Pattern Is Fundamental
By pattern here I don't mean shape or form per se, but rather the end result of a dimensional leap from the collection and arrangement of categorical parts to a whole.
Parts > Integration > Unity
A "simple group" (Group Theory, Modern Algebra) is characterized by its nondivisiblity; its only homomorphisms [structure-preserving maps, structure definition determined by respective geometry] are itself and its identity element. When mapping to itself under a homomorphism each element is placed into a separate equivalence class. There are no elements or subset of elements which can be 'removed' from the simple group without violating its integrity as a whole unto itself, maintained, necessarily, for the sake of its identity. O.K.
Now, a 'soliton wave' is composed of different frequencies nonlinearly feedbacking, tansmuting and interacting. [This is similar in concept to the proton-neutron interchange creating the nuclear force]. If one frequency were to be 'removed' or undergo an amplification, breaking the bonds of the feedback constraint, the entire wave would lose integrity and dissipate, releasing the energy it embodied.
How are the notions of a simple group, a soliton wave, and a geometric whole composed of rationally discernable elements or parts, similar or parallel? And what, if anything, does this imply or mean in the context of Nature?
The defining features of each are left invariant under structure maintaining maps. Each frequency of a soliton is itself composed of sub-frequencies. This layered effect represents an infinitely regressive nonlinear dynamic. In algebra, a composition series can be defined as a hierarchy of homomorphic images each one of which is a simple group, another layered effect of wholes within wholes.
If we consider the set of possible frequencies as a spectrum of possible quantum paths, each one of which is vital for the existence and nature of the whole in a nonlinear environment, then in the act of perception 'it may be' that we infuse sufficient energy to dissipate the wave and flatten the quantum curve. In effect, we linearize a previously, and by nature, nonlinear field. If we focus on a point of a circle, the point becomes manifest and the circle disintegrates as such.
Moreover, from chaos theory we have the notion of an attractor. This is a living breathing 'thing' unto itself. Bifurcation into a new pattern emerges under positive amplification. For a given attractor to maintain integrity, the parts or vortices, if it represents a turbulent system, must remain in a state of nonlinear negative feedback, in dynamic harmony with its environment. To cast any such system into a linear mold is to extract its unpredictable and spontaneous aspects, its complementary tendency towards disorder necessary for it to be what it is.
This spontaneity and capacity for symmetry breaking would seem to be an essential ingredient for anything natural, including subatomic forces, to be. This asymmetric quality allows for natural processes to change and evolve over time. An element or part of a whole, rationally apprehended intellectually as a 'thing in itself,' lacks this all essential characteristic of an asymmetric proclivity. As such, as a thing, it can only 'exist' in the realm of the abstract, as a mental construct.
The soliton and the simple group, as a static model, cannot be subdivided without losing definition and integrity. Reduction to a surmised 'smallest thing' is nothing more that an exercise in abstract analysis, assigning an ad hoc label to that which, out of context, has no physical reality.
In Nature, a 'part' of a dynamic nonlinear system has no meaning or being except and only as it relates interdependently to other connected 'parts' by affinity and continuity; collectively they manifest as a distinctive whole pattern. In the act and action of this relating is imparted the potential for spontaneous behavior.
Each individual psyche, separate and complete unto itself, finds meaning and being by virtue of membership in the human experience. Collective consciousness, delineated dynamically as nonlinear in nature, design, and composite behavior, reflects this cohesiveness of separate parts as a layer of identity. Layers of nondivisible psychic functions and variegated features generate whole forms forever fine-tuning: an individual, species, ecology, planet, universe. The common ground parameters infusing this hierarchy of embedded identities, in which we share intimately and in detail, formulate an ecology of not only interdependent ideas, but also, interdependent realities.
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