In the template, if we make the following assignments, A = whole, B = rational, C = irrational, D = imaginary, as we go deeper we find that all can operate within the context of the other:
A B C D (function) - dependent 'variable'
- - - -
A A A A (context) - independent 'variable'
There are characteristics common to each group that lead to implications about the roots of systems and the development of hierarchies. Calculus, for example, develops hierarchy through the use of the derivative where position --> velocity within position --> acceleration within velocity etc.
Common patterns across systems emerge for example, where, by considering the method of getting a 'real' number using logarithms we find it analogous to the way of getting 'real' numbers using complex numbers:
to get this from this we need this
real log antilog
real a + bi a - bi
This suggests a common mental structure/processing area for both. Implied here is that 'real' numbers are the more generic in that, say, a+bi 'means something' and a-bi 'means something' and that they symbolize opposite sides of the same coin; name the coin. In this context the name will always be a 'real' number and that this number is at a more 'generic' (and thus less abstract) level. This correlates with the discussion on dichotomy where the marker of seperation '/' in fact marks a point of emergence once the dichotomy is taken past 1:1, we enter 'middle space' -
log / antilog
^
real numbers
a + b1 / a - bi
^
real numbers
And since contract/expand can be associated with -/+, then there are, for example, sixty-four 'types' of mathematical numeric expression at level T6 on the template. (we can extend this but the 'noise' may become too much).
When we move into considering logs and exponentials we move into the dynamic process of converting products into sums (logs) and sums into products (exponentials).
We can see from this that different number types emerge as we attempt to 'cut the whole', and therefore the possible number of types and their mixing is in fact 'infinite' although the resolution power of our tools of analysis can introduce limitations and we find ourselves faced with a continuum. The observer/observed dichotomy, and the manifestation that it is in fact a continuum and therefore we must include observer and observed as a whole, is not just restricted to quantum mechanics. It is an 'attribute' of ANY dichotomous thought processing.
Another example of the emergent properties is the Mandelbrot set. This set is a mapping of the function f(x) = x^2 + C where all the variables (x,C) are complex numbers. If we set C to 0 then we find that, iterating the result of the function (making the result feedback into the equation), we generate a sequence of points in the complex number plane. Plotting this sequence, if the initial point is less than 1 unit from the origin then the sequence becomes smaller and is 'attracted' to zero. If the initial point is greater than 1 unit from the origin then the sequence becomes larger, it is attracted to the complex equivalence of infinity.
When the initial point is exactly 1 unit from the origin (i.e. it lies on the unit circle centered at 0) "...the sequence never leaves the unit circle. Thus the unit is a *boundary* between two domains of attraction, the one governed by 0, the other by infinity." p92 [emphasis in original] (Devlin, 1988).
This is another example of 'middle space', for Mandelbrot shows that for values of C other than 0, a whole new world emerges. The dichotomy at the start is infinity/0. Leaving C at 0 is the equivalent of not going past level 1 in the dichotomy map. Any other values of C replaces 0 as the attractor and we enter the dichotomy map of infinity/C, with a world emerging at the boundary, or cutting, point, symbolized as /. What has happened is that we have moved from a one:one position into a one:many position (here treating infinity as 1), or, (treating infinity as many) from a one:many position to a many:many position.
As we zoom-in on the boundary we enter the world of fractals, and we also find, using graphics, repetitions of the whole, which is a characteristic of the dichotomies tree. (visually, it is as if we are viewing a distortion of the unit circle - which we are (Affine transformation)).
This is analogous to the single slit experiment in quantum mechanics, where the dichotomy is open_slit/closed_slit, and the wave pattern emerges at the boundary, or cutting, point of open/closed (see Part II).
Mandelbrot's work demonstrates behaviour in the complex plane, but it rests on work done by Verhulst in the world of dynamic systems and real numbers.
Verhulst developed an equation for measuring rates of growth:
x = x + r * x * (1 - x)
This format can be typed into a computer program within a loop
causing iteration:
for y = 1 to 1000
x = x + r * x * (1 - x)
next y
If we set r, the rate of growth, to a number less than 2, the result is 1, or close to. The moment we take r beyond 2 we enter a world of complexity manifest by period doubling. This is manifest graphically by the presence of 'attractors'. As we take r to approach 2.57 we find that we approach infinity and chaos:
# of attractors
r = 0 to 2 1
2 to 2.5 2
2.5 to 2.55 4
2.55 to 2.565 8
2.565 + 16
......
------> 2.57 infinity
These attractors manifest the number of possible states at a specific level of a dichotomy tree. The original dichotomy is that of 0/infinity (2 is the cutting boundary), which becomes 2+/infinity; at 2 is the split. (There is a constant associated with period doubling, it is called the Feigenbaum number : 4.6692016609. Each set of attractors is related to the previous set by a factor of 1/4.669 times the size of the previous set).
Considering the concept of the Context Ratio introduced earlier, we here see what happens when we go beyond a development factor of 2 in that the result is more than the sum of the previous parts and a degree of increasing instability emerges when we try to predict states.
The difference between Mandelbrot and Verhulst is that Mandelbrot's world is the complex number plane where as Verhulst's world is the number line, but in both can be seen the generation of a dichotomies style 'map'.
Using the attractors concept, each state in a dichotomies tree acts like an attractor in that events develop a bias to an attractor. Each level is therefore the emergence of more attractors (as well as the possible emergence of repellors).
Following this we find that copies of the whole emerge within the system, and the development of an infinite number of attractors, WITHIN THE OVERALL CONTEXT, introduces a continuum.
These dichotomies trees demonstrate the results derived from some fundamental proofs of mathematics.
In 1930 Kurt Godel developed the incompleteness theorem, a complex 'proof' that, for mathematics to be consistent it cannot be considered complete; the studied dichotomy emphasized complete/incomplete and as such would automatically cause the emergence of a dichotomy tree the moment we move beyond level one and into the middle world.
In 1963 Godel's concept was developed for any form of mathematics by Paul Cohen, who demonstrated that there existed mathematical statements that were undecidable rather than True/False.
In both of these, the mere consideration of dichotomies, e.g. T/F, causes the emergence of the dichotomy tree, where once again we enter the world of the middle. When seen in the eyes of T/F it seems that some parts are 'undecidable', especially when basic mathematics emphasizes a one:one approach. The dichotomy tree shows that any level of mathematics that goes beyond the one:one dichotomy enters the middle world, and this suggests why, in esoteric areas, the integers have always taken-on a 'special' quality; a 'wholeness' unlike other numbers with prime numbers being 'super' wholes.
The moment a relationship is taken beyond level 1, we enter the middle world, the world containing irrational numbers and the transcendentals and the many forms of infinity, as we move from wholeness, to parts, to relations, to transitions and transformations.
Implied in this is that Godel & Cohen may be 'wrong' since their analysis was based on dichotomy and as such limits their scope, although it is hard to imagine anything else at this time (due to the bias to dichotomous thinking). However, developing deeper levels of analysis will enable a more refined degree of mathematical 'wholeness' but the question arises as to whether the derived facts have any worthwhile 'value'.
It is interesting to note that early mathematics (BC into AD) had a strong sense of cardinality but only a gross sense of ordinality. What emphasized ordinality was the introduction of arabic symbols, as well as zero. Using Roman numerals, for example, it was possible to express some numbers without order, e.g. XXX, which is the same value read from the left or right (or middle). The use of zero as a marker for 'no value in this position' forced a strong sense of ordinality into mathematics by explicitly fixing numeric positions, where the numbers became counters of the position 'type', where the types were units of a specific base. In integer mathematics, for example, the first position is that of units, whereas the second position is that of tens of units.
This enforcement of ordinality brought mathematics to life, giving us the tool we have today and removed the previously nebulous nature of numbering symbols. It also strengthened the letter/number dichotomy, although in some languages this nebulousness continues. In Hebrew, for example, it is context that determines whether a symbol is either a letter or a number.
If mathematics is founded on serialization and dichotomies, then it's success as a tool of prediction is understandable, but that does not suggest that mathematics is 'out there' in the form we know it; the above outline has I hope helped to emphasize the point that mathematics is a metaphor.
The fact that we are constrained by our own senses is often 'glossed-over'. These constraints are made up of neurological, sociological, and psychological forms, and are in fact built-in to our testing equipment in that we design the equipment to extend our senses. This implies that dichotomous thought is externalized and the apparent patterns that we detect (i.e. wave/particle) are possibly derived from dichotomy and therefore are not necessarily 'universal'.
The only way in which a link can be established between inner and outer is if we work on the supposition that the brain has adapted to it's environment by internalizing the environment's characteristics. In a 'gross' sense, this means the Space/Time dichotomy in the form of the SpaceTime Continuum. If this were so, we could then only state that the brain has adapted to *local* conditions thus still leaving an opening for other forms but at the same time 'allowing' our maps to have some value outside of our heads.
Evidence suggests that the highest levels of brain function, the tertiary sensory areas in the forebrain may in fact show a level of sensory system hybridization, where, predominately, the auditory and visual systems share neurons. The areas concerned, the frontal lobes for example, are highly abstract when compared to the primary sensory areas. Hybridization would allow for neural mixing such that the wave analysis of the audition system is used visually. We find that synesthesia, the mixing of senses, is common and creative. (See Stein & Meridith (1993) for a review)
In summary, mathematics serves a a metaphor for the the manipulation of levels of dichotomous analysis and thus we can include mathematical constraints as part of the list of characteristics of dichotomous analysis. Thus any dichotomy-based system cannot be wholly encapsulated by the methodology (incompleteness concept); all that can be achieved is a refined degree of analysis that in the everyday world is acceptable as a 'complete' description of the whole. (Lao Tzu knew this in 450 BC where he emphasized that you cannot cut the whole - see chapter 28 of his "Tao Te Ching")
Furthermore, any dichotomy-based system will show characteristics that can be deemed complex and/or chaotic and thus has emergent characteristics. These characteristics emerge the more we analize more than one serial thread of the dichotomy-tree at a time.
Having developed an idea of the characteristics of the sense of dichotomy, the next section outlines the emergence of dichotomy trees, and their inherant 'meanings', in Psychology in the form of the various typologies of human personality. Since these systems often lack any formal mathematical source, the emergence of the trees suggests an unconscious use of dichotomy when classifying. The more academic tests are based on dichotomy (BIG-5 etc) since they are biased to scientific analysis (and thus explicit dichotomization) but the more esoteric typologies seem to have arisen 'intuitively'.
My conjecture here is that the continued finding of 'meaning' in these systems is due to their serving as metaphors for dichotomy; they resonate with the template rather than they having any 'intrinsic' value.
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