Topology of consensus -- and bonding

The Consensus Delusion (Part #13)

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Geometry and topology offer tools for exploring the illusions which sustain consensus delusion. As argued above, this is most evident with respect to people variously positioned on a sphere and the conflicting claims they would then variously make -- if unaware of the geometry. The question is what surface facilitates consensus -- or rather to what degree any "habitable" surface facilitates consensus of a particular degree.

Flat: Whilst a flat surface would seem to do so, as seemingly advocated by Friedman (2005), it does not however honour those "on the other side". In practice they may well be demonised -- "there be monsters" -- as with the interaction between adherents of the World Economic Forum and the World Social Forum. The perspective of the other is then held to be irrelevant and even a fantasy, or both.

Sphere: It would seem that "global" civilization has as yet to catch up cognitively with the sense in which people live on a sphere -- and with the cyclic dynamic which ensures the coherence of that understanding (or sustains the respective delusions, failing awareness of its global implications). Ironically the dynamic is associated with "spin" -- and metaphorically with reinforcement of the delusion. A number of mathematicians have offered fictionalised accounts of the cognitive challenges encountered by dwellers on "flatland" in comprehending a third dimension (***).

Spherical polyhedra: However the very fact that mathematics has explored more complex forms suggests that these may be vital to civilization for the heightened levels of consensus required by the emerging challenges of psychosocial organization and governance. The question may be addressed in terms of the kinds of cognitive and social bond implied by "consensus". On a flat surface the dwellers will tend to form "two-dimensional" relationships, possibly mapped as flat networks, as is now so often done. In three dimensions, networks become more complex and can notably be mapped as polyhedra, where the robustness of consensus can usefully be related to spherical symmetry (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008).

Torus: It would seem however that even such potential robustness of bonding does not respond to the more complex challenges of consensus characterized by more radical disagreement. This could be understood as being sustained by lack of awareness of the more complex forms of "globality" -- the sphere being the simplest variant. In terms of topology, the sphere is in fact a "degenerate" form of torus -- which can itself have many higher dimensional forms. A torus allows for more complex forms of bonding and therefore for consensus of greater coherence, as separately discussed (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: Transforming a matrix classification onto intertwined tori, 2006). As noted in that argument, the torus has particular implications for more coherent argument, transcending the (typically unexamined) constraining assumptions associated with description on a flat surface.

Paradoxical forms: Comprehending the consensus potentially associated with the patterns of bonding of a torus is necessarily more challenging than that on a simple sphere. Hence the value of exploring simple forms which hold the level of paradoxical complexity undermining consensus and effectively honour and "legitimate" the associated delusions.

The simplest form of this kind is the Möbius strip -- a band so connected that it has only a single side. For a dweller at any point on such a band there is indeed another "side" -- but traveling along the band, the distinction between the "sides" is recognized as an illusion. A more complex variant is the Klein bottle, which appears to have an "inside" and "outside", but again these are not distinct -- in four dimensions however. Both the Möbius strip and the Klein bottle offer surfaces which respond to the divisive nature of the delusion that consensus can be effectively associated with a single side. Such possibilities are discussed separately (Metaphorical Geometry in Quest of Globality, 2009).

Mandelbrot set: Mathematics has explored numerous forms of greater complexity with potential psychosocial implications -- although these are of no interest whatsoever to the mindset of that discipline. Many may be understood as offering insights into forms which could sustain higher orders of consensus and bonding -- whether cognitive or relational. Striking examples are offered by symmetry groups as mentioned above, and by the Mandelbrot set (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005). The latter is especially significant in terms of the elegance of its visual rendering and the variety that may be explored within it.

"Rosetta Stone"? Speculatively it is appropriate to ask, from the design perspective explored by Christopher Alexander, what might be the topological "design" of a "Rosetta Stone" or a "Philosopher's Stone" of requisite complexity -- as metaphors of relevance to the level of "non-delusional" consensus required by the future (Systemic Crises as Keys to Systemic Remedies a metaphorical Rosetta Stone for future strategy? 2008; Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

As implied with respect to spherically symmetrical polyhedra, the development to higher and more sustainable degrees of "consensus" would seem to be associated with a process of cognitive embodiment involving higher degrees of bonding. A useful comparison could be made with the various forms and bonding possibilities of of carbon -- including diamond and fullerenes. Diamond, in contrast to other crystals, is especially useful because of the value attributed to it due to its optical properties (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002).

"Strange loops": The inadequacy of crystal forms in holding the "essential" qualities of consensus is that it readily reinforces the objectification associated with more constrained forms of consensus -- consensus of low degree. It is in this sense that the self-reflexive explorations of Douglas Hofstadter are so helpful (Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979; I Am a Strange Loop, Basic Books, 2007). Appropriate to this argument is his further work with Emmanuel Sander (Surfaces and Essences or the The Essence of Thought, 2011).

Exploiting the Möbius strip and the Klein bottle, their "strangeness" with respect to consensus lies in how they imply self-reflexivity to a degree which challenges conventional approaches and allows for the mirroring to which reference was made above in connection with Maya. Consensus may then be explored in terms of an unconventional approach to community (Sustaining a Community of Strange Loops: comprehension and engagement through aesthetic ring transformation, 2010).

Use of the Möbius strip may be taken further to hold more explicitly the paradoxical relationship between subjectivity/objectivity and static/dynamic in any consensual relationship (¡¿ Defining the objective ∞ Refining the subjective ?! Explaining reality ∞ Embodying realization, 2011). Related use may be made of the Klein bottle (Intercourse with Globality through Enacting a Klein bottle: cognitive implication in a polysensorial "lens", 2009).

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