Introduction

Transpositional Analogical Value Theory is my probably unfortunate coinage, but it was forged out of a cognitive journey with three unwitting mentors, who may stand in for the "Wise Men."  While we did simultaneously share this planet, we never met, as far as I know.  Even so, I begin this Thought Experiment by imagining them standing around their proverbial, yet amalgamated, "Elephant."

Buckminster Fuller thinks of an overly large organic system “elephant”, 4-dimensionally swallowing-up a too-small room, representing his heart-felt concern about our Species' "Irrationality."  Fuller wants to know why so many mathematicians would lead the way in a direction that culturally prefers algebra's irrational and infinite variables as a more significant metric language system than geometry.  This is, to him, so obviously an immense elephant of an irrational choice because it results in a STEM-ordered Species that is ineffably and inarticulately blind to true, right, just, ethical, good, and even beautiful, proportion (as he so humbly finds them synergetically "sync"ed, if you will).

Second, Thomas Kuhn becomes absorbed in wondering how the "Scientist" could know if we are ever really talking about the same objective elephant, or any other paradigm one might choose, due to the historic, cultural, and linguistic-faced "Problem of Incommensurability." Kuhn became persistently focused on why so many scientists and researchers, and lesser mortals of all types, sometimes change their paradigmatic assumptions to accommodate a revolutionary hypothesis, even before its potential for increased explanatory and predictive value has been experientially vetted.  We can deductively explain paradigm shifts when a new theory has been tested against experience, given "business-as-usual" time, but a "revolutionary" flurry of excited enquiry occasionally erupts into specialized paradigms prior to normative, evolutionary discovery.  Why?  Further, why can we recognize these paradigm speciation events only in hindsight? This remains a generic evolutionary question, emerging pervasively throughout the "evolution" paradigm itself, with the sole exception of Supervenient Evolutionary Theory (e.g. Arthur Peacocke, but also Wayne Teasdale's, Ken Wilber's, et. al.  "Interspirituality" thesis).

How, exactly, do we choose "commensurable" rather than "incommensurable" Elephant definitions without anything more than faith in some "Theory" that someone(s) has posited as "not irrational?"

Third, Julian Jaynes would see the relationship between Kuhn's Problem of Incommensurability and Fuller's Irrationality complaint as the analogically-rooted elephant in our claustrophobic room. Jaynes recognizes the primordial analogy:

rational language is to mathematicians/scientists as

"communication" is to "community".

Analysis

Jaynes was adept with perceiving analogy because he recognized that our cultural evolutionary journey began in the older Right hemisphere and gradually shifted to Left hemisphere dominance.  Measurement and calculation evolved from logic which, in turn, evolved from analogy, with analogy emerging as proportioned and well-worn Species memory.  Identity, or space/time commensurability, emerges with the entrenchment of confluent perception patterns.  The less cognitive dissonance, or stress, the more likely we, individually and collectively, are able to distinguish  relationship and pattern, rather than perceiving "chaos" or "incommensurability" or "irrationality".  "How on earth did we ever get such a very large elephant through such a very small, impenetrable, door, and what would be our motivation for achieving such a disturbing disproportion?"

Split-brain research has supported Jaynes' observation that, without capacity to learn from the Left hemisphere's experience, we have surprising cognitive lapses due to the Right hemisphere's linguistic blindness.  If handicapped by only Left visual field capacity, after corpus callosotomy, linguistic capacity is deactivated.  However, the left hand can signal recognition of an identified memory pattern. The Left-processor is calculating and linguistic; the Right-processor's information receptors operate analogically, spatially, and proportionally.  In short, geometric information is progenitive to numeric  information, as analogic pattern is progenitive to calculating with linguistic-based symbols.  Spatial pattern comparison preceded enumeration and use of numeric language in human cognition.  The Left hemisphere's role is newer, and conclusive; synaptically discovering symbolic labels for the Right hemisphere's pattern-analogy sets.

Fuller's and Kuhn's observations are parallel, that the evolution of Information structure ("communication" systems to a now Left hemisphere dominated culture) took a curious Shut Door, "dissonance" or "stress", turn when Homo Sapiens switched to an algebraic metric system as a normatively expressed Values-structure.  On what basis can we universally trust as consistently "True" a language that gives no preference to rationality, or commensurability?  While algebra is historically rooted in geometry, it originated the distinction between "Real" and "Natural" numerals.  This distinction is a level of abstraction that segregates "integers" as the even-numbered symbols for geometry's Natural Law of discernible ratios.  Integers become "Natural" and "Natural" numbers become a subset of "Real" numbers, by apparent fiat.

Fuller complains that algebraic systems, by comparison to geometry's capacity to explain and predict proportion and logic, the irrational choice, both literally and figuratively, culpable for leaving behind the vast majority of the human species.  Math, and perhaps therefore "science," "education," and "technology," usually feel alien to our confluence-seeking, stress-reducing Bicameral Species.  Kuhn's historical analysis of "scientific revolutions" suggest that eruptions into business-as-usual are best described as collective intuitions, not quite visible to the Left-brain's labeling capacity.  An adjusted paradigm structure theory "feels" confluent, like it shows potential for clearing out problematic chaff in the existing paradigm structure. At the same time, no significant dissonance (systemic "stress") is perceivable in these revolutionary moments.  Kuhn belabors his initial surprise that nothing in his observations of scientific revolutions throughout history would suggest that these rapidly accepted open doors of hope are accelerated because of anything like mathematical, including statistical, "proof." That is, we can use math and deductive logic to describe why speciation occurred after the fact, but but we have no historical precedent for believing math/logic can predict scientific speciation, "revolutions".

I was reminded of all this while reading a homily entitled "Open and Shut Doors" by Unitarian Reverend Julian Clifford Jaynes, Psychology Professor Julian Jaynes' father, who died when Julian Jr. was only two years old. Reverend Jaynes observes that our Species' (and individual) "Open Door" successes and "Closed Door" disappointments are of equal information value.  Open Doors to learning, inclusive of learning mathematics, are those that move us further along a sustainable confluence balance between our Left and Right hemispheres.  In Jungian-compatible terms, a learning opportunity is one that attaches a symbol to an analogical set-pattern.  The symbol "Rationality" becomes associated with "Language" is to mathematics as "Linguistics" is to communication, as "Predictability" is to cognition.

Open Doors, returning to Rev. Jaynes, are the experience of "power-with" the environment.  A newly-cognized labeling confluence between Left and Right hemispheres normally trends downward as time passes, as new questions, newly perceivable anomalies emerge over time.  Diverse levels of dissonance/stress emerge gradually.  The promise of Einstein's mid-life successes in what became Quantum Physics led to his subsequent two decades of business-as-usual gradually settling in.

Einstein's subsequent twenty years of disappointment with failure to find his Golden Fleece Unified Field Theory is what Rev. Jaynes homiletically labels the "Shut Door" emotional economy.  This information trend has its own reverse, or "transposed" peaks. A pattern of trials without the predicted, calculated, Left hemisphere Truth-search results results in a slower evolutionary trend toward gradually increasing dissonance, but has its own softer peaks of confluent balance as yet another idea emerges, hope-filled, on the cognitive horizen. Thomas Edison describes his discovery pattern toward an efficient light bulb in this way.

Rev. Jaynes would have us compare these "two disappointments...the disappointment that follows success [Open Door] and the disappointment that follows failure [Shut Door]. They are practically the same." (1922, p. 208) In both cases, the Left/Right hemisphere confluence/stress value economy is moving toward longer-term sustainable balance; neither unsustainably "peaked" at either extreme of Open or Shut.  Buddhists and Taoists remind us that neither the Yang-excitement of passion (power-with), or the Yin-engorged serenity of unitive enlightenment (power-over-attachment) can be globally sustained in any species' over the long term.  Irrationality ensues sooner or later, has its place in the learning process, but is optimally understood when it is in balanced harmony with rationality.

Rev. Jaynes explains that Open and Shut Doors are "practically the same", defined as "Both are educative; both are expansive in their power; both are prophetic of something larger and better beyond." (pp. 208-209)  This faith is substantively questioned by Julian Jaynes, his son. but his father roots his faith in the long-term balance of our Species' Right and Left hemispheres.  This faith is, for Rev. Jaynes, a Tradition-enriched Faith in the cooperative power-from-within of bicamerally balanced Homo Sapiens.  The wise in our Species' midst balance their Left hemisphere Truth satellites with what another Unitarian, Buckminster Fuller, referred to as our collective consciousness of "Spaceship Earth," or, more geometrically, "Galactic orientation."  A bit later, William Thurston defines rational Universe-orientation as any logically consistent 8-fold, "Geometrization Conjecture."  A revolution in mathematics and metaphysics continues apace.

It was Rev. Jaynes' Tradition-rooted faith (Right hemisphere dominant) that optimized Justice, Goodness, and Beauty is our Species' evolutionary purpose. This cognitive development is predicted to arrive in our Permacultured Paradise as our Left and Right hemispheres achieve Yang/Yin sustainably equipoised confluence.  Balanced stress/confluence boundary Doors are empowered to open fully, or close fully, by remaining ajar. These Boundary Doors are empowered by balanced binary-hinges to open in or out, up or down, back in time, or forward in time. While Doors are 3-dimensional, our Species' power to perceive paradigms and symbols, including Doors, as potentially revolving gives us power-from-within to perceive 4-dimensionally, with an internally-consistent symbol system.

The Left hemisphere's value for Hope is expressed as the expectation that Yang/Yin balance, neither overly Yin-expressed "Open" or too Yang-expressed "Closed," is sustained as a permacultured steady-state system.  In other words, the Left hemisphere's "Hope" value for Truth-through-discovery continues to evolve toward optimal confluence with the older Right hemisphere's enculturated memory consciousness, emotively "Faith"-resonant with holistic Justice, Goodness, and Beauty. Accessing Information encoded in RNA, inherited through our generations, the Left-hemisphere is most likely to recognize Truth-optimization values upon perceiving richly resonant, balanced proportion analogy patterns. When a newly and successfully Information Boundary revolving Door (binary-hinged, or mutually-iterative, Left to Right to Left again) iterates, then the potential for revolutionary "sync" is optimized at the Species, and individual, level: the Right-brained goodness value (Faith) and the Left-brained truth value (Hope), synaptically (mutually) trigger, thereby increasing our global power-from-within potential for future integrity, love, learning to learn, prediction, and expectation.

This admittedly tradition-rich, exegetical, Right hemisphere dominant hypothesis is cognitively rooted in a Prime Relationship, where Right hemisphere values translate Left hemisphere experience, defining Right hemisphere cognition, as "me" defines "I" defines mutually-defining self-awareness, and Yin defines "Yang" defines Tao.

Should this Jaynesian trans-generational and exegetical hypothesis have predictive and explanatory analogical merit, then we would expect, with Fuller, that the historical-cultural gradual trend toward algebraic systems as our normative calculation structure was anomalous.  It is a blind-alley metric-system choice that is culturally expressed as "enumeration" rather than its more deeply rooted, mutually perpetuating, Prime Relationship as Tao.  In a Tao-Vectored Universe, "0" might be cognized as equipoised form/function (Yang) and proportion (Yin). For an optimized internally-consistent metric language, the Prime Root Assumption is the verb of mutually-defining iteration; not any noun, or numeral. To Descartes we cooperatively respond, "Yes, but there is no symbol "think" unless "I AM" is equally True, by mutual (not unilateral) definition.

Going further out on this intricately latticed limb, let us hypothesize that our Species' "power from within," transposes as "within power," and reiterates as "therefore, 'without' power."  This reflective Information Theory reaches back to Proto-Indian-European (PIE) wisdom, "As within, so without" and "as without, so within."  In Information Language, this translates as a Prime Tao=Trinity:  +1/-0 = [1 + (-1) = 0], therefore [0 = 1 + (-1)].  There can be no language, no cognition, without at least one Prime Relationship (Root). This mathematical observation goes back at least as far as Pythagoras and remained obvious through Taoism, Buddhism, Judaism, Pythagoras, Christianity, Ptolemy, and the early evolution of Islam.

In the 3 spatial dimensions, +1 = (-1). In the Time dimension, "within-power" = Polynomial Time (P), while "without-power" = Non-Polynomial Time (NP).  Indeed, this ancient analogical Prime Root Assumption concludes that P=NP in a Yang/Yin rational Universe. This analogy is consistent with William Thurston's more contemporary 8-manifold rational Universe Geometrics, as well as Fuller's Synergetic, mutually-empowering, Universe.

When we re-emerge inside our Elephant-filled room (those few who have survived this reiterating journey) after traveling through a revolving door, we do not re-emerge as exactly the same Left-hemisphere dominant person, if we have any memory of the exterior view as we approached, and returned, from the 180-degree position.  However, the Right-hemisphere would not recognize any change in our identity.

Our shared Right hemisphere RNA-Information sees only proportional inter-relationships. Our suspected blind-alley may feel like a detour around an Open Door to confluence, but "algebra" merely disguises the "Real" problem.  It was algebraic numerology that took us to the Left hemisphere distinction between "Real" and "Natural" proportion.  There can be no such distinction in our Right hemispheres.

"Fractions" are naturally confluent proportions.  There could not be an irrational "ratio." Historically speaking, "irrational ratio" would be an oxymoron, not available to our shared cultural experience prior to Left hemisphere dominance.  Jaynes ballparks this emergent phenomenon at about 500 to 1,000 A.D.

Traveling back further to the age and cultural center of PIE, the "Golden Ratio" and the "Golden Rule" would have been mutually self-reflective as the Natural Law of "as within, so without; as without, so within." (And, Homer's epic search for the Golden Fleece may be an analogy for our continued search for the headwaters of the Golden River (Styx), where the mutually defining relationship between Golden Ratio and Natural Rule may be remembered as holonically expressed. But that is another journey.) This Natural Law might have emerged from measuring any internally radiant radius.  There are 2 "radiants" of equal length emanating from a Core Vector (borrowing, perhaps anachronistically, from Fuller), creating 1 dia-meter.  It would follow the "as without, so within" Golden Rule that this must also be true within any symmetrical structure, in any of the four dimensions. Two-over-one was probably the Golden Ratio gestalt behind pyramid design (for more on the Golden ratio, and P.I.E. linguistics, easily accessible, www.wikipedia.org is to the point).

The two-dimensional symbolic proportion of radius to circumference, drawn on a flat surface, was a more contemporary Left hemisphere inquiry.  It seems likely that "pi" emerged as a symbol for the rationally-accessible ratio of three symmetrically equidistant radiant lines, forming the outside vector of an equilateral triangle.  "Pi," in PIE, is a radiant ratio of 3 to 1  The circular version would have followed by analogy, not by calculation.

Rev. Jaynes observes that "it is difficult to see a promise [Left hemisphere "noticing"] in the blows of defeat." (p. 209)  In our Irrationality case, the defeat appears to be our continued Left hemisphere fascination with resolving the Right hemisphere's sense of "chaos" and "complexity".  Mathematicians refer to this dis-functionality as "Non-Polynomial."

Information Systems measure Polynomial-Time in a prime-root mutually defining "0" and "1" Language. Perhaps it is this metric Language structure that is best suited to "adjust" our metric symbol evolutionary journey. but it is "difficult to see" this promise. To do so would require a universal willingness to return to a "commensurable" and "rational" Supervenience Theory. The most Prime-Relationship symbol for Fuller's and Thurston's Core Vector structure is a Prime Trinitarian, mutually, equally proportioned, and confluently calculating "1/0". Polynomial values, (basically "integers") whether temporal or spatial, must always be "Natural." A prime-root 8-fold (Thurston) frequency structure corollary is that the set of integers is with distinguishable ("cognizable", as per Jaynes) proportions, and equitably balanced human measurement functions, in any 4-dimensional Universe obey prime-structured "Natural Law." Positive and Negative are always, and at all moments in Time, equivalent.  There can be no more Non-Polynomials than there are Polynomials, inclusive of Universal Information System and "time" metrics.

It has been the "Prime" Assumption of our cognitive evolutionary journey, tracking back at least as far as Ptolemy, flowing fully on through the early Right hemisphere, Old Brain, dominant symbolic metrics, that P and 0, are equal to NP and 1, not only across the four dimensions as a whole, but also within each of the four dimensions.  "As within, so without. As without, so it is within."

Admittedly this adds a metric frequency to Arthur Peacock's Supervenient (Evolutionary) Theory, metaphysically, culturally, neurally, and perhaps genetically. It seems to be supportive of his Christianity informs Science thesis, and perhaps adds very few Information-bits to the linguistic intent of Christian theology from Origen through Thomas Aquinas. At the same time, this Analogical Information System Theory explicitly includes all the much older and more diverse Traditions, ancient symbols, aboriginal cultures, and early geometrics.

Global Value Implications

The idea that NP, translated as stress, dissonance, "Boundary Experience," non-predictable, chaotic, and, yes, irrational and incommensurable, might not be equally proportioned with Polynomials in each of the four dimensions is not really the most significant STEM hypothesis that we could choose at this critical juncture, for optimizing the future of our children's children. Far more promising, based on the Geometrization Conjecture of Thurston, the Synergetics paradigm of Buckminster Fuller, the Right hemisphere long-term sustainable "Goodness" and "Proportion" agenda, as per both of the Jaynes, and our Binary-Cameral Species' 4-dimensional optimization stress/confluence equity value agenda, is the question of how to trend confluence proportions 1/0 and P/NP and +/(-) such that: 0 balances 1, Yang-e balances Yin-e, and +e balances (-)e.

When universal (Left hemisphere "hope) + (Right hemisphere "faith") = -1/()/+1, then "as perceived without, so within", is confluently balanced "power-from-within".  After that collective "Aha!" moment, I could well imagine Fuller, Kuhn, Jaynes (Reverend and Psychologist), the Elephant, and their Golden Room as happily ever after.

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Bateson, Gregory, Mind and Nature, A Necessary Unity, 1979, E. F. Dutton, New York.

Drees, Willem B, Religion and Science in Context: A Guide to the Debates, 2010, Routledge, London.

Fuller, R. Buckminster, Synergetics, 1975, MacMillan Publishing, New York.

Fuller, R. Buckminster, Synergetics 2, 1979, MacMillan Publishing, New York.

Jaynes, Julian Clifford, Magic Wells: Sermons…, 1922, George Ellis Co, Boston.

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