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.]

Views: 1532

Reply to This

Replies to This Discussion

Yes, where do we stop? This stuff does go on forever. Here is a version of a traditional yin yang next to a fractal, recursive version. This figure will appear in my book in press on Clinical Intuition in Psychotherapy (Norton's Interpersonal Neurobiology series). Tom, it's very cool that you bring intuition into your discussion of the quantum stuff as well.

Thomas said:

I want to give an example of self-similarity, this root-level fractal, as it appears in Bohr's quantum physics.  In several previous posts on other threads, I showed how Bohr's quantum thinking is intuition-based.  Intuition, for me, and as operating in the realm of thinking, is right-brain wholeness thinking.  Intuition is a form of tacit getting-it, and getting-it, as I've described previously, is to me an act of whole-understanding appearing.  This is a non-continuous, quantum happening, a quantum leap.


Bohr and Heisenberg used to think about quantum physics together, and they would do so, often, by talking up a storm, then leaving the matter---not talking about it---for some time.  Sometimes after this time, an answer to the particular question they were troubling over would appear, there, fully formed, whole.  Dürr, who worked closely with Heisenberg for 17 years, says this was his and Heisenberg's standard way of thinking physics.


Now compare my above description of intuitional thinking to a quantum event.  An event happens in a measurement situation.  An electron or photon is released to interact with some measuring device, like a photo plate or receptor, etc.  That "interaction" is a quantum happening.  A quantum happening, says Bohr (Heisenberg, Dürr, Zeilinger, etc.), is a whole change of state.  Thus if an atom with two full electron shells is radiated, it will eventually, with sufficient radiation, jump into the whole state of an atom with three electron shells.  The new atom appears whole, just like an answer appears whole from intuition.


This insight that what happens in quantum thinking is a self-same fractal, is identical, to the fractal pattern in what happens in nature viewed from a quantum angle---this self-sameness between thinking-process and matter-process is what led Bohr to say the observer is the observed.  This intuitive insight of Bohr's is the basis for all the wrong-headed understandings that "a conscious observer collapses the Schrödinger waveform."  Bohr said to that, "That's not what I meant."


Bohr said in view of quantum physics, the distinction between living and non-living is no longer tenable, at all.


Compare that to a few quotes from Max Planck, who also got quantum thinking:


Science cannot solve the ultimate mystery of nature. And that is because, in the last analysis, we ourselves are part of nature and therefore part of the mystery that we are trying to solve.


(Get what he's saying there?  Very subtle.)  Here's another:


As a man who has devoted his whole life to the most clear headed science, to the study of matter, I can tell you as a result of my research about atoms this much: There is no matter as such. All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.


Sure, that makes sense.  Matter-process is self-same with conscious process.  Ergo, it must be the same stuff.


Apart from that inference, let's see how self-sameness begets a non-linear view of development and relatedness.  A quantum jump, or an intuitional jump, is an appearance NOW of something whole.  That appearance is discontinuous: there is no "getting to getting it," there is only the singular happening of ok, got-it.  Because this non-linear jump is self-same---because my getting it fractals nature-happenings---development, or movement, is a self-same repetitional process of difference appearing same.  This is what Bohr meant by saying quantum thinking is non-conceptual and complementary (ie, contradictory).


Thus non-linear development is an understanding of self-same differencing.  This is a fully contradiction-internalized view.  It is a view that, to me, adequately explains the common human sense of always having been me.  That's true, says the non-linear theorist.  I have always been self-same I differencing over time in a difference wrapped by same.  I have changed, but not.  Emptiness proponents can tend to say there's no essence.  Well, I disagree.  The fractal called self---which is an autopoeitic appearing, to reference Varela---is the root fractal.


This selfing is the objective aspect to reality.  Objective is an absolute-pole indicator.  So is essence, and identity, and isness, and Now.  Selfing as the root fractal is a fractal of essence and identity---sameness, unity, oneness, intimacy.  The universe is objectively intimate.  Quantum physics evidences as much by its 14 decimal places of intimacy accuracy.


Just as a side note, I prefer self-same to self-similar in the description above where I differentiate between absolute (self-same) and relative (differencing) poles in the description.


And by the way, Bohr took into a coat of arms bestowed on him the yin-yang symbol.  The root fractal being Self, that "s" looks to me a lot like the "s" of the yin-yang symbol.  I mean, once you start with this stuff, where do you stop? 

Sounds like Roger Bacon to me...

Thomas said:
Now add to this mix that intuition is regularly regarded as feminine.  What might that imply?  That quantum thinking and nature are feminine?  (Who ever said nature was feminine?)  That a man, no matter how hard he tries, cannot figure out a woman on a linear basis?  (Whoever said anything like that?)  Any guesses?
Terry might have been referring to Bacon's idea that nature was a woman to be tortured into yielding her secrets...?  Oh, no!  That's Francis... Never mind.  :-)
Beautiful poem. Thanks for sharing.

Thomas said:

Yes, sex, the ur-duality of wholeness.  The self-saming sexual spiral called development where I manifest only relative to you.


Let's get a little personal-psychological about this.  Relationship events---feelings---co-happen. My feelings are twin-born with yours, regardless what those or my feelings are. If I feel anything, that feeling is irreducibly intermixed with the situation called you and what you are feeling at the same time. This is duality in wholeness: we co-appear each other: we is whole.  On a personal level, this self-same co-appearing pattern of always-my-feelings-and-yours aids me to deeply understand that my pushing your buttons is your pushing mine.  And this co-appearance, this I manifest you and you manifest me, implies: we need each other.  Surely that couldn't be some or another meaning of sex aka making love?  What is love but wholeness?  And knowing I need her and she needs me at this deep level is intimate. That intimacy considerably dulls the edge of ever wanting to criticize my beloved, be because be as bare infinitive takes no divisioning subject or object.  It stands solitary, whole.  Be! Follow the continuity trail:


be you be the be I see

the gift of the feel
of the wind
gently cool sweeping
bright long shadow
lined late light slanting red
leaf trodden city
walk place path
way beautiful
canopy treed
corridor whispering
autumn’s faith
like trustful promise to
be here again
moves me
the feel of the gift
of your life’s
year’s movement’s
delicate tentative
here too
before me
beloved woman

Sure, but its directness is nice too :)

Thomas said:
Thank you, Nicole.  I've added some shape to that poem.  Could be a touch subtler.  : )

Nunez's article is a shortened version of Chapter 8 of WMCF. From the latter:


“The uniqueness of the final state of a complete process is the product of human cognition, not a fact about the external world.... The basic metaphor of infinity maps this uniqueness property...onto actual infinity.... What results from the BMI is a metaphorical creation that does not occur literally: a process that goes on and on indefinitely and yet has a final resultant state” (160).


The discussion that follows that quote in interesting in that outside math, applied to philosophy, the BMI was responsible for a categorization hierarchy up to an end state category, the highest state of Being. And that these categories were mistaken as ontological realities, things-in-themselves. Within religion the highest category is God, an actual being, not just a metaphor.


Nunez concluded from his referenced paper:


The nature of potential and actual infinity can be understood not in terms of transcendental (or platonic) truths, or in terms of formal logic, but in terms of the manipulation of meaningless symbols in human ideas, and human cognitive mechanisms.”


Of course one issue is what is “just” a metaphor? Terry noted earlier in this discussion that because the observer is in the observed our metaphors are part and parcel of the observed object so therefore this relationship entails an ontological status as well as an epistemological one to actual infinity. It would appear though that Nunez might disagree, and both he and Lakoff are in the general pragmatic school on intersobjectivity that sees the relationship of subject/object as interactive (i.e. intersobject).


Also an infinite regressive (or progressive) series such as in fractals, what is termed potential infinity by Nunez, is “resolved” in an actual infinity by our metaphorical processes. And yet metaphors arise from our embodied interactivity in the world via a continuous process (the very process interpreted as a potential infinity) so therefore metaphors are a part of that intersobjective “field” and thus the resolution seems to come from a “nondual” way of interpreting the relational field so that there is no separation between subject/object. And yet as we can see above Nunez makes the distinction that actual infinity is “not a fact about the external world, it does “not occur literally.”


So it seems there is a con-fusion then with inside/outside and with analogies such as if it occurs in the whole it must also occur in the part. And that there is a final, resultant “whole,” a holon of the uni-verse, the highest category in the hierarchy. Such confusions are examples of metonymy, one such metaphor. Useful and highly practical, of course. A statement on the true nature of reality, not in Nunez's (or L&J's) view. On the nature of nested hierarchy and its con-fusions, see the real and false reason thread based on L&J's work, particularly some conclusions on p. 7 with reference to one our embodied image-schematic basic categories, the part-whole gestalt. This is why I'm also interested in Joel's work, since he uses nested hierarchies while yet undermining their very premise (and referenced in the link above).



Correction: I mixed my metaphors. I meant synecdoche instead of metonymy.

Hi Theurji,


Of course one issue is what is “just” a metaphor?


Great post, and yes, the above troubles me also.  It reminds me of the status of mathematics that you are also discussing, and that we often forget how our "knowledge" is influenced by our culture and biology.


Lately, I've been thinking of it this way: if we take three bananas, we can extrapolate the number 3, right?  Not necessarily.  Our words do not carve nature at its joints, as Plato suggested that good language should.  There are no pre-packaged objects out there for us to find.  Bananas (and all other "objects") should be understood as co-constructed suobjects.  So whether one person sees one object or another sees three is somewhat arbitrary.  Take away our shared bio-social environment and the logic-mathematics goes with it.


We can't go beyond "just" metaphor, because we'd have to use metaphor to do it.

This post highlights some of the issues in defining "embodied realism" according to L&J. (The first part of the post is about Mead, also relevant.) It also helps to define what I mean by (post)metaphysics, another ongoing issue in several threads. The third aspect by L&J seems acknowledged by Terry in that we cannot know the world-in-itself completely, though we do get some knowledge of it by virtue of our embodied relation with the world-in-itself (realism). Also highlighted in several other threads is the notion that we gain more access via aesthetic means, but how much more is a legitimate question.

I recall an interview with Daniel Dennet where he argued that mathematics has a sort of Platonic quality.  Not in the metaphysical sense, but in the way that basic mathematical principles will constantly be rediscovered because they are so useful.  So if we discovered an alien culture, it seems likely that it would have some knowledge of mathematics -- perhaps not a base-10 system, but something at least.


Do you buy that argument?  What implications does it have for an embodied philosophy of mathematics?

Implausible, IMO. As much as I love mathematics, I believe we as humans are so wedded to it that we miss that there are other possible ways to be in the universe.

infimitas said:

I recall an interview with Daniel Dennet where he argued that mathematics has a sort of Platonic quality.  Not in the metaphysical sense, but in the way that basic mathematical principles will constantly be rediscovered because they are so useful.  So if we discovered an alien culture, it seems likely that it would have some knowledge of mathematics -- perhaps not a base-10 system, but something at least.


Do you buy that argument?  What implications does it have for an embodied philosophy of mathematics?

I actually think that Dennett is correct here, to a degree.  Not that all alien species will come to the same exact mathematical structure, but that it's very likely that it represents an attractor in the morphological space of mathematical representation.  But it would be one of many such attractors, which could be arranged in developmental charts or trees of various sorts.  This is because representation, in my view, is a self-similar echo of reality itself, and we all live in the same reality (ultimately, though we may never realize it).  Mathematics, being the most generalized aspect of representational systems, has a certain empirical embodiment with a deep education in evolution.  It is evolutionarily informed, and has its own self-similar recapitulation of the basic embryogenetic form.  This is why reality appears "mathematical", as mathematics is generalized empirical and representational relation, and reality is the real relation being generalized.  So while I don't think there is any attractor on the base of the system (e.g. binary vs. decimal), though Fuller's work suggests some micro-attractors, I do think the general exfoliation from a general form of conceptual "closure" is likely.  This is simply because reality is nucleated and self-similar in a real infinite capacity and unfolding in a real immanence and transcendence.  The generalized self-similar form of identity itself, the number one, will thus also be nucleated and closed in this originary immanence and transcendence, and begin its machinations agglomeratively in transitivity, with the original utility of counting.  And the embryogenetic unfolding dance of number and operation (noun and verb) will commence and oscillate into new levels of each, as it moves from transitivity to transcendent integration in its recursive return to immanence.


So, there is indeed a Platonic aspect to mathematics, in the sense that it makes sense to speak of the space of morphological possibility, and certain key attractors which unfold from the self-similar nature of reality as it echoes into representational capacities.  I don't think the current mathematics as it exists on earth is the only one, or the pinnacle, by any means.  I mean we already know of alternative criterion for evolving mathematics aside from a focus on closure, such as Lebniz's infinitessimals and the granulars.  And I certainly don't think that aliens will necessarily end up in a 3 dimensional rectilinear universe as we have.  Rather I demonstrate that Fuller's Synergetic Geometry represents a higher level in geometric or dimensional (metamathematic) systems.  But it's likely that Fuller's system also represents another attractor in the morphological space of dimensional systems.  I show that Fuller's system can be categorized as generalized structural relation (and at the rational level of development), while the 3D rectilinear (Cartesian/Euclidean) system can be considered merely transitive and representational, and hence pre- or at best proto-rational, in terms of integrated core axes.  One could categorize this, along with Fuller, as a function of the prevailing senses, such that it's likely that an alien species which lacked sight but had touch would end up orbiting the Synergetics (structural) attractor, while a species which relied primarily on sight would end up focusing on linearity, with dimensional orthogonality implicitly defined as a function of the crossing of lines at right angles.  But again there are many other attractors in this space, and it's even hard to know if these are the main ones, though I'd guess that they are close to it.

Reply to Discussion


What paths lie ahead for religion and spirituality in the 21st Century? How might the insights of modernity and post-modernity impact and inform humanity's ancient wisdom traditions? How are we to enact, together, new spiritual visions – independently, or within our respective traditions – that can respond adequately to the challenges of our times?

This group is for anyone interested in exploring these questions and tracing out the horizons of an integral post-metaphysical spirituality.

Notice to Visitors

At the moment, this site is at full membership capacity and we are not admitting new members.  We are still getting new membership applications, however, so I am considering upgrading to the next level, which will allow for more members to join.  In the meantime, all discussions are open for viewing and we hope you will read and enjoy the content here.

© 2023   Created by Balder.   Powered by

Report an Issue  |  Terms of Service