Geometry and Culture: 'Burying The Hatchet'
May 15, 1998
"Thirty spokes share the wheel's hub;
It is the center hole that makes it useful.
Shape clay into a vessel;
It is the space within that makes it useful.
Cut doors and windows for a room;
It is the holes which make it useful.
Therefore profit comes from what is there;
Usefulness from what is not there."
--- Lao Tsu, "Tao Te Ching"
If we separate time from space, what we take away from this poem is a snapshot of a spoked wheel, empty pot and abandoned villa. A static collection of objects which we can manipulate in space as we might a jigsaw puzzle or the shapes in an 'IQ' test.
Viewed in terms of space-time flow, the pull of the future reaches through the emptiness of the hub to bring the scene to life, moving the wheel forward (perchance to carry goods to market) in a self-referential vortex as supply engenders its own demand. This geometry, in which 'empty' space is an active participant, exemplifies 'the circle of life', a forward-pulsing ontogeny whose purpose is to respond to its own natural need.
Our flat-space perceptual skills blind us to the curved space story, and we perpetuate this 'mastery of the wrong problem' by blinding our children as we have been blinded. While the aesthetic saddle-and-lobe spiral of an ammonite may touch us, we seem unable to grasp its curved space message that we must become our own ontogeny.
This essay is about geometrical assumptions and their impact on personal and societal development. It is about geometry-induced organizational high performance and dysfunction and how to cultivate the former while starving out the latter. The opening poem is to illustrate how euclidian 'flat-space' conceptualization, the mainstay of the west, induces 'realities' which differ radically from those inspired by the non-euclidian 'curved space-time' conceptualization of eastern, Celtic and Native American cultures.
In the face of rising societal dysfunction, we seem to be caught in 'the hell of frenetic passivity' . Our history of avoiding confrontation and our 'intolerance of different fundamental structures of experience' (major cultural change) is a matter of record. In fact, our change-resistive values of maintaining concensus and 'burying the hatchet' (underscored by our 'burning' of heretics) are far better developed than our ability to bring about cultural change, particularly in emergency conditions. It seems that the impression that "We are all in the same boat in a stormy sea, and we owe each other a terrible loyalty.", as expressed by G. K. Chesterton, is part of the problem. Such a mindset seems to emerge from anthropocentric incest, as indicated by David Abram; "Today we participate almost exclusively with other humans and withour own human-made technologies. It is a precarious situation, given our age-old reciprocity with the many-voiced landscape. We still 'need' that which is other than ourselves and our own creations."
It is our assessment that it is too late in the day to let confrontation-eschewing geometries such as 'bury the hatchet' interfere with an aggressive pursuit of solutions to our burning socio-cultural issues. Moreover, our unwitting participation in deep-seated cultural pathology has had us 'bury the hatchet' squarely in the backs of our own children and youth in general, in retribution for the heresy of being born with 'non-euclidian' vision.
What we CAN do, however, is to work towards removing that hatchet from the bleeding back of youth and helping them to make it work FOR them, rather than against them.
Our purpose in writing this essay is NOT to present a new theory of social systems management, nor to speak to those who are not already struggling with these issues. Our intention IS to offer our ideas, the most well-reasoned and heartfelt we are capable of, as a 'guide' for concerned youth and others who, like ourselves, believe that aggressive action is essential if we are to rid our society of deep-seated pathology. This guide is intended ; (a) to help cultivate sharper vision in discerning systemic sources of social dysfunction and to develop skills to avoid involuntary participation/victimization, and (b) to help heal the hatchet wounds and arm youth with awareness of this 'vicious circle' and its points of vulnerability to put an end to it.
The ideas in this guide are not difficult but are in some sense 'holographic', emanating from a confluence of disparate areas of study. We think it's unlikely that they will be absorbed without the intensity of interest which comes from the feeling of having 'skin in the game'; i.e. an 'involved' interest which is beyond the purely academic or pecuniary.
With this contextual backdrop in place, we can explore the geometric notion of 'a hatchet in the back of youth.' The following quote from Jules Henry's 'Culture Against Man' captures the essentials;
"In a society where competition for the basic cultural goods is a pivot of action, people cannot be taught to love one another, for those who do cannot compete with one another, except in play. It thus becomes necessary for the school, without appearing to do so, to teach children how to hate, for our culture cannot tolerate the idea that babes should hate each other. How does the school accomplish this ambiguity? Obviously through competition itself, for what has greater potential for creating hostility than competition? One might say that this is one of the most 'creative' features of school. Let us consider an incident from a fifth-grade arithmetic lesson.
At the blackboard:
Boris had trouble reducing '12/16' to the lowest terms, and could only get as far as '6/8'. The teacher asked him quietly if that was as far as he could reduce it. She suggested he 'think'. Much heaving up and down and waving of hands by the other children, all frantic to correct him. Boris pretty unhappy, probably mentally paralysed. The teacher, quiet, patient, ignores the others and concentrates with look and voice on Boris. She says, 'Is there a bigger number than two you can divide into the two parts of the fraction? After a minute or two, she becomes more urgent, but there is no response from Boris. She then turns to the class and says, 'Well, who can tell Boris what the number is?' A forest of hands appears, and the teacher calls Peggy. Peggy says that four may be divided into the numerator and the denominator.
Thus Boris' failure has made it possible for Peggy to succeed; his depression is the price of her exhilaration; his misery the occasion for her rejoicing. This is the standard condition of the American elementary school, and is why so many of us feel a contraction of the heart even if someone we never knew succeeds merely at garnering plankton in the Thames: because so often somebody's success has been bought at the cost of our failure. To a Zuni, Hopi or Dakota Indian, Peggy's performance would seem cruel beyond belief, for competition, the wringing of success from somebody's failure, is a form of torture foreign to those non-competitive redskins. Yet Peggy's action seems natural to us; and so it is. How else would you run our world? And since all but the brightest children have the constant experience that others succeed at their expense they cannot but develop an inherent tendency to hate, to hate others who are successful, and to be determined to prevent it. Along with this, naturally, goes the hope that others will fail. This hatred masquerades under the euphemistic name of 'envy'.
What is really going on here?
Henry further comments;
"Looked at from Boris's point of view, the nightmare at the blackboard was, perhaps a lesson in controlling himself so that he would not fly shrieking from the room under enormous public pressure. Such experiences force every man reared in our culture, over and over again, night in, night out, even at the pinnacle of success, to dream not of success, but of failure. In school, the external nightmare is internalized for life. Boris was not learning arithmetic only; he was learning the 'essential nightmare also. To be successful in our culture one must learn to dream of failure.'
R. D. Laing ('Politics of Experience') comments on these findings by Henry; "It is Henry's contention that in practice education has never been an instrument to free the mind and the spirit of man, but to bind them. We think we want creative children, but what do we want them to create? --- 'If all through school the young were provoked to question the Ten Commandments, the sanctity of revealed religion, the foundations of patriotism, the profit motive, the two party system, monogamy, the laws of incest, and so on ....' --- there would be such creativity that society would not know where to turn.
Children do not give up their innate imagination, curiousity, dreaminess easily. You have to love them to get them to do that. Love is the path through permissiveness to discipline; and through discipline, only too often, to betrayal of self."
* * *
The 'hatchet in the back' is the use of parental love to extort the child's 'buy-in' to a pathological system, exposing them to a 'betrayal of self' in the process. The pathology derives, incipiently, from deep-seated rationalist assumptions that have us view the world through soul-dessicating 'euclidian' lenses; assumptions which have been shown by science to be seriously incomplete, but whose hard euclidian threads weave through the fabric of our thoughts to form an envelope extremely resistant to break-through.
Meanwhile, it is no accident that the comparison with native american culture arises, since these native cultures, along with eastern and Celtic cultures, have escaped the flat-space infection. Recovering from this affliction involves a return to our intuitive, nature-based, circular-dynamic conceptualization of reality, i.e;
You don't have to be good.
You do not have to walk on your knees
for a hundred miles through the desert repenting.
You only have to let the soft animal of your body
love what it loves.
Tell me about despair, yours, and I will tell you mine.
Meanwhile the world goes on.
Meanwhile the sun and the clear pebbles of the rain
are moving across the landscapes,
over the prairies and the deep trees, the mountains
and the rivers.
Meanwhile the wild geese, high in the clean blue air
are heading home again.
Wherever you are, no matter how lonely,
the world offers itself to your imagination,
calls to you like the wild geese, harsh and exciting _
over and over
announcing your place
in the family of things.
'Wild Geese' by Mary Oliver
The call of the 'wild geese' is a reminder that life is a unifying, circular dynamic, and brings also to mind Heraclitus' words; 'The unity of things lies beneath the surface; it depends upon a balanced reaction between opposites.'. This view resonates with our natural experiencing of gravity, pulling us into the future as our inertia has us cling to the past; a 'balanced reaction' which gives rise to the relative motion of sun and planets, and thus to the 'heading home again' of the geese.
Heraclitus further asserted that 'the total balance in the cosmos can only be maintained' ... 'if there is unending strife between opposites'. And he rebuked the author of the phrase; 'Would that strife might be destroyed from among gods and men' : saying, 'for there would be no musical scale ('harmonia') unless high and low existed, nor living creatures without male and female, which are opposite.' The euclidian notion of the mutual exclusion of opposites such as 'good' and 'bad', a degenerate 'special case' of the yin-yang 'unifying dynamic of strife', deludes us into believing that we must march along the line from bad to good to where all notes will be in sweet soprano, burying our hatchets along the way.
So today, children of the west like Boris are taught to see the world in superficial, soul-void terms, as a sweet and peaceful world where love and cooperation is the order of the day, and to pretend, at the same time to be blind to the underbelly of that world, where thrice-denied strife in the form of ruthless win-lose competition becomes the bootleg determinant. This situation we have come to regard as 'normality'; as R. D. Laing says; 'What we call 'normal' is a product of repression, denial, splitting, projection, introjection and other forms of destructive action on experience.' What we see before we drown in indoctrination, as Laing says, is that; "They are playing a game. They are playing at not playing a game. If I show them I see they are, I shall break the rules and they will punish me. I must play their game of not seeing I see the game."
Education aims to build wheels to transport youth into their rightful future. But in our euclidian way, we focus on the shape, size and color of yang components, the spokes, the rim. We celebrate and reward those with the biggest and best wheels as they go careening on like McGoo, leaving a trail of destruction in their wake, living someone else's life, out of tune with themselves and the world around them.
Education needs to build wheels which will carry youth into their own unique ontogeny --- the theme of Lev Vygotsky, Maria Montessori, A.S. Neill ('Summerhill School') and other lateral-thinking psychologist/educators --- to cultivate the ability to tune in to the dynamic space-time relationships within and around, using the full senses; "The eyes, the skin, the tongue, ears, and nostrils --- all are gates where our body receives the nourishment of otherness." . Such an education can develop our innate potential for non-euclidian curved space vision, a prerequisite for an aware co-evolution of one's authentic 'self' in harmony with both culture and nature.
How can we then, assist youth in removing the hatchet we have plunged into their backs, transforming it into an instrument of emboldenment to assist them in liberating society from its euclidian pathology?
An answer to this question which emerges from our explorations comes in a form which is simple yet high dimensional; --- we must invert our primacy of purpose from its current 'euclidian orientation of measured performance first, ontogenetic development second, putting it back into its natural orientation, restoring primacy to ontogenetic development needs, at both the level of the individual and the organization. This does not imply an abandoning of 'yang' material pursuits, but rather that these must be put in the service of ontogenetic development. This is in fact what occurs in the spontaneous emergence of exceptional, high performance teams, Apollo 13 being a familiar example. The Apollo program itself emerged from the commitment (articulated by JFK) to correct the pathological inversion in materialist geometry (i.e. wealth as 'ends' and people as 'means') and restore "people [their ontogenetic development] as 'ends' using wealth as the 'means' of cultivating their 'becoming'.
This story of geometry and culture which flows through our experiential explorations into the text of this essay, we hope will flow on to merge with many other such rivers of conscious experience embodied in those who care and are not afraid to care, in a rising tide which needs neither argue nor negotiate with barriers in its path.
Space and Time and Space-Time
Our story of the socio-historical origins of western 'psychology' (i.e. our mental functions rather than the disciplinary study of it), and its co-evolutionary dysfunction begins with the rising awareness of space and time or 'space-time' in the ninth to sixth century BC. This is the period which history usually associates with the 'birth' of western philosophy. It is the time of Pherecydes and Thales of Miletus and the consolidation of theologic myth leading towards a quasi-rational view of the world; i.e. a view which was not solely dependent on the Gods.
At this time, the ancient peoples of the Mediterranean region tended towards a 'structuralist' rather than 'relational' view of nature, anthropocentrically associating the structure of nature with man's own structure; e.g. the historic separation of 'heaven and earth' was depicted in terms of the severing of genitals from a male god followed by activity in the inerstitial gap between heaven and earth to mechanically re-assemble the pieces.). This anthropo-structural view gave way during this period, particularly in the philosophy of Heraclitus, to the notion of a fundamental "geometry" or 'Logos' (i.e. relational pattern and dynamic ordering principle) which pervaded all of nature. This reduced the 'anthropo-structural' view from it's primal role to that of a specific instance of a more complete natural relational geometry of nature. [This structural-relational trade-off seems congruent with quantum wave-particle duality, where interference terms in the relational or 'wave' view allow the consideration of harmony and dissonance, dynamic entities which link not just to senses but to 'spirit' or 'emotion'.].
Through this transformation, men or gods ceased being seen as structural elements 'responsible' for the whole of reality in a causal sense but were now seen as being contained within the overall geometrical-relational schema of reality. Heraclitus' statement "Wisdom consists in understanding the way the world works." represents a shift in ethics, for the first time, from theology to physics. Until this time, all of life was ordered with the object of following god. More than this, the 'physics' of Heraclitus was not an 'objective physics' about the world 'out there', but a physics of the whole of the cosmos of which man was but a part.
Thus, the shifting philosophical currents of this time brought the possibility of a new behavioral 'ordering principle' for individuals and society based on physics and geometry (the 'Logos') rather than the relationship with the divine. In addition, the notion of 'top-down' (holistic) rather than 'bottom-up' reasoning was introduced as an explicit reasoning option. Until this time, reasoning had been ultimately rooted in the 'bottom-up' geneology of the gods, or 'divine causality', if you like.
The reasoning of Heraclitus started from the assumption of 'flux' which meant that there was no 'foundation' or documented 'initial conditions' from which to build a 'bottom-up' explanation; ... one was therefore compelled to adopt a 'top-down' or 'bootstrapping' reasoning approach, a notion which implied that understanding of the world could be developed, rather than from assumed axioms or structural causality, by the relationships between things; i.e. by 'bringing a multiplicity of real or imaginary sensory experiences into connection in the mind' . This was the reasoning approach used many centuries later by Albert Einstein, which enabled him to discover the principles of relativity. Einstein discusses this 'non-axiomatic' geometrical approach and its critical role in his discoveries in his 1921 essay, 'Geometry and Experience'.
The evolutionary flow model of reality of Heraclitus, and the associated 'top-down' reasoning approach, did not depend on any axiomatic geometry such as 'euclidian geometry' --- a geometry which as Henri Poincare noted " ... is distinguished above all other imaginable axiomatic geometries by its simplicity", and as Einstein noted, "if we admit non-inert systems we must abandon Euclidean geometry."
[It is worth noting here that the geometry we 'think in' in the west, and the only geometry which is taught in most schools, is 'euclidian geometry' with its inert 'things' separated by 'void', its flat planes, and rectangular measurement grid which extends on out to infinity.]
The assumption of detached 'things' and 'void' essentially equates to 'euclidian geometry'. Geometry is thus intertwined in knowledge, in that knowledge implies a subject and object (i.e. a 'knower' and something 'known') and a relationship between the two (e.g. see Erich Jantsch, 'Design for Evolution'). In the 'divine causality' model of the Greeks, the gods were self-contained agents which 'caused' things to happen and to come into being. It is easy to see the correlation between this 'simplest' (i.e. euclidian) geometry and the 'divine causality' model.
Meanwhile, the Celts, Egyptians and other mythopoeic peoples did not espouse the notion of gods as discrete beings and, in a manner similar to Heraclitus, associated divinity with nature. There are many references to the Celtic General Brennus mocking the Greeks when he (briefly) took Delphi in 278 BC and found that the Greeks had 'fixed' (euclidianized) the gods in human shapes, as the following web note on Celtic Cosmology notes; "The Celts do not seem to have had a hierarchy of divinity in the sense of a coherent pantheon dwelling in some remote place. The human world and the Otherworld formed a unity in which the human and divine interact. Each location has numinous powers which are acknowledged by the people as we can see by their naming of mountains, rivers and other natural features many of which have associated deities. When the Celts invaded Greece in 278 BCE, Brennus entered the precinct of Delphi, saw no gold and silver dedications and only stone and wooden statues and he laughed at the Greeks for setting up deities in human shape. Caesar mentions that the Germans worship forces of nature only." 
As Caithlin Matthews observes in 'The Celtic Tradition', "For the Gods [of the Celts] are nothing but the forces of those laws [natural laws] in manifest form, and the magic of the aos dana is their gift and means of communication between themselves and humankind." Thus there is a 'perceptual reciprocity' between a person and the 'holy ground' in the Celtic tradition, similar to the Native American traditions of the Lakota and Omaha  where even a rock may be addressed with the respect and reverence that one pays to an ancient elder; unmoved ... from time without end ... you rest ... in the midst of the paths ... in the midst of the winds ... you rest ... covered with the droppings of birds ... grass growing from your feet ... your head decked with the down of birds ... you rest ... in the midst of the winds ... you wait ... Aged one."
Clearly, in the 500 BC era, western philosophy was at a crossroads, not only with respect to shifting the ordering principle or ethics of society from the divinity to physics and nature, but also with respect to the choice of a 'geometry' for nature; i.e. the inert euclidian space with its detached 'things' and void, where change was explained in terms of a causal ('bottom-up') geneology, and a 'non-euclidian' active space emanating from the concept of 'evolutionary flow', where change was explained in terms of 'bootstrapped' relationships. As pointed out by Frankfort et al, the latter geometry, espoused by Heraclitus led to "... to a way of thinking in which clarity and precision were no longer dependent on the 'concreteness' of geneological constructs. [i.e. while Heraclitus' geometry avoided the reductionist 'burden of concreteness' of euclidian geometry, as do the Riemannian non-euclidian geometries needed to explain relativity, the western world opted for euclidian geometry and the information overload we suffer today is largely attributable to this choice.]
Thus, by the time of Heraclitus of Ephesus (540 - 480 BC.) and Parmenides of Elea (515 - 460 BC.) there were two competing ways of looking at space and time; via a euclidian geometry which looked at space independent of time or a non-euclidian geometry which look at 'space-time' as an evolutionary flow.
I.e., Heraclitus saw space-time as a continuum or 'flow', with 'objects' being illusions in the manner of a candle flame which preserves its shape although it is essentially no more than a 'stationary' oscillation emanating from the underlying reality of a continuous 'flux' or energy transformation. In the modern era, scientists have described the human body in a similar way, since the molecular constituents of our body flow through us in much the same way as the combustible fuel flows through the candle flame. This Heraclitian 'flux' perspective suggests that empty space, rather than being an inert void, is a participant in natural phenomena [i.e. transformation involving the latent, relational energy in space underly natural phenomena]. This active view of space places in question the relative role of material and intermaterial space in shaping our reality, and undermines the notion of 'causality'.
While, as will be discussed shortly, the western world opted for (and still clings to) the euclidian geometry assumption, this 'Heraclitean' space-time view connects with the Hegelian dialectic, Nietzschean philosophy, the psychology of Vygotsky etc. It is also congruent with space-time notions of Einstein and his Mathematics (geometry) professor, Hemann Minkowski, who asserted in a famous 1908 lecture; "Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of unity between the two will preserve an independent reality."
While, as will be discussed shortly, the western world opted for (and still clings to) the euclidian geometry assumption, this 'Heraclitean' space-time view connects with the Hegelian notion of dialectic within a totality (although Hegel was a 'idealist/rationalist whilst Heraclitus did not make a distinction between rational- and phenomenal worlds), the 'Middle Way' (at the interface between the manifest world and the hidden or 'otherworld'), Nietzsche's philosophy, the psychology of Vygotsky etc. It is also congruent with space-time notions of Einstein and his Mathematics (geometry) professor, Minkowski, who asserted in a famous 1908 lecture: "Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of unity between the two will preserve an independent reality."
[Minkowski's remark seems to imply an optimism for our culture's ability to swap out geometries, that was shared by Einstein, who summed up his 1921 lecture on 'Geometry and Experience' with the comment; " My only aim today has been to show that the human faculty of visualisation is by no means bound to capitulate to non-Euclidean geometry." Thomas Kuhn, in 'The Copernican Revolution' and 'The Structure of Scientific Revolutions' sheds light on why changes in our geometric models play themselves out so slowly.]
The 'doomed' inference underscores the fact that western philosophy opted for the implicitly euclidian model of Parmenides which was based not on flux but on stasis --- the passive existence of 'things' within a non-participating void. This geometry was formalized by Euclid (ca. 300 BC.) and began to subtly and pervasively infuse our western philosophy, psychology, mode of systems inquiry and language.
The clash of geometric models underpinning the philosophies of Heraclitus and Parmenides is well-described by Frankfort et al; "In the writing of Heraclitus, to a larger degree than ever before, the images do not impose their burden of concreteness but are entirely subservient to the achievement of clarity and precision." ... "... for Heraclitus fire is purely a symbol of reality in flux; he calls wisdom "to know the thought by which all things are steered through all things."
Heraclitus rejected the emphasis on knowledge-as-content, saying "The learning of many things teacheth not understanding, else it would have taught Hesiod and Pythagorus." [i.e. Heraclitus subscribed to a 'relational' view of knowledge/understanding rather than the 'structuralist' view which we still cling to today.] This follows from Heraclitus' grounding of his views in nature; i.e. from his observation that nature is a developmental (evolutionary) dynamic which transcends materiality. Parmenides, and later Plato and Aristotle, putting more faith in their rationality than their senses, argued (indirectly) for euclidian geometry.
Meanwhile, Parmenides denied the possibility of motion or 'becoming'; "How, then, can what 'is' be going to be in the future? Or how could it come into being? If it came into being, it is not; nor is it, if it is going to be in the future. Thus is 'becoming' extinguished and 'passing away' not be heard of."
Frankfort et al, describe this philosophical standoff as follows: "Heraclitus had declared 'being' a perpetual 'becoming' and had correlated the two concepts with his 'hidden attunement'. Now Parmenides declared the two to be mutually exclusive, and only 'being' to be real." History records that the ideas of Parmenides, as later incorporated in the works of Euclid, Plato, Aristotle etc., prevailed.
And as Kirk et al note: "Parmenides' arguments and his paradoxical conclusions had an enormous influence on later Greek philosophy; his method and his impact alike have rightly been compared to those of Descartes' 'cogito' [cogito ergo sum : 'I think therefore I am']. That is, the Parmenidian concept of something either 'existing' or 'not existing' (independent objects in 'inert' space), not only leads to the notion of 'mutual exclusion' and 'bivalent logic', but also to a fundamental 'dualism' between mind and matter, subject and object, man and world, right and wrong, good and evil - notions which dominate our mode of perception and inquiry to this day.
So the story of the development of our conception of space and time or space-time in the west goes back to these beginnings and the struggle between Heraclitus' 'nature-grounded ' view of reality, and Parmenides 'logical' view. Whereas, Heraclitus flux-based view is consistent with the modern science idea of a space-time continuum [which has not yet been taken up by our culture], and with eastern, Celtic and native american philosophies, Parmenides view of a 'thing-and-void' space, understood independently of time, has been shown to be both inconsistent with our observations of nature, and incompatible with the modern theories of quantum mechanics, general relativity etc.
Euclidian space is implicit in the Parmenidian view, a view which asserts by logic (finite systems of logic have been shown to be incomplete by Goedel, Wittgenstein, Chaitin and others) that the senses are completely fallacious, while non-euclidian space, grounded in nature and natural experience and validated by modern physics, is implicit in the Heraclitean view.
Before leaving the story of the historical origins and 'passage' of euclidian and non-euclidian space-time geometries, it is useful to consider their relative implications on reasoning and knowledge. I.e. motion introduced 'on the top of' the euclidian space view leads to the concept of 'causality'; i.e. the detached objects of euclidian space essentially replace the fixed-persona gods of the Greeks who were responsible for the state of the world. The Heraclitean view, which stems from motion or flux, because of its time-irreversibility aspect; i.e. "One cannot step twice into the same river, nor can one grasp any mortal substance in a stable condition, but it scatters and again gathers; it forms and dissolves, and approaches and departs."  leads to a non-euclidian or 'curved' space-time framework, where space is a 'participant' in natural phenomena (as Einstein 'puts it' in a later citation). Out of the notion of curved, participative space-time and time-irreversible progression, come the notions of 'ontogenesis', self-referentiality, and 'goal-orientation' as are observed in evolution and complex systems behaviors.
There is an important perspective dealing with problem generalization associated with the Heraclitean - Parmenidian geometry 'story' which must be kept in mind to make sense of the overall story in this essay; the first is that the Greek attribution of the works of nature to fixed, personified gods was a generalization of massive proportions which provided the foundation for the euclidian space assumption and which represented a major bifurcation in thinking between the Greeks and Celts (and other divinity in nature oriented cultures). William Blake spoke to this in his work 'The Marriage of Heaven and Hell', as follows;
The ancient Poets animated all sensible objects with Gods or
Geniuses, calling them by the names and adorning them with the
properties of woods, rivers, mountains, lakes, cities, nations,
and whatever their enlarged & numerous senses could perceive.
And particularly they studied the genius of each city &
country. placing it under its mental deity.
Till a system was formed, which some took advantage of &
enslavÆd the vulgar by attempting to realise or abstract the
mental deities from their objects; thus began Priesthood.
Choosing forms of worship from Poetic tales.
And at length they announced that the Gods had ordered such
Thus men forgot that All deities reside in the human breast.
Plate 14, from "The Marriage of Heaven and Hell"
--- William Blake 1757-1827
Blake's philosophy, as expressed in this engraved plate from "The Marriage of Heaven and Hell", is essentially the same as that expressed above by the modern day Celtic writer/historian/philosopher, Caitlin Matthews. Blake's overall argument was that we have tried to do something very unnatural --- to generalize a multivalent world in terms of bivalency, splitting everything into 'good' or 'bad' or 'high' and 'low' etc.. In so doing, we are destroying the natural harmony in the world.
Blake further wrote; "To generalize is to be an idiot. To particularize is the alone distinction of merit knowledges are those knowledges that idiots possess. The indication is that Blake is thinking along the same lines as Heraclitus, that nature is, innately, 'flow', driven by the 'attunement of enfolded opposites. As Blake says, "Opposites: Without Contraries is no progression. Attraction and Repulsion, Reason and Energy, Love and Hate, are necessary to Human existence." and with respect to the 'unnaturalness' of stasis, "Opinion: The man who never alters his opinion is like standing water, and breeds reptiles of the mind.".
[Components of systems which involve 'whole-in-part' unity through yin-yang like mutual enfoldment geometry cannot be generalized since whatever subset is examined can never be independent of the portion of the unity that lies outside of it. Blake's lines "To see a world in a grain of sand And a heaven in a wild flower, Hold infinity in the palm of your hand And eternity in an hour." clearly indicates an appreciation of the 'holographic' character of nature, and thus the beauty of the 'particular', the natural grace between the 'thing' and its components which scales up to include itself. The discrete 'thing and void' character of the euclidian space assumption, which our culture conditions us to take for granted, is a necessary condition for generalizing 'things' in a statistical sense. Meanwhile the 'top-down' geometric reasoning implicit in Heraclitus' reasoning makes no assumptions whatsoever as to any structural granularity (thing-base) in nature, but works by 'bringing a multiplicity of real or imaginary sensory experiences into connection in the mind', a process which brings rational, sensory and emotional (harmonic) experiences into a unity. An approach which was clearly subscribed to and advocated by William Blake.]
Clearly, Blake did not mince his words on the issue of geometry and generalization and his prime target was 'euclidian geometry' although not in these terms.
The generalization which folded the complexities of nature into fixed theistic persona, divine geneology and thence on to an inert euclidian-causal view of reality, set the stage for2500 years of 'mastery of the wrong problem' . Ian Stewart's formulation of Poincare and others warning on approximating the problem and thenceforth asking the wrong questions and coming up with the wrong answers comes to mind here; i.e. 'the exact solution to an approximate problem is not the same as an approximate solution to the exact problem.' No Virginia, there is no Santa Claus in mathematics.'
The approximation or generalization of space as euclidian dropped out some rather important stuff; i.e. 'harmony' or the wave aspect of nature . Harmonic content was fundamental to the Heraclitean view, as it was to Pythagorus. Pythagorus, purportedly guided by the Orphic cosmogenies and some ancient books coming from farther east had, through the concept of 'harmonia' succeeded in linking the rational abstraction to both the senses and the 'spirit'. As Frankfort et al put it, while Heraclitus was content to include the hidden 'attunement', ... "the Pythagoreans were anxious to determine it quantitatively'. The starting point for their enterprise was a remarkable discovery by Pythagoras. Measuring the lengths of the string of the lyre between the places where the four principal notes of the Greek scale were sounded, he found that they had the proportion 6:8:12. This harmonic proportion contains the octave (12:6), the fifth (12:8) and the fourth (8:6). If we attempt to regard the discovery naively, we shall admit that it is astonishing. It correlatesmusical harmonies, which belong to the world of the spirit no less than to that of sensual perception , with the precise abstractions of the numerical ratios."
Clearly, the directly observable and even measurable harmony in nature, an important connective principle, became a casualty of the generalization leading to the euclidian view of space and time which is the approximated 'problem' we continue to try to 'master' today.
It's worth noting here that much of what was known in the time of Pythagorus and before has been falsely attributed to the later works of Newton and others. In the 300 year commemorative of Newton's Principia (July 5, 1686), "Let Newton Be" (Oxford University Press, 1988), the writer researchers review Newton's access to 'the wisdom of the ancients'. This access included the knowledge that "Thoth, the Egyptian Hermes or Mercury, had been a 'believer' in the Copernican system', while Pythagorus and Plato had 'observed the gravitation of all bodies towards all'. In addition, "Gregory came into possession of a set of 15 folio sheets in which Newton assembled a formidable array of classical authorities to support key propositions of the 'Principia'. In logical sequence, they showed these ancients as having held that;
o The Moon was like another Earth, and itself like other celestial bodies; therefore, the planets and their satellites were heavymasses of the same substance as the Earth.
o Bodies in the solar system exercised a mutual gravitation that extended to all other systems (each fixed star being the centre of a gravitational system of planets), that force itself extending indefinitely in all directions.
o All matter was made of atoms, and they were hard, solid, and immutable; gravity accrued to both atoms and to the bodies they composed; gravity was proportional to the quantity of matter in every body.
o The proportion in which gravitational power decreased with distance was analogous to the law governing tension and pitch in a string instrument, and was concealed in the ancient concept of the 'harmony of the spheres'.
o The cause of gravity was explained in the 'mystical philosophy' of the ancients by their supreme divinity, Pan, playing on his pipe --- that is, by the direct exercise of God's divine power in all bodies whatsoever."
The Pythagorean item on string tension and pitch was particularly noteworthy (Pythagorus had, by hanging weights on strings, essentially determined the inverse square relationship which applied to the 'harmony of the spheres' in that a quadrupling of the weight (tension) was required to double the frequency of oscillation). The notion of harmony, omitted in the Parmenidian view of space and time which we in the west inherited and included in the Heraclitean view of space-time flow along with the idea of the hidden pervasiveness of harmony in nature inherited by the east, Celts, Native Americans etc. comes back to us again in modern physics, i.e. "Since the Pythagoreans taught that the order of notes of a vibrating string pervades the whole universe, they would have been moved to learn, two thousand years later, of the work of Nambu (the theory of strings also contained contributions from Leonard Susskind at Yeshiva University and H. Neilson of the Neils Bohr Institute in Copenhagen), who was able to show that the hadrons, or strongly interacting particles, are also quantum manifestations of this Pythagorean ideal."
So, the generalizations leading to the Parmenidian - euclidian space assumption excluded the important notion of harmony, and thus 'set up' the western culture for the 'master of the wrong problem'. One might ask why a mastery of this 'simplest of axiomatic geometries' has been so successful, setting the stage for a technological revolution that has put western culture into a globally dominant position. The answer is that euclidian approximation assumes no space-time interaction other than simple linear cause. This approximation is only valid over small regions of space-time and as the actions based on euclidian thinking propagate over large times and distances, undesirable side effects are likely to emerge. For this reason, society's overemphasis of euclidian space is leading to 'runaway feedback'  wherein our evolutionary fitness is decreasing as a direct result of 'selection' (i.e. the increased influence of western culture and technology in the world).
This 'suboptimized fitness' is clearly seen from the discussion in the introduction, in terms of the choice between quantitatively measured performance as a social/organizational ordering principle, which goes along with the euclidian space and time view, and ontogenesis (personal development) as the meta-quantitative ordering principle, which goes along with the non-euclidian, evolutionary, space-time flow view. Ontogenesis in the sense used here includes the environmental interplay, which is a major if not primary developmental influence, and thus human systems oriented to their own ontogenesis will stay in 'co-resonance' with their environment rather than becoming increasingly isolated in structure of their own making. David Abram's point that we need 'the nourishment of otherness' (in the introduction) is reinforced by artificial life simulations, which show that progressive incestuous over-specialization can lead to extinction.
In this brief history of our conceptualizing of space-time, it is worth noting that our current western (mainstream) culture continues to base its perception and systems inquiry on the Parmenidian dualist assumptions of mind/matter split, the independence of space and time, the innate bivalency of reality (e.g. 'good and evil'), causality etc. Whereas scientific evidence overwhelmingly supports the 'flux' model of Heraclitus which assumes unity through opposition, the mutual enfoldment of space and time (i.e. space-time seen as a continuum), the innate multivalency of reality (e.g. 'beyond good and evil' harmonies and dissonances), circular causality or 'goal orientation' etc. Generally speaking, the euclidian attributes of the Parmenidian model, rather than being in fundamental conflict with the non-euclidian attributes of the Heraclitean model, are 'contained' with the Heraclitean model as a 'special case'; i.e. flat euclidian space is a special case of curved non-euclidian space, bivalency (good/bad mutuallyexclusive opposites) is a special case of multivalency (harmony/dissonance mutually inclusive opposites), linear causality is a special case of 'circular causality' (i.e. goal orientation) etc.
The historical choice of geometries has had a profound effect on most aspects of our lives via their different implications on subject/object relationships, reasoning approaches, epistemological concepts etc. In addition, it has led to philosophical divisions in the world: for example, the Heraclitean (non-euclidian) model is consistent with the eastern (and Celtic, Native American etc.) notion of divinity in (a dynamic) nature, a view which entails 'metamorphosis' and the mutual enfolding of matter and participative space, as in the 'yin-yang' geometry of the east. The Parmenidan (euclidian) model, meanwhile, gives rise to a 'good' or 'bad' and measured view of things. As mentioned earlier, the ordering of social behavior stressing one or the other of these set of attributes can be expected to give rise to very different evolutionary consequences.
This continuing essay-story will try to illustrate the profound manner in which these alternative geometries feed into and influence our current systems inquiry, psychology, philosophy, knowledge management and language.
A 'Killer Kindness'
In this last leg of our story, on 'space-time' models, we indicated that the euclidian 'flat' space and independent time concept was contained within the non-euclidian 'curved' space-time concept as a special case. In the west, shortly after the 'birth' of western philosophy, we opted for the former 'rationalist' model, which we have retained to this day. In the east, and among the Celts and Native Americans, the 'rationalist' euclidian model was never an issue, thus naturalist views prevailed, with their notions of circular dynamics, yin-yang etc.
Whether or not people are guided by a bivalent sense of 'good' and 'bad' versus a multivalent sense of 'harmony' and 'dissonance' can lead to huge differences in individual and cultural behavior. Differences are amplified by the complicating fact, as discussed in the introduction, that those of a euclidian 'good/bad' perspective tend to think in mutually exclusive terms and see only 'good' OR 'bad' in any given situation. In situations perceived as 'good', (e.g. Peggy's 'win' in math's class), the 'bad' consequences (e.g. Boris' 'loss') are ignored. In the case of Peggy and Boris, the teacher, by focusing exclusively on 'successful' performance, delivers a kind of 'killer kindness' by which Boris is devastated even though the demeanor of the teacher is always positive and kindly.
This 'killer kindness' was inflicted on the aboriginal peoples of the world as Europe intensified and extended its colonizing activities, some four hundred years ago. Meanwhile, the ancient notion of yin and yang --- the mutual enfoldment of good and bad --- a notion which could have greatly reduced the lethal aspect of western colonialism --- had been subordinated to the bivalent concept of 'good' and 'evil' forming an unnatural, 'tangled hierarchy' (the multivalent concept of yin-yang contains the bivalent good-bad as a special case, where we reduce the dimensionality of such high dimensional entities as people or groups of people (e.g. aboriginal people) down to two, characterizing them along a linear axis between two extremes or opposities. However, one cannot contain a high dimensional view within a linear view, hence the 'tangled' aspect of the hierarchy).
Today, though science has torn the blindfolds from our eyes to see through this linear self-deception, the dazzle seems to make us cross-eyed and our blindness to the non-euclidian yin/yang geometry of nature persists. The price of our blindness is a western psychology which '... makes hungry where most she satisfies'; ... a psychology of compassion for the casualties of dysfunction, but a psychology which itself breeds dysfunction and the growth of casualties. Like Cleopatra, the love given is free-flowing and true, and like a pre-Pasteur physician reaching out to heal the wounded, the more touched by this love, the more the infection.
In this 'Say's Law' economy, where supply engenders its own demand, western psychology cultivates a thriving commerce in psychic dissonance which has evolved into a powerful breeder of global dysfunction. Not only the aboriginals but all children, in their school years in particular, are cast into a sea of psychological 'infection'. It is not the case that our lack of vigilance has allowed this dysfunction to flooded the cultural landscape, rather, it is WE who are the witless procreators of this dissonance.
'Wait a minute!', you say, 'western psychology may not be perfect, but it's not that bad!'
... And the 'grammatical error' implicit in this judgement-based protest is where our story starts, 2500 years ago, as we opt, arbitrarily, for a flat, inert, euclidian geometry to conceptualize our reality. Bundled with this simple option came the bivalent, mutually exclusive logic of 'things' and 'void', a perceptual schema which, through rampant runaway feedback, severely suboptimized our evolutionary future as it steam-rollered the multivalent, mutually inclusive logic and unified evolutionary flow perspective of the mythopoeic era. The tenacious retention of non-euclidian perspective by the eastern and aboriginal cultures has since been vindicated by the findings of twentieth century science.
Through our western euclid-lenses, 'psychology' or any'thing' else one would like to assess can be judged as either 'good' _OR_ 'bad', but not both at the same time, notwithstanding our experiential history with 'Cleopatras' and 'pre-Pasteur physicians'.
This essay reconciles our 'euclidian option' with the 'path not taken', the non-euclidian world 'beyond good and evil', a paradigm which does not 'do away with' the judgemental notion of 'good and bad' but which subsumes it as a special case of the higher dimensional notions of 'harmony' and 'dissonance'; i.e. a 'quantum wave view' of reality which 'contains' the euclidian particulate view.
But this essay is NOT about the aesthetics of alternative ways of perceiving and inquiring into reality, it is about coming to terms with how WE, in our blindness and denial of the ramifications of our euclidian choice, continue to infect our children and the globe with dysfunction-inducing psychic dissonance. This essay is about breaking our addiction to looking at life as a linear progression and re-learning how to experience it as a continuously enfolding subsumation of historical experience, wherein we, like the ammonite and all natural systems, 'become' our own ontogeny.
Part I. Homecoming
Over the past 2500 years, we have had a split perception of the world, as if through 'crossed' stereo-lenses. Our perception as we look out from our place in the human community has diverged from our perception as we look out from our place in the natural landscape --- a divergence which has been the source of mind-bending, heart-rending stress. Culturally, we have been 'lost in euclidian space'; --- immersed in technologies, roles and scripts which have sprung not from a resonance with nature, but from arbitrary intellection, and most of all from an historical accident in which we chose to see our containing 'space' in terms of an inert, detached and non-participating 'euclidian' geometry. Disoriented by this psychical dissonance, our ability to co-resonate with the symphonics of the natural landscape has been seriously injured.
The poets, artists and musicians and the present day elders of the pre-euclidian cultural traditions have been the narrators of this long journey through the euclidian desert. In parallel, the innovative, pre-geometrical inquiry of individuals such as Poincare, Bohr, Einstein, Nietzsche, Vygotsky, Wittgenstein and Prigogine have given us scientific mappings of the same territory; --- an understanding of how our 'euclidian choice' has woven itself into the evolving substrata of our science, philosophy, psychology, and language to seed and perpetuate a painful psychical dissonance. Two stories of the same long journey from two very different perspectives.
The bad news is that the detour has taken us deep into the heart of a non-natural scientific landscape which both threatens the artist and seduces the scientist in us. It is clearly the 'euclidian scientist' in us which has a headlock on the navigational apparatus essential to rediscovering our natural unity. This socio-historically induced 'intellectual virus' has in an insidious and transparent manner, invaded our psychology and become an incipient breeder of psychic dissonance and social dysfunction.
The good news is that cognitive synergies from multiple fields of 'non-euclidian' scientific inquiry are providing a more consistent view of the overall landscape which promises to relax that headlock. We are beginning to be able to articulate in the language of reason, what we could previously only express through poetry and art. Guided by new, more natural geometries of reason, the path is rising up to meet 'both of us', repair our split vision and take us back home to our place in the natural 'family of things'.
Two and a half millennia of having our minds enslaved to a rectangular pseudo-reality has eroded our ability to use the self-enfolding circles and metamorphosing spheres of nature to guide our 'becoming', as they have been overprinted at every turn by the rigid stasis and discordant detail of euclidian lines and boxes. Clearly, we have been 'lost' in euclidian space, and finding our way 'back home' will involve a rediscovery of old wisdom abandoned in our pursuit of euclidian abstraction. In the native tradition of the pacific northwest, captured in poetic form, the following counsel for is given for recovering one's calm and one's bearings if lost in the wilderness;
Stand still. The trees ahead
and bushes beside you
Are not lost. Wherever you are is called Here.
And you must treat it as a powerful stranger.
Must ask permission to know it and be known.
The forest breathes. Listen. It answers.
I have made this place around you.
If you leave it you may come back again.
No two trees are the same to Raven.
No two branches are the same to Wren.
If what a tree or a bush does is lost on you.
You are surely lost. Stand still.
The forest knows
Where you are. You must let it find you.
'Lost' by David Wagoner
The feeling of detachment from nature seems to be intensifying as divergence grows between the respective evolutionary directions of corporate and social systems, and social and environmental systems. This tug-of-war between the mutually antagonistic 'wild horses' of rational and natural evolution, which threatens to dismember the dualist society cultivating the antagonism, dates back to the time of Parmenides (ca. 500 BC). One hundred years prior to this (ca. 600 BC), as the birth of philosophy signalled the approaching end of the mythopoeic era, western civilization --- the egyptians, mesopotamians, the celts --- were all agreed "in the fundamental assumptions that the individual is part of society, that society is imbedded in nature and that nature is but the manifestation of the divine."  But that agreement was soon overtaken by a mind-bending 'trip' on the mechanical horse of rationality, deep into the frozen space of euclid. Only now is the smokescreen of denial lifting and the admission of having lost our way finally surfacing.
The story of this historical diversion, or 'perversion', has been told many times in the language of myth and poetry, and it is a convincing story, yet the ambiguity inherent in myth and poetry seems to render it incapable of swaying established scientific reasoning which our culture is so heavily invested in. Exploration into this curious state of affairs reveals that euclidian geometry has sown the seeds of its own self-preservation into; (a) our systems inquiry or 'science' tools, (b) our philosophy, (c) our psychology and, above all, into (d) our language.
[[2500 years seems a rather long 'diversion', particularly when one considers that the euclidian 'low road' we have been driving along is only a flower-filled meadow away from the non-euclidian 'mother' road. The source of staying power of the 'euclidian attractor' which keeps us on this low road is revealed in the adage, "united we stand, divided we fall". This adage speaks not only to the fragmentation inherent in physical euclidian geometrical constructs but gives poignant reminder to the power of linear abstraction in inducing intellectual 'feudalism', a self-preserving trait of euclidian geometry which emerges self-referentially. Thus we are caught, as Vygotsky suggests later in this essay, in an ontogenetic 'circle of life' even as our culture mocks the 'circle of life' beliefs of the aboriginal cultures.]]
While the dysfunction-inducing folly of mentally internalizing euclidian space-time notions has been blatantly exposed: i.e. --- in systems inquiry or 'science' by Poincare, Minkowski, Bohr, Einstein, Prigogine and others; --- in philosophy by Nietzsche; --- in psychology by Vygotsky, and; --- in language by Wittgenstein (to give a few exemplars), the euclidian paradigm has persisted almost unscathed. As long as we are immersed in the naturally fragmentative euclidian paradigm, disciplinary insights into the euclidian source of our dysfunction, which could through natural transdisciplinary co-resonance powerfully resolve themselves, fall victim to a consummate INTRA-disciplinary focus wherein debate over euclidian alternatives holds primacy, though such alternatives have far more to do with the abstractions of the discipline than with the unity of nature. To unravel this paradox of our continuing proclivity for elevating the mechanical over the harmonious in spite of our desire to the contrary, it is going to be necessary to look at systems inquiry, philosophy, psychology and language as one interdependent system.
[[Johannes Kepler spoke to this selfsame asystemic disciplinary issue as he strove to debottleneck century-long efforts to move from Ptolemaic to Copernican thinking; "... it frequently happens that the things which have to be chosen are not those which are most true but those which are most easy. And by that division in things which makes different people form different judgements, it so happens that certain people are in error contrary to their own opinion." [Epitome of Copernican Astronomy, 1621].]]
The persistence of the euclidian paradigm is, as mentioned, a 'systemic' problem and it cannot be solved theoretically with resultant 'solution' being imposed on the system. It must instead be solved 'ontogenetically' or 'developmentally' (Vygotsky). This means, in the case of language, that though some such the poet in us may complain about the lack of dimensionality in our euclidian language used in direct inquiry mode, our language as used in inquiry mode is an integral part of the systemic problem and however adeptly we can make 'end runs' to the solution using poetic or mystical language, this does not touch the heart of the systemic issue. Historically, the choice of a euclidian view of space enabled and encouraged science to sacrifice linguistic dimensionality to consistency and we are now paying a heavy price for it. But the answer is not to seek an 'antidote' for dimensional deficiency and abandon consistency in the process, (e.g. by a simple regression to the mythical and poetic), it is instead to find a way to enrich our language so as to satisfy both the needs of dimensionality and consistency. This is, in effect, an evolution which is naturally underway in the sciences of complexity.
This essay, then, attempts to give an accounting of our 'bogus' euclidian journey in the 'flat-space' language of euclid, the lingua franca of our western culture. Since euclidian terms and phrases have insufficient reasoning dimensionality in themselves to expose the euclidian source of social dysfunction, and since (non-euclidian) ontogenetic inquiry compels us to stick with euclidian language, the needed leverage must come from the illuminating power of geometric consistency resident in transdisciplinary reasoning. Such an approach can also illuminate how euclidian seeds of self-preservation are sown into the individual disciplines and our 'general psychology' and how this 'parasitic networking' arrangement provides a powerful defense mechanism to protect the euclidian space paradigm against encroachment from non-euclidian paradigms.
In other words, in our attempts to 'trap' and address the flaws in our euclidian ways, we have been stymied by having to use euclidian fishnets whose webbing is coarser than the non-euclidian fish we are trying to catch. For example, our use of statistics-based 'performance' measures to assess the quality of our business and educational efforts is inherently euclidian, and attempts to deduce the overall behavior of the system by reifying and assessing its various attributes. In 'complex systems' involving non-euclidian space-time assumptions, it is not possible to deduce overall system behavior from the properties of the reified attributes or 'parts' of the system.
[[We cannot classify 'things' according to statistical generalization, attribute by attribute, if 'ceteris paribus' does not hold. That is, the assumption that a particular systems attribute can be taken out of the context of the whole system, 'reified' (regarded as a 'thing') and measured as it occurs in an individual, relative to the overall mean of a population of multi-individuals is innately euclidian. It is this assumption which makes us laugh in films like 'What about Bob?' when the psychotherapy patient (Bill Murray) demonstrates an integrated, or situational 'knowledge' of psychology far superior to the statistically assessed intellectual knowledge of psychology of his psychologist who enjoys 'high priest' status in the world of psychology. The assumption of a hard boundary between a single reified attribute and other systems attributes does not hold in a curved space paradigm. That is, 'statistics' is the 'euclidian' special-case of attribute 'generalization' or regression assessment, where the flat-space assumption ensures continuing homogeneity and stationarity --- i.e. the absence of self-referential recursion as in curved space-time]].
Our 'either/or' choice of rationalist or naturalist philosophies to support our inquiry is also inherently euclidian. That is, non-euclidian philosophies, such as the philosophy of Nietzsche, implies mutual inclusion and ontogenetic recursion.
[[The recursive dynamical relationships which characterize complex systems cannot be understood in terms of hard closure and independence; i.e. the 'closed systems' view which derives from the euclidian space notion. The rational abstraction of 'closed systems' and the finality of 'scientific laws', which are enabled by euclidian space assumptions, represent 'the exact solution to an approximate problem' rather than an explanation or understanding of nature. Nietsche in 'Will to Power' points out how such assumptions are built into our philosophy and distort our inquiry into reality. For example, he points out that the euclidian space derived concept of linear causality is incompatible with our observation of attraction and repulsion;
627 (from 'Der Wille zur Macht')
" 'Anziehen' und 'Abstossen' in rein mechanischem Sinne is eine vollstaendige Fiktion: ein Wort. Wir koennen uns ohne eine Absicht ein Anziehen nicht denken, --- Den Willen, sich einer Sache zu bemaechtigen oder gegen ihre Macht sich zu wehren und sie zurueckzustossen --- das 'verstehen wir': das waere eine Interpretation, die wir brauchen koennten.
Kurz: die psychologische Noetigung zu einem Glauben an Kausalitaet liegt in der UNVORSTELLBARKEIT EINES GESCHEHENS OHNE ABSICHTEN: womit natuerlich ueber Wahrheit oder Unwahrheit (Berechtigung eines solchen Glaubens) nichts gesagt ist. Der Glaube an causae faellt mit dem Glauben an [telos] (gegen Spinoza und dessen Kausalismus).
i.e. in english;
"Attraction and repulsion in a pure mechanical sense is a complete fiction: in a word. we can't think in terms of an attraction without a purpose. --- the will to possess a thing or to defend against its power and push back against it --- that we can understand; that would be an interpretation that we can use.
In short, the psychological necessity for a belief in causality lies in the unimaginableness of events without purpose; the truth (the validity of such a belief) or falsehood of which is naturally not discussed. The belief in cause collapses with the (emergence of a) belief in purpose.
This 'passivity' is a prime characteristic of flat euclidian space; whereas self-referentiality which relates to the notion of 'purpose' or 'circular causality' is implicit in curved non-euclidian space, and satisfies our observation of 'goal-oriented' systems as observed in nature. Nietzsche further criticizes the euclidian paradigm (in 'The Gay Science') in no uncertain terms; "But how could we possibly explain anything? We operate only with things which do not exist: lines, planes, bodies, atoms, divisible time spans, divisible spaces. How should explanations be at all possible when we first turn things into an 'image', our image!" ... "An intellect that could see cause and effect as a continuum and a flux and not, as we do, as an arbitrary division and dismemberment, would repudiate the concept of cause and effect and deny all conditionality."]]
Our dependence on a performance-orientation in understanding the psyche and its cognitive processes is also inherently euclidian. That is, non-euclidian psychology, such as Vygotsky's, entails a developmental, evolutionary or 'ontogenetic' orientation). That is, the performance measurement orientation which is our culture's standard practice in education, business, and all aspects of life, implies the validity of making independent measurements of a system's (or person's or organization's) attributes as a means of determining its overall 'quality'. This approach implies a reification ('euclidianization') of the attribute being measured; i.e. it implies that attributes can be regarded as independent, isolable, causal agents or 'things'.
[[Vygotsky criticized the 'reification' (euclidianization') of the psyche by psychologists of his day (a practice which has prevailed to the present). To Vygotsky, this objectified the psyche and isolated the discipline of psychology (i.e. reification opened the door to seeing psychology in 'structuralist' terms and the science of psychology as the study of the structural components of the psyche) instead of seeing psychology in terms of a suite of functions which were co-evolutionary with cultural and social phenomena. Vygotsky saw the psyche (i.e. the socio-historically evolved psychological functions) in a non-euclidian, self-referential light, using the earth as a model. In Alex Kozulin's 'Vygotsky in Context' forward to Vygotsky's 'Thought and Language', he says; "Vygotsky further elaborated that 'the structure of behavioral development to some degree resembles the geological structure of the earth's core. Research has established the presence of genetically differentiated layers in human behavior.' The older layers do not die out when the new emerges, but are superseded by it. The conditional reflex, for example, is 'copied' in intellectual action, simultaneously existing and not existing in it.' Vygotsky's framing metaphor Had Vygotsky been privy to plate tectonic, which was firmly established by the 1960's and described how the earth's surface is formed by materials which continuously upwell, spread and subduct, the consistency with his theory would have jumped up several more orders of magnitude.]]
In addition to the above-mentioned 'euclidian' assumptions, our use of a noun-orientation (i.e. 'thing'-orientation) in language whereby we 'reify' all manner of objects and object structures, making them 'things in themselves' ('Dinge an sich'), and distinct from the surrounding space with which they engage is inherently euclidian. That is, non-euclidian languages, such as celtic, aramaic and native american involve the use of substantives whose boundaries are blurry with respect to the action-space they are immersed in. Put another way, non-euclidian languages are not characterized by 'hard' boundaries between 'ends' and 'means'; i.e. the strong warrior is seen not only in terms of a 'thing' but also as a potency in the space he is occupying or as an influence in the space he historically occupied. Again, this notion of seeing things in historical or 'ontogenetic' terms comes into play; ... a geometry which characterizes non-euclidian, self-referential forms.
[[The boundary-blurring in non-euclidian languages appears congruent with the situation in physics where there is an innate relativistic blurriness as to the proportion of energy invested in mass and motion relative to the potential energy of the containing and 'participating' space.]]
The difficulty in using euclidian language as a medium for conveying non-euclidian understanding is implicit in the above comments; i.e. the invoking of statistical, bivalent, mutually exclusive, or quantitative performance-oriented imagery and values in the course of our linguistic discussion subtly pulls us back into the euclidian paradigm. Perception and inquiry on the non-euclidian plane means that we cannot implicitly develop any argument using the ordering principles of bivalent (good/bad) judgement, quantitative measure or statistical correlations, since these are the products of 'special case' euclidian geometry.
Since the perspectives in this essay are grounded in investigations into the source of high performance and dysfunction in collaborative social systems; i.e. by empirical observations of fully functioning 'complex systems' (e.g. 'teams' and 'organizations') rather than by out-of-context theoretical derivations or projections, the observations and conclusions are to some extent indemnified from the maze of euclidian pitfalls which most inquiry methods are exposed to. As just indicated, insights have been 'bootstrapped' from congruencies straddling the disciplines of systems inquiry/science (complexity, relativity, quantum mechanics), philosophy, psychology and language. This multi-dimensional relational view not only converges in a very convincing manner, yielding a clear view of the dominant role of space-time assumptions in western cultural dysfunction, it underscores the valuable potential of aboriginal cultural traditions and 'remanent' cosmologies wrt. the restoration process.
Systems Inquiry/Science: The Pivotal Role of Space and Time
So where did the notion of euclidian space 'come from'? We can imagine ourselves in Greece, around 600 BC, sitting outside our sun-whitened bouganvillia'd villa, deliciously framed against cool blue mediterranean skies, dreaming of a new and more explicit understanding of the world around, ... an understanding more satisfying than the ubiquitous, implicitous myth of Zeus and the theistics. In the dazzle of the afternoon sun, set against deep shadow, the tangible, measurable lines, rectangles and cubic volumes of the villa burned their imagery into our brain, became frames for our vision, and when we next turned our face, we entered the euclidian otherworld.
The self-similarity between form and content proved to be instantly addictive, linear content linearly projected onto all of space, modulating every aspect of our reality. The 'linear' becoming the general case and the natural 'non-linearity' of nature now having to be viewed through linear eyeglasses. An absurd polar reversal prompting one present day scientist to observe; if we had taken this approach with zoology, we might have started by selectively studying a special case, such as the elephant, and referring to the study of the rest of the animals as 'non-pachydermology', describing, for example, 'mice', as being 'tiny elephants with shrunken ears and lacking a trunk'.
Paul Pietsch, neuroscience professor emeritus at Indiana University, maintains that it is impossible to understand the mind-brain using euclidian 'flat-space' concepts --- i.e. understanding the mind-brain [and, one can assume, natural self-referential phenomena in general] requires the notion of curved space, a la George Friedrich Riemann, whom Pietsch quotes as he scenarizes how it came to be that we have taken euclidian space for granted:
"It is well known," he [Riemann] continued, "that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It gives only nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relation of these presuppositions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible."
Pietsch continues; "In the detached bon ton of scholars then and now, Riemann in effect was telling his august audience (including no less that his mentor, Karl Friedrich Gauss) that mathematicians and philosophers had flat-footedly assumed that space is just there. Like the gods in The Iliad who had an external view of the mortal realm, mathematicians and philosophers had inspected space in toto, had immediately brought rectilinear order to the nullity, with the flat planes of length, width and height, and thereby had known at a glance how every journey on a line, across a surface or into a volume must start, progress and stop. Geometric magnitudes--distance, area, volume--plopped inexorably out like an egg from a laying hen's cloaca."
Returning to our sunny afternoon in 600 BC Greece, concepts of space and time were pivotal in the shift away from the theological monopoly on explanations of reality and the bringing forth of a split playing field of rational inquiry plus theology. That is, space-time concepts were caught up in the very birth of philosophy. Plato put it [i.e. divided it up] as follows;
"... for had we never seen the stars, and the sun, and the heaven, none of the words which we have spoken about the universe would ever have been uttered. But now the sight of day and night, and the months and the revolutions of the years, have created number, and have given us a conception of time, and the power of enquiring about the nature of the universe; and from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man."
---- from 'Timaeus'
A contrast between Plato's linear (euclidian) view of the cosmos, with its implicit 'either/or' mutual exclusivity, devoid of the 'circle of life' (non-euclidian) view of the mythopoeic peoples, with its implicit 'and/and' mutual inclusivity, is starkly evident.
Like the single sand grain that triggers an enormous avalanche, Greek 'rationalism' shifted the psychology of 'nature' and 'self' from 'mutual inclusion'; i.e. a unifying, containing and participative mother-child fractal-like relationship, to 'mutual exclusion'; i.e. a detached, 'dualistic' geometry in which all 'things' stand independently and alone in their own right. The choice of space and time geometry is clearly a powerful modulator of the practical understandings of nature we acquire through physics and the 'natural' sciences. The understanding of complex systems (as sought by the sciences of complexity), with their 'emergent behaviors' and 'structure-embodied know-how' are particularly dependent on space-time geometry assumptions.
Clearly, there is overwhelming scientific evidence which supports the superiority, in describing natural phenomena, of the non-euclidian curved space-time, over the euclidian space assumption, yet the latter, because it is so deeply woven into our science, philosophy, psychology and language, resists being subsumed by the former. That is, euclidian space need only be subsumed and not abandoned --- the psychological-perceptual change that is required is that we no longer think in terms of euclidian space being 'the way it is', but see euclidian space as an expedient means of perceiving certain aspects of reality, acknowledging that a far more consistent way to perceive reality is in terms of a curved, non-euclidian space as was (implicitly) the practice of the mythopoeic peoples and as is still the custom of those who have retained a 'celtic consciousness' or native american cultural tradition, and those who follow buddhist, sufi etc. traditions. It is important to note that we are not talking vague or mystical notions here but mathematically definable space-time concepts.
Thomas Kuhn, in 'The Copernican Revolution', has observed that science evolves by breaking and extending the current paradigms; ... paradigms characterized by a high degree of internal self-consistency, but 'incomplete' with respect to delivering an understanding of complex natural phenomena. The resistance of existing paradigms to being broken is bolstered both by this internal self-consistency and, as Kuhn points out, by the degree to which space-time concepts permeate the way we perceive and inquire into systems, i.e.; "Fundamental astronomical concepts had become strands in a far larger fabric of thought, ... The story of the Copernican Revolution is not, therefore, simply a story of astronomers and the skies."
Kuhn's point was that it can take many years for the old paradigm to give way to the new. According to Kuhn, it took more than a century for Copernican sun-centricity to subsume Ptolemaic earth-centricity. It has been said that technology is holding the euclidian paradigm in place, even though a century and a half has passed since it was shown to be the special, 'zero curvature' instance in a sea of curved space possibilities, some of which had been shown to be far more consistent with the findings of modern phsyics.
The nature of space, when considered only with respect to fixed material structures in local regions of space and time (i.e. assuming a passive (time independent) and 'empty' role for 'unoccupied' space), took on its flat euclidian attributes very naturally from the notions of length, width and height used in commonplace scenarios such as house construction. Using these local material attributes as axioms, the rigid, infinite and unbounded characteristics of euclidian space emerged as corollaries; a linear projection of linear content.
Meanwhile, the subsuming of the euclidian paradigm by its curvaceous riemannian 'mother-paradigm' seems to be playing itself out as in an old movie where the mortally wounded hero indulges in a seemingly unending monologue prior to final adieus. Many of us have a soft spot for grand performances, and it is on the basis of 'performance' that we seem to be clinging to this dysfunction-inducing view of space. Unfortunately, even as it 'performs' in a technological sense, it introduces dysfunction in a 'complex systems' sense.
Here we have all the ingredients of a 'catch 22' situation in that our euclidian culture embraces a performance-oriented approach to learning and valuing. Performance measures involve an assumption of homogeneity and the use of measurements and statistics to establish a 'best' and 'worst' performance framework. In other words, the psychology underlying the relative valuing of systems (including alternative paradigms) is, in the case of space-time paradigms, incestuously emanating from the resident paradigm! The alternative, 'non-euclidian' psychology of Vygotsky, based on the nurturance of ontogeny (as put into place in education, for example, by Maria Montessori, an anthropologist and respected contemporary of Vygotsky's) has been embraced only by a small minority of psychologists and educators.
Thus our culture has evolved a dominant performance-orientation in education which infuses in the student a performance-oriented value system, all of which derives from the euclidian paradigm and is the continuing source of pro-euclidian values and tools which we use to assess alternative space-time paradigms!
In the non-euclidian view, we see ourselves in terms of an overall (mind and body) ontogeny; i.e. a 'self-organizing' phenomena in which one is pulled towards a future state by one's natural desire (i.e. by a 'will to power' in Nietzsche's terms). This view, of course, comprehends our connection to the collective or 'evolutionary flow' since it is a fundamental aspect of our psychological ontogeny (i.e. development). In the euclidian view, we see ourselves in terms of a spreadsheet of 'performance' ratings, measurements which are statistically 'calibrated' from the existing cultural paradigm.
Vygotsky's argument that we must flip from (i.e. reverse the primacy from) a performance to ontogenetic orientation in psychology and education is seconded by current era Vygotsky reviewers, such as Cole and John-Steiner , who state; "His legacy in an increasingly destructive and alienating world is to offer through his theoretical formulations a powerful tool for restructuring human life with an aim toward survival."
The current situation wherein the euclidian space assumption underpins the psychological and educational orientation to 'performance' and therefore delivers paradigm assessment tools biased to its (the euclidian assumption's) own self-preservation, can be viewed in terms of 'fitness peaks' and 'runaway feedback'. That is, we set up a virtual fitness peak based on euclidian performance (i.e. based on how well we manage materials in an inert space independent of time) along with an eminently euclidian support mechanism (technology). Thus we have contrived a carrot-and-mule geometry which is pulling us into a runaway feedback fitness 'death spiral'. Lifting this shadowy threat would appear to hinge on our shifting from 'performance' to 'ontogeny' (development) in our values and educational approach. There is nothing new in the positive side of this recommendation; i.e. John Kennedy proposed in a1962 'state of the union' address; "Wealth is the means and people are the ends. All our material riches will avail us little if we do not use them to expand the opportunities of our people."
The difference in perspective which emerges when one comprehends the 'geometric' supporting rationale, is that JFK's proposition that we shift from a 'performance' to a 'development' orientation now appears as a two-sided coin where the degree to which we do not re-orient our values and education to human ontogeny will be matched by the growth of dysfunction in our society.
II. The Alternative Geometries of Space-Time
Returning to the intertwining, evolving histories of science, philosophy and space-time conceptualization. In the mid eighteenth century, Emmanual Kant was maintaining that euclidian geometry was transparently true and that no alternative system of geometry was conceivable to the human mind. As for time, Kant defined it as; "A category allowing one to order events in a before-after-relationship"; i.e. a definition conceived with anthropic utility in mind and one which simplistically complemented three dimensional euclidian space in the manner of a fourth, independent dimension.
A century or so later, Minkowski and Einstein were radically revising the definitions in the direction of a flexible space-time continuum, with Einstein asserting that; "time is what i see on my watch" and space, a participant in natural phenomena rather than a passive container. Einstein said of Riemann (1826 - 1866) who developed the curvature-based framework for non-euclidian space; "Only the genius of Riemann, solitary and uncomprehended, had already won its way to the middle of the last century to a new concept of space, in which space was deprived of its rigidity, ... in which its power to take part in physical events was recognized as possible.". Max Born reinforced this unrestrained praise, saying of Riemann's spherical space, "This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived."
While euclidian space was the mother of mutual exclusion and bivalent rigidity (the eternal 'things' or 'not.things' [void] of Parmenides) and established linear measurement coupled with 'good/bad' judgement as the prime ordering principle, ... non-euclidian (curved) space-time opens the door to mutual inclusion and multivalent flexibility (the eternal 'flow' of Heraclitus), provides the basis for 'complementarity', a dipolar mutual-enfolded complementarity of opposites reminiscent of the ancient eastern symbol of 'T'ai Chi Tu' or 'yin-yang', and establishes 'tuning-in to harmony'  as the prime ordering principle; i.e. in the sense of 'tuning-in' to naturally resonant evolutionary flow and consequence. In all cases, the euclidian attributes are 'special cases' of the non-euclidian attributes; i.e. mutual exclusion is a special case of mutual inclusion (where the permeability of systems' boundaries goes to zero), bivalent logic is a special case of multivalent logic (where fuzzy sets become crisp), and ordering according to a linear good/bad measurement scale is a special case of 'tuning-in to harmonious evolutionary consequence' (where the phase component of the wave view goes to zero; i.e. where spatial latencies are ignored).
The yin-yang geometry ("Where there is low, there is also high. Where there is night, there is also day. Where there is birth, there is also death") symbolized, for Neils Bohr, the principle of complementarity (the dual wave and particle nature of space/matter which together constitute a unified whole), and when he was knighted for his contributions to physics in 1947, he chose the yin-yang symbol for his coat of arms. This space-time geometry 'fuzzifies' to a considerable extent, via it's harmonic or wave properties, the rigid, bivalent measurement-oriented systems perception and inquiry which persists as the dominant organizing paradigm in today's western culture. 
Clearly, 'yin-yang' can be viewed as a general 'geometry' describing a 'dipolar' relationship where polar opposites are mutually enfolded in a complementary relationship, and it will be used in this 'geometric' (rather than mystical) sense throughout this essay.
While non-euclidian, curved space is the general case of which flat (zero curvature) euclidian space is just one instance (a passive and rigid instance to be sure), it similarly turns out that aristotelian logic is just one rigid instance of the more general case of 'fuzzy logic'. Fuzzy logicians such as Bart Kosko, refer to the 'yin-yang' equation (A _And_ Not.A) in describing 'mutually inclusive' logic which more accurately models natural (complex) systems than does the 'mutually exclusive' (A _Or_ Not.A) 'excluded middle' logic of aristotle and the modern analytical method.
Kuhn's model wherein the persistence of the old paradigm is due to ideas having become 'strands in a far larger fabric of thought' can be examined with respect to the persisting primacy of the paradigm of euclidian space, mutual exclusion, bivalent judgement etc. In fact, this model would appear to be the field-of-ideas equivalent to the concept of 'runaway feedback' in evolution, as described by Robertson and Grant , wherein the fitness of the possessor of the 'fit' attribute is reduced by over-exercising the attribute, and the possessor is forced to make a painful descent from its over-populated local space-time fitness peak and rebegin the ascent on the flanks of a more generalized space-time fitness peak.
With respect to space-time geometries themselves, it appears that the euclidian model, used as a basis for systems perception and inquiry, performs rather well when dealing with issues which are constrained to small, confined regions of space-time but falls seriously short when dealing with issues of extended (unbounded) space-time (i.e. as is the case with natural or complex systems). That is, the euclidian model while providing a good basis for, for example, developing a chemical to solve the problem of insomnia (e.g. thalidomide) or insect infestation of crops (DDT), works less well in addressing the evolutionary outcome of systems with broad space-time extension, hence the unanticipated birth defects from thalidomide and environmental problems with DDT.
It appears that it is already past time for our western culture to 'put on the white belt of the beginner', 'empty our cups' and begin the ascent into the new space-time paradigm. While the application of the old paradigm has brought us a cornucopia of technology-based benefits, it has sewn some threads into the fabric of our society and its natural container, which are radically reducing our evolutionary fitness.
Seen from the vantage point of standing within the new (to-the-west) yin-yang paradigm, euclidian mutually exclusive bivalency (i.e. matter and space seen as two independent states) is now just a special case of non-euclidian mutually inclusive multivalency (matter and space are mutually enfolded aspects of a unified evolutionary space-time flow). It also appears as if we have over-played the 'yang' aspect of the yin-yang unity. That is, while 'yin' corresponds to the latent 'integrative' energy of space-time, 'yang' corresponds to the 'assertive' material aspect of space-time; a relationship wherein 'yang' is pulled into existence by 'yin'.
We can examine this 'nested' relationship both from the perspective of general relativity, and the ancient eastern tradition.
General relativity suggests that the potential energy of 'space' is ultimately the mother of matter since matter or 'mass', rather than being an 'absolute' entity, is only an ambiguously determinable notion, being based on measurements of (necessarily relative) motion. As Richard Feymann has pointed out , general relativity theory must be arranged so that it will hold for all moving observors, and thus Einstein's 'field equation', specifying the local curvature of space, must be specified in terms of total energy [i.e. Einstein's field equation states that the excess radius of curvature of a spherical volume of space will be G/3c**4 times the total energy content inside the sphere; ... where G is the gravitational constant and c is the speed of light]. Thus it appears as if there is a natural primacy of 'yin' over 'yang' in that while the geometry of the polar opposites of 'yin' (integrative energy of space) and 'yang' (assertive matter) is one of complementary mutual enfoldment, it seems also that 'yin' (latent energy of space) gives 'birth' to 'yang' (matter).
This 'mother-child' relationship between 'yin-integrativeness' and 'yang-assertiveness' is equally suggested in the domain of information and knowledge management where a 'one-to-many' (expert or boss-oriented) assertiveness (i.e. 'yang') leads directly to hierarchical or boss-oriented knowledge management structures, and where 'many-to-one' (customer or 'systems-oriented') integrativeness (i.e. 'yin') leads directly to peer-peer network-oriented knowledge management structures.
In the ancient eastern tradition, Lao Tsu puts the relationship between space ('yin') and matter ('yang') in a kind of naturally complementary 'supply and demand' context.
"Thirty spokes share the wheel's hub;
It is the center hole that makes it useful.
Shape clay into a vessel;
It is the space within that makes it useful.
Cut doors and windows for a room;
It is the holes which make it useful.
Therefore profit comes from what is there;
Usefulness from what is not there."
--- Lao Tsu, "Tao Te Ching"
Lao Tsu and the ancient eastern tradition did not stop with this 'static' imagery of the yin-yang relationship, but went on to describe the natural harmony of its dynamic; a harmony in which the assertive 'yang' dances to the latent 'seduction' of 'yin'. This opening poem in the Tao Te Ching not only brings out the complementary aspect and unity of nature's yin-yang geometric dynamic, but also the 'inarticulability' of the pull of 'yin space' (reminiscent of Wittgenstein's comment that; "what we cannot speak about, we must pass over in silence.")
"The Tao that can be told is not the eternal Tao.
The name that can be named is not the eternal name.
The nameless is the beginning of heaven and earth.
The named is the mother of ten thousand things.
Ever desireless, one can see the mystery.
Ever desiring, on can see the manifestations.
These two spring from the same source but differ in name;
this appears as darkness.
Darkness within darkness
The gate to all mystery
--- Lao Tsu, 'Tao Te Ching'
The geometry of this harmonic relationship between the 'yin-pull' of space and the assertiveness of matter and motion was brought out by Johannes Kepler in 'Harmonies of the World". In his summary remarks concerning the overall harmony of the solar system, he says; "... while the harmony does not adorn the termini, i.e., the single movements [yang-assertive aspect], in so far as they are considered in themselves but only in so far as by being taken together and compared with one another [yin-integrative aspect], they become the object of some mind; ...". Newton, of course, paid no heed to this multivalent geometric view of Kepler and he exploited Kepler's three laws (which were all in terms of pure 'yin' geometry rather than 'yang' materiality; i.e. they never mentioned 'mass') in a 'yang'-only sense, to achieve the culturally demanded 'closure' (the certainty which gave a sense of security) at the expense of a natural unifying balance and harmony between 'yin' and 'yang'.
For Kepler, 'geometria' was the source of nature's mystery and divinity (Kepler once asserted; "Why waste words, geometry existed before the creation [yang], is co-eternal with the mind of God, 'is God himself') and the uncertainty associated with its multivalent harmonies and self-referentiality was an innate source of beauty in nature. Kepler quoted Virgil in regard to the elusive absence of finality in astronomical space-time relationships; "Galatea seeks me mischievously, the lusty wench; she flees to the willows, but hopes I'll see her first."
Newton, exhibiting a considerably less 'romantic' view of nature and a one-sided 'yang' focus, reduced Kepler's multivalent, mutually inclusive view of space-time to a bivalent, mutually exclusive formulation, more consistent with Aristotelian logic and euclidian space which provided the rigidity needed for 'closed' laws. Kepler, meanwhile, was a Neoplatonist who, together with Tycho Brahe, was steeped in hermetic tradition and its 'law of correspondences' --- "as above, so below" thought to underly all things, ... a 'spherical geometry' with the mutually enfolding characteristics of yin-yang.
Interestingly, the yin/yang geometry of space-time which only began to surface in physics in the twentieth century, was already being re-threaded into philosophy through the works of Friedrich Nietzsche(1844 - 1900) and into psychology through the works of the Russian psychologist Lev Semyonovich Vygotsky (1896 - 1934).
David Abram "The Spell of the Sensuous" , speaks of the fact that we turn to psychology to help us disentwine subjectivity from overall experience so as to discover how our subjective awareness and perception imposes its own 'coloring' on the pre-objectified 'object' of our awareness; "In psychology, however, we discover nothing of the sort. Instead we find a discipline that is itself modeled on the positivism of the 'hard' sciences, a science wherein the psyche has itself been reified into an 'object', a thing to be studied like any other thing in the determinate objective world."
The process of 'Reification' is equivalent to 'euclidianation', bringing the object of our attention into the rigid logical euclidian space which Kant wrongly maintained was the only mental manipulation workspace available to us. And once we euclidianize the psyche, it seems quite natural for us to begin to disassemble it into parts; the conscious, the unconscious, the collective unconscious etc., and to think in terms of recovering lost thoughts by seeing them as fully formed, euclidianized objects which we must dredge the murky depths for. And to close the mechanical, euclidian loop, once we think we have dredged up such a thought, we see it as the linear-causal key to our present distress. Thus the lost images or illusions of lost images, of sexual abuse or satanic rite, once recovered, supposedly give causative meaning to current psychical 'pathology' and what remains for healing is to forgive and 'let go' (or sue, depending on one's disposition).
As the non-euclidian view would indicate and as David Abram suggests; "The living pulse of subjective experience cannot finally be stripped from the things that we study (in order to expose the pure unadulterated 'objects') without the things themselves losing all existence for us." That is, if subject and object are tied together by an evolving self-referentiality, then their disentwinement is no more than a figment of our euclidian mindset, and since linear causality depends in a fundamental way on such disentwinement, it too is no more than a deceptive, approximative abstraction.
Fortunately, there has been at least one psychologist who resisted the euclidianizing trap, and that is Lev Semyonovich Vygotsky (1896 - 1934). According to Alex Kozulin , "...Vygotsky never believed that psychological inquiry should be considered as a goal in itself. For him, culture and consciousness constituted the actual 'subject' of inquiry, while psychology remained a conceptual tool, important but hardly universal."
Vygotsky, in critiqueing the 'method' of psychology, introduced the notion of 'theoretically laden facts' where what is conceived as a 'fact' in one discipline may simply disappear in another. This reasoning dysfunction is alive and well today, as evidenced in the growth of the "The Diagnostic and Statistical Manual of Mental Disorders" (DSM). When published in 1952, the DSM listed 60 disorders, the second publication in 1968 listed 145 while the fourth and latest version (1994) lists 410. While psychiatrists regard as 'fact' that a patient exhibiting certain symptoms is afflicted with one of these disorders, the disorder is an abstraction which owes its existence to the specifics of the disciplinary theory. Culturally, such 'facts' are marketable as 'the real thing' in courts of law, with insurance companies and in society at large.
In treating this issue of a reasoning dysfunction in the very establishment of psychological methodology, according to Kozulin, "Vygotsky made it clear that only epistemologically competent metapsychological analysis of the current state [1920's] of psychological knowledge could provide a genuine general methodology --- what he called 'general psychology. To the question, Where may the resolution of the crisis come from?, Vygotsky gave a dialectically sharpened answer: From the crisis itself!
Clearly, Vygotsky was trying to 'get beneath' the starting assumptions implicit in the methodologies of psychology. I use the phrase 'get beneath', though it is itself linear and euclidian, because our language does not have a concise way of expressing Vygotsky's intent. What he intended was that the disagreement between the various schools of psychology; behaviorism, reflexology, psychoanalysis, Gestalt psychology etc. , the yin-space between the yang-assertions, if you like, was where the 'true' understanding of the psyche lies. This active or participatory view of space is, however, incompatible with the euclidian perspective which was woven into the development of the psychological methodologies.
Vygotsky's inherently non-euclidian perspective predisposed him to Hegelian and Marxist dialectics. The continually evolving sequence of thesis, antithesis, synthesis = new thesis, antithesis, synthesis ... is essentially a notion of the non-euclidian relational 'space' between 'things' as 'they' 'morph' in an ever-changing evolutionary dynamic.
"Vygotsky applied epistemological analysis to distinguish two major poles of attraction dividing all psychological systems: the naturalist and idealist worldviews." ... "Objective historical development turned out to be a hidden source of the crisis." (Kozulin).
Vygotsky maintained that the essence of any phenomenon could be apprehended only through a study of its origin and history. For this reason, he coined the term 'developmental psychology' (geneticheskii) to connote a methodology which looked at the mutually enfolding ontogenetic and socio-historical evolution of mental functions. [elsewhere in this essay, I use the term 'ontogenetic' in its broader sense of including the influences of environmental interplay.].
What Vygotsky was putting together here was an essentially non-euclidian package where our understanding of the psyche must come from a mutually-inclusive relationship between the material (organismic) development and the idea-al or imaginary (socio-historical) development of the individual. Here we have a situation where the information or understanding we are looking for is not 'additive' as in the case of euclidian 'things', but is instead embodied in the historically-evolved structure of intertwining opposites (i.e. the 'real' and the 'imagined' or the 'yang' and the 'yin').
Kozulin describes how Vygotsky viewed this structure; "Vygotsky suggested that the new developmental approach must be built upon three concepts; higher mental functions, cultural development, and mastering one's own behavioral processes. Vygotsky further elaborated that "the structure of behavioral development to some degree resembles the geological structure of the earth's core (sic). Research has established the presence of genetically differentiated layers in human behavior." The older layers do not die out when the new emerges, but are superseded by it. The conditional reflex, for example, is 'copied' in intellectual action, simultaneously existing and not existing with it. Thus psychology is faced with a double task; to be able to distinguish the lower stages imbedded in the higher, but also to reveal how the higher stages mature out of the lower ones. One and the same psychological formation (for example a concept) may have a number of "geologically" different layers, and will play different roles depending on which layer is activated."
The multivalent, self-referential geometry in which our understanding of the psyche (i.e. of higher human mental function) resides, is clearly non-euclidian. As R.D. Laing ("Politics of Experience") observes, "psychology is the logos of experience", and we cannot perceive that 'behavior' is the outer component and 'experience' the inner. Laing, instead speaks of 'modalities of experience' (Vygotsky's geological layers), "none more 'inner' or 'outer' than any other". This, again, is a multivalent geometry and leads directly to the self-referential or 'mutually enfolding', yin-yang, fractal type of structure as connoted in Laing's s statement; "The Life I am trying to grasp is the me that is trying to grasp it."
To cut to the quick and prevent this part of the discussion from becoming a re-iteration of the work of Vygotsky and Laing, the fact that the psyche cannot be understood in terms of euclidian geometry, a conclusion which is clearly evident in the statements and metaphorical models of Vygotsky and Laing, if not in their terminology, gives birth to some important corollaries.
By now in this essay, it should be clear that (non-euclidian) 'space' can be regarded as an 'active entity', as a 'participant' in phenomena. As Einstein put it, Riemann's non-euclidian geometry was; "... a new concept of space, in which space was deprived of its rigidity, ... in which its power to take part in physical events was recognized as possible." The model of yin (the latent integrative energy of experience) pulling yang (tangible behavior) into existence is thus a scientifically sensible way of perceiving psychological phenomena however, 'uncomfortable' it may feel to our euclidian-inculturated psyches. The advantage of this 'inverted' mode of perception is evident in the history of the development of the 'transistor' where the critical insight was triggered by, instead of thinking in terms of yang-electrons assertively swimming from high to low potentials, thinking in terms of the holes (representing electron deficiencies) being sucked through the electron-poor semiconductor. The perspective now becomes top-down, goal-oriented and the ordering principle is 'need' or 'desire' (evolutionary consequence-oriented), instead of bottom-up, linear causal with the ordering principle being bivalent 'judgement' (i.e. if_then_else).
Coming from this non-euclidian 'yin-pull' view, it is easier to understand Vygotsky's argument against a 'performance' based model for learning (as currently dominates our entire way of western life; education, business, social relations etc.). In Vygotsky's model, learning occurs when new understanding is internalized in such a manner that, rather than replacing old understanding, it subsumes the old in a progressively enfolding geometry (and one which is ubiquitous in nature). To understand this in a bit more detail we need to invoke Vygotsky's notion of distinguishing between a 'pseudoconcept' and a 'concept'. A pseudoconcept is simply a web of relational associations (a 'geometry') in the mind which we develop through our real or imaginary experiences. An example is when a child sees a duck and associates it with the word 'quak'. In fact the child may also associate water with the word 'quak' and he may say 'quak' as he looks at the milk in his bottle. Similarly, if he sees an eagle on a coin, he may say 'quak' and begin to associate 'quak' with round objects such as coins. He thus evolves an expanding repository of webs of association or 'pseudoconcepts' and it is from this repository that he forms new concepts ('learns').
New experiences, real or imaginary, as they are accreted on existing pseudoconcepts, engender their own natural integrative pull. That is, as a new association is made in the mind, extending a pseudoconcept associative web and subducting it into the interior, it may give birth to a new concept which asserts itself by upwelling to the surface of one's awareness. Vygotsky referred to this as 'spontaneous learning' where new yang-assertive concepts are the 'children' of yin-integrative pull.
If Vygotsky's model is consistent with what actually occurs, we can see that this type of learning is naturally and harmoniously integrated with all other aspects of the psyche; hence the success of the 'situational' learning approach of Maria Montessori, A. S. Neill ("Summerhill School") etc. Conversely, we can see that 'performance-driven' education, which looks at learning out of the context of natural integrative pull, is likely to infuse the student with the type of 'psychic dissonance' which we discussed in the preface to this essay, since such learning will not be harmonically integrated with the other psychological functions.
There is yet another aspect to this problem of 'yang performance-driven' learning, and this is that the statistics-based measurement of performance is innately euclidian as well. That is, there is an implicit assumption of homogeneity (and stationarity) in the statistical approach. What we measure when we measure learning performance is the ability of the individual to demonstrate their understanding of concepts in an intellectual, rather than a situational sense. The assumption of homogeneity or 'ceteris paribus' basically says that we can ignore the relative degree of 'integration' of the learned concepts. Thus, not only are we exposed to infuse 'psychic dissonance' into the individual (or organization) through performance-based learning, we are ignoring the issue of how well such learning can be situationally applied.
The reason why the film "What about Bob" is so hilarious is that it puts the finger precisely on these issues. When it comes to the intellectual demonstration of conceptual mastery, Bill Murray, the psychotherapy patient, lags miles behind Richard Dreyfuss, the psychologist. But when it comes to situational application, Dreyfuss strikes out while Murray excels. Clearly, our performance based system rewards and cultivates intellectual rather than situational learning and knowledge application, a situation which gives rise to psychic dissonance and inadequate application in a 'death spiral' type of evolutionary geometry.
Instead of including our low level assumptions (e.g. euclidian geometry) in our problem-solving reflections, we design yet another intervention on the basis of euclidian, performance-based intellection rather than non-euclidian, situational, spontaneous learning.
Non-Euclidian Knowledge Management
The notion of euclidian space gives us the solid and secure sense that there is an innate 'origin' or a 'center' to all types of knowledge, and that we are relentlessly progressing towards the 'kernals' of wisdom, the (local) 'theories of everything', which sit within those centers, stewarded by the high priests of the disciplines, like pots of gold guarded by leprechauns. There is a mounting body of evidence, however, which suggests that like the ant meandering on the surface of a spherical non-euclidian space, we will find no beginnings, no ends and no centers.
Meanwhile, our implicit euclidian belief that our current repository of knowledge contains a continuously refining knowledge 'concentrate', the rich tarry residue we obtain by fractionally converting data to information to knowledge to wisdom, induces us to severely discount and discard maturing 'experience'.
So what does the non-euclidian perspective have to say about this?
On the cover to one of the editions of Michel Foucault's 'The Architecture of Knowledge', we see the increasingly ubiquitous image of an ammonite, with its spiralling and finely divided lobe-and-saddle shell structure --- a picture of a unified whole which is its own ontogeny. Like Vygotsky's model of the development of thought as multivalent 'geological' layers enfolded within layers, Foucault's 'decentred' model has to do with the 'envelope' of knowledge evolution rather than its euclidian structure; i.e. it has to do with the meta-geometry of knowledge rather than 'knowledge' per se. The Foucault model brings to mind Heraclitus' assertion that 'the knowledge of many things does not teach intelligence.'
The entogenetic envelope of knowledge, rather than being the current culmination of one's (or an organization's) knowledge, is the enfolded history of its evolution. In complex systems terminology, (Kaufmann), 'The structure of the organization is also the record of the embodied 'know-how'.' This places the value of knowledge clearly on 'experience', or the evolutionary record of the learning process as captured by the individual and/or the employee collective (i.e. the 'organization'); a valuation which flies in the face of knowledge management practice associated with mergers, acquisitions and downsizings (this is perhaps the implicit reason behind management researchers referring to these practices, collectively as 'MADness'). Once again the familiar pattern emerges wherein the euclidian 'thing'-oriented view (i.e. of knowledge-as-content) appears as a special case contained within the more general, ontogenetic view. This view shifts the value upstream from the euclidian 'content' to the historical 'development' of that content; i.e. the old 'story-telling' sessions in the pub 'contain' the knowledge particulars (in an enfolded form) just as quantum waves contain particulate matter in a phase-encoded form. The ontogenetic encapsulation, however, retains the situational phasing of the knowledge (the importance of relative timing) which is generally as, if not more important than the 'knowledge in itself' ('Ding an sich').
Our pre-occupation with the 'yang-assertive' or 'one-to-many' form of knowledge as opposed to the yin-integrative or 'many-to-one' 'envelope' which gave it birth is the source of pervasive continuing dysfunction in knowledge management. Reconciled with the findings of Vygotstky on the inadequacies of 'performance'-oriented development (education) (i.e. that there are innate deficiencies in the internalization of non-ontogenetically oriented learning which undermines the ability to situationally apply the knowledge), we see a massive overbalance of yang-assertive knowledge production in the universities, corporations and all types of organizations, RELATIVE TO yin-integrative or 'systems thinking' capacities which come in the form of 'experience'.
A. J. Berkhout of Delft University , in an invited submission to the OECD, (Sept. 1997) speaks to the issue in these terms; "Successful interaction at the two principal interfaces of the innovation system requires two types of knowledge workers at each interface; the knowledge disseminators [disciplinary specialists or one-to-many 'yang-asserters'] and the knowledge integrators [systems thinkers or many-to-one 'yin-integrators'].
Clearly, our performance-oriented culture and education disproportionately rewards and promotes the successful 'marketers' of new concepts and technologies, entirely out of context with ontogenetic need and associated integrative requirements. While the responsibility of those in possession of new concepts and products to disseminate them is understood implicitly, requirements in the integrative domain are commonly left flapping in the breeze. This gives rise to accusations, particularly in high tech environments, of 'too many solutions looking for problems' and to the Say's law type of economy where once the initial supply is disseminated (e.g. a new computing widget), it engenders its own continuing demand which, in dollar terms, may rapidly rise to levels several times the initial outlay.
Once again, this yang-assertive knowledge overbalance is systemically tied to management oriented to performance, rather than to natural 'ontogeny' or 'development'. This is as true in business as it is in education.
Euclidian knowledge management thinking which gives rise to the notion of a unidirectional refining column to mechanically convert reified data to information to knowledge to wisdom, turbocharged by performance-based rewards, leads directly to 'information overload' and the devaluing of experience. Experience, as just noted above, which comes complete with relative 'phase' information as to the 'rhythms and harmonies' which bring out the full value of knowledge content; i.e. higher dimensional rhythms and harmonies which, like those involved in riding a bicycle, are only experientially accessible. The current impetus towards reification of knowledge in terms of 'intellectual capital', while shifting the focus from capital to human resources, appears to be regression to euclidian 'structuralist' mode. Sustainable development will require a shift in focus away from the 'trees' towards the 'forest ecology', ... i.e. a bringing into conscious awareness of the ontogeny of knowledge on both the individual and organizational levels. This is indeed what happens naturally in the evolution of exceptional, high performance teams; in this environment, the primacy of a boss-asserted command structure for 'how things are done' gives way to a 'customer'-pulled demand primacy which provides integrational orchestrating and amplitude-modulating 'ordering principles' for the operation.
[[  A framework for looking at information in motion in a non-euclidian curved space context is to envisage it in a plate tectonics type of geometry involving a sphere whose surface represents the manifest component, in a continual state of evolution due to continual surficial upwelling and subduction into the latency within the sphere. In this imagery, 'yang' (assertiveness) corresponds to the upwelling; this can be pictured in terms of outward-pointing radial arrows (of a single color) on the surface of the sphere, emanating from the point of emergence of the upwelling (a 'one-to-many' geometry). 'yin' (integrativeness) corresponds to the subduction; this can be pictured in terms of inward pointing radial arrows (of diverse colors) on the surface of the sphere, being sucked inward towards the point of subduction (a 'many-to-one' geometry). From these simple geometries, considerable complexity can emerge since the upwelling yang is mutually enfolded with the subducting yin. And since the surface of the sphere represents time = the present, an unattainable, limiting abstraction according to Heisenberg's uncertainty principle, the counter-intuitive suggestion is that the surface manifestation is an abstraction or virtual image, much as a flame is with respect to continuously flowing fire (as was suggested by Heraclitus).
In this model, reified knowledge comes across as 'virtual' (abstraction) while latent knowledge potentials can be seen as the more substantial 'mother' of the transient manifest forms of knowledge. Michel Foucault  describes this situation using a word geometry which closely parallels that of Vygotsky in the domain of 'developmental psychology', "Behind the visible facade of the system, one posits the rich uncertainty of disorder; and beneath the thin surface of discourse, the whole m ass of a largely silent development (devenir): a 'presystematic' that is not of the order of the system; a 'prediscursive' that belongs to an essential silence. Discourse and system produce each other --- and conjointly --- only at the crest of this immense reserve."
The issues of language have woven themselves into all of the above discussions on systems inquiry, philosophy, psychology and knowledge management. In fact, it is obvious that our language serves euclidian concepts far better than non-euclidian; i.e. our language is essentially bivalent and single-issue at a time (it direct usage mode, at a given point it can deal with only one 'article', surgically extracted from its ecological relationships.).
[[Wittgenstein who claimed that his philosophical works were "struggles against the limits of language" (Ankaempfen gegen die Grenzen der Sprache') speaks to this issue, as follows; "There is a truth in Schopenhauer's view that philosophy is an organism, and that a book on philosophy, with a beginning and an end, is a sort of contradiction [Elsewhere Wittgenstein quotes Heraclitus "everything is in flux" on this same problem of being forced to capture a complex continuing dynamic by 'parts']. One difficulty with philosophy is that we lack a synoptic view. We encounter the kind of difficulty we should have with the geography of a country for which we had no map, or else a map of isolated bits. The country we are talking about is language, and the geography its grammar. We can walk about the country quite well, but when forced to make a map, we go wrong. A map will show different roads through the same country, any one of which we can take, though not two, just as in philosophy we must take up problems one by one though in fact each problem leads to a multitude of others. We must wait until we come round to the starting point before we can proceed to another section, that is, before we can either treat of the problem we first attacked or proceed to another. In philosophy matters are not simple enough for us to say 'Let's get a rough idea', for we do not know the country except by knowing the connections between the roads. So I suggest repetition as a means of surveying the connections." ]]
Wittgenstein speaks to the innate bivalent or 'euclidian' nature of language (and of philosophy which depends upon language). Nature, meanwhile, is innately multivalent or 'harmonic', entailing geometries such as the dipolar yin-yang geometry in which multiple dynamics are occurring at the same time. While Shakespeare could get away with saying 'she makes hungry where most she satisfies', a clearly multivalent or harmonic (interference-based) notion, the imagery so produced is engendered within the reader and is not fully 'grounded' in the words themselves. So while a literary document may be designed for the feelings and interpretation it invokes within the reader, a philosophical or mathematical document, to ensure its internal consistency, is more dependent on the direct signification of the words and how they are connected (as opposed to how the reader interprets and connects them).
This presents a formidable challenge when using western languages since non-euclidian (e.g. self-referential, harmonic) notions cannot be directly developed from closed form 'nouns'. Wittgenstein attempted to get around this problem, of our language's inadequacy in dealing with non-euclidian notions, by 'surveying the connections'. What emerged from this approach, which tried to preserve the consistency of rationalist philosophy and capture the harmonies of nature (and art) at the same time led him to comment; "My style is like bad musical composition." (Mein Stil gleicht schlechtem musikalischen Satz.)
Clearly, the medium of western 'euclidian thing'- based language is a poor medium for the self-referential concepts of non-euclidian geometry. While mythopoeic era languages possessed a non-euclidian base, the shift away from ontogenetic (developmental) learning to the embrace of material performance driven organizational learning (i.e. trade and commerce), led to the rise to primacy of the phonetic, noun-based structure which is currently in use (with exceptions such as Celtic and Native North American languages which escaped this transition because their cultures escaped the shift from ontogenetic to material performance driven learning and the stewardship of their languages did not pass from the priests, shamans and poets to the traders and merchants.)
[[While a backward looking euclidian 'causality' would have it that language has transformed culture; e.g. Oswyn Murray cites Goody and Wart in their statement on the effects of literacy in early Greece; "A famous article published by the anthropologists Goody and Watt in 1963 gives the clearest statement of this problem, both in general and in relation to Greece. The study was heavily influenced by the general approach of the 'Toronto School', the best representative of which is Harold Innis and the most notorious, Marshall McLuhan: in the work of this group, communication is seen as a fundamental force in society, and changes in the mode of communication are taken as the central catalyst, altering both social and individual relations; 'the medium is the message' --- new forms of communication, widespread literacy, the dissemination of the book through printing, or television, so alter our perceptions that they replace social, economic or religious factors as the primary theoretical explanation of change in society.'
This above viewpoint starts on a plane above where the effects of one's basic geometric conception of reality are formed. It was by purposeful design that Egyptian hieroglyphics had the capability for both phonetic constructs (22 of the 750 pictographs were required to reproduce the vocal constructs of the egyptian language) as well as ideographic constructs (which utilized the remaining 700 plus pictographs). Thus, the egyptians had the capability in their written language of simultaneously rendering yang-assertions and yin-integration; i.e. non-euclidian geometries. Yet after the reign of Persian conqueror Darius I, who ruled Egypt in the 520 - 480 BC era and not only recognized the power of hieroglyphics but planned to introduce them into persia, hieroglyphics were overtaken by phonetic Greek, because of its trading and commercial value, and went into decline.
Had Oswyn Murray taken a broader base of observations which included the differing space-time assumptions of the peoples of the region and the evolutionary history of hieroglyphics, it may have cast a different light on the linear-causal assumption that transformation of the language 'caused' cultural transformation. In the non-euclidian goal-orientation of complex systems, Nietzschean, Vygotskyan etc. philosophies, future vision reaches back to determine current behavior (e.g. the desire for harmonious evolutionary consequences reaches from the future back to the present to shape current behavior; i.e. a happy outlook induces virtuous behavior rather than vice versa).
Murray points out that the amazing material affluence of the Phoenicians is denounced by the Old Testament prophets in the sixth century, citing Ezekiel; "Tarshish (in Spain) was a source of your commerce, from its abundant resources offering silver and iron, tin and lead, as your staple wares, Javan (Ionia, the Greeks), Tubal (in Cappadocia) and Meshech (Phrygia) dealt with you, offering slaves and vessels of bronze as your imports. . . Rhodians dealt with you, great islands were a source of your commerce, paying what was due to you in ivory and ebony . . . Dealers from Sheba (Aden) and Raamah (S. Arabia) dealt with you, offering the choicest spices, every kind of precious stone and gold, as your staple wares. Harran, Kanneh and Eden (in Mesopotamia), dealers from Asshur (Assyria) and all Media, dealt with you; they were your dealers in gorgeous stuffs, violet cloths and brocades, in stores of coloured fabric rolled up and tied with cords; your dealing with them were in these. (Ezekiel 27.12-24)"
Since both phonetic and ideographic languages were already in place in the sixth century BC, the birth of western philosophy and associated propagation of a euclidian view of space may have more to do with the shift from ontogeny to rampant materialism as the ordering principle in western society' evolution. The psyche of the western world, to use the Vygotskyan framework, was undergoing a socio-historical adaptation resulting from the philosophically induced notion of euclidian space.]]
Scholars of the Aramaic language spoken by Jesus note that it is in the non-euclidian [our term] mold of the Celts etc. and much of the self-referential meaning has been laundered out through translations from the greek (euclidian) into english (euclidian). Neil Douglas-Klotz  points out (as mentioned earlier) that aramaic, unlike english, did not use the 'immiscible' approach of english, squeezing all materiality into closed form nouns and leaving an empty 'playing field' with which to give linear causal descriptions of noun-precipated action. Instead, aramaic blurred the boundaries between a 'thing' and its spatial presence. As Douglas-Klotz puts it, "Unlike Greek, Aramaic does not draw sharp lines between means and ends, or between an inner quality and an outer action. Both are always present. When Jesus refers to the 'kingdom of heaven', this kingdom is always both 'within' and 'among' us. Likewise, 'neighbor' is both inside and outside, as is the 'self' that we are to love to the same degree as our 'neighbor'. Unlike [euclidian] Greek, [non-euclidian] Aramaic presents a fluid and holistic view of the cosmos. The arbitrary borders found in Greek between 'mind', 'body', and 'spirit' fall away."
The geometry so-described, in terms of physics, is one of interplay between the latent potential energy of space and the materiality induced by spatial latencies, a geometry reminiscent of non-euclidian space, self-referentiality, general relativity theory, yin-yang etc. This notion of space allows for the geometric notion of divinity (as held by Kepler and Heraclitus etc.); i.e. a divinity (or 'heaven') which is the fundamental ontogenetic flow of the living cosmos.
The point here is not to get into religious debate, simply to observe that different space-time assumptions; i.e. euclidian and/or non-euclidian assumptions, can be 'built in' to the language base, and that this can have a significant impact on how we perceive and inquire into the reality around us. This takes us back to Wittgenstein's suggestion that we can be 'captive' of words or 'bewitched' by language. Certainly, Douglas-Klotz' interpretation of aramaic suggests that speakers of aramaic in the time of Jesus might have 'imaged' his words in a 'divinity in nature' context.
There are many more citings of how the nature of language gives rise to different 'psychology', such as with regard to native american poems and their 'circle of life' themes. The following poem was in this case attributed to Dayton Edwards, however, by its 'geometrical signature', it could have as easily come from a Celtic as a Native American source;
The Circle Again: Birth and Death
Do not stand at my grave and weep.
I am not there; I do not sleep.
I am a thousand winds that blow;
I am the diamond glimpse in the snow;
I am the gentle autumn rain;
I am the sunripe golden grain.
And when you wake in the morning, hush.
I am the swift uplifting rush
of circling birds and circling flight;
I am the soft starlight at night.
Do not stand at my grave and weep.
I am not there; I do not sleep.
One can find variants of this poem, whose 'divinity in nature' theme predates our euclidian space oriented western philosophy, in Christian, as well as Celtic and Native eulogies around the globe. Apparently it is geometry of choice to 'die by' if not to 'live by'.
The impact of space-time assumptions built into language emerges in the case of 'treaty negotiation. The following comments  are excerpted from Chief Roy Whitney's preface to 'The True Spirit and Original Intent of Treaty 7' (This 1877 treaty, involved the First Nations of the Alberta region; i.e the Bloods, the Peigan, the Siksika (Blackfoot), the Stoney, and the Tsuu T'ina (Sarcee), and the Canadian government. The book captures the passed-down impressions of tribal elders as to Treaty 7's intent.
* * *
"This volume thus represents a collective effort rather than one that can be attributed to only a few individuals. It is a remarkable achievement for five nations representing three distinct language groups to produce such a book.
However, we recognize that there may be problems with what we have produced. For example, meanings may not be accurately conveyed where there are no Blackfoot, Nakota, and Tsuu T'ina (Dene) words that correspond to English words or concepts - or the reverse. More generally, First Nations peoples have expressed the belief that our languages represent a significantly different world view from that of the newcomers, one that cannot be glibly translated without an understanding of the context and environment in which the languages were created. "Verb-centred" Native languages and "noun-centred" European languages arose out of radically different contexts and environments. As Sakej Youngblood-Henderson has recently pointed out in an article, these fundamentally different languages are not compatible with simple translation.
* * *
The First Nations agreed to share the land with Canadian newcomers in return for the Crown's promises, which entailed annuity payments, education, medical care, ammunition, assistance in farming and ranching, and assurance that we would be free to continue to hunt as we always had. All of the First Nations elders said that their people did not give up the land; in fact concepts such as "surrender" and "cede" had been untranslatable to them. What is clear from the elders' testimony is that our people would allow newcomers to farm and to use the topsoil of the land. Our elders from each of the nations were adamant that there was no discussion of surrendering the land.
* * *
Just as a modern day Christian reading the aramaic words of Jesus translated through greek into english may 'image' them very differently than a contemporary of Jesus' whose native language was aramaic (due to the differing inbuilt space-time assumptions), the First Nations Chief, discussing treaty 7 with a Christian negotiator, through interpretors, is unlikely to converge on the same imagery. In fact, given the nature of the different space-time assumptions incorporated in the respective languages, it would be far more likely for the First Nations Chief and the negotiator to converge on a common imagery, if the negotiator was fluent in the aramaic of two thousand years ago.
The main observation in this discussion of non-euclidian language is that those languages which have not converted their language base to euclidian concepts are far better equipped, in philosophical or systems inquiry mode (which demands high levels of self-consistency) to deal with complex natural phenomena. Euclidian geometry influenced languages, such as english, while internally self-consistent in a rational sense, cannot easily capture and convey the self-referential patterns inherent in complex natural systems such as collaborative human structures.
On the artistic side, had Shakespeare been a violinist, he most likely could have achieved virtuoso performance with a single string, since music and art are much more about the 'yin-space' between the notes than about the 'yang-notes' themselves.
Bridging the gap between art and science will require the development of language which has high levels of consistency in 'yang-assertive' mode (high generalizability yet is capable of complex multivalent concepts) together with high levels of 'yin-integrativeness' (high specificity). Since the 'yin-integrative' aspect is embodied in the geometry of the spaces between the structural components of language, rather than being directly conveyed by the components (words, sentences etc.), the basic structural (yang) requirements of the language are less demanding with respect of cultivating 'yin' effect. Nevertheless, the power of Shakespeare's prose seems to derive from his skill in using the yin-yang combination in the most synergistic, mutually complementary manner, and thus achieving great specificity at the same time as great generality.
From observations of high performance and dysfunction in teams, brought into connection with transdisciplinary findings in systems inquiry, philosophy, psychology and the study of language, a clear picture emerges, of psychological dissonance and social dysfunction emanating from the arbitrary historical assumption that space is euclidian.
The alternative assumption, held on an implicit basis prior to the birth of philosophy by all of the worlds major civilizations and retained throughout history by many eastern and aboriginal cultures, that space is curved or 'non-euclidian', has been proven by twentieth century science to provide a more consistent conceptual base for perceiving and inquiring into natural phenomena, including complex systems such as human organizations.
Since in a geometric sense, the euclidian assumption is a special case (curvature = zero) which is contained within the non-euclidian, curved space-time assumption, the corrective action must be to subordinate those aspects of our system inquiry approach which derive from the euclidian assumption, to those which derive from the non-euclidian model. This suggests that measured performance-based learning (personal and organizational), the dominant approach in education, business, and government, must be subordinated to 'ontogenetic' or 'developmental' learning; --- that 'good/bad' bivalent judgement must be subordinated to an intuitive 'tuning'-in to harmonious evolutionary consequences; --- and that our language must be extended in such as way as to be able to deal in a scientifically consistent, well reasoned fashion with higher dimensional phenomena heretofore only aesthetically articulable via the arts. In short, 'cause' needs to be subordinated to ontogenetic 'purpose'.
The euclidian space assumption has led to particularly insidious forms of dysfunction in which corrective interventions have exacerbated rather than alleviated the dysfunction. 'Killer kindnesses' fall into this category, whereby compassionate efforts to assist others emanating from truly humanist values are themselves a source of pathology. The solution here is re-synch one's gyros to the harmonies of nature, rather than to the errant constructs of euclidian rationalism.
A number of scientist-authors feel that a change in this state of affairs is in the making. Fritjof Capra, in 'The Turning Point' quotes the I Ching in this regard; "After a time of decay comes the turning point. The powerful light that has been banished returns. There is movement, but it is not brought about by force. . . . The movement is natural, arising spontaneously. For this reason the transformation of the old becomes easy. The old is discarded and the new is introduced. Both measures accord with the time, therefore no harm results." 
The revisions in scientific thinking which support this re-enlightenment emerged in the late nineteenth and twentieth centuries; i.e. deterministic chaos, relativity and quantum physics. Our notion of space-time is now open to reinterpretation in terms of an evolving continuum, wherein material structures may be 'seen' as being precipitated out of a 'participative' (quantum) energy field through the pull of 'attractors'. This reformulation of space-time, centered around the curved (non-euclidian) space-time assumption is far more consistent with scientific observations and sensory experience than was the earlier euclidian assumption. The 'non-euclidian' assumption liberates perception and inquiry from an exclusive focus on material structures, and extends inquiry to self-referential and harmonic issues such as the 'imaginary' (purely geometric) space-time envelopes or 'ontogenies' within and through which material structures come into being. Ontogenetic envelopes form out of an oscillating dialectic between attractor-pulled growth and environmental growth-constraints. Such envelopes are often visible within a plant or organism; examples include the Fibonacci spiral structures in sea-snails and ammonites, sunflower seeds, etc. Similar non-material envelopes are associated with the development of human experience (Vygotsky) and language and are available to us through our cognitive processes.
The western 'build' on top of this flawed base commenced some 2500 years ago, as the philosophical ideas of the ancient Greeks and contemporaneous developments in written language incorporated the euclidian space concept in an implicit, dependent manner. Though the inadequacy of the euclidian space-time geometry has been recognized by western science for over a century, it has been woven so deeply and pervasively into the fabric of western thought, that a shift to the non-euclidian paradigm can be seen to be of similar or greater order of magnitude as the shift from the Ptolemaic to Copernican world view.
While the magnitude of the 'real' structural adjustment associated with this paradigm shift may be huge, the timing of the 'imaginary' ontogenetic adjustment can be as brief as the time needed to strum the next chord in a musical progression, an inversion from 'cause-over-purpose' to 'purpose-over-cause', and it is our imagination that determines our feelings of self-worth, happiness and optimism for the future.
... Ted Lumley / Martine Dodds-Taljaard
Return to 1998 UPDATE Page
 I Ching, cited in Capra, Fritjof, "The Turning Point", Bantam Books, 1982
 Henri Frankfort, H.A. Frankfort, John A. Wilson, Thorkild Jacobsen, William A. Irwin, "The Intellectual Adventure of Ancient Man", University of Chicago Press, 1946
 Wittgenstein, Ludwig, "Philosophical Investigations" rem. 115, 109.
 Jantsch, Erich, "Design for Evolution", 1975
 Vygotsky, L.S., "Thought and Language", The MIT Press, 1986
 Abram, David, "The Spell of the Sensuous", Vintage Books, 1996
 R. D. Laing, 'The Politics of Experience', Pantheon, 1967
 Abram, David, 'The Spell of the Sensuous', Vintage Books, 1997
 Einstein, Albert, 'Geometry and Experience', Address to the Prussian Academy of Sciences in Berlin, January 27, 1921
 The Pomegranate, http://www.interchg.ubc.ca/fmuntean/POM2a1.html
 Kahn, Charles H., 'The Art and Thought of Heraclitus', Cambridge University Press, 1979
 D.S. Robinson and M.C. Grant, "Feedback and Chaos in Darwinian Evolution, Part II", Complexity Volume 2, Number 2, 1996
 E. David Peat, "Superstrings and the Search for The Theory of Everything",