure, just as the idealized points in geometry have no internal structure. This is in direct con-trast to set theory where the objects are sets, and except for the empty set, every set has internal structure in the sense of its mem-bers. In category theory there are objects (possibly void of internal structure) and morphisms, represented as directed arrows from one object to another object. Thus, if A and B are objects in a category, then there may be a morphism f: A→B. We name the morphism by the letter f, as shown above. This has the same form as the notation for a function from one set to another, but it can denote nothing more than a directional re-lation from A to B.

« 31 » If I have a morphism f: A→B and another morphism g: B→C, then it is an ax-iom of category theory that there shall be a composite morphism fg: A→C. You should think of fg as obtained by going from A to B by f and then going from B to C by g. (I will use the ordering fg here so that we go from left to right lexicographically to compose the morphisms.) We can depict this relationship as shown in Figure 2.

fg

fg

Figure 2 • Composition of morphisms.



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« 32 » The next axiom of category theo-ry concerns three morphisms.

f: A→B, g: B→C and h: C→D.

« 33 » now we have (fg)h and f(gh) by performing the composition of f and g first and then composing with h, or by perform-ing the compositions of g and h first and then composing with f. See Figure 3.

« 34 » The second axiom of category theory asserts that there is an equality of morphisms (fg)h = f(gh). We say that com-position of morphisms is associative.

« 35 » Finally, the third and last axiom of category theory asserts that every object A has an identity morphism 1A: A→A. This is a morphism from A to itself, such that any composition with it by another morphism, leaves the other composition unchanged. For example, if f: A→B, then 1A f = f.

« 36 » Any directed graph G generates a category. We let the nodes of the graph be the objects and we make each edge of the graph into a morphism. We add an identity mor-phism at each node (object) and define each directed path in the graph to be a composite morphism. Paths are composed in the usual way by having two paths such that the start node of one is the end node of the other. The composite path is obtained by walking along the paths consecutively. Thus, we obtain from a directed graph G, a category Pa(G), the path category of the graph. There is a complementarity (in the sense of mutually related conceptual domains) between graphs and categories in the following sense. We can consider the collection of all graphs as a category in its own right where the objects are individual graphs. Let us call this catego-ry of graphs Graphs. We can also consider a category of all (small) categories. Here the objects are categories and we will call this category of categories Categories. Then we

have functors (maps of categories preserv-ing their structure) Pa: Graphs→Categories, and F: Categories→Graphs. The first functor is our path method of converting a graph to a category. The second functor F forgets the category structure and just sees the objects as nodes and the arrows of the category as directed edges. This back and forth associa-tion between graphs and categories formal-izes the complementarity and is an example of an adjoint pair of functors. The reader will, in fact, find a lucid description of ad-joint functors in relation to complementar-ity in Chapter 10 of Varela (1979: 97). Varela made a good beginning in this work on a categorical analysis of the meaning and pos-sibility in the concept of complementarity at many levels. He used this formalism of cat-egories to think about many specific aspects of biological organisms such as the way the immune system operates and at another level how mind, language and body interact. At another level, complementarities occur between the very concepts of process and object and he managed to summarize these with a precision that was bolstered by the underlying category theory.

« 37 » Category theory is a way to ap-proach mathematics and its applications with attention to concept and meaning. Consider the simplest category. This catego-ry has one object O and one morphism 1O : O→O that composes with itself to produce itself. In other words, this simplest category just has one entity and the barest sort of self-reference of this entity to itself, in the form of the morphism 1O.

« 38 » What is the next simplest cat-egory with only one object? We can have another morphism f: O→O that is distinct from the identity morphism. note that dis-tinct means that we declare that f is not the identity morphism. The morphism f can be

distinct from the identity even if its compo-sition with itself is equal to either the iden-tity (one possibility) or itself (another pos-sibility). We need not put any restrictions on the compositions of f with itself. Then f is seen to be distinct from the identity mor-phism in that ff is not equal to f, while the composition of the identity morphism with itself is the identity morphism. Then we have many generated morphisms that are the finite compositions of f with itself: f, ff, fff, ffff, and so on. We have an infinity of such morphisms, and they correspond to going around the arrow f that points from O to itself some number of times. In the usual category theory we do not allow in-finite compositions of morphisms. Having an infinite composition of morphisms is distinct from an infinity of finite composi-tions of morphisms. This category with one object and one non-identity arrow embodies the concept that “self-reference is infinity in finite guise.” The arrow f makes a reference from O to O, and the iteration of this refer-ence gives an associated infinity of compo-sitions. These compositions are the paths in the graph that is indicated by the one node O and the one directed edge f. Thus, this simple category indicates the adjoint rela-tionship between the path category and the simple graph of self-reference. See Figure 4 for an illustration of this category.

« 39 » The usual category theory allows only finite compositions of a morphism with itself or other morphisms. We can go further than the usual category theory by consider-ing the infinite composition C = fffffff… of the morphism f with itself, and we then ar-rive at a first eigenform, for it is the case that fC = f(fffff…) = ffff… = C, since one more f does not change the infinite composition. C is unchanged under the application of f. The infinity generated by the self-reference has

fgf

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f

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Figure 3 • Associations of morphisms.

Of

Figure 4 • Self-reference in a simple category.



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returned in the form of self-reference as an eigenform.

« 40 » There is more to be done in us-ing categorical ideas in understanding cy-bernetics and biological autonomy. This is an aspect of Varela’s work (and his collabo-rations with Joseph Goguen) that can still be investigated by mathematically oriented cyberneticists. It is not my intent to suggest a program of research here. nevertheless, even the remarks that we have made above (in relation to Figure 4) about the relation-ship of Laws of Form and circularity in re-lation to categories with a single object can be taken as a beginning. We will explore this further in other papers.

« 41 » In following the epistemology and the fundamental notions of cybernetics, one should, in my opinion, start not with categories, but with Laws of Form. In Laws of Form we have a simplest possible mathemat-ical formalism, a symbol < > that represents a distinction between its outside and its in-side. (Here I will use brackets. In Spencer Brown’s book a partial box, the mark, is used to the same purpose.) Even without any further axioms for using the mark, one can proceed to find a multitude of iconic forms:

< >, <<>>, <<<>>>, <<<<>>>>, …

« 42 » Here I have only indicated the simple nesting that we have discussed ear-lier. By taking an infinite nest as in

K = <<<<<…>>>>>

we obtain the reentering mark with K = <K>. Other self-referential forms can be con-structed by similar recursions. For example, we can have F so that F = <<F> F> and G so that G = <G G>. We have almost at once an arithmetic of infinity and reentrance ema-nating from the idea of a distinction.

« 43 » The next remarks are a descrip-tion of one way to think about the elements of Laws of Form. A single distinction (a first distinction if you like) is given and this giv-en distinction has two parts, sides or states. One state is said to be marked. One state is said to be unmarked. The mark < > indicates the marked state and can be seen to be the result of crossing from the unmarked state. The icon < > for the mark can be seen as an operator that transports from the un-marked inside to the marked outside of a given initial distinction that can be taken to

be the mark itself. That is, <E> represents the state obtained by crossing from the state E. If E is unmarked then the state obtained by crossing from E is marked and so <E> = < > is marked. If E is marked, then <E> = <<>> and is unmarked. Seeing the mark as an op-erator, we then see that < < > > can be in-terpreted as a passage from a marked state (on the inside). Crossing from the marked state yields the unmarked state. Thus < < > > represents the unmarked state and we write

<< >> =

where I place nothing to the right of the equals sign since the icon for the unmarked state is no icon at all. This is the law of cross-ing.

« 44 » By the same token, one can un-derstand that two adjacent marks

< > < >

stand just for the marked state since each mark can be seen as the name of the other mark. Thus, we can write

< > < > = < >.

This is the law of calling. note that the mark can be interpreted as both an operator (of crossing) and a value (marked).

« 45 » These two equations (the laws of calling and crossing) complete the construc-tion of the mark as a logical particle and form the beginning of the mathematics and epistemology of Laws of Form. I recommend that anyone looking at my short exposition read the original work of Spencer Brown and come to his or her own understanding of the calculus of indications generated by the mark.

« 46 » To this day we continue to work on interweaving the fundamental simplicity of Laws of Form with the more complex lev-els of recursion, reentry, reflexivity, reflexive domains, eigenform, category theory and other aspects of mathematical modeling.

« 47 » It is in following the epistemo-logical track of autonomy that Varela (1979) was led to consider mathematical domains that included distinctions (Laws of Form), recursions, category theory, lambda calcu-lus and eigenforms in his book Principles of Biological Autonomy. All of these math-ematical domains have stood the test of time in relation to cybernetics and yet all of them are at the beginning of a development that

can bring forth the potential that cybernet-ics has a nexus where the observer and the observed are not one nor are they two. The cybernetic nexus where the observer plays hide-and-seek in the relationships that be-come an observed world is our subject / ob-ject matter in working with this field. Varela had the courage to go into the mathematics and begin a necessary exploration of formal language in relation to biology and the na-ture of the organism.

Eigenform and reflexivity

« 48 » I want to summarize very briefly how I work to explore the mathematics of reflexivity in relation to cybernetics. Varela considered this theme in his work as well, and the difference between the approach I shall outline and his is more a matter of taste than anything else. The reader should com-pare with Chapter 13 of Principles of Biologi-cal Autonomy (Varela 1979).

« 49 » I say that a set D is a reflexive do-main if every element of D is also a transfor-mation of D to itself. Thus, given an object or element a in D there is a corresponding function (morphism) a: D→D. Here the role of a as a morphism is that it is a function from D to D. This means that if b is an object in D then ab will be both a new object in D and a new mapping of D to itself. We then further assume that if we define a mapping from D to D using these morphisms, then the new mapping corresponds to an element of D. For example, if I define fx = (ax)(bx) for any x in D, then there is an element f in D with this very property. Here x denotes any element of D and ax is the application of a to x giving a new mapping of D to D. Then ax is applied as a mapping to (bx). The alternation of roles is always such that XY means X as a mapping is applied to Y. The parentheses are crucial in this formalism. In the reflexive do-main, every “person” is also an “agent,” who transforms the whole space D to itself. There is a complementarity in a reflexive domain between its objects and its morphisms. This complementarity is axiomatic as explained here, but such domains can be constructed via the work of dana Scott (1980) who uses topological limit constructions in a hierar-chy of languages. Varela was well aware of Scott’s work and gave a sketch of it in his



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Principles of Biological Autonomy. One can also take the pragmatic point of view that a reflexive domain corresponds to an expand-ing language of actions and objects. This viewpoint is also discussed by both Scott and Varela.

« 50 » A warning to the reader: Uncon-trolled reflexive domains can produce logi-cal fixed points. For example, suppose that negation (~) is an operator in a reflexive do-main D. Let Rx = ~(xx). Then, substituting R for x, we have RR = ~(RR). Most logicians do not want fixed points for negation and so will take great care in controlling their re-flexive domains. Cyberneticists should be used to this kind of circularity and allowed to freely explore the concept of reflexive do-mains, finding their own constraints. This is a manifesto for freedom of foundations.

« 51 » A key theorem about reflexive domains is this following result (Baren-dregt 1981). (Here phrased in the language of reflexive domains as above.) This is my rephrasing of a fundamental theorem of Lambda Calculus, discovered by Alonzo Church and Haskell Curry.

Church-Curry Fixed Point Theorem. In a reflexive domain D, every element F has a fixed point. That is, if F is in D then there is g in D such that Fg = g. In other words, every F in D is the transformation for an eigenform that corresponds to F.

Proof of the Theorem. define Gx = F(xx). Then there is a G in D with this property, by the assumption that D is a reflexive domain. Therefore GG = F(GG) follows, by letting G act on itself. But this means that we can take

g = GG and then Fg = g. This completes the proof.

« 52 » The remarkable fact about this construction of eigenform is that it does not involve an excursion to infinity. Look at a special case.

Define Gx = <xx>.Then GG = <GG> and so if J = GG, we have J = <J>.

« 53 » I have produced the reentrant mark without the “construction” of infinitely many marks as I did in the introduction, and as I did in the section on category theory, where I went around the circle of self-refer-ence infinitely many times. note that if we simply posit an entity K with K = <K>, then we also do not have to use the notion of in-finity. The present discussion explains why the concept of reflexive domain is important in biology and cybernetics. This was under-stood by Varela.

« 54 » With this concept of reflexive domain, we can continue the discussion and extend it beyond biology to the many reflex-ive domains that appear to us once we adjust a cybernetic lens and keep the observer as an actor in the system that engulfs us.

conclusion

« 55 » Let us give Varela the last word(s) in this essay by quoting a passage from Sec-tion “16.3 Linguistic domains and Conver-sations” of his Principles of Biological Au-tonomy:

“ From another perspective, if we consider a con-versation as a totality, there cannot be a distinc-tion about what is contributed by whom. Linde and Goguen (1978) […] in their careful descrip-tions of the structure of discourse, […] found no evidence that the text, as a coherent entity, could be attributed to separate speakers, but it was an al-loy of their participation, and exhibited rules and laws that are not reducible to the separate contri-butions. A similar basic methodological principle is behind Pask’s approach to teaching machines, where a conversation is a coherent recursive ag-gregate. […] These ideas are precisely in line with the central theme of this book: that every autono-mous structure will exhibit a cognitive domain and behave as a separate, distinct aggregate. Such autonomous units can be constituted by any pro-cess capable of engaging in organizational closure, whether molecular interactions, managerial ma-nipulations, or conversational participation. I am saying, then, that whenever we engage in social interactions that we label as dialogue or conver-sation, these constitute autonomous aggregates, which exhibit all the properties of other autono-mous units.” (Varela 1979: 269)

« 56 » In this way, Varela produced a coherent theory, mathematical at base and based on fundamental notions of distinc-tion and autonomy. This theory has enor-mous reach and we are only at the threshold of beginning to appreciate it and understand that his conversation is our conversation in an exchange without end, in the wholeness of our conversational domain.

Received: 23 May 2017 Accepted: 11 September 2017

louIs h. KauFFManhas a BS in mathematics from MIT and a PhD in mathematics from Princeton University. He is Professor of Mathematics at the University of Illinois at Chicago. Kauffman works on topology, knot theory, quantum topology, form and cybernetics. He is the founding editor and editor-in-chief of the Journal of Knot Theory and Its Ramifications and the editor of the book series on Knots and Everything. He is well-known for his discovery of the bracket state sum model of the Jones polynomial and for his discovery of a generalization of the Jones polynomial called the Kauffman polynomial, for his discovery of virtual knot theory and other topological structures. Kauffman is the recipient of the 1993 Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natural Philosophy Association for his work on discrete non-commutative electromagnetism. He was president of the American Society for Cybernetics from 2005 to 2008. Kauffman plays the clarinet in the Chicago-based Chicken-Fat Klezmer Orchestra.

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Monologic versus Dialogic Distinctions of selvesKlaus KrippendorffUniversity of Pennsylvania, USA klaus.krippendorff/at/asc.upenn.edu

> upshot • This commentary contrasts the monologic accounts of self-reference pursued in Kauffman’s target article with natural language notions of refer-ring to one’s self, drawing distinctions, and constructing identities collabora-tively. It suggests that mathematical cal-culi, which assume the perspective of ob-servers, are fundamentally incapable of accounting for how the selves of interac-tively involved participants in social sys-tems come to be. It questions the claim that a mathematical notion of “self” can shed light on the being of humans and argues for dialogical distinctions of iden-tities. Finally, it suggests that publishing and enacting monological conceptions of selves can have undesirable social consequences.

« 1 » What I appreciated most in Fran-cisco Varela and Louis Kauffman’s works is their making George Spencer Brown’s eso-teric Laws of Form clearer to all of us. To me their work is important for two reasons. First, it brought me closer to understand-ing that we draw distinctions when using language. We are taught to think of words as symbols of what we are talking about. This representational theory of language has gotten us into numerous epistemological pa-thologies from the narrow linguistic defini-tion of language as grammar, semiotic con-ceptions of two distinct worlds, that of signs

and that of referents, to Cartesian dualisms. Second, it gave me a basis of conceptualiz-ing human agency: making distinctions is an act that creates the very differences we observe and talk of. This insight offered an important qualification of Gregory Bateson’s notion of information as the differences that make a difference. To me this also enabled me to talk of accountability, otherwise as-sumed given.

« 2 » I once took these rather funda-mental insights to heart and suggested an epistemology for understanding human communication (Krippendorff 1984). Fol-lowing Varela, I realized that all acts of drawing distinctions create differences among the parts they leave behind. On a blank piece of paper, the drawing of distinc-tions is largely arbitrary (save for a piece of paper’s finite extensions) and this is what the Laws of Form start out with. However, when distinctions are drawn in domains that possess a modicum of organization, as for surgeons trying to solve a medical prob-lem, for participants in conversations, and for politicians making decisions affecting an economy, distinctions may well reveal previously unrecognized connections that underlie the domain in which distinctions are made. Some distinctions create parts that simplify understanding of how they are connected, while other distinctions make the distinguished parts incomprehensible. Surgeons avoid cutting through organs whose unity would be difficult to compre-hend when cut in half. not caring for the re-lationships among distinguished parts may impoverish understanding of the domain to which they belong. For example, a list of the words occurring in an essay omits their context and makes what competent readers take away from the essay not recoverable

from that list. drawing distinctions with-out regard for the relationships laid bare thereby can hound us later. This also applies to larger domains. The boundaries drawn in the Middle East after WW I failed to recog-nize culturally, religiously, and linguistically unlike communities and a century later we experience the turmoil caused by these dis-tinctions.

« 3 » nobody can entirely escape grow-ing up in the categories of a natural lan-guage. For biologists it made sense to limit their domain of interest to living organisms. While distinguishing organisms defines a productive field of inquiry, it renders invis-ible the interactions among these organ-isms, which makes them viable. Jakob von Uexküll (1934) recognized this long ago and Humberto Maturana and Varela (1987) acknowledged as much by saying that or-ganisms are structurally coupled with their environments and that humans live in lan-guage. But merely acknowledging these omissions is a far cry from understanding humans as social beings, as participants in communities, constituents of social institu-tions, and creators of technologies to live with. My 1984 paper had the limited aim of suggesting that communication is an ex-planatory social construct that compensates for the inadequacy of conceiving human be-ings as individuals, a distinction deeply en-grained in Western individualism, theorized in psychology and radical constructivism, and built into our legal and political system. Communication accounts for why human beings cannot exist alone, as biological or-ganisms or cognitive systems.

« 4 » Mathematicians think with pencil and paper and use programmable comput-ers. Therefore, I did not expect that Kauff-man, in his excellent presentation of Varela’s

open Peer commentarieson louis Kauffman’s “Mathematical Work of Francisco Varela”



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and his own work, would get into the messy world of scientific distinctions. However, his review addresses their application to biolog-ical organisms and describes their accom-plishments in such human terms as “self-,” “acts of drawing distinctions,” and “coopera-tion.” For me, this raised old flags.

« 5 » Let me start with a pertinent con-ception that predates Spencer Brown by two decades: self-organization. Ross Ashby (1947) wrote about its principles in the late 1940s. The concept did not receive much traction until the 1959 conference devoted to it (Yovits & Cameron 1960), which Heinz von Foerster described to me as a quan-tum leap in the history of cybernetics. That conference dealt with biological, cognitive, physical, and computational systems with-out addressing the constructive effects of participants’ language. Subsequently, Ashby (1962) revisited his earlier paper and, to his credit, struggled hard to find a meaning-ful definition of the self of self-organizing systems. He was justly dissatisfied with the then common definition of self-organiza-tion as developing indigenous structures. He had already shown that all mathemati-cal transformations, repeatedly applied to any network of relations, eventually either converge to their own eigen-structures or exhibit positive feedback and cause run-away instabilities. This prompted him to add two characteristics to their definition. Self-organizing systems had to develop not just indigenous but foremost good forms of organizations, measured by resisting exter-nal perturbations. And they had to exhibit circular dynamics. For Ashby, the self of self-organizing systems revealed itself in the organizational properties they preserved. Formally:

O = T n (O, P)

where O refers to a network of relations, the organization; P to perturbations from the system’s environment; and T

n to a repeated transformation of both. Applied to its own products, T represents the circular dynamic of information flows as long as it exhibits negative or stabilizing feedback. To the ex-tent that the dynamics of self-organizing systems is immune to a range of perturba-tions from their environment, such systems can be interpreted as closed to organization and information but remain open to pertur-

bations, which Varela later described as “in-formation,” and energy. From the perspec-tive of an experimenter, defining the self as what self-organizing systems preserved was pretty convincing. It recognized the difficul-ty of manipulating self-organizing systems from their outside. While observers always assume considerable freedom to describe such systems in any terms they please, I was never happy with imposing selves on mind-less systems.

« 6 » Although self-organization is not mentioned in the target article, I started my commentary with this conception for two reasons. Ashby’s explorations predate the current discussion by half a century and his formal definition is virtually identical to Kauffman’s statement of self-reference visu-alized in Figure 4 – except for limiting it to only one category, not referring to extrane-ous circumstances under which an identity persists. Also, his arrow probably did not in-tend to represent a transformation in time. My uneasiness with assuming that selves act within mindless self-organizing systems also applies to Kauffman’s conception of self, whether he talks of biological organisms or marks on paper.

« 7 » In §2, Kauffman argues that dis-tinctions are foundational to biology and the biology of language. If he means to say that distinctions are fundamental to the dis-course of biologists, I could not agree with him more. To me, distinctions are founda-tional to all uses of language, to any dis-course. But Kauffman extends an observer’s ability to draw distinctions to what biologi-cal organisms do when he writes that

“ (a)n organism is seen, by an observer, to make a distinction. By starting with a distinction we un-derstand how […] the organism […] becomes an exemplar of the living. This notion of distinction is […] a joint creation of the organism in its envi-ronment and the observer. Together they give life to the organism.” (§2)

« 8 » Frankly, I cannot see how observ-ers participate in giving organisms their life. Organisms do their living regardless of how biologists conceive of life and regardless of the distinctions they make – unless they actively intervene in their lives. The history of our conceptions of life is long and largely at odds with how we conceive of life today.

That history had no effect on the evolution of living organisms. Maturana and Varela’s autopoiesis did not change what organisms do either, only how biologists reconceptu-alized their objects of attention. To me, the boundaries of organisms are drawn in the discourse of biologists. Ecologists, by con-trast, have little interest in organisms. They draw boundaries around species of organ-isms. Whatever is inside observer-drawn boundaries does what it does. I doubt that biological organisms (as described in the discourse of biologists) know what living means. I doubt that species have a sense of the ecology in which ecologists see them play a role. I doubt that biological organ-isms draw distinctions the way speakers of a language do. I cannot endorse the claim that organisms draw distinctions, and collabo-rate with their observers on how they distin-guish themselves from their environment. Their identity, their “self ” is an observer’s construction.

« 9 » I had several occasions to men-tion my misgiving about imposing selves on observed system without a clue of whether they were capable of having one to Varela. I was pleased to sense his uncertainty as well. Although he could not rewrite his by now well-known calculus of self-reference, his terminology shifted to talking of the au-tonomy of biological organisms. To me au-tonomy is a more appropriate term for what was heretofore called self-organization. It denotes the operational independence of systems from certain variables. It suggests that experimenters who face autonomous systems have limited abilities to intervene into what goes on inside them, having to treat them as black boxes. And it does not imply autonomous systems to have selves they could articulate.

« 10 » Kauffman notes Varela’s shift to-wards talking of autonomy but in §8, he pro-ceeds to associate it with self-reference. He argues that any sign can be seen as standing for itself, as self-referential. He suggests that it not only stands for a distinction, the act of entering it amounts to making a distinc-tion in the plane of the writing. He is cor-rect, of course, when acknowledging that a mark entered by an active observer is not autonomous in the sense of making its own structure, but to me this acknowledgement is not strong enough.



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Monologic versus Dialogic Distinctions of selves Klaus Krippendorff

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laws of Form and Paraconsistent logic Jean Paul Van Bendegem

« 11 » Taking the monologic of distinc-tions seriously, I would have to say that an observer’s recognition of a mark on paper presupposes her ability to distinguish it against the background of the paper that bears it. But reading a mark as a sign goes beyond that distinction. It is an interpreta-tion, based on the preconception of signs as having referents. Unable to find a referent should lead observers who own their inter-pretations to abandon this preconception. This, however, is not what Kauffman does. He seems not only to attribute sign qualities to the mark, but lacking an obvious refer-ent, he claims that the mark refers to itself – without acknowledging this to be his projec-tion. This is a rhetorical move, and only that, of granting a mark on paper an undeserved observer-independent agency: that of a self-referring self.

« 12 » On close examination, I find this logic hard to swallow. We could not possibly get an answer from a sign, or from biologi-cal, ecological, or computational systems to the question of whether they refer to them-selves, what their selves consist of, what they seek to preserve. I am suggesting that claims of systems to have selves should be limited to those capable of articulating their selves by answering the question of who they are, the identity they own, and for what actions they claim responsibilities. Systems able to answer such questions have to be able to express themselves in ways that observers can interpret. In language, the most obvi-ous self-reference is made by speakers us-ing “I” in a proposition of who “I” is, what “I” does, says, believes, or strives for. Such self-references are not limited to individual speakers. The self of social organizations can become evident when their members refer to themselves as its participants, are articu-lating what “we” aim at, or what “our” mis-sion is. The US constitution starts with “We the people …”

« 13 » I am suggesting that granting sys-tems the ability to define their own identi-ties and organize themselves as distinct from how other systems conduct themselves – regardless of how observers categorize them – is made impossible by the mono-logic nature of mathematical formalisms. By monologic I mean the reliance on a single perspective, that of observers who describe but do not intervene in their observations;

on a consistent logic of explanations that has no place for multiple and conflicting voices; and is unable to conceive of realities as under continuous construction and re-construction in language, to which the very publication of such monological accounts may well contribute. This is not to suggest that mathematics could not provide diverse formalisms and support alternative con-structions of reality. Mathematicians have a wealth of formal systems at their disposal. But I claim that the formalisms they pursue do not account for the effects of being pub-lished, read, enacted, and participating in what they describe.

« 14 » needless to repeat, I admire the work of Ashby, Varela, and Kauffman and have no objections to inventing interest-ing formalisms, exploring where their rules lead to, or conducting computer simula-tions. When applied to systems that are un-able to respond to how they are observed and theorized, describing them in anthro-pomorphic terms may not harm them. However, the monological assumption of superior observers becomes no longer justifiable when dealing with social sys-tems, which are constituted by people who speak a language and may be able to read mathematical accounts, especially when explained in socially relevant terms: refer-ences to selves, purposes, abiding natural laws, and conforming to theories. Publica-tions offering such interpretations may be rejected by readers but can also encourage enacting and reifying them. Theorists of so-cial worlds can hardly avoid participating in their constructions.

« 15 » One of Varela’s later contribu-tions, not mentioned in Kauffman’s article, is his acknowledgement that language is enacted (Varela, Thompson & Rosch 2016). Saying “I am …” refers to a speaker’s em-bodied self, “I feel …” articulates his or her experiences, “I did …” narrates what he or she enacted. Unlike a mark on paper, the articulated self reflects a speaker’s corporal participation in a context in which the use of language matters. This context is what I claim escapes monologic accounts of obser-vations.

« 16 » Moreover, in the social world, self-references occur almost exclusively when conversing with someone else. “I” does not only refer to its speaker’s identity.

Wittgenstein would say that its meaning be-comes evident in the response it elicits. In a job interview, “I” means something un-like when testifying as a witness in court, or declaring one’s love to someone. The self to which “I” refers is fundamentally relational. It requires acceptance by, if not mutual ac-commodations among, present addressees. If I claim to be someone I am perceived not to be, I could end up in jail or be declared insane. If I speak to my boss as I would to a baby, I might be fired. If I tell someone who he or she really is, I might trigger violent objections. If I refer to others in derogatory slurs, I might incite a riot. Only unques-tioned and clearly circumscribed authori-ties – coaches, medical doctors, or judges – could conceivably get away with catego-rizing people without their consent. In the social world, the “I” is rarely ever a mono-logic self. Kauffman exemplifies what a cal-culus of self-reference leads to by quoting von Foerster, in §24, who wrote: “I am the observed relation between myself and ob-serving myself.” In stark contrast, I hope to have convincingly argued that my own self, who I am at any one moment, has much to do with whom I am with. I can hardly talk without at least a sense of whom I am ad-dressing, without expectations of who my partner thinks I am, without expectations of how the other responds to what I say, without being willing to adjust my identity to continue the conversation, and without my expectation of my partner’s willingness to do the same given what we have learned of each other. Even in the medium of writ-ing, authors can hardly avoid imagining who their readers might be.

« 17 » In the social realm, I suggest re-placing the abstract monologism of math-ematical formalism by dialogical construc-tions of actual language use. Social realities are fundamentally dialogical. Assigning selves to systems, whether they can or can-not articulate them; attributing self-refer-ence to marks on paper that cannot act; and distinguishing selves from their negation always privileges observers’ categorizations over joint conversational accomplishments. Monologic accounts are antithetical to dia-logical conceptions of social phenomena. Importing them into the world governed by the use of language can become the source of ethnic prejudices and oppression. They



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justify the construction of inferior others, unworthy of respect, and can cause horrible things that human beings can do to one an-other.

« 18 » For me, the fundamental self-references occur in the consensually alter-nating roles of speakers and listeners, in Martin Buber’s (1958) “I” and “Thou.” none of us can maintain independent selves. Our identities are complementary to each other and context sensitive. Buber contrasts I-Thou with I-It distinctions on the ground that the former entails reciprocity of respect whereas the latter does not – as when inter-facing with technology or natural systems. I-It distinctions have plurals. The We-They distinctions render our own identity un-problematic but assign collective identi-ties to outside others without giving them a chance to articulate their own. This eth-nocentric monologism is the source of rac-ism, gender inequality, slavery, and violence done to declared enemies.

« 19 » Buber is not the only proponent of dialogical conceptions of identities. Er-win Goffman (1959) studied how individu-als perform themselves in the presence of others, in public. John Shotter (1984), relat-ing selfhood to being accountable to oth-ers, conceived of selves as dialogically con-structed. Mikhail Bakhtin’s (Todorov 1984) dialogue embraced the identity of voices that speak through authors and participants in conversations. And the later Varela’s selves reside in embodied narratives. These are just a few dialogical companions.

« 20 » Our social world is badly dam-aged by the dominant pursuit of monologic constructions. I am suggesting that we rec-ognize the fundamentally dialogical nature of distinctions in the domain of the social.

Klaus Krippendorff is the Gregory Bateson Emeritus Professor for Cybernetics, Language, and Culture

at the Annenberg School for Communication, University of Pennsylvania. He has published widely on communication theory, cybernetics,

social science methodology, discourse and content analysis, and social constructions of realities. As a critical scholar he investigates how the use of

language can entrap communities in burdensome conditions and explores for emancipatory paths.

Received: 16 October 2017 Accepted: 25 October 2017

laws of Form and Paraconsistent logicJean Paul Van BendegemVrije Universiteit Brussel, Belgium jpvbende/at/vub.ac.be

> upshot • The aim of this commentary is to show that a new development in formal logic, namely paraconsistent log-ic, should be connected with the laws of form. This note also includes some per-sonal history to serve as background.

some personal history« 1 » One of the attractive features of

Louis Kauffman’s article is that, apart from many other considerations, it focuses on his personal and research connection(s) to Francisco Varela. In the initial paragraphs of this commentary I would like to do the same as I believe it to be relevant for the comments that follow. Having been a stu-dent of the philosopher and logician Leo Apostel at Ghent University – in those days the Rijksuniversiteit Gent – we, as doctoral students, had the great luxury of profiting from his knowledge, which covered many fields and was always at the forefront. Giv-en the combination of his interests in the work of Jean Piaget – I had the pleasure of accompanying Apostel to one of the sum-mer schools at the Fondation Archives Jean Piaget, in 1980, in Geneva – and in the broad area of systems theory, initially fo-cused on Ludwig von Bertalanffy and his general systems theory, we were made fa-miliar with the work of Heinz von Foerster, Ernst von Glasersfeld, Humberto Maturana and, of course, Varela. The Principles of Bio-logical Autonomy was on our desks pretty soon after it was published.

« 2 » I am no longer sure whether I was already familiar with Spencer Brown’s Laws of Form before I read Varela’s book or whether it was this very book that intro-duced me to the curious and rather amaz-ing calculus of distinctions. I do remem-ber being impressed by the steps made by Varela beyond Spencer Brown and I made several attempts, rather unsuccessfully I am afraid, to contribute to the further develop-ment of their ideas. I should add here that a large part of my interest was also gener-

ated because of my doctoral research, the central theme being the development of a strict finitist form of mathematics. In a cer-tain sense, I wanted to repeat the exercise done by L. E. J. Brouwer, founder of the intuitionist school in the foundations of mathematics. The major difference, how-ever, was that it should not take place solely in the mind, as he proposed, but rather in an extremely concrete fashion by signs constructed by an agent. Hence the use of marks or distinctions on paper became the cornerstone of the whole enterprise and the connection with Spencer Brown should be clear.

« 3 » It was also through Apostel that I got to know the Alternative Natural Phi-losophy Association (AnPA). The original members were Clive Kilmister, Ted Bas-tin, Pierre noyes (who came up with “bit-string physics”), Fred Parker-Rhodes (who created the “combinatorial hierarchy”) and John Amson. It was, incidentally, at the 9th Annual International Meeting, in Cam-bridge, in 1987, that I met Kauffman for the first time and where I had the occasion to present my findings on strict finitism (and where he tried to convince me of the importance of knots, and rightly so). One of the outcomes was a publication in the ANPA-West journal (Van Bendegem 1992), where I had the chance of corresponding with Tom Etter on finitist matters.

« 4 » All that being said, this commen-tary is not meant as a form of criticism of the target article as it is rather a historical overview and a presentation of Varela’s work. It is therefore much more appropri-ate to propose suggestions for further elab-orations. In addition, this would also be something that Apostel would have appre-ciated, as one might expect from the previ-ous paragraphs. More precisely, I will focus on the curious phenomenon that formal logicians (as part of the philosophical com-munity) have hardly taken any interest in, in the Laws of Form. What I want to claim is that a recent development in formal logic, namely the study of paraconsistent logic, invites us to make a connection. Let me add that, unsurprisingly, I learnt about these new logics through Apostel.



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Paraconsistent logic and imaginary values« 5 » It will be useful, I assume, to first

say a few words about paraconsistency. Think of classical propositional logic. If p, q, r, … indicate basic statements, and ˅ (dis-junction), & (conjunction), → (implication) and ~ (negation) indicate connectives and ⊢ stands for “there is a proof,” then it can be shown that the following holds: p, ~p ⊢ q, known as the infamous “ex contradictione sequitur quodlibet” (ECQ). Here is the shortest proof known: given p and ~p, p ˅ q follows from p, but p ˅ q together with ~p leaves us with q. In more common language: a contradiction is fatal for classical logic be-cause it leads to triviality, as any statement q is now provable. Any logic that eliminates ECQ is called a paraconsistent logic. note that this only means that the presence of a contradiction is no longer fatal, it does not mean that the logic should contain or effec-tively contains a contradiction.

« 6 » To make the connection with the laws of form, let me reformulate Spencer Brown’s calculus. A distinction I will indi-cate with D, the empty space with I and the re-entry form with K (to follow Kauffman’s notation). The concatenation will be repre-sented by ⊕ and the crossing over by ⊗. So, the two basic rules for the calculus of dis-tinctions then become:

D ⊕ D = D, and

D ⊗ D = I.

« 7 » Since we have three “states” in to-tal, we can ask the question whether we can “complete” the ⊕ and ⊗ operations? Take ⊕ first: it seems more than plausible to accept the equations I ⊕ I = I, D ⊕ I = I ⊕ D = D. This leaves us with K. K ⊕ I = I ⊕ K = K, which is acceptable, as is K ⊕ K = K. Two cases are now left: K ⊕ D = ? and D ⊕ K = ? One might suggest that the answer should be K, as an additional distinction cannot make the re-entry disappear. If that sounds reasonable, then we find the following matrix:

⊕ D K I

D D K D

K K K K

I D K I

« 8 » A similar exercise for ⊗ will lead to the following matrix:

⊗ D K I

D I K D

K K K K

I D K I

« 9 » That I ⊗ D = D ⊗ I = D and I ⊗ I = I seems plausible enough. I did take some liberty here as far as K is concerned. That K ⊗ D = K is guaranteed by definition and that K ⊗ I = I ⊗ K = K is defensible, but that leaves us with K ⊗ K and D ⊗ K. I am aware that we can have a nice (and necessary) discussion about whether these statements even make sense but, as can be seen from the matrix, I have assumed that the “absorbing” power of K remains. So, both expressions are equal to K.

« 10 » Here is the upshot: these ma-trices are familiar to a non-classical lo-gician. They correspond to the matrices for disjunction (⊕) and the negation of equivalence (⊗) in the three-valued Bo-chvar logic, B3, that is, if D is interpreted as 1 (truth), I as 0 (falsity) and K as ½ (paradoxicality). See Alexander Karpenko & natalya Tomova (2017) for a detailed analysis of this logical system. disjunction seems quite plausible as the concatenation of two marks often has been and still is in-terpreted as a disjunction, but the result for ⊗ is rather surprising (and, to be hon-est, it is completely unclear for me how to make sense of this). But there is more. In this formal logical framework it is impor-tant to talk about “designated” values. Usu-ally the designated value par excellence is 1 (representing truth), as one is, classically speaking, focused on conservation of truth. Thus, given p ⊢ q we want the truth of p to guarantee the truth of q. But in paraconsis-tent logic ½ is also included in the set of designated values. So not only truth, but also paradoxicality is conserved. Therefore, if p and ~p have both a value ½ and q has a value 0 then we can reject the ECQ and the acceptance of contradictions no longer poses a problem.

« 11 » I do realize that the above para-graphs presented nothing but a single rather specific example. At the same time,

it might also have the status of an exemplar. There is a wealth of paraconsistent logics at present – see Graham Priest, Koji Tanaka & Zach Weber (2017) for an overview – so many connections can be investigated in order to better understand both the laws of form and the logician’s approach to in-consistency. That being said, there is an important caveat to make: if the interest in Spencer Brown’s Laws of Form was mini-mal on the side of the classical logicians, see as an example Bernhard Banaschewski (1977) for a rather unkind evaluation, so it is equally minimal on the side of the para-consistent logicians. The former attitude is to a certain extent understandable: once the re-entry form is introduced, ECQ trivi-alizes their logic. The latter’s position how-ever is rather bizarre and I know of only one logician who has written about it and that is Walter Carnielli. In Carnielli (2009) he remarks that the third “imaginary” truth value, i.e., K, surpasses the capacity of the Boolean universe and “has some obvious connections, never explored, to paraconsis-tent thinking” (ibid: 207, my italics). Such a statement sounds like an invitation to start that exploration. Given the formidable task that Varela started so many years ago, it seems obvious to me – and I think the same holds for Kauffman, given his efforts to continue the Varelian research program – that whoever can participate in it, should do so. After all, it is about understanding life, the universe and everything through constructing the required distinctions that make sense of it.

Jean Paul Van Bendegem is professor at the Vrije Universiteit Brussel, Belgium, director of the Center for Logic and Philosophy of Science at the same university,

http://www.vub.ac.be/CLWF, and guest professor at the University of Ghent, Belgium. His research

is in the domain of the philosophy of mathematics, especially the study and philosophy of mathematical

practices. He teaches logic and philosophy of science.




http://www.vub.ac.be/CLWF

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the Mathematics of autonomy arthur M. collings

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the Mathematics of autonomyArthur M. CollingsIndependent Scholar, USA otter/at/mac.com

> upshot • I strongly affirm the author’s reflections on the mathematical contri-butions of Varela, and also offer related observations about the mathematics of autonomy.

Varela’s principles of biological autonomy« 1 » I heartily endorse Louis Kauffman’s

effort to rekindle interest in the mathemati-cal works of Francisco Varela – and equally his focus on Varela’s 1979 book Principles of Biological Autonomy, a work that is deeply mathematical, uncompromisingly cybernet-ic, and highly deserving to be back in print. Kauffman’s account is engaging and directly addresses the key mathematics at the core of Principles in a series of personal, histori-cal, and conceptual reflections. The points I will make affirm the author’s viewpoint, pro-vide additional context, and are organized around these themes in the subject article:

� distinction and the existence of imagi-nary logical values;

� Categories and complementarity; � Eigenbehavior, and reflexive domains.

« 2 » Varela has generalized the con-cept of autonomy from living, autopoietic systems. Unlike autopoiesis, autonomy ap-plies not just to living systems but to non-biological domains such as conversations or social systems (Varela 1979: 57). Autonomy means self-guidance, or “the assertion of the system’s identity through its internal func-tioning and self-regulation” (Varela 1978: 77). Autonomous systems are organization-ally closed, exhibit structural coupling and recursion, and have boundaries that are de-fined by observer-participants.

Distinctions and imaginary logical values« 3 » Self-reference and circularity are

central to organizationally closed systems, and were driving motivators in Varela’s (and Kauffman’s) research dating from the 1970s. distinction and indication, as described by George Spencer Brown in Laws of Form (Spencer Brown 1969), play a key role in

Varela’s epistemology: in fundamental rec-ognition that the boundaries of a system are actively drawn by the choices of observers, and because the mathematical formalism provides a basis for recursion and self-ref-erence. Kauffman and Varela both express keen interest in imaginary logical values, which Spencer Brown insightfully describes as reentering forms that are analogous to the imaginary numbers, especially when the lat-ter are regarded as solutions to the equation x = –1 / x. Varela states:

“ By allowing an antinomic form (from the point of view of logic), we have constructed a new, larger domain akin to the complex plane, where new forms can be lodged […] Again, rather than avoiding the antinomy, by confronting it we enter a new domain. This intercrossing of phenomenal domains at the point of self-reference is of course encountered repeatedly in nature; its typical pro-cess is the emergence of autopoietic systems.” (Varela 1979: 169)

« 4 » Varela introduced an extension to Laws of Form that added a third value, the re-entering mark , the shape and behavior of which resonate with the cybernetic im-age of a snake eating its tail (Varela 1975). As Kauffman makes clear (§3), Varela’s in-tuition is that the re-entering mark is syn-onymous with autonomous systems.

« 5 » The introduction of imaginary logical values intends to resolve the prob-lems that arise with self-reference. As it stands, the variety of representation for these values includes Spencer Brown’s … , Varela’s autonomous value , and Kauff-man and Varela’s phased pair of imagi-naries I and J (Kauffman & Varela 1980). Research continues, and to these I have recently added a fourth representation, in a calculus called Brown-4 or B4 that re-calls Kauffman’s earlier idea of the “square root of a distinction” (Kauffman 1987). B4 employs a right-facing mark with modu-lus four cancelation, = , in which two nested right marks are equivalent to the fa-miliar left mark = . B4 is a functionally complete, fully expressive system in which Aristotelian and non-Aristotelian charac-teristics coexist (Collings 2017). Kauffman and I have subsequently jointly extended this approach to a 16-valued system based on Hamilton’s “Quaternions.”

categories and complementarity« 6 » The principle of complementar-

ity is vitally important to Varela, for whom autonomy requires the openness to regard opposing concepts as complementary rather than mutually exclusionary.

“ A reductionist attitude is strongly promoted, yet the analysis of a system cannot begin with-out acknowledging a degree of coherence in the system to be investigated: the analyst has to have an intuition that he is actually dealing with a co-herent phenomenon. Although science has publi-cally taken a reductionist attitude, in practice both approaches have always been active. It is not that one has to have a holistic view as opposed to a re-ductionist view, or vice versa, but rather that the two views of systems are complementary.” (Va-rela 1979: 104)

« 7 » We can see in Kauffman’s account that the idea of complementarity exists pro-totypically in the concept of distinction. When we draw a distinction, the boundary fits the bounded like a glove; what is marked is defined by what is unmarked (§16).

« 8 » Category theory, which Kauffman has chosen to focus upon, embodies the concept of complementarity mathematically. For Varela, categories and “adjunct func-tors” provide a precise mechanism to express complementarity in autonomous systems. As Kauffman relates, every category is asso-ciated adjunctly with a directed graph, and vice versa (§36). For Varela, the adjunct rela-tion between networks and trees is especially important, and corresponds to what he sees as the complementary relation between au-tonomy and control (Varela 1979: 91).

« 9 » Consider the diagram in Figure 1, which contains a simple set of instructions to bake a cake. As drawn, it reenters itself with the reflexive instruction “eat enough cake to keep eating cake.” The diagram describes an autonomous system: a primitive cake-eating entity whose “life” consists solely of baking/eating cake, and is structurally coupled with the supermarket. The life arc of this entity is subject to external perturbations such as inventory depletion at the supermarket and ants in the cake mix, but is autonomous and exhibits closure. However, from a comple-mentary but non-autonomous perspective, the entity simply receives external inputs and responds with outputs.



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Eigenbehavior and recursive domains« 10 » Varela and Kauffman each ex-

press their thinking in terms of eigenbe-haviors and reflexive domains. Eigenbehav-iors are iterative, circular actions in which each transformation, whatever variation it entails, leaves some aspect of the system

invariant as a fixed point. Reflexive do-mains are systems in which every element is transformation of the domain itself – an agent “who transforms the whole space […] to itself ” (§49). For Varela, the ideas of autonomy, reflexive domain, recursion, and eigenbehavior intersect in a network description:

“ The idea of a solution of an equation over the class of infinite trees is an appropriate way to give more precise meaning to the intuitive idea of coordinations and simultaneity of interac-tions. The self-referential and recursive nature of a network of processes, characteristic of the autonomy of natural systems, is captured by the invariant behavior proper to the way the com-ponent processes are interconnected. Thus the complementary descriptions of behavior/recur-sion (cf. Chapter 10) are represented in a non-dual form. The (fixed point) invariance of a net-work can be related explicitly to the underlying recursive dynamics.” (Varela 1979: 170)

« 11 » Varela saw a need to “diversify,” to push the theory of autonomous systems beyond a strictly indicational approach (Va-rela 1979: 169). In this, he turned to a col-laboration with the mathematician Joseph Goguen, in which they generally followed and greatly refined the path-breaking work of another mathematician, dana Scott. Scott’s work resulted in establishing a sound model for the “untyped” l-Calculus (Lamb-da Calculus), and revolved around proving a way to construct reflexive domains.

« 12 » The l-Calculus is an abstract model for computation, developed by Alonzo Church in the early 1930s. The l-Calculus is known to be equivalent to a universal Turing Machine and thus theo-retically capable of executing any com-putation that can possibly be performed. Expressions in the l-Calculus are based on three basic concepts: variables, arguments, and “l expressions” (l expressions define functions). Here are several simple exam-ples of functions defined in the calculus:

F1(x) = (lx. x) yaya → yaya

F1(x) simply replaces the variable x with the argument “yaya,” which it returns.

F2(x) = (lx. 2·x) 5 → 10

F2(x) accepts the argument 5, which it mul-tiplies by 2, and returning 10.1

1 | The definition of F2(x) is given here to il-lustrate the concept simply but takes considerable liberties. neither numbers nor arithmetic opera-tions have primitive definitions in λ. Both must first be defined using λ-expressions.

While the mix is

lumpy, stir with fork

Pour smooth mixinto pan and bake

Cook for 25 minutesand eat

Always eat enoughcake to keepeating cake

To bake a cake,follow the instructions

Do I have cake mix?

Combinecake mixwith H20

Obtaincake mix

from supermarket

NoYes

Figure 1 • Instructions for baking a cake.

d

∞…20 1

c

10

b

0 1

a

1 2 3

Figure 2 • “Lifting” a set.



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F3(x) = (lx. x x) (lx. x x) → (lx. x x) (lx. x x)

F3(x) results in the creation of a simple loop, essentially similar to Kauffman’s single loop in §39. Here, F3(x) replaces each of the two instances of ‘x’ with the argument, (lx. x x). Thus, F3 returns (lx. x x) (lx. x x). Remark-ably, this simple looping mechanism forms the basis for constructing arbitrarily com-plex, recursive functions in the calculus.2

« 13 » Scott’s concept of “flow dia-grams” gives an intuitive way to understand the problem he addressed. Flow diagrams are directed graphs that represent the flow of states and operations in a computer pro-gram (Scott 1970). The diagram in Figure 1 represents an example of a flow diagram and illustrates the concepts of an “if-then-else” construction and a “while loop.” Scott’s theory embraces the idea that while loops can loop forever, with the construction of a particular class of infinite lattices he calls d∞, which constitute reflexive domains.

« 14 » A simple way to describe the no-tion of a lattice is using the concept of “lift-ing” a set. Given a set such as {1, 2, 3}, {0, 1}, or even the infinite set {0, 1, 2, …} we imag-ine the existence of a new element, called ⊥ (the “bottom”) that is less than every ele-ment in the set. Figures 2a and 2b show ex-amples of lifted sets. Similarly, we can also imagine the existence of another element ⊤ (“top”) that is greater than every element, and thereby constitutes an “upper bound” for the other elements. Figures 2c and 2d show a four-element and an infinite exam-ple. In every case, the numbered elements are considered to “incomparable” to one an-other in the lattice ordering. The lattice in 2c, which Scott calls the ⊤ lattice, is used by Scott to define the representation of a flow diagram’s if-then-else construction. nuel Belnap later used the concept of the ⊤ lat-tice to define what is now called a “bilattice” (Belnap 1977). Bilattices have known ap-plication to logic programming using fixed points (Fitting 1991), and this author’s B4 calculus is equivalent to the four-element bilattice (Collings 2017).

« 15 » Scott shows that the infinite lat-tice d∞ has the special characteristic of be-

2 | A “typed” version of the calculus would not let a function take itself as its own argument.

ing a “continuous lattice,” which means that successive values and successive states of an iterating while loop can be approximated by a limit-based approach. Consequently, there is a meaningful way to compute recur-sive expressions in the calculus. For Varela this result is a deep, mathematical and com-putational confirmation of the coordinated, recursive eigenbehaviors that he alludes to (§10 above).

conclusion« 16 » near the end of his target article,

Kauffman demonstrates that starting from the standpoint of a reflexive domain, it is possible to derive (rather than assume) the principles of autonomy. He suggests that there may be advantages in doing so, such as expressing fixed points without the need for excursions to infinity (§§49–54, pre-amble). This very interesting idea deserves further consideration, and seems to reflect the notion that languaging constitutes a re-flexive domain, even though few of us will have thought to know that.

« 17 » Varela’s theory is important both as a biological theory and as a general math-ematical theory of recursive domains. Fol-lowing publication of Principles of Biological Autonomy, Varela authored over 20 papers on the network theory of the immune sys-tem and is credited with being one of the originators of the theory of artificial im-mune systems. Varela expressed optimism that Scott’s approach to theoretical com-putation could be applicable to modeling living systems. Evaluating specific results is outside the scope of this commentary, but an extremely interesting range of opinion has been expressed by his former collabo-rators (Bersini 2002; Coutinho 2003; Vaz 2011).

art collings is a cartographer and conservation planner by profession, and a many-valued mathematical

logician by avocation. He lives in New York’s Hudson Valley in the town of Red Hook, and is the Treasurer of the American Society for Cybernetics.


Moving toward a Paradigm shift by Developing that Paradigm shiftRobert J. MartinTruman State University, USA rmartin/at/truman.edu

> upshot • Kauffman’s target article ex-plicates Spencer Brown’s Laws of Form and Varela’s Calculus of Indications as a way of thinking about the observer and the observed. This commentary points out that thinking about observer and observed in this way can also be a way of thinking about learning and meaning.

Process and object« 1 » I relate Louis Kauffman’s target ar-

ticle to areas outside mathematics, focusing on three statements (and their development) that offer anti-intuitive insights into our experience of the world. By anti-intuitive I mean statements that contradict a realist way of looking at the world. In each of the follow-ing statements, the clause before the colon is my interpretation; the statement after the co-lon is from Kauffman’s text. Each statement (and its development in Kauffman’s article), offers an exploration of ideas about the pro-cess of constructing the world – ideas also developed by Jean Piaget, Heinz von Foer-ster, Humberto Maturana, Ernst von Gla-sersfeld, and elsewhere by Kauffman himself:a Object and process cannot be separated:

The re-entry mark K “can be seen to cross the boundary between object and process. It is at once both an object and a process […].” (§8)

b We define and understand what some-thing is by understanding what it is not: “[…] any given ‘thing’ is identical with what it is not.” (§16)

c Observer and observed cannot be sepa-rated: A development “that can bring forth the potential that cybernetics has a nexus where the observer and the ob-served are not one nor are they two.” (§47)

Re-entry« 2 » In the remainder of the commen-

tary I connect the three statements above, as well as the idea of re-entry, to perception



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and learning. These ideas from the math-ematics of Kauffman, Francisco Varela, and George Spencer Brown are a good fit with key constructivist ideas about perception and learning. In particular, they provide ways of thinking with non-Aristotelian logic that fit constructivist thinking about autopoiesis, learning, and other topics not covered here. Traditional ways of thinking about perception and knowing (including scientific knowledge) are rooted in Aristote-lian logic and a realist view of the world. For every statement in Kauffman’s article there could be a statement contrasting it with traditional logic or traditional ideas of the world as a world of objects that exist inde-pendently of us.

« 3 » The marked state, the unmarked state, and the reentering mark provide a compelling set of metaphors for the process of learning and meaning-making through repeated entering into interaction within a domain. Meaning arises from entering into a process again and again; we interact with someone or something – something/some-one that may not yet make much sense to us – until we invent a way to make this new experience meaningful. Initially we do not perceive that something as an object that has the attributes we later come to expect it to have. Our naive experience is very different: someone points to an object and we perceive it. For example, students may think that they see poison ivy when the teacher says, “This is poison ivy; remember and avoid,” but they do not see it as a member of the cat-egory “poison ivy plant” and they cannot identify it in other situations without being put through processes such as picking it out among a group of plants, drawing it, learn-ing the difference between leaves and leaf-lets, and so on. Further, as Jerome Bruner (Bruner, Goodnow & Austin 1967, see also Martin 1970) pointed out, names for objects (for example “tomato” or “poison ivy”) are categories we use to identify things; there are no categories in the environment; they exist as abstractions within us. Still further, our perception of distinctions is always in relationship to ourselves. What we consider the properties of objects – their form, color, taste, smell, hardness, and so on, is always in relation to our own perceptions. For exam-ple, a good musician knows that the secret of playing loud (forte) is to play soft (piano):

without playing something that is not loud, you cannot give the impression that you are suddenly playing loudly.

« 4 » One conclusion from the above: The word “perception” privileges the idea of apprehending something external to us; but when we finally are able to perceive “poison ivy,” we have supplied the process of catego-rizing that allows the object to appear in the environment. We do this through repeated reentry via processes that include (in ad-dition to looking) conversation, reflection, writing, and through practice in a domain – whether it be gardening, plant identifica-tion, or neurosurgery. Making re-entry both an object and an action is a way of hold-ing the ambiguity of perceiving objects in suspension: observing an object cannot be resolved into either an object or a process. In perception, the process disappears from view, along with a realization that repeated reentry has allowed us to distinguish the object from what it is not. A constructivist description allows us to re-capture what has disappeared: Observer and observed are not two; neither are they one. These insights not only allow us to better understand percep-tion and learning (and perception is already a result and also a cause of learning), they also provide a powerful tool for designing learning and other related processes, such as designing itself and creativity in general. Kauffman (as well as Spencer Brown and Va-rela) did not invent the constructivist view, but their mathematics and language provide another description that fits well with the biology of cognition and with the cognitive sciences, and that provides another vantage point from which to explore and develop the constructivist project to create a more com-plete paradigm for how we understand our-selves and our relationship to the universe.

We are seldom without meaning« 5 » Traditional ideas of learning as-

sume a bifurcation between teacher and student and between knowledge and ig-norance, but research suggests that human beings developing normally are almost never without understanding of the world in which they live. That is, we always perceive a world that is whole and entire. As our ex-perience grows, we continue to perceive a world that is whole and entire, only larger or different. We may be confused about

things we perceive, but overall, regardless of age, we find ways of making sense of the world in which we find ourselves. Even in a new environment, as long as we can en-compass the new environment within our existing understanding, we feel no loss of continuity. Spend time with a two-year-old, and you find that the two-year-old has no more problem of making sense of the en-vironment than we do. It is a different un-derstanding at a different level of complexity (the child’s understanding is qualitatively different from the adult’s understanding), but it is usually whole and entire. Further, the two-year-old, even with her limited vocabulary, has little difficulty expressing herself. She makes use of the language she has to deal with her world. She makes her way about in her world without difficulty (assuming she is not in a situation that pro-vokes fear) in a complete and satisfying way – all with very limited experience and lan-guage. That is, like the ouroboros, from birth we are whole and entire just as we are, even as we are developing our schema for per-ceiving and acting. At each moment we pro-cess the world (which is also ourselves – and not ourselves) through the circular process of perceiving and acting. (Piaget describes this as the process of building and refining schema through a process of assimilation and accommodation in which behavior and perception refine themselves through their interaction.)

seeing by learning to act, acting by learning to see« 6 » All of the above are reflected in

von Foerster’s (1973) aphorism, “If you de-sire to see, learn how to act.” We might also add a second, “If you want to know how to act, learn how to see.” The process is circular and cannot be separated. Earlier this sum-mer I set up a catamaran for the first time in two years – a task that required interacting with the boat, with a setup manual, and with my companion. On the day prior to setting up the catamaran, we watched a video on how to set up the boat. The next day, with frequent references to the manual and con-versation, we got the boat set up. In the eve-ning of that same day, we again watched the video showing how to do the setup – and I immediately caught several points I had missed when watching the same video the



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previous day – no doubt because I had held the equipment in my hands and performed the actions necessary to set up the catama-ran, had had conversations that enabled me to reflect on the process, and, as I watched the video, reflected on what I had seen and done. The problem is that we all learn in this way, but then we forget how we learned. We ascribe meaning to a speaker or a text and not to the continual interaction of percep-tion and action – which often include re-flection and conversation. What I want to happen is for all the insights of Kauffman, Spencer Brown, Varela, and others, to al-low human beings to view the world in this deeper way so as to increase the number of choices available in seeing the world and acting in it.

Paradigm shift as learning« 7 » A deeper understanding of how

objects arise and become clear through in-teraction affords the design of contexts and opportunities for these processes to take place. designing does not provide control; designing can allow a space where conversa-tions and practice can involve learning. For example, we can see that Kauffman’s target article is designed to involve the reader in thinking through the processes that the ar-ticle presents.

« 8 » By explicating Spencer Brown and Varela, Kauffman also explores and develops the constructivist paradigm of the observer and the observed – a paradigm that contradicts the existing, fiercely held paradigm of observations (e.g., scientific observations) as objective. The paradigm of the observer and the observed as nei-

ther two nor one arouses an intellectual and emotional resistance that is difficult to overcome. Known ways of presenting ideas in a way that allows them to be considered by those with an otherwise intractable re-sistance include storytelling, humor, and (I suggest) formalisms. The beauty of stories and formalisms (Kauffman has also used humor, though more in other papers) is that they require neither belief nor acceptance for us to understand them. They can be con-sidered on their own terms. As Kauffman points out, mathematics is an invention; it does not have to correspond to anything we know or experience.

« 9 » Kauffman enables readers to un-derstand the formalisms of Spencer Brown and Varela-formalisms that provide entry into the thinking that is embedded in the bi-ology of cognition (Maturana 1970) regard-ing the observer and the observed. As we have discussed in this commentary, it is not the presentation of ideas that allows us to understand them, but the repeated reentry into a domain that allows us to understand the ideas that constitute it. Kauffman does not present his ideas in the usual way, he in-vites the reader to think through the math-ematics that constitute his understanding. The reader willing to undertake the effort to join him in this process develops an under-standing of the domain that constitutes the ideas. This is a powerful tool to introduce constructivist ideas without creating the usual resistance from non-constructivists. As soon as readers begin to grapple with the formalisms Kauffman presents, they are working with the constructivist and second-order cybernetics ideas.

« 10 » In concluding, I point out that Kauffman’s mathematics do not prove any-thing beyond their own mathematics (nor does he mean them to); they are explora-tions of ways of thinking with mathematics that are also relevant to how we think about observers, cognition, and other ideas central to the biology of cognition, constructivism and second-order cybernetics. Kauffman never allows realist language to sneak into his formulations. Kauffman is consistently careful in his language to make clear that his explications are meant to explore some-thing about the nature of mathematics and how we think; he carefully avoids assump-tions about these explorations as revealing something about the universe. I appreciate this subtle difference and recall the Sufi no-tion of bliss: Heaven is a subtle conversation.

Robert J. Martin is a composer, psychologist, and professor emeritus at Truman State University. He completed a doctorate in educational psychology

at the University of Illinois at Urbana-Champaign with an interdisciplinary thesis guided by Heinz von

Foerster and Herbert Brün. He has a life-long interest in composition, creativity, learning, psychotherapy,

constructivism, and cybernetics/systems science. He has composed music for a variety of solo instruments

and ensembles including two releases by Parma Recordings on the Ravello label. He maintained a private practice as psychotherapist, working with

children, adults, and families. A long-term member of the American Society of Cybernetics, he is currently

serving as ASC Executive Board Secretary.




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author’s Responseself-Reference and the selfLouis H. Kauffman> upshot • In this response, I revisit the themes of my article on the mathemati-cal work of Varela and I attempt to clarify the linguistic and human meanings of the terms “self-reference” and “self.”

« 1 » I shall begin by addressing issues raised in the commentary of Klaus Krippen-dorff who wrote:

“ [Kauffman] seems not only to attribute sign qualities to the mark, but lacking an obvious refer-ent, he claims that the mark refers to itself – with-out acknowledging this to be his projection. This is a rhetorical move, and only that, of granting a mark on paper an undeserved observer-indepen-dent agency: that of a self-referring self.” (§11)

« 2 » I wish to clarify my position and to assure readers that I most certainly do acknowledge this to be a projection of “my-self ” or a description of what I believe an observer of the mark does. The mark is an abbreviated box and is understood to in-dicate an inside and an outside. As such it is seen, by the reader (who might be my-self or another) to make a distinction in the notational plane. When I say that the mark “makes a distinction” I am indulging in a shorthand that could be expanded as “the mark and the reader (observer) come to make a distinction that appears to be located in the perceived notational plane.” I use this kind of shorthand throughout in order to avoid the pedantic. I see now that it can lead to serious misunderstanding, but what can one do? The mark is seen by the observer to make a distinction. The mark in the calculus of George Spencer Brown is intended to refer to a distinction, not neces-sarily itself. But the mark can be taken (by the reader) to refer to itself. In this sense, the mark can be regarded as self-referential. Self-reference requires an observer (e.g., a reader).was highly aware of this point and made his awareness quite explicit in his Laws of Form, where he says: “We see now that the first distinction, the mark and the observer are not only interchangeable, but, in the form, identical” (Spencer Brown

1969: 76). By the time this identification has occurred, the mark is not just a mark on the paper. The mark has become a “sign of the self ” in the sense of Charles Sanders Peirce (Singer 1980). The mere mark on the page has been promoted to the mark, standing for and being identical with the observer and with the first distinction. As the reader can see, in the end I must plead guilty to Krip-pendorff’s accusation, and I do not repent.

« 3 » “This sentence has thirty-three letters.” The sentence in quotes is self-ref-erential as indeed it does contain 33 let-ters. This use of the word self-referential is standard in the English language. There is no metaphysical, social or emotional self that is involved with the sentence except for the reader’s. In this use of the word self-reference, I do not invoke a self except that there must be an observer, a reader of the sentence, to bring forth the actual refer-ence of the sentence to itself. The necessity of the reader occurs throughout all work with texts and with mathematics with its formal systems and rules. Whether I care to invoke the essential identity of the self that arises in each distinction that occurs, or to just make external references to the char-acters in the sentence, there is nevertheless not one dot, one iota of written expression that has any meaning or existence without the context and presence of a reader or an author. Of course, it is natural to assume that the physical details and relationships of the text persist when our attention is diverted. We make no mistake when creat-ing a symbol or a sign that points to itself, but we must understand that the pointing is done by us. The distinction is something that I mean.

« 4 » We say that all is well just so long as we understand that everything is seen in a context with the eyes of an observer. But we do not know who the observer is, nor do we know who the reader is. And lacking an obvious referent, we claim that the observer refers to herself – without acknowledging this to be the projection of herself. Alas, even if we acknowledge this self to be the projection of herself, we have not found her. She is nowhere to be found. I cannot con-struct the reader, the author or the observ-er. The observer arises with any distinction and that is it. She is a pole of the circularity when she insists on a polarity.

« 5 » In §12, Krippendorff says:

“ I am suggesting that claims of systems to have selves should be limited to those capable of artic-ulating their selves by answering the question of who they are, the identity they own, and for what actions they claim responsibilities.”Yes, indeed. none of this discussion makes any sense unless the text is coupled with a “sophisticated self,” and she is not you and not me and not Krippendorff. She is what arises in the distinction that appears in rela-tion to the text, and she is the text, as well. does the text insist on metaphor, or do you, the reader, resist this interpretation? Who is writing this text and who is reading it?

« 6 » I make a sign , and I ask you the reader to imagine it to be an abbreviated box . As such, the mark or the box makes (you make it make) a distinction between inside and outside in the plane. I invite you to interpret the mark as a sign that refers to this distinction. The distinction to which the mark refers is the entire process of you, the reader, interacting with the indicational plane of your page or word processor to come to find this distinction. It requires ev-erything that you are. It is a rhetorical device to say that there is a mere mark on the page. The existence of that mark requires all that you are in the whole universe that you have. The universe has opened and produced you, the mark, the page, the world. The mark has produced you and all. You have produced the mark and all. All has produced you and the mark.

« 7 » now repeat the previous para-graph without the mark. Use the word “nothing” instead of the mark: The nothing has produced you and all. You have pro-duced the nothing and all. All has produced you and nothing. Universe has emerged from nothing in relation to nothing, and form is emptiness, emptiness is form.

« 8 » I have taken the trouble to task my word processor with producing all of this to help Krippendorff to see that it is not just a monologue, this monologic that would dis-cuss everything with the help of nothing. In §17 he says:

“ In the social realm, I suggest replacing the ab-stract monologism of mathematical formalism by dialogical constructions of actual language use.”



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combined References louis h. Kauffman

Enaction


I say there is no monologue. Mathematics is actual language use. It is an extra distinction to cut the formal away from the creation of the formal. All is dialogue and mathematics per se is a part of the human conversation.

« 9 » It is time to move on to the very interesting commentary of Jean Paul Van Bendegem, about paraconsistent logic in re-lation to Laws of Form. He points out very perspicuous formalisms that can put the Varela calculus in the context of logics that would allow inconsistency. We have earlier placed our calculi in contexts of multiple-valued logic where one will not insist that “P or not P” is necessarily true. And I have discussed in a number of places the Flagg Resolution, where a self-contradictory state-ment (~K = K) is handled with kid gloves so that if one replaces K by ~K somewhere in an expression, then it must be so replaced everywhere. This enables us to keep the dia-logue, even in the presence of apparent con-tradiction. The suggestion to use paracon-sistent logics is very useful and I am looking forward to more dialogue about this. I am sure that some paraconsistent logic would aid in sorting out the present response to the commentaries.

« 10 » art collings adds very fine clarifi-cations about the role of complementarity, imaginary values and the lambda calculus. I agree with him that we need more explo-ration of the relationship between lambda calculus and the language that we speak, in

which the dialogue is apparently identical with the meta-dialogue. I suspect that we have not yet fully understood what is hap-pening in language that allows this to oc-cur without leading us into serious confu-sion. We may have become paraconsistent reasoners of a high order. This also speaks to Krippendorff’s worries about the rigidity of the usual mathematical approach (that makes perhaps too many artificial distinc-tions).

« 11 » Finally, I wish to thank Robert J. Martin for his very beautifully articulated words about how these matters of distinc-tion, reference and self-reference, and in-deed the self, come about in the actual dis-course of our lives. In his §3 he says:

“ The marked state, the unmarked state, and the reentering mark provide a compelling set of metaphors for the process of learning and mean-ing-making through repeated entering into in-teraction within a domain. Meaning arises from entering into a process again and again […]”He just says, and I agree, that these “math-ematical entities” can be seen to be meta-phors. I further assert that we must allow ourselves the luxury of speaking in meta-phors. We have to be able to say that the marked state is a metaphor. We need to say that the mark, the observer and the first dis-tinction are identical in the form. The saying is itself a metaphor. In making a metaphor

such as Shakespeare’s “Juliet is the Sun,” we assert the identity of what has been taken to be distinct. We assert the connection of what has been put asunder. We make the distinction that joins and in so doing we cre-ate a universe.

« 12 » Let us now turn back to Varela’s work. The question of autonomy must be separated from the question of self and selves. The re-entering mark can be seen to point to itself, to be self-referential, but this does not give that mark a self. The mark, as a mark on a page, is not a person. On the other hand, we interact with texts, and in our interactions come to endow these texts with meaning and reference, and the texts come alive for us (in a metaphor that must not be denied). It is in the autonomy or au-topoiesis of a biological organism that the possibility of a self can arise. The possibility can become an actuality, as it does for us in our interactions with other human beings through dialogue, through the coordination of coordinations that constitutes shared lan-guage. Here the delicate relationship of liv-ing selves and their support in the autonomy of organisms comes into full play and leaves us in the domains that Varela spent his life exploring (Maturana & Varela 1987; Varela 1979).

Received: 2 november 2017 Accepted: 5 november 2017

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