-- / --
* Two-body problem
* Three-body problem
* N-Body problem
* "Two-body" terror experience
* "Three-body" terror experience
* "N-body" terror experience
Finite vs Infinite
Terror of the sublime
Varieties of dynamic resolution of faith-based terrorism
* 1. One-body "solutions"
* 2. Two-body "solutions"
* 3. Three-body "solutions"
* 4. Configurative "solutions"
* 5. Emergent behaviour "solutions"
* 6. Embodied "solutions"
* 7. Peripatetic "solution"
The suggestion here is to focus on the the number of elements in a variety of systems in the light of the long-explored mathematical distinctions between one-body, two-body, three-body and N-body problems. The assumption is made that, as a pattern, the latter constitute a useful template through which to explore n-body systems of other kinds.
This investigation is made in the context of the framework of general systems as currently sustained by the International Society for the Systems Sciences, following the early initiatives of the Society for General Systems Research. General systems recognizes a degree of isomorphism between systems of apparently different type.
The purpose of this exploration is to use the ordering of the more tangible systems to gain greater insight into the challenges of the less tangible systems. In particular the concern is with perception of threat (especially that associated with terrorism) arising from views sustained by fundamentally irreconcilable religions (such as Christianity, Islam, and Judaism). The aim is to suggest that the set of worldviews may be seen as either a one-body problem (where there is exposure to only one worldview), or a two-body problem (where there is exposure to two worldviews), etc. The two-body case offers a description of the dynamics of the relation between God and Satan, for example -- or between any two religions that tend to demonize each other. The three-body case might then offer insights into a more fruitful dynamic between the Abrahamic religions, for example.
In experiential terms, the terror associated with the other in a two-body case, might then be understood in terms of the dynamics of a form of "two-body terror". That between the three Abrahamic religions might be understood as "three-body terror". It is assumed that the faithful of particular religions are effectively terrified by the perspective of any alternative.
The merit of such an approach is that it offers guidelines for the avoidance of inappropriate oversimplification, whatever particular resolutions are considered applicable under certain constraints. It also points beyond the tendency towards the search for structural static resolutions to the need to explore dynamic resolutions -- as possibly being the only ones offering sustainable relations between the "bodies"
The following table has rows associated with different "systems", of which the first are those well-recognized by mathematics and physics. The structure of the table draws attention to the possibility of recognizing isomorphic conceptual challenges within non-physical systems, whether biological, social, conceptual, or experiential. Such isomorphism has long been a theme of general systems research (cf James Grier Miller, Living Systems, 1995).
|Characteristics of different systems -- by number of elements|
(Sun, Moon, Earth; triple stars)
(cf coaction cardioid)
("clash of cultures")
(wave vs particle)
("crisis of crises")
The purpose here is simply to point to the complex formal descriptions of the dynamic relationships in each case. Note that simpler satisfactory solutions may be found in each case if particular constraints are allowed. Many mathematicians have given considerable attention to the solution of the equations of motions for n gravitationally interacting bodies.
Two-body problem: The two-body problem is the challenge of determining the movement of two rigid point masses in mutual orbit about each other [more]. It can be solved analytically. In mechanics, the two-body problem is a special case of the n-body problem that admits a closed form solution. This problem was first solved in 1687 by Isaac Newton who showed that the orbit of one body about another was either an ellipse, a parabola, or a hyperbola, and that the center of mass of the system moved with constant velocity. If the common center of mass of the two bodies is considered to be at rest, each body travels along a conic section which has a focus at the centre of mass of the system. If the two bodies are bound together, they will both trace out ellipses. If they are moving apart, they will both follow parabolas or hyperbolas.
In a nice analogy with planetary motion, the quantum two-body problem is exactly solvable. The most commonly encountered version of the problem, involving an inverse square law force, is encountered in celestial mechanics and the Bohr model of the hydrogen atom.
For an illustrative applet, see Joachim Köppen (Orbits of One Body around Another, 2001)
Three-body problem: This is the problem of determining the mutual gravitational interaction of three massive bodies [more]. It involves finding the positions and velocities of the three bodies, which are interacting with each other gravitationally -- at any point in the future or the past, given their present positions, masses, and velocities. It has been demonstrated to be impossible to solve in the general case, namely when the three bodies are in a random configuration. The resulting motion in most cases turns out to be chaotic, making it impossible to predict precisely what paths those bodies would follow.
The problem can only be handled in certain special cases. Joseph-Louis Lagrange showed that there were at least some solutions to the three body problem if the three bodies move in the same plane and it was assumed that the mass of one of them was so small as to be negligible. In the solution which bears his name, the three bodies move in unison, always maintaining the same positions relative to each other. If we think of a two body system (like the simplified Earth and Sun system) then the 5 points at which a third small body may be found are now known as the Lagrange points (L-5 points). These mark positions where the gravitational pull of the two large masses precisely equals the centripetal force required to rotate with them. Three are unstable and two are stable. The unstable points lie along the line connecting the two large masses. The stable Lagrange points form the apex of two equilateral triangles that have the large masses at their vertices [more].
As noted by James P. Sethna (The Restricted Three Body Problem, 1996): "After Newton solved the problem of the orbit of a single planet around the Sun, the natural next challenge was to find the solution for two planets orbiting the Sun. Many of the best minds in mathematics and physics worked on this problem in the 19th century". More generally, only the planar restricted three-body problem can be simply treated, although even this case has proved surprisingly difficult to solve even though the three masses move in a common plane.
The gravitational three-body problem has been called the oldest unsolved problem in mathematical physics. Unlike the two-body problem, there is no closed analytical solution. Numerical orbit integrations have to be used to determine the evolution of a typical three-body system. Most of these are unstable. In astrophysical systems, they decay either into three separate stars moving away to infinity, or into a binary star and a single star. There are however some stable configurations. Recently a new category of stable three-body orbits has been discovered in which the three stars chase each other in a figure-eight orbit.
Eugene Butikov (Collection of remarkable three-body motions) has provided a series of applets which provide some understanding of the "fascinating trajectories of three-body motions that delight the eye and challenge our intuition".
For an illustrative applet, see Joachim Köppen (Three Bodies in Gravitation, 1997)
N-body problem: The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e. Newton's laws of motion and Newton's law of gravity (cf K R Meyer. Periodic Solutions of the N-Body Problem, 1999) [more]
The many-body problem in quantum mechanics in quantum mechanics, is posed as the question of solving for more complex problems than the hydrogen atom, for example, the chemistry of all realistic molecules
|It would indeed be remarkable if Nature fortified herself
against further advances in knowledge
behind the analytical difficulties of the many-body problem. (Max Born, 1960)
"Two-body" terror experience: The suggestion here is that there is a case for the experiential framing of the encounter with a single source of terror as having dynamic properties analogous to the encounter with:
In each case the "attractive" properties recall:
People may be effectively petrified by such "encounters", as explored elsewhere (Thinking in Terror: Refocusing the interreligious challenge from "Thinking after Terror", 2005).
To avoid such terror, and any associated danger (notably to one's "identity" or to one's "soul"), it is understandable that the authorities of a locally dominant belief system consider themselves totally justified in eliminating any possibility of exposure to such threats by placing emphasis on:
As a "two-body" problem, there are various "solutions", in the light of the mathematical model:
These options describe situations in which the terror source is in effect static, and imply that one is oneself (as the other body) relatively small in attractive capacity. The situation may however be different:
Both the above descriptions obscure qualities of terror associated with existential exposure to what may be experienced as the "field effects" of the terror source -- perhaps when exposed to those who experience the world in radically different ways from oneself. In this case one has a more or less direct experience of the geometry of those fields, notably how one may be twisted within them to experience reality differently. An instructive description of the "traffic on this geometry", and of how the geometry is "felt", is provided by mathematician Ron Atkin (Multi-dimensional Man: can man live in three-dimensional space? 1981) as summarized elsewhere Comprehension: social organization determined by incommunicability of insights. This notably points beyond the two-body case.
"Three-body" terror experience: In this case, there are two sources of terror external to oneself -- and one's own reactions may make oneself become a source of terror. Managing this experience is much more complex. The dynamic is illustrated by:
It is possible that the dynamics of the exposure of an individual of one faith (eg Christianity) to those of two other unfamiliar faiths (eg Islam and Judaism) might be experienced as terrifying in similar ways. In this sense Jerusalem might be understood as a focus of terror.
"N-body" terror experience: In this case, the situation appears even less orderly -- and is typically experienced as chaotic. Managing this experience is even more complex. The dynamic is illustrated by:
There are interesting assumptions that tend to be embedded in the challenge of addressing a two-body, three-body or n-body problematique. These are associated with understandings of what is:
It might be asked whether the inability to provide satisfactory three-body solutions, except under particular constraints, is due to the manner in which "body" and "solution" have been locked into a particular logic. This gives special meanings to "body" and "solution" -- despite their abstract representation in mathematics -- that may not be as general as circumstances require. Aspects of the challenge have been highlighted in the incompleteness theorems of Kurt Gödel (see below). The inability is therefore a consequence of a logic imposed upon what is perceived and conceived -- a logic that is inadequate to more general cases that may be characteristic of much experience. Any conception is memetically constrained. A current challenge relating to the issue of terrorism is whether al-Qaida is a "body" and how eliminating "terrorism" and "extremism" are to be understood as "solutions".
Interesting examples of other kinds of "solution" to the n-body problem are, for example:
In such cases it might be said that the "solution" is effectively "embodied" by the dynamics of the phenomenon. In computing terms, such a solution might be understood as an "analogue" solution in contrast with a "digital" solution.
Under a system of faith-based governance, whether religious or scientific, these issues become especially significant:
It is possible that the "body-centered" approach needs to be complemented by one which emphasizes the significance of the dynamic that enables perception of the "body", which effectively constructs it. In the contrast between particle and wave theories of light, for example, it is the waveform which calls for greater attention. Any understanding of "community" may perhaps be more significantly described in terms of interweaving waveforms -- as is notably the case with atoms or molecules.
In this sense a distinction can be made between a faith-based system as a singular "body" (a "massive particle") and a faith as a set of interweaving waveforms.
The terrifying relationship between incommensutable faith-based systems, whether disciplines (C P Snow's "two cultures": sciences, humanities), or religions (Huntington's "clash of civilizations": Christianity and Islam, notably), may be better understood as the relationship of sets of waveforms -- perhaps implied by "movements of opinion". The emphasis is then on the "dynamic" rather than on the "body" -- an emphasis carried to a higher degree by current usage of "community". The respective "bodies" are effectively dynamically gated conceptual communities (cf Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004).
The excessive focus on the distinct "body" in any of these faith-based cases ensures that the experience of other "bodies" is associated with a form of terror -- terror at any dynamic that is not associated with the description of the "body" with which one is familiar, or with which one identifies. Using an astrophysical metaphor, terror is created by exposure to external gravitational effects. At the same time, there is the irony that all faith-based systems claim to seek ways of transcending the "body":
As the prime source of terror at this time, the intractable relationship between the Abrahamic religions (Christianity, Islam, Judaism) is insoluble as a three-body problem, despite references in the Qu'ran to The People of the Book, viewed in Islam both as an articulation of tolerance (Qur'an 29:46) ) and as an expression of an adversarial relationship (Qur'an 5:51) with Christanity and Judaism. As a three-body problem, particular cases might be usefully explored. What has not yet been explored is the nature of the dynamics of their existing relationship. In effect that relationship is a form of "solution" -- whether we like it or not. It is terrifying because each "body" has a heavy conceptual investment in a stasis (or internal dynamic) that is brought into question by the exposure to other forms of stasis (with their own internal dyanmics) -- with which it is then obliged to engage in a terrifying dance. Hence perhaps the folktales about appeasing the terrifying Beast by entrainment -- entrancing it through music and, by implication, dance (cf Attitude Entrainment: Communicating thrival skills and insights, 2004). Hence, also the prohibition against music and dance by fundamentalists in various religious faiths.
The definitional challenges of the previous section can be related to two well-known challenges in mathematics and to understandings of mathematical theology:
This argument therefore suggests that there is a need to "turn the problematique round" -- precisely by giving form to a dancing relationship. Perhaps current attempts might be contrasted with this approach as efforts to "fit a square peg into a round hole" -- or at "squaring the circle".
The role attributed to divinity in this context has been neatly expressed by Gregory Bateson (They Threw God Out of the Garden, CoEvolutionary Quarterly, Winter 1982) argued:
... Original Sin was the discovery of planned purpose; and that, following this discovery, Adam and Eve expelled God from the Garden.... The general notion was that God symbolized the systemic and cybernetic nature of the environment which inevitably took vengeance on man's short-sightedness. It occurs to me now that this little parable can be considered to be a serious truth -- especially if we turn it upside down.
I suggest that one of the things that man has done through the ages to correct for his short-sighted purposiveness is to imagine personified entities with various sorts of super natural power, i.e., gods. These entities, being fictitious persons, are more or less endowed with cybernetic and circuit characteristics. In a word, I suggest that the supernatural entities of religion are, in some sort, cybernetic models built into the larger cybernetic system in order to correct for noncybernetic computation in a part of that system. I do not believe anybody has said this, but I do not think that this view of religion contradicts what has been said by others -- the religious, the mystical, and the scientific.
It is readily assumed that any relationship with divinity is free of any degree of terror. The fear associated with any encounter with God is however widely recognized -- God may indeed be terrifying (Is God a Terrorist: Definitional game-playing by the Coalition of the Willing?). Fear of God has been acknowledged down the ages [more more]. As noted by by Charles H. Hinnant (Schiller and the political sublime: two perspectives, Spring, 2002):
More relevant to the theme of this exploration is the probability that any rapprochement with a divinity, other than that central to one's faith, may be experienced as even more terrifying. The relationship between faiths is therefore necessarily associated with a degree of terror -- a terror of otherness -- seldom acknowledged in interfaith dialogue (cf D Tracy, Dialogue with the Other: the inter-religious dialogue, 1990). Each faith may therefore be understood as offering a window on a different kind of terror of which it is effectively the guardian. This contrasts with the belief of co-religionists that their own religion is in no way associated with terror.
The terror of the divine, as a form of the "other", is readily transformed into terror of any "other", as noted by Reverend Tom Goldsmith (American Funk, November 1995):
Now there is so much terror of the other: the person of color, the immigrant, the homosexual, that we seek communities which preserve the narrow focus of ourselves.
| I believe that the 21st century will probably
be a century of exploring the mechanism of "becoming." It is indeed rather
sad that even if you can imagine that cosmology, or the origin of life are
associated to successions of bifurcation, we know very little about the
mechanism of bifurcations. We may safely assume that everything in our universe
is evolving in the same direction of time: rocks evolve in the same direction,
stars, galaxies, supergalaxies, all objects evolve in the same direction.
We age all together. We can only conclude that our universe seems to be
ruled by a semigroup with broken time symmetry. It is an open world in which
the direction of time plays a central role.
Ilya Prigogine (The Arrow of Time)
In the light of the above, there are possibly seven distinct approaches to resolution of the faith-based dynamics that engender, or are experienced as, terrorism. Meaningful solutions may indeed be understood as different forms of structured dialogue (cf Anthony Blake, N-Logue: an introduction) and may be explored in the light of intractable conflict (cf Guy Burgess and Heidi Burgess, The Beyond Intractability Knowledge Base Project, Conflict Research Consortium). The focus here is however on the nature and dynamics of the terror which informs interfaith dynamics -- consideration of which tends to be avoided in efforts to frame the situation positively and in a spirit of mutual tolerance.
1. One-body "solutions": Any such solution involves some form of special identification with the divine or the transcendent. This is the ultimate aspiration of any faith-based agenda -- whether understood in mystical terms as the union with God or as a Theory of Everything.
The key issue is the reconciliation with the terror associated with that identification and its experience -- knowing the "will of God". The solutionbs might then take the following forms:
These "solutions" tend to evoke questions about what is not absorbed into the ultimate synthesis. The answers may be expressed unsatisfactorily in "weak" explanations of "Acts of God" or of the "collateral damage" associated with technocratic understandings of the Theory of Everything.
2. Two-body "solutions": Such solutions are related to the well-recognized challenge of dealing with "otherness" and terror of the "other", especially the collective or corporate other . Alterity, namely the otherness of the "other" is recognized in philosophical and theological thinking. "Fear of the other" and the challenge of otherness are frequently cited in relation to:
The question is whether the mathematical approach to framing the stable solutions to the relation between two bodies offer unexplored insights into the challenge of the other.
3. Three-body "solutions": These could result from careful exploration of the conditions for viable restricted solutions of the three-body problem. They are necessarily dynamic rather than static, namely they are dependent on the different bodies -- in the light of analogues to the gravitational effects between them.
Eugene Butikov (Collection of remarkable three-body motions) provides applets to illustrate what are effectively some of the special case "solutions" to the three-body problem (from which insights might be derived for the relations between any set of three interacting belief sysrtems):
Many other examples of orbital motions can be found in an extensive package of educational software developed by Eugene Butikov (Planets and Satellites) and distributed by the American Institute of Physics. The software enables students to explore the application of classical dynamics to stars, planets, natural and artificial satellites, and manned and automatic space vehicles.
It is possible that such visualizations could trigger insights regarding new kinds of relationships between the Abrahamic religions, for example.
It is ironic that much may also be learnt about such solutions from the experience of fairground amusement rides. These include a wide variety of devices found in funfairs and amusement parks designed to appeal to various senses of riders -- and notably to "terrify" them (cf 25 Years of Orbiting, Fairground Mercury, 2001; Christine Hahn, Fairground Machines, 1979; The Philip Bradley fairground collection, 2005). It is interesting that interactive software has now been developed to encourage children to conduct research into the design of ride layouts [more]. There is continuing interest in the imaginative design and development of new rides and the experience they offer (cf Edward M. Pribonic, Twisting and Turning: simulating the stresses of a thrill ride tests the real-world soundness of a cinematic illusion, 1999). Specific attention is given to these through computer modelling for safety engineering [more]. Online tutorials exist to enable mathematicians to display the forces experienced in various fairground rides [more].
In these cases the "terror" is transmuted into the kinds of terror experienced through the dynamics and gravitational effects of fairground machines. This raises the question of the nature of the interface between "thrill" and "terror" -- and how this relates to "risk" (An Assessment of Risks at Fairground Rides: A report produced for the Health and Safety Executive, 1990).
Here the "terror" is transmuted into that which might be associated with (possibly high-density) three-dimensional traffic with a degree of centro-symmetric ordering, in contrast to the terror which may be experienced in high-density traffic in an urban grid-type system of roadways and junctions.
A suggestive pointer with respect to configuration is provided by the language of a patent that effectively addresses the issue of compositions of fundamentally irreconcilable materials, including "matter" and "anti-matter" (Composition of matter having defined energy states (EP443836: Software Patent, 1991)
A composition of matter C, prepared by combining a composition of matter A with composition of matter B, at precisely defined energies, and wherein said energies of A and B are defined by the solutions to relativistic two-particle wave equations utilizing the transverse photon exchange interaction. Compositions of matter A and B each have intrinsic spin quantum number equal to 1/2. Composition C has utility in numerous energy, communication and defense applications.
5. Emergent behaviour "solutions": In the light of the emergent behaviour that can be associated with the n-body situations of much-studied flocking behaviour, there is a case for exploring the minimum set of rules required of bodies to engender emergent order in the dynamics between them.
As noted above, flocking behaviour has been the subject of extensive study in the artificial life focus of complexity studies -- initiated by the work of Craig Reynolds with his Boids simulation (Flocks, Herds, and Schools: A Distributed Behavioral Model, 1987; Boids: Background and Update + dynamic visualization, 2001). This simulation of the relative movement of simple agents is based on the following basic rules governing their individual movement:
With these three simple rules, the flock moves in an extremely realistic way, creating complex motion and interaction that would be extremely hard to create otherwise. The emergent behaviour is like that of a flock of birds, a school of fish, or a swarm of insects. More complex rules can be added, such as obstacle avoidance and goal seeking. Recent studies and links may be found in a paper by Lee Spector, Jon Klein, Chris Perry and Mark Feinstein (Emergence of Collective Behavior in Evolving Populations of Flying Agents, Proceedings of the Genetic and Evolutionary Computation Conference, GECCO-2003).
The question is whether such "basic rules" can be understood in terms of the most basic value elements of a "global ethic" governing the relation between religions (or between any belief systems) [more | more]. It is possible that efforts towards such a global ethic have been effectively "over-designed", to a degree that has alienated many religions -- in practice if not in principle. Very simple "dynamic" rules might be less offensively constraining and might avoid calling into question "static" principles of particular religions. It is interesting that "flock" is a common descriptive metaphor in various religions, notably the Abrahamic -- although none stresses the dynamics of a flock, preferring to focus on the role of the shepherd.
The rules engendering flocking behaviour might then be interpreted in faith-based terms as:
Does the boid visualization provide insights into interfaith systems of relationships -- or into interdisciplinary systems of relationships? Bram van Heuvelen (Emergence and Consciousness: Explorations into the Philosophy of Mind via the Philosophy of Computation, 2000) successfully explores the relevance of such boid-like emergent behaviour for consciousness.
It is important to note that the emphasis is on "neighbouring". The rules govern local behaviour not the emergent global behaviour of all bodies. Aspects of this argument have been explored elsewhere (Human Values as Strange Attractors: Coevolution of classes of governance principles, 1993; Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004).
In these cases the "terror" is transmuted into the kinds of thrills analogous to those associated with the risks and skills of piloting and navigation in aerobatics [more]. More problematic however are the negative interpretations of the metaphor -- with the emphasis on "herding" a flock, notably by the suitable positioning of threats. Conspiracy theorists would readily expect the suitable planting of "terrorist" bombs as a 21st century simplification of the challenges of global governance (cf Promoting a Singular Global Threat -- Terrorism: Strategy of choice for world governance, 2002).
Boid-like behaviour has proven to be of considerable interest to the US military (cf Nancy J Wesenstein et al. Cognitive Readiness in Network-Centric Operations, Parameters, Spring 2005). A particular interest relates to swarm scenarios (Joshua J. Corner and Gary B. Lamont, Parallel Simulation of UAV Swarm Scenarios, 2004). The focus is on the ability to achieve a form of n-body control over automata that perform surveillance, defensive or attacking functions -- all of which may be experienced as terrifying to those exposed to them. The nature of such scenarios has been extensively and imaginatively explored in media representations of vast "dark hordes" controlled by unseen "dark forces". Ironically it is boid-type simulations that ensure the visual realism of such effects.
The military and imaginative activation of n-body terror could usefully justify an adage such as the following: "If you are unconcerned with the n-body problem, be aware that some terrifying n-body solutions have you specifically in mind"! The "more complex rules" added to govern boid-like behaviour can be readily modified from defensive "obstacle avoidance and goal seeking" into an attack mode of which the citizen is the focus.
6. Embodied "solutions": The "flocking" approach to ensure emergent order suffers from a major disadvantage in that it is based on a set of external rules -- even though these may constitute a consensual ethic.
A somewhat different approach is to ask how such rules can be internalized and to look for models of how this is accomplished. Individual belief systems might refer to this process as education, inculcation or indoctrination -- with respect to their own belief system. It is far less evident how such education is achieved with respect to behaviour of individuals or groups in distinct faith "vehicles" on different pathways, however interwoven. The desperate attempts by the UN, through UNESCO, to promote a "culture of peace" are an indication of how the challenge is currently framed.
Perhaps the most obvious model is then that of drama (cf Aesthetics of Governance in the Year 2490, 1990; Participative Democracy vs. Participative Drama: lessons on social transformation for international organizations from Gorbachev, 1991). Traditionally the relations between the gods have been described in dramatic terms, whatever values were exemplified by that process -- whether creative or destructive. Understandings of the movement of bodies in the solar system have traditionally been conflated with the dramatic relationships among the gods. Pantheons of gods might then be analyzed as n-body problems (or solutions).
In this dramatic sense the perceptible relations between the Abrahamic religions, and the bloody violence and terror that they engender, are "solutions" to a many body problem. Just as fundamental particles may collide with the release of destructive energy, or molecules may encounter each other with analogous outcomes, the terrifyingly destructive drama between religions is indeed a "solution" -- however apparently primitive.
What is missing from this dramatic perspective is the description of it as drama. Dramatization of particular stories, about Israelis and Palestinians for example, may be produced -- evoking great controversy. But the dramatic interpretation of the sweep of historical relations between the Abrahamic religions is not attempted -- so as to evoke recognition of how dramatically impoverished it is and to identify ways in which the dramatic expression could "move on", to embody a richer aesthetic, rather than being stuck in uncreative "sitcom" cycles of violence.
7. Peripatetic "solution": Another "solution", with a long tradition, may perhaps be derived from that associated with the peripatetic school of philosophy founded by Aristotle. The peripatetic axiom, later formulated by Aquinas, was that "Nothing is in the intellect that was not first in the senses." (Latin: "Nihil est in intellectu quod non prius in sensu"). The name derives from the process of walking whilst philosophizing -- or perhaps in order to philosophize effectively. "Walking the talk" and "Talking the walk" are interesting variants. The classicial context for doing so, the rectangular covered walkways of the Lyceum (peripatoi), may suggest a "rectangular" constraint on the aristotelian perspective.
Iimportance has long been attached to the spiritual insights associated with labyrinths, especially with how these are traditionally triggered through walking them (cf Lauren Artress, Walking a Sacred Path: Rediscovering the Labyrinth As a Spiritual Tool, 1991; Patricia Telesco, Labyrinth Walking: Patterns of Power, 2001) [more | more]. It is therefore worth considering how this process might be the basis for a form of "solution" -- notably to the three-body problem constituted by the Abrahamic religions.
The great proportion of labyrinths which are walked in this way are centro-symmetrical. As such they may be considered as offering insight into the one-body problem of engaging with the higher dimensionality of the spiritual insight of a single body of faith -- if only in one's own identity. The frequent need to change direction -- possibly understood as an "initiation" -- in order to move towards the centre is a significant trigger to reflection on what is not obvious, however simple it may be claimed to be.
Of more interest to the three-body challenge is the possible existence of labyrinths with an interwoven three-fold symmetry. An early, and relatively unique example, is given below. It was notably used by the foundation Mankind 2000 in its catalytic role in promoting the emergence of international futures research and what became the online Encyclopedia of World Problems and Human Potential.
Pre-historic representation of a pattern
of relationships i
|Neolithic engraving at the Passage Tomb at Newgrange, Ireland|
In a sense the "gravitational" effects, which are challenging to sustainability of relationships in the dynamic models described earlier, are effectively "insulated" here by the structure of the pathways between the kernels of each of the "bodies". As attractors, the potentially terrifying core insights of each are not directly "exposed" to those of the others. The terrifying monster is locked within the labyrinth -- at each centre? As a labyrinth, these insights can only be sequentially explored by a particular pattern of walking. The need for such a damping effect recalls the need for graphite rods for neutron damping in a nuclear reactor to control the level of reaction and ensure stability.
The emphasis in this solution goes beyond that of embodiment as described above. It is best associated with the work of Francisco Varela and others (cf The Embodied Mind, 1991; Laying Down a Path in Walking: essays on enactive cognition, 1987) and the understandings of enactivism. [more]
Eizo Akiyama and Kunihiko Kaneko. Evolution of Cooperation, Differentiation, Complexity, and Diversity in an Iterated Three-Person Game. Artificial Life 2: 293-304 (1995) [text]
Anthony Blake. N-Logue: an introduction [text]
Gary E Bolton and Axel Ockenfels. Self-centered fairness in games with more than two players [text]
J. Allan Cheyne and Donato Tarulli. Dialogue, Difference, and the "Third Voice" in the Zone of Proximal Development, Theory and Psychology, 9, 1999, 5-28 [text]
Gregory of Nyssa. On "Not Three Gods" [text]
Charles H. Hinnant. Schiller and the political sublime: two perspectives. Criticism, Spring, 2002 [text]
Kenneth Humphreys. Christianity's "civil war": the struggle for power [text]
P Hut. The Three-Body Problem in Stellar Dynamics, 1984, in The Big Bang and Georges Lemaitre International Symposium G. Lemaitre, ed. A. Berger (Dordrecht: Reidel), pp. 239-256.
Steven Johnson. Two ways to emerge, and how to tell the difference between them. Extreme Democracy [text]
George Lakoff and Rafael E. Nunez. Where Mathematics Comes From: How the embodied mind brings mathematics into being. Basic Books, 1992
M Matsushima and T Ikegami. Evolution of strategies in the three-person iterated prisoner's dilemma game. J Theor Biol. 1998, Nov 7, 195, 1, pp. 53-67. [abstract]
Brian Paul. Looking for GOD in all the wrong places... like KOLOB [text]
Anatol Rapoport. Two-person game theory. University of Michigan Press, 1966
William Riker. Bargaining in a three-person game. American Political Science Review, 1967, 61(3): 642-656.
Stewart Shapiro. Thinking about Mathematics: the philosophy of mathematics. Oxford University Pres, 2000.
Mauri Valtonen and Hannu Karttunen. The Three-Body Problem. Cambridge University Press, 2005
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