The Observer in the Observed:

Fractal Dynamics of Re-entry

Terry Marks-Tarlow
Santa Monica, California

“Do what you will, this life’s a fiction/And is made up of contradiction.” – William Blake


Most of us take for granted the ability to distinguish between ourselves as observers and what we observe in the world. Outwardly our skin seems visible proof of a clear boundary that encases and protects our organs. Inwardly our sense of self, when intact, also feels like a relatively clear boundary, at times even to the point of isolation from others. Yet whether we consider our bodies or minds, the subjective experience of closed boundaries rests precisely on the opposite state of affairs – wide-open portals that continually allow transaction between inside and outside, body and world, self and not-self.

 Open portals are evident in our “posthuman” existence (see Hayles, 1999), where the interface between human being and machine presents boundaries which have grown ever more complex over time, with each technological advance. We plug our consciousness into virtual realities, as we augment, even invade our bodies with the presence of machines. This intense exchange between flesh and mechanism demands nothing short of a redefinition of human subjectivity.

 Mystical poets, like William Blake in the above epigraph, allude to life as fiction inherently made up of contradiction. Hinduism offers the concept of Maya to describe the false perceptual veil by which we shield ourselves from an ultimately mysterious reality. At higher levels of cognitive organization, psychologists study related phenomena. For instance, Shelly Taylor (1989) identifies self-deception in the form of “positive illusions”, those overly optimistic attitudes and expectations towards the future that may be entirely unrealistic, but nevertheless her research shows they can help us to beat the medical odds.

 Despite the complexities of our alleged posthuman existence, most of us live as if consistency, certainty, predictability and clear boundaries, especially between truth and falsity, reside at the base of things, from the workings of our bodies and minds, to those of the universe at large. Boundaries are everywhere, yet most are permeable. By focusing on this highly contradictory state of affairs that extends invisibly under the surface, I follow Blake’s lead to explore the paradoxical dynamics embedded in the very fabric of existence.

 This paper traces a line of logic, begun by George Spencer-Brown and continued by Francisco Varela, which puts paradox at the heart and seam of things. I place Varela’s ideas about re-entry within the context of a branch of contemporary mathematics called fractal geometry. I argue that a deep understanding of fractals helps to illuminate the profound yet invisible paradoxes that permeate ordinary life.

 To set the stage historically, I will explain the cybernetics revolution and how reflexivity first entered the social sciences. Into this historical context, I then place the primitive logic of George Spencer-Brown, plus the extensions added by Francisco Varela. Together, their dynamics of re-entry articulate paradoxical foundations not only for logic and but also for the creation of all structure. Next I connect these logical assertions with mathematics of the complex plane, where imaginary numbers are used to model extra or hidden dimensionality. Imaginary numbers provide the bridge to fractal geometry, whose mathematics involves recursive iteration of simple formulas on the complex plane.

Fractals are dynamic process-structures that etch time into space. They are boundary keepers that negotiate spatial and temporal interfaces between different forces and dimensions of being. My thesis is that fractals provide the paradoxical foundation by which different levels of nature both connect and separate. Every boundary becomes a door, every border a portal. Because the same dynamics hold inside as well as outside the psyche, fractal geometry provides a bridge and language for linking inside and outside worlds. Whether they occur in nature, our bodies or minds, fractal separatrices or boundaries reveal infinite, hidden frontiers in the space between ordinary, Euclidean dimensions.

 I conclude this paper by examining the mechanics of fractal production to reveal a new twist in the reflexive march of science. In a world filled with fractals, not only is the observer detectable in the observed, but the observer is also embodied there, in a primordial, concrete way. Natural fractals, like shorelines and mountain scapes, reveal how the embodiment of the observer in the observed paradoxically precedes the presence of conscious observers.  


The period following World War II was a time of tremendous intellectual growth in America. Emerging from technology developed during the war, a number of trends converged to legitimate the scientific merit of psychology, including the birth of cybernetics, the science of information. This new field, spearheaded by the mathematician Norbert Wiener, mushroomed out of the interdisciplinary Macy Conferences held yearly between 1946 and 1953 (See Heims, 1991). Cybernetics brought a new metaphor of the mind as mechanism. Roots of this idea extend at least as far back as Renaissance times, when the natural sciences, one by one, split off from philosophy. As more empirical studies began, the heart resembled a pump, the body a machine, and the whole universe little more than clock works.  

The cybernetic association between mind and machine made in the mid 20th century proved a boost to the neurosciences, when neural loops in the brain were modeled as logical chains. This association also ushered in the cognitivist revolution, as activity in the psyche was likened to information processing in computers. Initially, the new metaphor of brain as computer was logically derived from the behaviorists, who compared human behavior to machine output based on environmental input. These stimulus-response relationships were both quantifiable and predictable, thereby turning the discipline of psychology into a fully-fledged behavioral science.

The more humanistically inclined raged against the cold, mechanical, and at times reductionistic views being espoused by behaviorists, psychoanalysts, and eventually cyberneticists. Meanwhile, within the Macy Conferences, protests of a different kind began to surface. Lawrence Kubie, a psychoanalyst and recent retread from the “harder” field of neuroscience, stimulated heated discussion among his colleagues by pointing to the problem of reflexivity (See Heims, 1991).

Reflexivity, by which an assertion points self-referentially to itself, e.g., “What I say now is false,” involves a confluence or melding between observer and observer. Reflexivity is inherent in the very subject matter of psychology. It occurs, for example, whenever researchers use consciousness to understand the nature of consciousness, narratives to study the narratives of others, or behavioral repertoires to examine behavioral responses in others. Research in psychology is like the mythical Uroborus, a snake eating its own tail. Despite every attempt to remain objective by sidestepping subjectivity, even behaviorists find little relief from the Uroboric beast of reflexivity.

During the Macy conferences, Kubie objected to the early cybernetic agenda of separating information fully from its material, embodied sources. The psychoanalyst protested that within any theory, even inside the “hardest” of sciences, reflexivity lurks and the observer lay hidden in the observed. Kubie claimed that all theories about the outside world say as much about the unconscious of the subject who espouses them as they do about the outside universe as consciously perceived. When it comes to theory making, no matter what is observed, the observer winds up implicated in the observed. Although Kubie’s protests were dismissed by most of his fellow scientists, his ideas about reflexivity later became ingrained within the history of psychoanalysis. Robert Stolorow and his colleagues (Atwood & Stolorow, 1979/1993), cofounders of intersubjectivity theory, argue that every theory of personality is self-reflexive in that it universalizes the therapist's personal solution to the crises of his or her own life history.

During the early years of the Macy conferences, the notion of science still rested upon the hitherto bedrock foundation of objectivity. By requiring a clear separation of subjects from objects, objectivity was a position that ran contrary to reflexivity. Because early members of the Macy conferences were interested in maintaining science as an explicitly objective enterprise, they chose to ignore Kubie rather than to revise their own ideas. Instead of including reflexivity within the rubric of science, they dismissed psychoanalysis as science.

Generally, during this first wave of cybernetics theory, the problem of reflexivity was successfully avoided by isolating pattern as a separate realm from which all others emanate. When the pattern of information reigns supreme, its material substrate can be first ignored and then eliminated from consideration altogether. According to this view, even without matter the pattern still matters. By removing information entirely from its material sources, the need for observers was also eliminated. We are left with only pattern as a virtual reality with neither observed nor observer. 

This strategy worked temporarily, but only until the whole enterprise of science began taking a reflexive dive. At the cosmic level of grand-scale events, Einstein’s earlier discoveries in physics destroyed the previously immutable framework of space and time. The notion of objective observation stretched and deformed, as relativity theory and the subjective stance of observers took center stage. Meanwhile, at the subatomic level of tiny, quantum events, another field spawned by Einstein’s work, consciousness began pushing its way self-reflexively into the middle. The still controversial Copenhagen interpretation asserts that at the quantum level, the very act of observation is necessary to materialize that which is observed.

Even mathematics was not immune from a reflexive fall. In the 1930s, an Austrian mathematician named Kurt Gödel used recursive methods in order to code numbers and then talk about them reflexively at a higher, meta-level of abstraction. In the process, Gödel proved that no single theory could ever provide a consistent, complete foundation to logic, annihilating any residual hopes for perfect objectivity within the mathematical underpinnings of science.

As reflexivity was seeping into the physical and mathematical sciences, a second wave of cybernetics arose between 1960 and 1985. Spearheaded by Francisco Varela, among others, information scientists became better prepared to embrace reflexivity (see Hayles, 1999). In fact, the very name of this new trend, “second-order cybernetics,” amounted to the recursive study of observers studying the higher order processes of observation: the observers observed themselves observing themselves.



Second-order cybernetics arose within a broad, societal sea change known as postmodernism. Over the years the use of this term has been stretched so far as to encompass practically everything, while being deconstructed so thoroughly as to mean almost nothing. For this reason, I beg to dismiss its broader definition in order to focus upon a single facet, its inherent reflexivity. In order to symbolize the postmodern imagination, Richard Kearney (1988) offers the recursive symbol of two mirrors reflecting one another. He contrasts this with the premodern imagination, symbolized by a mirror, in which human creativity reflects God’s creation, as well as the modern imagination, symbolized by a lamp, in which human creativity is illuminated from within.

Because of its reflexivity, the posthuman imagination becomes lost inside an infinite regress of imitations, copies and simulacra. With origins deconstructed into dust, the postmodern being is often portrayed as rootless, wandering inside a mechanical, artificial desert of re-production. Within this bleak frontier, on the one hand, the demise of human creativity and originality is decried. On the other hand looms the cybernetic threat of machines usurping the very autonomy, indeed existence, of their humanist creators.

In How We Became Posthuman, English professor Katherine Hayles (1999) details the threatened demise of embodied existence. She analyzes cyberpunk novels with heroes that evaporate into virtual reality, as their consciousness becomes thoroughly enmeshed and encapsulated within machines. The flip side of this futuristic nightmare portrays machines sophisticated enough to take over the evolution of life itself. As artificial intelligence becomes increasingly equipped with emotion, creativity, the capacity to learn, self-repair and self-generate, this sci-fi genre depicts humanoid machines that threaten to extinguish carbon-based evolution as we know it, replacing it with the far-superior, silicon-based life forms. 

As posthuman boundaries have become more blurred and human beings self-reflexively entangled with facets of their own technological production, lines between observers and observed continue to grow more complex. As reflexivity is integrated more and more consciously into cybernetics, one positive outcome is that the door is thrown open for the scientific study of subjectivity. Since subjects can now study their own subjectivity, consciousness itself has recently regained status as a legitimate and serious object of scientific study.

In the postmodern view, reflexivity is often viewed as a by-product of modern technology constructed in the context of particular social and economic trends. Contemporary methods such as neural feedback even allow us to become active observers of our own brain processes. Inarguably, computer-driven, cybernetic extensions of our perceptual and conceptual apparatuses do help us to detect, direct and even create reflexivity with greater ease. Yet I believe that the roots of reflexivity are much deeper and more organic than social and historical trends suggested by postmodernism. I maintain that the discipline of fractal geometry provides evidence that reflexivity is intrinsic not only to human-made productions, but also to nature at large. 

Fractals help us to advance beyond the cybernetic metaphor of psyche as mechanism to the more organic one of nature, including human nature, as fractal. Here mechanistic means of computer simulation reflexively guide us beyond mechanism, as we circle back to a different kind of origins, for both human and machine, in fractal bases of nature. Before turning to fractal geometry itself, the section to come presents Spencer-Brown and Varela’s logical underpinnings for reentry dynamics as they are embedded in the very fabric of creation.


A great truth is a truth whose opposite is also a great truth.

– Neils Bohr


When developing his “Laws of Form,” mathematician and logician George Spencer-Brown (1969; 1979) tried to specify how we create “some-thing from no-thing” in consciousness (See Robertson, 1999). Spencer-Brown used a 2-valued system that consisted only of “marked” and “unmarked” states plus two axioms. From these simple bases, he derived a calculus of first distinctions. Although it is commonly believed that George Boole (1958) developed the most basic form of logic, Spencer-Brown disagreed, claiming his own calculus is so primordial as to provide a cradle not only for logic itself, but also for the basic structure of any universe.

Within Spencer-Brown's system, in order to distinguish marked from unmarked states, value must be attributed to one state over the other. This act of marking or making a distinction requires an observer. We can readily understand this requirement for logic: in order to make a mark, apply a set of axioms, or distinguish truth from falsity, a conscious observer must be present. But how does this process of valuation apply for more primitive levels of a system that supposedly precedes logic and even people? Is an observer implicated along with the observed there too?  I will return to this issue in my subsequent discussion of fractal geometry.

As Spencer-Brown progressed with his work, he used basic axioms to derive higher degree equations. But then something strange began to happen: anomalies appeared; re-entry of equations back into themselves sometimes resulted in paradox. This occurred when marked states became equated with unmarked ones. Spencer-Brown offered an interesting interpretation. Rather than to view this as the simultaneous presence of contradictory states, he suggested an alternative. Maybe the system was oscillating between opposite states in time. If so, then self-reflexive acts of re-entry, or self-indication, would add the dimension of time to that of space already implied by first distinctions. Given enough time, both marked and unmarked can exist in the same space.

Neuroscientist and researcher Francisco Varela was intrigued by Spencer-Brown’s ideas, especially by his explanation for the dynamics of re-entry. Varela (1975; 1979) developed “A Calculus of Self-Reference” to extend Spencer-Brown’s work. In so doing, he took a bold, if not radical leap. Rather than to conceptualize re-entry as characterizing higher degree equations only, Varela proposed that re-entry be added at the ground floor, as its own term, along with the other two marked and unmarked states.

This simple difference made all the difference, as Gregory Bateson might have said. It signaled Varela’s departure from Aristotelian logic, which had held an iron grip around philosophers and logicians for millennia. Varela abandoned Aristotle’s dichotomous system, where all propositions are either only true or false; its law of identity, where A can never equal not-A; as well as its law of the excluded middle, where the space between truth and falsity is pristinely empty.

By adding reentry as a third term, Varela opened up an infinitely deep, Pandora’s box of middle ground filled with fuzzy grays, lost identity, and unfathomable complexity. Here not only can something be true and false simultaneously, but even more, Varela actually believed that the existence of autonomy in nature depends upon this contradictory state of affairs. Varela and his mentor, Humberto Maturana, coined the term “autopoeisis” to explain how biological systems self-organize (Varela, Maturana and Utribe, 1974). With re-entry dynamics at the core, autopoeitic systems embody paradox at their boundaries, expressing their autonomous functioning through remaining functionally closed, yet structurally open.

By asserting reentry as a third value in its own right, Varela agreed with Spencer-Brown that self-referential dynamics establish the presence of time. But he went even further, to assert that paradox becomes embodied at the most basic level, in the very form itself. Whether in organic or inorganic forms, autonomous systems appear supported by inherently contradictory underpinnings.

[The rest of the essay is continued here.]

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Rosch however, co-author with Varela and Thompson in The Embodied Mind, also maintains the ontological status of reality as such. We discussed her on p. 3 and following of the essence and identity thread. For example this from p. 4, noting the results of a meditative beginner's mind: “They find not only what they themselves really are already but what the world actually is.” It seems both Rosch and Varela were both students of the “shentong” tradition she describes so perhaps Varela also held this view, although as I said I'm not sure. But they both appear adamant that it is not through representational mind but through pure nonconceptual awareness wherein the “nondual” lies. It that way they seem more akin to Tom, as discovered in that thread.

Whereas for me I'm more in agreement with Joel in that the “nondual” resides in a conceptual “mind,” not in a pure nonconceptual realm (if I understand him correctly?). And yet my Lakoff et al. inspired version does not have this “mind” as representational, the latter being false reasoning. Lakoff and Rosch seem to agree on that point, though differ in terms of a pure nonconceptual awareness.


I know that Lakoff, Johnson and Nunez do not see this connection, that “representation,” which is indeed grounded in basic image schemas and metaphors, is not akin to the structure of reality itself. Rather they assert that the representation paradigm is a non-embodied “false” reasoning

Let me turn the question around. Can reality have a structure, is there a gap?  the reality we experience is it structural ? is the physical universe a structure or an experience? Is structure which is probably necessarily open interpreted as closed ? there you could have the meta contradiction - the fabric of structure. or, structure is the contradiction


since finite and infinite are self similar, is polarity between open and closed, or you have  experience which is an interpretation of open as closed. and I can feel that I have captured this intimately but it is eternally open. Then ultimate polarity is the collapse of polarity.  whole is non, non is whole . From terry’s paper – a great truth is a truth whose opposite is also a great truth – Neils Bohr


I don’t know about fluid dynamics, should look it up – but the term fluid refers to reality in a more realistic way – I guess what in this case a science might authenticate


I get fuzzy, I can’t remember where I was ^..^


Hi theurj and valli, I'd like to get back to your questions, and I was beginning, but combined events have found me breaking into fluidity once more and finding new ground in the process, so it's taking much longer to finish or complete or reach closure than I had anticipated.  But I thank you graciously for your stimulation here as it's helped me resolve some issues that were not clear.  I'm hoping I can distill it to respond here, perhaps in a week.  Thanks again. I wish I had the time like I used to to participate steadily in this forum.  So many great discussions going on here!


Hi Joel, cool how you find new ground. Look forward to your responses, when you find the time :)

I’ll pick up on a couple of issues above and Edwards points. If the nature of bodymind or embodied is considered, as an interpretation (or construct/open or closed/ or contradiction) that could be the difference between real and false reason.

Between body and mind , the functional separation is clear enough. As between the organism and representation/ conceptuality /consciousness . What the self similarity of fractals (and quantum physics, fluid/reentry dynamics (?) etc) tells us is that these boundaries remain structurally open. The seamlessness here is what I meant – beyond the finite/infinite divide  – openness implies not only between structures, but also between structure and emptiness/the unknown . It appears open or closed is contradictory, but open and closed is not, it moves beyond the prototype or constituents

Conceptuality as real reason gets closer to bodymind as an interpretation/contradiction. With embodied itself being true and false as a contradiction. False is false, and the livin is easy . but the trip is, open – true and false – false – and back (That’s sort of easy too, if I examined my head. Gol) then real reason always has to consider or be the contradiction, and false reason is not contradictory in itself (unless its abused !) amusement is written in the fabric of existence?


OK, I'll just kinda summarize things that I've been thinking so as not to take up too much time that I never get back to it, as I think it's an important question theurj asked about the relation between mathematics, representation and reality, and in particular the differences Terry or I may have with the views of Lakoff, Johnson and Nunez.  Now I can't speak about Johnson, because I haven't read him, nor can I speak for Terry, of course.  But I did recently read Where Mathematics Comes From (WMCF), and I can speak from my reception at this interface.  


In general my impression was simultaneously one of exhilaration at the possibilities of cross-fertilization with the information, and one of exasperation at the linearity of history.  Mostly I just felt like the book was the missing body of empirical data needed for my more intuitive groping in my own experiential recesses.  But the material presented seemed to call out for a deeper framework to make sense of things, and there was an occasional such issue here and there. But only a few centered on the issue of the relation between reality and mathematics.  


I think the issue is mainly this.  In my work I drill down to what appears to be some of the primitives of conceptual and mathematical thought.  At this level they are one with the evolutionary roots in perceptual instinct or the primitives of sensory-mnemonic intelligence within the brain emergent from its recapitulation through embryogenesis.  I also demonstrate that the conceptual evolution of mathematics is in itself a fractal, and that (now, here mostly in my second book, on Sorce Theory, which was to be part of SZII) it recapitulates some of the key forms of ontology and an emerging nondual meta/-physics.  


But more importantly, and this I am just beginning to realize, thinking of the brain/mind (spirit, soul, ...) itself as fractal, as is becoming more clear by the hour...this realization of the Fractal Nature of the cosmos itself, this Fractal View of mind radically changes everything.  This is because along with the fractal comes the recognition of self-similarity, and new dynamic concepts like reflections at boundary conditions, attractors, structural couplings, etc.  And so, no longer do we have to say that mathematics holds some Platonic absolute form of reality or that it absolutely doesn't.  It's recursive, similar and emergent novelty.  We can understand the art aspects of mathematics, even the fractal itself, while still understanding the deep relation in the similarity in the recursion, and we can begin to see this duality itself as a bifurcation at boundary conditions, albeit of a more abstract sort, perhaps, but also likely with a very real counterpart in a dynamical process(s).  So, simply put, with these ideas we finally have a modeling language complex enough for the task of deepening the analogies in understanding the evolutionary and embryological embodiment of some of these core 'mental' constructs/inventions/discoveries, like Mathematics.


So from this view, then, Representation, indeed Mathematics as well, is "a self-similar echo of reality itself."  Indeed it is nothing other.  It is also a structural coupling of subject/object and ontic/epistemic, and hence of Reality and Representation (in a local pocket) through the process of evolving intelligence and consciousness.  A feedback loop, involution in evolution, as the edges of Mathematics finally reach into its foundations at the Fractal (and indeed there is a strange epistemic, laser-like resonance or autocatalysis that occurs at this stage, it seems).  This is precisely the reason for the uncanny effectiveness of Mathematics in dealing with reality.  The real empirical/perceptual traction at the heart of the proto-conceptual infrastructure of Mathematics and indeed relation itself.  


And in the sense that a child has a certain purity, so too is Mathematics founded on a similar purity at the heart of relation.  There is a certain perfection and deep and real possibility at this level.  And even more clearly, this core abstraction is the simple intuitive/instinctual understanding of dynamic boundary.  It shows up as the primitive number itself, and recapitulates through a process of expansion (evolution) and contraction (involution) into the closure of a new set.  Closure is this dynamic boundary (a recapitulation of the ego itself), this transcendent drive for comprehensive or exhaustive expansion or evolution of the forms of the dance of Operation and Number (the verb/noun, object-event/dynamic-boundary itself) contracting or involuting into a new unity or set.  When you look for it, it's really a very fractal form of the exploration of the "boundary conditions" of boundary itself, or simply put, Mathematics is the conditions of boundary.  And at the same time it is a strange inversion or mirror image (reflection and bifurcation) at and as these boundary conditions of boundary at the interface of the cogito itself, as it magnifies up into the collective as the ontic-epistemic interface, which is yet again another layer (noosphere) in the deepening self-similar holonion. 


And this brings us back to your question, theurj.  You gave the following quote in contrast to my quote that "Representation ... is a self-similar echo of reality itself." 

“Varela, Thompson and Rosch argued that the standard division between pre-given, external features of the world and internal symbolic representations should be dropped, as it is unable to accommodate the feedback from embodied actions to cognition via the actions of a situated cognitive agent.” (Stanford encyclopedia)


...and I actually think these quotes are in agreement, though not explicitly so.  I am indeed arguing that the standard division should be dropped, mainly because it is not informed from the Fractal View.  It's an either/or proposition.  Either Mathematics is fundamental or absolutely real, or it is fundamentally fantasy or art and only approaches reality in some respects or capacities.  The Fractal View allows us to see the false dichotomy here, or at least their integration as the bifurcation at the boundary conditions of the new layer of abstraction which is the heart of embodied Mathematics.  Mathematics is Art and Science of Pure Relation, is how I expressed it in SpinbitZ, which is the integration here of this bifurcation at the heart of Mathematics, in that it can explore new worlds in its capacity as "disembodied", emergent, or abstracted relation, and at the same time it is a new and deeper layer of this real relation. Mathematics is dealing with metaphysical principles, which are at once immanent to Representation itself, as they are founded in the primitive operational elements of consciousness itself, in its dynamic coupling with reality through evolutionary entrainment.


Well I know this is rambling, but I felt this question needed some clarity, and I deeply enjoy exploring here, as I had to take the time to do recently, and so I thought a quick summary of some of the points I'm touching could help make some sense of this complex relationship. 




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