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{{AnthonyJudge}}: I confess that it is my view that the absolute truth that is absolutely true is not the absolute truth that we are able to talk about when we talk about absolute truth. This would preclude any further insight in the thousands of years to come on many ordinary matters as illustrated by the illusion that the sun does rise.

It is interesting that those who specialize in the study of time now accept a variety of understandings of time. I would argue for a variety of understandings of truth. The criticism of relativism is useful because the requisite complexity of a relativistic understanding of truth has not yet been clarified. I am personally interested in complementarity between different forms of truth rather than deducing from such complementarity an absolute truth which I can possess. It is a good question whether one possesses truth or is possessed by it.

Anyway my writings on truth are:

[ Complementary Truth-handling Strategies: Mediating the relationship between the "Last class" and the "Liar class"]

[ Complementary Patterns of Meaningful Truth and the Interface between Alternative Variants ]

[ Politicization of Evidence in the Plastic Turkey Era: al-Qaida, Saddam, Assassination and the Hijab]

[ Warping the Judgement of Dissenting Opinion: towards a general framework for comparing distortion in rules of evidence]

[ 12 Complementary Languages for Sustainable Governance]

===AnthonyJudge offers a challenge===

AnthonyJudge: I truly appreciate your initiative. From a methodological perspective I would offer the challenge of envisaging the mathematical/topological structure in which every perspective positioned in it would be right -- including the perception that others are wrong. A good example is when I phone my mother in Australia and she swears it is midday and I swear it is night. We are both right and both wrong -- the resolution is through the space on which we are respectively positioned.


I feel blessed that we have among us Tony Judge and his mind's boundless energy. In my understanding, he's dancing through all of intellectual space, fleshing out the various kinds of structural thinking that can be found in our lives. I myself tend to focus on how to sort through all that and put it all together. I suppose I would say that Tony is pre-sorting and it would be good for me to learn how to connect my efforts with his.

Recently, he's sharing his draft on "Extremism". One thing that struck me are his thoughts on "degrees of extremism". For example, how might we rank Muslim extremists? This kind of thinking shows the problem of a simplistic approach to addressing extremism. It also reminds me of the picture I use for thinking about dissident movements. There it is important that supporters fill in the entire spectrum of dissidence, from most mild to most defiant. That makes it impossible for the tyrant to find a natural cutting point at which to isolate the extremists from the mainstream. It also makes it a moral matter for people to take risks in support of what is right no matter where they may sit on the spectrum. This kind of moral calculus makes the extremist outlook very natural and potent even when the issues at hand may be questionable.

Personally, I have a deep sympathy for extremists. All of my life I was raised to understand the significance of Lithuania's quest for freedom in the face of Soviet oppression, and I am ever amazed by the miracle that we became free in my lifetime. Our whole lab is based on the observation that independent thinkers always end up in the periphery because the center is taken by those who accomodate each other superficially.

For me, nonviolence is part of my faith. So I look at violence with respect as I would a person of a different faith. In particular, the idea of noble action seems important in trying to make sense of violence. As Tony writes, there is a sense in many cultures of "rooting for the underdog". I was raised in American schools and Lithuanian Saturday schools that it was noble for snipers to shoot redcoats, or to dump the tea in the Boston Tea Party, or to say "Give me liberty or give me death!", or to drop the bomb on Hiroshima, or for Lithuanian warriors to kill themselves, their women and children rather than give them up as slaves to marauding Christian knights. And it was noble for Lithuanian guerillas to blow themselves up with hand grenades so their faces would not be recognizable in the market square when the Soviets lay them and force people to look so they could see who would cry. And it was noble to kill as many Soviet soldiers as you could in the process. It was noble for Muslims to use US Stinger missles to shoot down Soviet aircraft. I don't see why it wouldn't be noble for somebody to shoot down Amercian aircraft if they felt that America was an oppresive power. I never understood why Pearl Harbor was not consider noble. I never understood why Iraqi military casualties don't weight on us but American military casualties do. I don't quite accept the distinction any more between soldiers and civilians. Why is it more noble to kill soldiers rather than civilians? Don't soldiers have children, wives, parents? Don't they die just as helpless? Can the decision to join an army (with all its implications) ever be fully thought out, ever be truly willful, when we are so ignorant of war and its effects? Isn't the source of the current strife the free trade between the consumers and producers of oil? Wouldn't it all stop if we chose (or required ourselves) to be self-sufficient in energy? And who is in charge and why aren't they taking responsibility? Isn't the gigantism of the oil industry and the resulting giant flows of capital precisely what is needed for rogue princes to fund themselves? As the cost of destruction shrinks, wouldn't we want the flows of capital to shrink as well? Wouldn't it make more sense to live in a land of lakes than on an ocean front waiting for the inevitable tsunami? And why are the lives in rich countries so vastly more important than those in poor countries? Why do a few thousand deaths get so much attention when a few thousand die every hour? When hundreds of thousands are washed away by a tsunami? Or millions die of painful diseases? "Saul killed thousands, but David killed tens of thousands" as the Bible says. Terrorism was a part of the struggle against apartheid in South Africa and the struggle of Jewish partisans against the British army in Palestine. There is a thoughtful articile at Wikipedia: Perhaps we might respect terrorism enough so that we might distinguish within it, just as we respect nuclear weapons programs enough so we might have degrees of acceptability. Perhaps that is what I mean by nobility - that improper means such as violence may yet, if accepted as such, allow us to look for other qualities that do have merit, including a truthfulness to one's self and one's cause. We may see shades of grey that shine even in the darkness. And that helps us see the spectrum of humanity as it extends even into violence. It gives us a way to engage each other, whereas vilification does not.

In pagan days, there was a Lithuanian duke Kestutis who would announce to the Crusaders when and where he would fight them. And he always showed up on time. And he would berate them if they showed up late. They ended up respecting him greatly. I think that all put their cause in perspective and helped point to a deeper humanity.

I share an excerpt from Tony's draft. It's wonderful that he places his work in the Public Domain!

I have also created a link from our wiki page:

Thank you, Tony!


Andrius Kulikauskas Minciu Sodas +370 (5) 264 5950 Vilnius, Lithuania

Anthony Judge wrote: > Greetings > > You may have some interest in the newly posted paper of mine > entitled: > > Global Struggle against Extremism: from "rooting for" extremism to > "rooting out" extremism > > > Regards > > Tony

Mandelbrot Set

See also: DissipativeSystems

[ Sustainability through the Dynamics of Strategic Dilemmas in the light of the coherence and visual form of the Mandelbrot set] by AnthonyJudge, 7 March 2005 | Draft

Methodological approach

The following points endeavour to provide a rationale for the approach taken:

  • Everything in the technical description that follows is the coherent expression of one "thing"
  • As a fundamental description of dynamic relationships, it is in a significant sense already "known" to the reader
  • As such it is characteristic in some way of human living and being
  • This is despite the curious mathematical technicalities through which it is described and which, as an artificial language, may be difficult to comprehend
  • To the extent that it is in some way already intuitively known and recognized by the reader, it may resonate with archetypal symbols and patterns of quasi-similar form from different cultures
  • Such resonance can be usefully distinguished from popular enthusiasms for "fractal thinking"
  • There is a case for exploring the more rigorous mathematical descriptions to determine whether particular features and properties support valuable insights relevant to challenging psycho-social issues and dilemmas

In support of this approach, for example, Chris C. King (Fractal and Chaotic Dynamics in Nervous Systems, 1991) presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel.

Features of the M-set

See web resources in the above table.

The M-set can be divided into an infinite set of figures (typically represented as black, as in Figure 1) with the largest figure (in the center) being a cardioid. An (infinite) number of circles are in direct (tangential) contact with the cardioid -- but they vary in size, tending asymptotically to zero. Each of these circles has in turn its own infinite set of smaller circles in contact with it, and these surrounding circles also tend asymptotically in size to zero. Repeatedly indefinitely, this branching produces a fractal. In addition the M-set is characterized by filaments or tendrils within which some new cardioids appear, not attached to lower level "circles". [more]

  • 1. ComplexPlane: The M-set is not represented graphically on a plane in a normal 2D space. The nonlinear dynamics to which it points can only be effectively represented on a complex plane. Mathematically one dimension is then "real" and the other "imaginary". These dimensions are discussed with respect to the axes, to be followed later by their possible psychosocial implications.
  • 2. Axes: The axes of the complex plane on which the M-set is represented may be usefully compared to the experiential significance of being "crossed". This expression tends to be used to describe the encounter with a mode of behaviour that is inconsistent with the logic which one normally used. It reflects a different mode of organization. This distinction might, for example, be used to describe the relationship between "right" and "left" in politics, or between "mainstream" and "alternative" development strategies -- or even between "female" and "male". It is important to recall Palmer's articulation above regarding the illusory quality of what is being described. The axes are given their significance by arbitrary convention regarding "x" and "y", "vertical" and "horizontal", "positive" and "negative".
  • 3. Points: The axes of the M-set permit various complex numbers (having a real and an imaginary component) to be positioned in relation to one another in a systematic manner. Specially named points include:
    • Origin: This is the point at which the axes cross, defined as (x=0; iy= 0) and on the basis of which the M-set is generated. It might be usefully compared with the birth place of person, one's home, the starting point of the dynamics between two irreconcilable positions, etc. It may be related to the centre of gravity of the body (hara), as in martial arts.
    • Complex points: These are the positions that complex numbers taken up in terms of the axes. They effectively mark the condition of a nonlinear dynamic relationship, such as the status of an argument or a relationship relationship
    • FixedPoints: In conditions of nonlinear dynamics, behaviour in a person's life may be governed by:
      • attracting fixed points: where behaviour tends to be attracted to a single fixed point of focus (family or job), holding the notion of eventually "returning home" or "coming back". In the complex plane, the origin (0) and 1 are considered special fixed points.
      • repelling fixed points: where behaviour is repelled by a single fixed point (a place, a person, a perspective, etc)
      • attracting periodic points: where behaviour tends to alternate between attraction to two or more such fixed points (family and job and sport)
      • repelling periodic points: where behaviour tends to alternate between repulusion by one or more such fixed points (family and job and sport)
      • Lyapunov-stable fixed points:
      • neutral (nonhyperbolic) fixed points: ****

Infinity is also treated as a fixed point since complex numbers near infinity (far from 0) stay near infinity (far from 0).

  • CriticalPoint: This is the starting point through which the dynamic function is tested to determine whether it results in connected or disconnected sets. It might be understood as the critical point through which the coherence of a discussion or an initiative is tested. It might also be understood in terms of kairos as the dramatic moment at which the future outcome of an interaction is effectively mapped out -- a moment of "destiny".
  • 4. Iterative generation of M-set:
    • Iterations: The maximum number of iterations (N) used in testing points in the generation of the M-set can be selected as desired, for instance 100. Larger N will give sharper detail but take longer [more]. Human life may be understood as characterized by iterative processes. Physiologically these include breathing and the pumping action of the heart. Vision is based on rapid eye movement (REM). The circadian rhythm of the waking/sleeping cycle can also be understood in this way, as can the cycle of consumption/excretion. Many habits are characteristically iterative, as is engaging in sex. The succession of human generations, through which society (and the planetary surface) is populated, may also be considered iterative. A number of religions hold strong convictions regarding reincarnation, itself an iterative process, through which eventually individuals evolve into "buddhahood", for example. Within society there are many regular processes that can be usefully seen as iterative: rituals, regular meetings, festivals, etc that provide benchmark points indicative of its status. The most fundamental debates have an iterative aspect as the same points are explored again and again. It might be argued that in order for sustainable consensus to emerge amongst divergent perspectives a pattern of iterations is required to engender a form isomorphic with graphical representations of the M-set.
  • 5. Surfaces and volumes: The M-set can be represented on a complex plane surface or on a complex sphere. Humans make extensive metaphorical use of a supposedly flat "surface", whether to describe the "domain" or "territory" to which they lay claim ("my land") or which forms part of their "homeland", "nation" or "empire". This usage is reflected in the competitive relations between corporations and (organized criminal) gangs -- even when the "territory" is a range of products and services rather than tied to a particular geographical surface. The territory may be effectively associated with a "sphere" of operations and may be understood in "global" terms. Such understandings of territory are fundmental to the highly dynamic relations between academic disciplines -- even with respect to their specialization into "fields". It is useful to stress that any such "surface" understood through such metaphorical usage is not as stable as is implied by efforts at demarcation by surveying and mapping techniques. As well demonstrated in the Middle East, even the demarcation of the land surface may be highly disputed. Such distorting dynamics are even more evident in competitive relations and conflicting between commercial interest groups, ideologies, religions or academic disciplines.
  • 6. Scope:
    • Size: ***
    • Boundedness: ***
    • Boundary zone: ***
    • Dimension: Human experience may be understood as lying within the world of polarization and duality. The coherence and integrity of human experience -- any sense of unity -- therefore emerge within the framework of that duality.
  • 7. Sets and connectedness -- J-set and M-set:
    • Julia set (J-set): With respect to human behaviour and understanding, a J-set might be usefully described as a "pattern". A distinction can then be made between three kinds of pattern:
      • 1. Essentially unstable patterns that persist (or exist) only briefly, if at all. These may include behaviours which seem to be part of an enduring pattern but more or less quickly prove not to be. Equally they are the modes of thought which may breiefly appear to be consistent, but quickly prove not to be.
      • 2. These are patterns which are essentially habitual and unvarying, consistent with a single general pattern of behaviour of which they are an exemplification.
      • 3.
        • Connectedness:
        • Mandelbrot set (M-set):
  • 8. Form
    • Self-similarity: The Mandelbrot Set and Julia Sets The Mandelbrot Set - Small Copies
    • Symmetry:
    • Other M-sets:
      • Cardioid: This is the main body of the set as represented in Figure 1. Attached to it are "bulbs".
      • "Vertical x-axis": Positive values of real numbers lie above the horizontal axis (negative below).
      • "Horizontal y-axis": Positive imaginary "y" values are *** left and right.
        • Single-factor / Integral
      • "Bulbs" (or "circles", or "disks") around the cardioid: triadic, quadrilemma, multi-set
      • "Head": binary thinking, dualistic, polarization
      • Filaments / Tendrils:
  • 9. Orbits: An orbit is the trajectory of a point through a succession of iterations. In the M-set representation, the outer zone represents unbounded orbits (escaping to infinity), the central cardioid zone represents fixed points (to which the orbit converges), the other circular features represent of distinct cyclic periods. The thin boundary zone around the figure represents chaotic orbits. The chaotic regions appear to be restricted to the boundary of the M-set and to a portion of the real axis (represented vertically at the top of the figure). [more] [demo]
    • Period of attracting cyle: This may be understood as the number of distinct states that an iterative system cycles through. Demonstrations are provided by interactive applets (James Denvir. The Mandelbrot Set Iterator; Alexander Bogomolny, Iterations and the Mandelbrot Set, 2005, after bypassing the sponsor's advertising). Iterations within the M-set evolve differently depending on the value of c. Where the starting point c, is within the cardioid, iterations converge -- period 1. For c inside the "head", the iterations converge to a cycle of period 2. For c inside each "bulb" attached directly to the cardioid, the iterations converge to a cycle whose period is determined by the corresponding "bulb". If c lies in the interior of a bulb, then the orbit of z0=0 is attracted to a cycle of a period n -- it is a multiple of n for c inside any smaller bulbs attached to the primary bulb. The behaviour of the iterations is related to the appearance of the Julia set associoated with c. [more]
    • Period doubling / Bifurcation: Moving along the real axis (vertically in the seated-orientation), the period of the iterates of the current point keeps doubling. This is admirably illustrated by an applet (Period doubling and Feigenbaum's scaling, 1999; Bifurcation diagram for quadratic maps, 2002; Universal period n-tuplings cascade of bifurcations, 2000)
    • Attractor / Limit cycle: This is the value to which a function may converge on iteration, irrespective of the starting point. The number of points in a limit cycle is called the period. Fractal shapes are depictions of attractors. A region of points with attractors, like the M-set, is termed a basin of attraction. This may be associated with processes such as temptation and conversion. A J-set is the boundary of all its attractor basins.
    • Repellor: At the boundary of an attractor basin, points inside the boundary are convergent and trapped by the attractor. Points outside the boundary are divergent and escape the attractor --the attractor then acts as a repellor. The boundary of a J-set functions like a repellor -- as the closure of all the repellors.
    • Mode locking: This is the tendency to fall back into the behaviour pattern, the attractor, even when external perturbations disturb the pattern momentarily.
    • Rotation numbers: (see Robert L. Devaney. Rotation Numbers and Internal angles of the Mandelbrot bulbs, 2000)
    • Unit circle: This is the circle of radius 1 centred on the origin. Any iterative seed on (or inside) the circle of radius 1 has an orbit that does not escape to infinity. All orbits of x2 that lie on the unit circle are those that behave in a chaotic fashion.
  • 10. Features: Many of the detailed features of the M-set have been given colloquial names, usually descriptive in the light of their resemblance to natural phenomena [more].
  • 11. Colour: Three main approaches to colour are used:
    • Exterior: The exterior of the Mandelbrot set consists of points for which certain iterations diverge. Normally colours are added to representation of the points that are not inside the set, according to how many iterations were required before the magnitude of z exceeded two. This creates concentric shapes, each a better approximation to the M-set than the last. Other colouring conventions may be available (as with Xaos) based in part on the real and imaginary coordinates.
    • Interior: Areas inside the set are usually filled in black, but this is only a convention. Again (as with Xaos), there may be many different ways to show the colour within the set, notably based on the real or imaginary coordinates of the latest orbit, or its angle.
    • Colour cycling: This is a technique to automatically shift or rotate the palette of colours through which the M-set is displayed, making the display strikingly dynamic.
  • 11. Mapping in higher dimensions: The classic M-set is a map in the complex plane. Software to explore the M-set typically allows the user to more complex fractal structures:
    • Polynomial maps: the power in the quadratic recurrence equation may be increased from the standard "squared" form (with one symmetry axis) to the cubic form (with two such axes), the quadric (with three), etc -- with any number of "poles" by suitable choice of exponent. [more] Including the Mandelbrot set (power 2, 3, 4, 5 and 6) in the case of Xaos. Also known as multibrot sets [more | more | more | more]
    • Complexifying the plane: The complex plane may be rendered more complex using a choice other common "surfaces"
    • Higher dimensions: Potentially the M-set can also be mapped into higher dimensions. for example, using quaternions the M-set is a 4 dimensional object [more].


See also: MetaphoricalThinking, {{Knottedness}}, {{Questions}}

[ Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness] by AnthonyJudge 23 October 2004 | Draft

This is an exploration of how significance is associated with being what is metaphorically described as "straight" (or droit in French) as opposed to being "bent", "twisted" or "warped" -- notably as in "going straight". Following the French, this may well be interpreted as "right-thinking". Specifically it is concerned with how higher degrees of "twistedness" are encountered in social interaction and in arguments in support of certain strategies. Is it reasonable to expect a straight anwer to a straight question under such circumstances? What might be the nature of questions and answers of higher order?

The aim in this exploration is to recognize how twistedness works and the conditions under which its complexity is of "positive", as opposed to "negative", significance. This argument aims to clarify the nature of more complex forms of understanding that may appear "twisted" to others and may, or may not, indeed be usefully associated with "richer" or "higher" forms of cognitive insight --whether exemplified by "holiness" or "perversion". The argument relates to structural insights summarized in a separate paper (Strategic Opportunities of the Twice Born: reflections on systemic camouflage of mass deception, 2004).

The approach here is through two lines of exploration. The first is through the extensive work on the phenomena of coiling and knottedness that are fundamental to magnetic, bio *** and specifically to DNA replication. The latter is related to the preoccupations of biosemiotics. In this connection, it is also appropriate to note here the importance of various forms of "coiling" in the symbolism of different cultures, whether the caduceus, the ourobouros, or more generally the coiled snake and its "satanic" associations dating back to Adam and Eve. The ambiguity of serpentine associations is illustrated by the symbolic status of Moses's Snake: "Just as Moses lifted up the snake in the desert, so the Son of Man must be lifted up, that everyone who believes in him may have eternal life" (John 3:1-16 ).

The second approach is through a review of current investigations into the taxonomy of questions and the insights of work of Arthur Young (The Geometry of Meaning, 1978; The Reflexive Universe, ***) as a template for interrelating higher order questions beyond conventional taxonomies..


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