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Recognizing a confluence of imaginative possibilities

Imagining Order as Hypercomputing (Part #11)

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Given the accumulating levels of global crisis, any exercise in re-imagining possible order merits consideration. In the light of the argument above, some threads which could be considered to be converging are indentified in what follows. Part of the challenge may lie in how they are configured or woven together (Interweaving Thematic Threads and Learning Pathways, 2010).

People are increasingly obliged to dwell in virtual worlds -- creations of their imagination (Living as an Imaginal Bridge between Worlds: global implications of "betwixt and between" and liminality, 2011). There is then a case for imagining how an imaginary society might be sustained (Manfred B. Steger, Globalisation and Social Imaginaries: the changing ideological landscape of the Twenty-First Century, Journal of Critical Globalisation Studies, 2009; Benedict Anderson, Imaginary Societies, 2003; Cornelius. Castoriadis, The Imaginary Institution of Society, 1987). Of particular interest is the framing of Imaginary Society cultivated by the Imaginary Foundation.

Technomimicry through technological metaphors: The argument for engines as a generative metaphor was made above. This could be extended to aircraft engines (as highlighted by references to the helicopter), to rotary engines (as with the Wankel engine), or to electric motors and dynamos. Also separately argued are the insights to be derived from nuclear fusion and the requisite toroidal reactor design (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006).

Deriving patterns from geometry and topology: here is a curious ambiguity between the unifying claims and seemingly divisive roles of belief systems in society. Great emphasis was placed above on the relevance of mathematics to eliciting more fruitful patterns of order through which the roles and relationships of belief could be comprehended -- with its indicative potential for hypercomputing.

The focus above was on 6-fold patterns, as with the I Ching and the Star of David. Especially problematic are of course the relations with Islam, potentially exemplified by the 5-fold pattern of the Islamic Star. Geometry offers an interesting ways of reconsidering that relationship, as argued separately (Middle East Peace Potential through Dynamics in Spherical Geometry: engendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks, 2012).

Especially intriguing, given the association of hypercomputing with analogy, are explorations of mirroring and the paradoxes of introversion -- given that they may constitute phases in a cognitive cycle (World Introversion through Paracycling: global potential for living sustainably "outside-inside", 2013).

Particular significance can be associated with the discovery of mathematical surfaces on which contrasting perspectives can be located such that each has the "superficial" sense of being "right" -- and reinforced in the impression that others, located elsewhere on the surface, are necessarily "wrong" -- even "upside down' . One of the most comprehensible surfaces of this kind is the planetary sphere, on which conflicting assertions as to whether it is "day" or "night" can be made at any one time. This example points to the possibility of working with disagreement, rather than endeavouring to eliminate it (Using Disagreements for Superordinate Frame Configuration, 1992). As suggested by the Star of David configuration, the 12 "tribes of Israel" may be variously oriented to one another -- if not orthogonally -- inhibiting agreement.

Memorable clusters: There is a marked tendency to order many kinds of information into sets exhibiting particular charcteristics (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980). This can be highlighted by the following simple multiplication table linking below to examples of global significance.

Examples of patterns of order associated with the simplest numbers
  3 4 5 6 7 8 9
3 9 12 15 18 21 24 27
4 12 16 20 24 28 32 36
5 15 20 25 30 35 40 45
6 18 24 30 36 42 48 54
7 21 28 35 42 49 56 63
8 24 32 40 48 56 64 72
9 27 36 45 54 63 72 81


  Governance groups Selected polyhedra for mapping Encodings and research
4 Tetrahedron (F/V)
5 UN Security Council
(Permanent members)
Triangular prism (F)
Square pyramid (F/V)
Wu Xing (5-processes cycle)
Hygieia (Pythagoreans)
6 Tetrahedron (E)
Cube (F)
Octahedron (V)
Triangular prism (V)
7 Szilassi polyhedron (F) George Miller (The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information, 1956)
8 Group of 8 (G8)

Cube (V)
Octahedron (F)
Truncated tetrahedron (F)
Square pyramid (E)

BaGua; Stephen Prothero (God Is Not One: the eight rival religions that run the world, 2011)
9 Triangular prism (E) Enneagram (A. G. E. Blake, The Intelligent Enneagram, 1996)
12 Various round tables Cube (E)
Octahedron (E)
Dodecahedron (F)
Icosahedron (V)
Truncated tetrahedron (V)
Cuboctahedron (V)
Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources (2011);
Multiple references
14 Cuboctahedron (F)
Truncated octahedron (F)
Truncated cube (F)
Szilassi polyhedron (V)
15 UN Security Council; Group of 15 (G15) Transformations identified by Christopher Alexander (Harmony-Seeking Computations, 2009)
18 Truncated tetrahedron (E) Multiple references
20 Group of 20 (G20) Dodecahedron (V)
Icosahedron (F)
Amino acids (as encoded directly by the genetic code)
21 Group of 21 (Non-Aligned Nations in the Conference on Disarmament) Szilassi polyhedron (E) Multiple references
24 Group of 24 (G24) Cuboctahedron (E)
Truncated cube (V)
Rhombicuboctahedron (V)
Snub cube (V)
Multiple references
26 Rhombicuboctahedron (F)
Truncated cuboctahedron (F)
Multiple references
30 Group of 30 (G30) Dodecahedron (E)
Icosahedron (E) Icosidodecahedron (V)
Goldberg Variations of Bach; Requisite variety of perspectives (Stafford Beer, Beyond Dispute: the invention of Team Syntegrity, 1994); Participants in the The Conference of the Birds by Farid ud-Din Attar.
32 Icosidodecahedron (F)
Truncated icosahedron (F) Truncated dodecahedron (F)
Multiple references
36 Truncated octahedron (E)
Truncated cube (E)
38 Snub cube (F)
42 Small rhombidodecahedron (F) Answer to The Ultimate Question of Life, the Universe, and Everything by Deep Thought
48 Rhombicuboctahedron (E)
Truncated cuboctahedron (V)

Snub cube (E)
Icosidodecahedron (E)
Truncated icosahedron (V)
Truncated dodecahedron (V)
Rhombicosidodecahedron (V)
Snub dodecahedron (V)
Small rhombidodecahedron (V)

Multiple references
62 Rhombicosidodecahedron (F)
Truncated icosidodecahedron (F)
Snub dodecahedron (F)
64 Hexagrams of the I Ching (Book of Changes)
72 Truncated cuboctahedron (E) Multiple references
81 Tetragrams of the Tài Xuan Jing (Canon of Supreme Mystery)
90 Truncated icosahedron (E)
Truncated dodecahedron (E)
92 Snub dodecahedron (F)
120 Rhombicosidodecahedron (E)
Truncated icosidodecahedron (V)
Small rhombidodecahedron (E)
150 Snub dodecahedron (E)
180 Truncated icosidodecahedron (E)

With respeect to the imagining of order framed as characteristic of hypercomputing, the above tables suggest how the variety of polyhedral patterns may be considered metaphorically as a form of cognitive gearbox. This is then indicative of pathways along which changing gear is viable between distinct patterns of order through compatible transformations. Especially relevant is the manner in which significance can then be variously attributed to faces (F), vertices (V), or edges (E). examples of such attributions are given separately (Mapping possibilities with regular polyhedra, 2008; Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008; Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009)

Metatheory, metamathematics and metaphysics: If hypercomputing is to be associated with meta-analogy, as argued above, there is a case for exploring understandings of other "superordinate" approaches. This is curiously associated with "omega" as originally highlighted by Pierre Teilhard de Chardin in terms of the Omega Point -- a maximum level of complexity and consciousness towards which he believed the universe was evolving.This can be compared to the logical arguments of Gregory Chaitin (Metamaths: the quest for omega, 2007; Thinking about Gödel and Turing, 2007; Mathematics, Complexity and Philosophy, 2011). As noted above, such preoccupations can be related to hypercomputing (Stephen Blaha, The Metatheory of Physics Theories and the Theory of Everything as a Quantum Computer Language, 2005).

There is some irony to the fact that the so-called super-Turing computer (or O-computer), should have been associated with an oracular function -- given that the I Ching has been widely valued as a form of oracle through its use in divination processes. Given the manner in which circular processes were highlighted above, similar irony can be explored with respect to occurrences of the O-ring (The-O ring: Theory, Theorem, Theology, Theosophy? a playful intercultural quest for fruitful complementarity, 2014).

Significance attributed to symbols of belief systems: As an extension of the use of polyhedral forms, symbols of a variety of belief systems, including the Christian Cross and the Star of David, can be interrelated in exploratory animations of greater complexity (Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation, 2008).

Given the focus on distinguishing the cognitive phases of imaginative hypercomputing, of particular interest is the order implied by traditional patterns especially associated with health in its most general systemic sense. In this respect a comparison was made separately between the Chinese 5-phase Wu Xing pattern and that of the Pythagorean Hygieia from which "hygiene" derives (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012). These can be provocatively compared with recent understanding of hypercycles (as illustrated above).

Hugieia Pentagram of Pythagoreans
Chinese 5-phase Wu Xing cycle Hypercycle
Hugieia Pentagram of Pythagoreans Chinese 5-phase Wu Xing cycle

Aesthetics, poetics and autopoiesis: As was briefly argued above, aesthetics can play a fundamental role in engendering and sustaining a pattern of associations potentially vital to the coherence of hypercomputing processes. This is especially evident in music and poetry and the value of their experience. Given the technical arguments for autopoiesis, it is curious to recognize the origins it shares with poesis and poetry-making. This merits consideration in the light of the extreme contrast between the logical encoding of I Ching hexagrams and the poetry through which commentary on them is presented.

In this sense the order engendered in hypercomputing can be understood as poetry-in-the-making, as can be variously argued (Being a Poem in the Making: engendering a multiverse through musing, 2012). The argument can be extended to mysuc and song (A Singable Earth Charter, EU Constitution or Global Ethic? 2006). It is appropriate to ask how the sustaining memorability of sung epic poems, like the Mahabharata -- could be understood in terms of hypercomputing. Of particular potential relevance is the signifiance associated with overtone singing.

With respect to arguments of cognitive psychology for the embodiment of mind, especially in movement, there is a case for recognizing the role of dance in sustaining a form of recognition -- or enaction -- of hypercyclic patterns.

Personal implication in hypercomputing: This speculative exploration was introduced with the indication that hypercomputing could prove to be much more intimately related to cognitive processes than is currently suspected (Radical Cognitive Mirroring of Globalization: dynamically inning the unquestioningly outed, 2014). Attention was drawn to relevant arguments of such as Hofstadter and Sander (Surfaces and Essences: analogy as the fuel and fire of thinking, 2013), notably in the light of the imaginative creativity of such as Albert Einstein. To what extent is the imaginative ordering of reality, in which many engage, to be considered as hypercomputing?

One valuable summary of speculation on these matters is offered by Selmer Bringsjord and Konstantine Arkoudas (The Modal Argument for Hypercomputing Minds, Theoretical Computer Science, 2004):

We now know both that hypercomputation (or super-recursive computation) is mathematically well-understood, and that it provides a theory that according to some accounts for some real-life computation (e.g., operating systems that, unlike Turing machines, never simply output an answer and halt) better than the standard theory of computation at and below the "Turing Limit". But one of the things we do not know is whether the human mind hypercomputes, or merely computes -- this despite informal arguments from Gödel, Lucas, Penrose and others for the view that, in light of incompleteness theorems, the human mind has powers exceeding those of TMs and their equivalents. All these arguments fail; their fatal flaws have been repeatedly exposed in the literature. However, we give herein a novel, formal modal argument showing that since it's mathematically possible that human minds are hypercomputers, such minds are in fact hypercomputers. We take considerable pains to anticipate and rebut objections to this argument.

Bringsjord has also articulated the argument otherwise (Selmer Bringsjord and M. Zenzen, Superminds: People Harness Hypercomputation, and More, 2003), described as follows:

This is the first book-length presentation and defense of a new theory of human and machine cognition, according to which human persons are superminds. Superminds are capable of processing information not only at and below the level of Turing machines (standard computers), but above that level (the "Turing Limit"), as information processing devices that have not yet been (and perhaps can never be) built, but have been mathematically specified; these devices are known as super-Turing machines or hypercomputers. Superminds, as explained herein, also have properties no machine, whether above or below the Turing Limit, can have... The final chapter of this book offers eight prescriptions for the concrete practice of AI and cognitive science in light of the fact that we are superminds.

As a metaphor -- about engagement with analogy and metaphor -- to what extent does the remark of Kenneth Boulding apply:

Our consciousness of the unity of the self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of a group, organization, department, discipline, or science. If personification is only a metaphor, let us not despise metaphors - we might be one ourselves. (Ecodynamics; a new theory of societal evolution, 1978, p. 345)

Or, as the poet John Keats puts it: man's life is a continual allegory - and very few eyes can see the mystery of his life - a life like the scriptures, figurative. Or again, as Gregory Bateson stated in concluding a conference on the effects of conscious purpose on human adaptation, is that: We are our own metaphor. (2004, p. 304).

42 as the answer of Deep Thought after millions of years
to The Ultimate Question of Life, the Universe, and Everything.
Illustrated by animation of two 42-faced polyhedra?
(produced with Stella Polyhedron Navigator)

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