Psychosocial and strategic implications of distinct octants

Comprehension of Requisite Variety via Rotation of the Complex Plane (Part #3)

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An earlier approach to this theme elaborated a fourfold framework to interrelate four contrasting categories of Problematique, Resolutique, Imaginatique and Irresolutique (Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007). The first two were introduced by the Club of Rome, but only the first is commonly used. Together they framed the question as to the extent to which "problems" and "strategies" could both be considered imaginary in some ways, or at least intangible concepts. The last identified the game-playing associatd with effective decision-making. A degree of relation to the complex plane was indicated through the use of an axis of explicit real and an axis of explicit imaginary -- thereby suggesting that the four quadrants of the complex plane could be used to explore that pattern.

The mutually orthogonal configuration of complex planes offers a framework inviting consideration of the distinctive octants so framed. The coding convention for the octants noted above suggests one approach. The axial distinctions between positive and negative extremes suggests another. Most intriguing is how the distinction between real and imaginary axes can inform that exploration, especially given the emergence of a third axis as a consequence of rotation of the original plane. Given the asymmetric positioning of the Mandelbrot rendering with respect to any axis, there is the further question of the chirality consequent on the direction of rotation.

In this preliminary discussion, it is not proposed to attempt any reconciliation of these possibilities -- necessarily presumptuous. Of potentially greater interest are the distinctions which may be framed by each octant as the locus for the articulation of strategy of a given (cognitive) style -- a "way". In developing that argument, there is a case for recognizing the original association of the Cartesian x-axis with real numbers and the y-axis with imaginary numbers -- thereby framing the question as to what a z-axis might be understood to imply as the third axis.

Imagination vs. Reality: It is of course strange that the use of "imaginary" by mathematicians in the study of complexity has virtually nothing to do with the imagination which is recognized as a valuable theme of both strategic thinking and of mathematics (Matthew Handelman, The Mathematical Imagination: on the origins and promise of critical theory, 2019; Ricardo Nemirovsky, Mathematical imagination and embodied cognition, Educational Studies in Mathematics, 70, March 2009; Edward Kasner and James R. Newman, Mathematics and the Imagination, 1940). How is that imagination to be understood as related to the reality with which governance is purported called upon to deal -- purportedly informed by the "real numbers" of statistics?

Curiously it is asserted with respect to an imaginary number, or imaginary unit (denoted by "i"), that despite the historically "imaginary" nomenclature (and initial scepticism in that regard), complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world. It is therefore unfortunate that mathematics has been unable to extend this comprehension to the imagination which is so fundamental to the manner in which the people of the world engage with reality -- a theme of relevance both to the reality of religion in society and to the relation between art and science, as addressed by Arthur Koestler (The Truth of Imagination, Diogenes, 25, 1977). With so many iconic mathematicians having been people of faith, it is also unfortunate that in a society wracked by religious violence that mathematics has found so little means of engaging credibly with the imaginative dimension which empowers such conflict (Mathematical Theology: Future Science of Confidence in Belief, 2011).

Potgentially fundamental to that argument is mathematical appreciation of the so-called Euler identity. As discussed separately, this equation has been named as the "most beautiful theorem in mathematics" and has tied in a nomination by mathematicians for the "greatest equation ever" (Robert P. Crease, The greatest equations ever, PhysicsWeb, October 2004). It may be presented as follows in two variants:

Euler Identity
some variants	legend
e π i + 1 = 0	e is Euler's number, the base of natural logarithms, π is the ratio of the circumference of a circle to its diameter i is an imaginary number defined by its property i2 = -1. This is consistent with 3 variants: i0 = 1,  i1 = i,  i3 = -i
e π i = -1

In a much quoted comment with reference to the above by Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth. In the light of the arguments of George Lakoff and Rafael E. Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000), metaphor has been exploited to facilitate its comprehension (Understanding Without Proof, 2004; Intuitive Understanding of Euler's Formula, 2010; Chris Fields, Metaphorical Motion in Mathematical Reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains, 2013).

The argument here exploits the mathematical formalism of imaginary numbers in order to give a degree of formal "legitimacy" to the role of the imagination, whether individual or collective, whether in governance or otherwise. Whilst this may be anathema to mathematicians, the question is what imaginative insights articulated by mathematics are of value to the engagement with the surreal challenges of governance at this time?

In the surreal condition of society calling for appropriate governance, if the question is to be understood as one of how to encompass both the "unimaginative" and the "unreal", can the distinction between real numbers and imaginary numbers offer an indication beyond the constraints of the mathematical formalism? Is there a strong case for "appropriate appropriation" of mathematical insights at this time?

Quadrants from Cartesian 2-fold axes: The surreal condition of the times can be fruitfully associated with the confusion of fake news and the polarization of discourse -- extending into the irrational world of blame-gaming and demonisation. Polarization can therefore be distinguished or conflated as:

real versus imaginary

positive versus negative

Conventional renderings of the Mandelbrot set then offer the distinctive quadrants indicated in the image above: of the complex plane.

Quadrants of complex plane		Quadrant notation
real negative / imaginary positive	real positive / imaginary positive	(- / +)	(+ / +)
real negative / imaginary negative	real positive / imaginary negative	(- / -)	(+ / -)

To that pattern may be added those of the tetralemma -- with the emphasis it offers as to the nature of the confusion of surreality. This is of concern in paraconsistent logic, namely the effort to deal with contradictions in a discriminating way (Hajime Sawamura and Edwin D. Mares, How Agents Should Exploit Tetralemma with an Eastern Mind in Argumentation, In: Barley M.W., Kasabov N. (eds) Intelligent Agents and Multi-Agent Systems. PRIMA 2004. Lecture Notes in Computer Science, 3371. Springer, 2004).

Tetralemma (or Quadrilemma)		Fourfold experiential strategic assessment
["IS NOT"]	["IS"]	realistic doom-mongering	realistic hope-mongering
["neither IS nor IS NOT"]	["both IS and IS NOT"]	fearful imaginings	emergency preparedness

Four-gatedness has been variously explored in fiction (Doris Lessing, The Four-gated City, 1969). The domain of "fearful imaginings" is exemplified by the description by Stanley Brunn (Gated Minds and Gated Lives as Worlds of Exclusion and Fear, GeoJournal, 66, 2006, 1/2).

Octants from 3-fold axes: With the rotation of the complex plane on the x-axis, such distinctions would appear to hold although the consequent incursion into the z-dimension implies the emergence of other considerations. It is the rotation of the complex plane on the y-axis that confirms or reinforces the existence of such a dimension -- whatever it may be held to imply in containing more appropriately the surreal or hyperreality.

In contrast with the 2-fold polarization, the z-axis could then be understood as indicative of what is variously and mysteriously indicated as a "third way" -- with which strategic initiatives have been variously associated, notably as articulated by Anthony Giddens (The Third Way: the renewal of social democracy, 1999; The Third Way and its Critics, 2000). From the perspective of the above argument, it has been unclear with what part of "strategic space" such initiatives are associated -- and how to distinguish it from other parts. Reference is also variously made to an elusive "middle way", notably the Middle Way described by Siddhartha Gautama as the path leading to liberation. Other variants include the philosophical golden mean between extremes, the Confucian Doctrine of the Mean, a Middle Way Approach advocated for Tibet (in China), and the political philosophy of Harold Macmillan (The Middle Way, 1938).

Despite the variety of such references, and the degree to which some have been related to the strategic challenges of governance, there is clearly a subtlety to any "middle way" which transcends the problematic dynamics of polarization. One "reasonable" approach to its clarification is through second and higher order feedback processes characteristic of cybernetics. It could then be inferred that the transcendence characteristic of a middle way would be associated with a self-critical capacity and some degree of enactivism. From a poetic perspective these could be understood in terms of the negative capability famously articulated by John Keats. Such subtlety could be recognized in interoception, as argued by Noga Arikha (The interoceptive turn is maturing as a rich science of selfhood, Aeon, 17 June 2019).

The eightfold pattern of octants indicated above suggests the possibility of a fruitful relation to classical Chinese thinking and its succinct use of the trigram notation of a pattern of 3 broken or unbroken lines.

Attribution of trigram codes and signs to octants
Adaptation of table above	Attribution of trigram codes	Octant sign convention
x z y I + + + II - + + III - + - IV + + - V + - + VI - - + VII - - - VIII + - -		real "third" imaginary I pos "pos" pos II neg "pos" pos III neg "pos" neg IV pos "pos" neg V pos "neg" pos VI neg "neg" pos VII neg "neg" neg VIII pos "neg" neg

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