Quaternity, quaternions and the unifying goal of partnership?

Eliciting Potential Patterns of Governance from 16 Sustainable Development Goals (Part #11)

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There is a curious sense in which the nature of a goal, especially the 17th Goal of the SDGs as the "Partnership of the Goals", is especially elusive -- despite the ease with which reference is made to it (Engaging with Elusive Connectivity and Coherence, 2018). Particularly problematic is a form of premature closure and definition associated with its identification, most notably through use of "target" (Health and sustainability misleadingly framed as target acquisition, 2012; Reframing a fundamental attractor as a target, 2014 Enhancing Sustainable Development Strategies through Avoidance of Military Metaphors, 1998). This is equally true of the many references to unity in a global context -- and the desirability of it.

Missing in such usage is recognition that the objectivity with which "goal" may be variously represented tends to obscure the subtlety of which it may be indicative, as exemplified by circular configuration of targets, dartboards, mandalas, codons and hexagrams (Objectively understood configurations indicative of fundamental cognitive implication, 2021; Tabling a motion according to rules of order in debate, 2016). This may well have contrasting aesthetic dimensions too readily set aside, as in the appeal of liminality (Liminality of betwixt and between, 2011).

The considerations of quaternity noted above, especially from a depth psychology perspective, can be readily recognized as equally subtle and elusive -- to be understood progressively (if at all) -- and individually rather than collectively. Somewhat ironically, this could also be considered the case with respect to insight into the degree of mathematical abstraction encompassed by quaternions.

Given their etymology, it is therefore somewhat surprising to find little exploration of how these disparate perspectives might be related. One exception is the brief commentary by Herb Klitzner (Welcome to the Culture of Quaternions: past, present, future, The Culture of Quaternions: the Phoenix Bird of Mathematics, 18 January 2015; Quaternions and the World of Carl Jung and His Followers, 1 March 2015; Quaternions, Cognition, Music, and 4D: a detailed exploration, May 2015).

Curiously there is frequent reference in a management context to "quarterly goals" (Jay O'Donnel, Quarterly Goals: How To Set Them And Why They Work, 2021; Tan Shirley, Visualizing Success: How To Set Up Effective Quarterly Goals, Business.com, 29 June 2022; Lee Garrett, The Power Of Quarterly Goals, Productivityist, 2022). The sense of "goal" does not however seem to feature in the other references to quaternity and the fourfold from the integrative psychological perspective (as noted above). The sense of a fourfold goal has however been presented as fundamental to the Christian mission (Joseph Babij, The Four-Fold Purpose of the Church, Calvary Community Church, 2019; The Fourfold Gospel, Reidsville Alliance Church). The Fourfold Gospel is the Christological summary on which the core values of the Christian and Missionary Alliance is based.

It is therefore ironic to discover that "goal" does feature in some applications of the algebraic abstractions framed by quaternions in robotics. The challenge in that context is how "goals" can be attributed through programming to robots and to artificial intelligence more generally (Ilian Bonev, How to Use Quaternions in Industrial Robotics, Mecademic, 18 February 2022). The focus could raise the question of possibilities of new insights into how goals can be attributed to humans collectively, or elicited from them -- as implied by techniques of motivation. Intriguingly Bonev uses the term "end-effector" rather than goal, suggesting another way of understanding the 17th SDG goal -- whether desirable or inherently questionable.

The nature of quaternions may be understood to a limited degree through comments such as:

• In mathematics, the quaternion number system extends the complex numbers, but in four dimensions instead of just two.
• Quaternions are hypercomplex numbers with 4 dimensions that can be used to represent 3D rotations -- used in graphics programming as a compact representation of the rotation of an object in three dimensions (Kenwright, Dual-Quaternions and Computer Graphics, 2020).
• Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic structure analysis. They have been applied in many areas including control theory, signal processing, number theory, flight dynamics and navigation systems of aircraft, orbital mechanics, bioinformatics, molecular dynamics, quantum mechanics, among others
• Quaternions are often the best choice whenever rotation or attitude representations are required. This includes robotics, aerospace engineering, video games, and the like (Understanding Quaternions, 3D Game Engine Programming, 25 June 2012). They are of particular use in optimal control or state estimation scenarios: they are often the representation of choice for the attitude of an object.

With its extensive use of computer graphics, the exercise above invites the question as to the extent to which it could be considered to involve quaternions -- ironically unbeknownst to the developer of the animations presented. References to quaternions make explicit reference to their relation to the rotational symmetry group of the regular tetrahedron -- the focus of the exercise.

Whether reference is made to quaternity, quaternions or polyhedra, there is clearly a challenge to representation of the subtle coherence associated with "goal" and its "unifying" function. In one sense, any preferred representation may well be essentially misleading. The corollary is however that recognition of the complementarity of disparate forms of representation may itself be a challenge.

There is even the curious possibility that correspondences between them may merit recognition (and deprecation) as in the case of the "moonshine theory" through which the "monstrous" nature of the most fundamental forms of symmetry -- the "monster group" -- has been discovered and explored in mathematics (Potential Psychosocial Significance of Monstrous Moonshine, 2007). The engagement with "correspondences" which this required is indiciative of the exceptional form of symmetry which may be characteristic of any Rosetta stone for cognitive frameworks (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). As "partnership of the goals" of global governance, the "monstrous challenge" of SDG Goal 17 would appear to merit corresponding appreciation -- avoiding tendencies to misleading oversimplification.

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