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 CYBERNETIC REVOLUTION

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.

 

POSTMODERNISM

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.

 CONTRADICTION BUILT INTO THE FABRIC

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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Oops, sorry Christophe and Bruce, please forgive me for mixing you two up!!

Terry Marks-Tarlow said:
Hi Bruce,
It's very nice to dialogue with someone able to engage deeply with the material. I'm at a disadvantage, because I am not familiar with many of the references and terms you cite. For example, what is the Deleuze reference? I'd like to read this. I don't know what a quasi-cause or phallic signifier is, so I can't respond to these ideas.
That said, I don't believe fractals are at the surface or only a mirror. That notion corresponds to some early critique of computer generated fractals plus their mimetic capabilities as merely "pretty pictures." What you say may apply to the Mandelbrot set itself, which bears little direct relationship to nature. But to me, naturally occurring fractals are a whole other matter. They are deep because they arise universally as deep attractors in self-organizing systems, both inorganic and organic. They are deep in the sense of unifying form and function in physiological systems. For example, fractal branches in the circulatory system correspond to power laws in heartbeat, shifting the action back and forth between spatial and temporal dimensions. As another example, Carl Anderson just sent me a paper entitled, From Molecules to Mindfulness: How Vertically Convergent Fractal Time Fluctuations Unify Cognition and Emotion. This kind of vertical alignment couldn't be deeper! To go from fractal fluctuations in molecular events like ion channel currents and synaptic processes right up to macro level events in the temporal lobees, brainstem and cerebellum during the expression of emotional memory to collective levels of social behavior, like internet use is quite extraordinary.
Best,
Terry
Ch
Hello Mrs. MarksTarlow,

No problem. My Name is Christophe, and I'm a psychologist from Germany. I'm sure Bruce will come back shortly and contribute to the discussion himself.

I'm sorry for using too much jargon in my post. It's just that I finished this fascinating Deleuze book recently, and my head is still filled with images from it. I mentioned the title of it earlier in this other thread, where you can also find the link to a Google Books Page.

And yes, I might need some lessons in mathematics - I can't really comment on the implications fractals have for mathematical models.

Namaste,
Hi, Terry, Christophe, all, I'm looking forward to finally catching up with this conversation. More soon.

In the meantime, if anyone is interested, here's an interesting web page on an intriguing fractal object, the Mandelbulb.
Yes, self-reference is extremely powerful all the way around. From a spiritual perspective, it can be translated into God looking at him/herself through humankind's consciousness. By the way, I love the Hoftsadter book you quote Wikipedia on above. This book was a huge influence for me in Psyche's Veil, where I aspired to form following content as well.

Bruce, I believe you left out the link to the Mandelbulb, which I agree is quite amazing. The more the Mandelbrot set gets fleshed out and dimensionalized, the more remarkable it appears.
Hi, Terry, I think I need to change the formatting on this site. The link is there (the words are in slightly lighter color: click on "here's an interesting web page"), but they are not distinct enough from normal font color to be easy to detect. I'll change that when I get home.

Thank you also for your patience with me! I've been swamped with work on returning from retreat and haven't been able to give your paper the time it deserves, although I really look forward to engaging with it. My fingers are crossed for this weekend.

Best wishes,

Bruce

P.S. On my TSK (Time-Space-Knowledge) forum, I also posted a thread on the Mandelbulb. Tarthang Tulku, the creator of TSK, wrote a book a number of years ago in which he attempted to create a sort of fractal geometry of perspectives -- not as a formal mathematical system, but as a means of facilitating inquiry and self-exploration -- and one of his images reminds me of what has now emerged as the "Mandelbulb."
Dear Bruce,
All is interesting! Please don't feel any time pressure on any of this. I too am wildly busy with things, so whenever you have a chance to respond in greater detail will be delightful. Meanwhile I'll go back and check out the things you mention.
All the best,
Terry

First off I'd like to direct discussion of Terry's paper, introduced on p. 24 of the QE thread, to this thread Balder originally started on Terry's work. Now I'd like to ask Terry, Tom and Joel (and any others) if they are familiar with this paper by Raphael Nunez, “Creating mathematical infinities,” and how it might or might not fit in with your mathematical ideas of infinity. Responding to the paper would clarify for me (and perhaps others) your own ideas.

 

Abstract:

 

The infinite is one of the most intriguing, controversial, and elusive ideas in which the human

mind has ever engaged. In mathematics, a particularly interesting form of infinity—actual infinity—

has gained, over centuries, an extremely precise and rich meaning, to the point that it now lies at the

very core of many fundamental fields such as calculus, fractal geometry, and set theory. In this article

I focus on a specific case of actual infinity, namely, transfinite cardinals, as conceived by one of the

most imaginative and controversial characters in the history of mathematics, the 19th century

mathematician Georg Cantor (1845–1918). The analysis is based on the Basic Metaphor of Infinity

(BMI). The BMI is a human everyday conceptual mechanism, originally outside of mathematics,

hypothesized to be responsible for the creation of all kinds of mathematical actual infinities, from

points at infinity in projective geometry to infinite sets, to infinitesimal numbers, to least upper

bounds [Lakoff, George, Nunez, Rafael, 2000. Where Mathematics Comes From: How the Embodied

Mind Brings Mathematics into Being. Basic Books, New York]. In this article I analyze the BMI in

terms of a non-unidirectional mapping: a double-scope conceptual blend. Under this view ‘‘BMI’’

becomes the Basic Mapping of Infinity.

An excerpt from the article above, setting the frame:

"At this point, and in order to avoid any misunderstandings about the goal of this article
and of the nature of mathematical idea analysis, it is very important to make clear that:

1. A cognitive analysis that takes into account the properties of mathematics described
above, and

2. the bodily-grounded nature of human cognitive mechanisms, such as conceptual
metaphors and conceptual blends,

provide a non-arbitrary explanatory proposal of the nature of mathematics. This nonarbitrary
approach radically differs from post-modern accounts, where mathematics is seen
as an arbitrary social text or as a mere cultural artifact. The position we will endorse here
recognizes the importance of culture and history in the emergence and development of
mathematical ideas, but explicitly rejects the claim that mathematics is arbitrarily shaped
by history and culture alone (for details see Lakoff and Nunez, 2000: 362–363). With this
perspective in mind, we are now ready to ‘approach’ infinity."

For example, here are a few sample excerpts from Terry's paper that might resonate with Nunez:

 

I offer up the fractal geometry as the underpinnings for a dynamic unconscious destined never to become fully conscious.

 

By occupying the infinitely deep 'space between' dimensions and levels of existence, fractal boundaries contribute to the notion of intersubjectivity, where self and other become most entwined.

 

Kauffman highlights paradox as he underscores the hidden action of the imaginary in bringing forth the “real” world of everyday distinctions.

 

Jung and his dedicated follower, Marie-Louise von Franz, were interested primarily in the counting numbers as symbols and founts of inexhaustible metaphor during the production of conscious experience.

 

Whereas Jung speculated about the archetypal significance of the natural numbers, in this essay I explore number as manifested naturally in the fractal geometry of nature. Whereas Jung speaks of number as an archetype of order which has become conscious, I present fractal geometry as deep order under chaos, which is yet to become conscious and impossible ever fully to do so. With fractals as a new metaphor of mind, we no longer need to deify or defile mechanism as metaphor.”

And this from Joel's Integral World article, which are excerpts from Spinbitz:

 

"Indeed, it will be shown herein how and why the positive infinity of the rational does not give rise to antinomies, as Kant thought, but that the necessities of logic and relation demand the existence of the infinite. Regardless of whether the infinite can fit into the finite or indefinite imagination of man, it can very simply fit into his logical and rational understanding, itself informed in deep evolutionary symbiogenesis with experience, intelligence and memory. To be sure, only through affirming and incorporating the necessity, in the rational, for the infinite can the nondual synthesis between the relative and the absolute be truly accomplished. Just as only through incorporating the necessity of Emptiness in form, at some level or zone, can the nondual begin to present itself."

I am a huge fan of Nunez, and the book mentioned, "Where Mathematics Comes From," dovetails with and has influenced my perspective greatly. While I have not read this paper on creating infinities, though intend to, Nunez's basic framework is that metaphor is deeper than mathematics, and that metaphor begins with the body's interaction with the real world. All abstraction is bootstrapped from there. This embodied and very concrete foundation is what provides a non-arbitrary foundation. This is very different from postmodern thought, where the relativity of all perspectives leaves all systems of thought completely untethered to truth. I don't abide by this. According to the perspective  set forth in my book, Psyche's Veil, among other writings, Self, World, and Other constitute three separate realms, each separated by fractal boundaries. This implies that there is an objective realm, but that we can never grasp it fully due to infinitely deep fractal boundaries that separate it from subjective realms. As individuals and as collectives, we can grasp at this realm of truth, but never fully capture it. As for Nunez, he and Lakoff point out that mathematical truths depend upon the underlying metaphor used. So, for example, there are at least three ways of defining a point in space, such as a member of a set or a point on a line. How we define a point will determine the truth set that surrounds the metaphor. In the case of infinity, how we define infinity leads to wholly different branches of mathematics, whose entailings sometimes even contradict one another. So here is another place where we can see contradiction built into the fabric of things, but in such a way that supports rather than negates the realm of truth.      

theurj said:

First off I'd like to direct discussion of Terry's paper, introduced on p. 24 of the QE thread, to this thread Balder originally started on Terry's work. Now I'd like to ask Terry, Tom and Joel (and any others) if they are familiar with this paper by Raphael Nunez, “Creating mathematical infinities,” and how it might or might not fit in with your mathematical ideas of infinity. Responding to the paper would clarify for me (and perhaps others) your own ideas.

 

Abstract:

 

The infinite is one of the most intriguing, controversial, and elusive ideas in which the human

mind has ever engaged. In mathematics, a particularly interesting form of infinity—actual infinity—

has gained, over centuries, an extremely precise and rich meaning, to the point that it now lies at the

very core of many fundamental fields such as calculus, fractal geometry, and set theory. In this article

I focus on a specific case of actual infinity, namely, transfinite cardinals, as conceived by one of the

most imaginative and controversial characters in the history of mathematics, the 19th century

mathematician Georg Cantor (1845–1918). The analysis is based on the Basic Metaphor of Infinity

(BMI). The BMI is a human everyday conceptual mechanism, originally outside of mathematics,

hypothesized to be responsible for the creation of all kinds of mathematical actual infinities, from

points at infinity in projective geometry to infinite sets, to infinitesimal numbers, to least upper

bounds [Lakoff, George, Nunez, Rafael, 2000. Where Mathematics Comes From: How the Embodied

Mind Brings Mathematics into Being. Basic Books, New York]. In this article I analyze the BMI in

terms of a non-unidirectional mapping: a double-scope conceptual blend. Under this view ‘‘BMI’’

becomes the Basic Mapping of Infinity.

Very nicely said!

Thomas said:

Terry, here are some random comments on your post above.

 

metaphor is deeper than mathematics, and that metaphor begins with the body's interaction with the real world.

 

The word metaphor is actually closely related to differ, suffer, infer, refer.  The -phor suffix is the -fer suffix in these words, which means "to bear," as in to bear a load or to bear (birth) a child.  Differ thus means to bear one's difference (suffer) and to give birth to the new.

 

Metaphor carries the dual meaning of meta as "beyond" and "through," such that metaphor, etymologically, is the beyond-birth-through-birth of reality, a paradoxical combination of objective (beyond) and subjective (through).  This beyond-through paradox seems to me a root fractal from which, as you say, all abstraction is bootstrapped, and looks to me to be something like what you say here:

 

This implies that there is an objective realm, but that we can never grasp it fully due to infinitely deep fractal boundaries that separate it from subjective realms.

 

I like what you say here:

 

So here is another place where we can see contradiction built into the fabric of things, but in such a way that supports rather than negates the realm of truth.

 

A contradiction-view, if you will, is to me positive.  It's a leap out of postmodern negation, but includes the insights gleaned by the postmodernist deconstruction of that denser Newtonian positing.  That inclusion, for me, is an internalization of polarity, of contradiction, whereby view becomes positive-negative paradox.  As I've attempted to illustrate with quantum physics, that view has greater intimate reach in something like a wholing of participation.

 

I draw a parallel here with Joel's positive infinity, which is antimony-free in the sense that antimony (contradiction) becomes embodied in a deep, tacit world-participation.  This, I think, is Jung's «objective psyche» formed, as it were, from archetypal patterns (fractals).

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