Sunday, August 21, 2011

Part 2 of 2 -- A Key Difference Between Ordinary, and F.E.D.-Dialectical, Algebra.


A Key Difference Between Ordinary, "Purely-Quantitative" Algebra, and F.E.D.'s "Purely-Qualitative" First Dialectical Algebra, that of the NQ Dialectical Arithmetic.


Note:  Throughout his blog entry, I will be using visible light spectrum color-order color-coding to call attention to dialectical, qualitative ordinalities:  "ROY G. BIV", connoting the qualities of First-ness, Second-ness, Third-ness, Fourth-ness, Fifth-ness, Sixth-ness, and Seventh-ness, respectively.



Dear Readers,

Below is the second and final part of the text begun in my blog entry # 21


The standard interpretation of an initially "unknown-meaning" symbol such as q/AA is that it stands for "the [<<aufheben>>] self-subsumption of A".

This often, most precisely, means the "self-meta-unit-ization" of the units that implicitly constitute the <<arithmos>>/ontological category, connoted by A, i.e., it signifies the formation of combinations of some of the units of which the "<<arithmos>>", or "number", of units, called A, is made up, into the "meta-units" of which the new, q/AA, or B, <<arithmos>>>, or "number", of [meta-]units, is made up.

So, methodologically-speaking, one asks one self, right after having "squared" A, this question:  "What known, or presently, to me, unknown, but potentially -- "combinatorically" -- predictable, phenomenon/process/"eventity", in the past, the present, and/or the expected future of the experienced [and/or measured] actuality of the dialectical  progression for which I am constructing this model, could the otherwise "unknown-meaning" symbol, q/AA, aptly stand for?"

Suppose that we are successful in identifying an objective correlative, in the experience of, and/or in the available data about, the actual progression being modeled, that corresponds fittingly and specifically to the generic, "self-subsumption"-in-general, meaning of the symbol
q/AA.

If so, we have "solved" the dialectical equation for the specific meaning of B in this equation-model so far --

A^2   =   A + q/AA   =   A + B [wherein the symbol '=' denotes the phase "is equal to by definition"].


Our next methodological, algorithmic step is to, (a.) (b.in turn, "[re-]square", or self-apply, that very result, [A + B], which is now a non-amalgamative sum, or heterogeneous sum, hence an ir-reducible sum, of two mutually qualitatively different <<arithmoi>>>; of two ontological categories, representing a minimal ontological "cumulum", or "accumulation"; for a "self-application" which yields a four-fold "cumulum" as its "[self-]product" --

[A + B]^2   =   q/A + q/AA + q/AAA + q/AAAA   =

   
A + q/B + q/BA + q/BB   =   A + B + C + D

-- and (b.) to try to figure out what the initially "unknown" symbols q/BA and/versus q/BB might "mean" -- to what they might correspond in our experience, and/or measurement, of the concrete actuality about which we are constructing this [meta-]model -- 


e.g., the observed historical progression whose past-to-present we are reconstructing, and whose likely future we are "pre-constructing", or,


e.g., the systematic categorial-progression of the "table-of-contents"-summarized "content-structure" of a dialectical theory presentation, such as Marx's Capital, or Hegel's <<Logik>>, 

which you are super-sum-marizing, and also super-mnemonizing, by means of a compact "dialectical equation" formula, e.g., of the Seldon-Function form --

)-|-(s    =    (A)^(v^s)

wherein v denotes, e.g., the value 2 for the "Dyadic Seldon Function", and the value 3 for the "Triadic Seldon Function".


The standard interpretation of an initially "unknown" symbol such as q/BA is that it stands for "the [<<aufheben>>] subsumption of A by an/its later-explicitly-arising, supplementary "other", B.


This often means a "hybrid" "unit-ization" of the units of A with the units of B, i.e., the formation of combinations of some of the units of which the "<<arithmos>>", or "number", of units, called A, is made up, with some of the units of which the "<<
arithmos>>", or "number", of units, called B, is made up, into the "hybrid-units" of which the new, q/BA, or C, <<arithmos>>, or "number", of [hybrid-]units, is made up.


So, methodologically-speaking, one asks oneself, right after having "squared"
[A + B], this question:  "What known, or, to me, presently unknown, but potentially, "combinatorically", predictable, phenomenon/process/"eventity", in the past, in the present, and/or in the expected future of the experienced [and/or measured] actuality of the dialectical  progression for which I am constructing this model, could the otherwise "unknown-meaning" symbols, q/BA, and q/BB, aptly stand for?"


Suppose that we are successful in identifying objective correlatives, in the actual progression being modeled, that correspond, fittingly and specifically, to the generic, "hybridizing subsumption"-in-general, and to me specifically previously-"unknown" meaning of the symbol
q/BA, and to the "self-subsumption"-in-general, and to me specifically previously-"unknown" meaning of the symbol q/BB.

If so, then we have "solved" the dialectical equation for the meanings of C and of D in this model equation --

[A + B]^2   =   A + B + C + D

-- and are ready to take our next methodological, algorithmic step: "squaring", or self-applying, the resulting four-fold "cumulum", 
[A + B + C + D], and trying to identify the specific meanings of each of the now four new of the eight total symbols [re-]generated by that second "self-<<aufheben>>" self-application. . . .

. . . And so on, until none of the new symbols generated by such "squarings" appear, to us, to have any concrete, specific meaning that we can identify [despite our deep knowledge and experience of the totality-being-modeled, and about any "measuremental" data which may be available about it], in the sense of matching specific objective correlatives -- in the actuality of that dialectical [sub-]totality whose emergence/becoming, or being, is thereby being chronicled, or presented, via our dialectical-algebraic meta-model -- that adequately "answer to" those newly-generated symbols that result from each such "self-iteration" / "[re-]squaring" of the previously "squared".



Examples: Specific Applications of General Method.

1.  Suppose, in the context of the diachronic dialectic, or historical Dialectic, of Nature as a whole, that the category we name "pre-nuclear "particles" ", e.g., quarks, should be the <<arche'>> ontological category of the universe, so that the "ontology possibility-state", or "possible ontological content", of the tau = 0 epoch of the total cosmological "meta-evolution" is symbolized by just the "singleton" n, not by even a minimal "cumulum" [a minimal
"cumulum" would mean a [non-amalgamative, "ir-reducible"] "qualitative sum", or <<synagoge'>>, of just two ontological "categorograms"].

Then, we obtain, for the "possible ontology-content" of the tau = 1 epoch , the "non-amalgamative, ir-reducible", two-component,
minimal "cumulum" expression, consisting of two mutually qualitatively different "numbers" of qualitatve units, i.e., of two "populations" of different kinds of individuals as units, or of two <<arithmoi>> --

n^2    =    n + q/nn

-- and, suppose, we determine that the "meta-unit-ization" of the pre-nuclear "particles" "<<arithmos>>", or "number" ["population"], is the "proto-nuclear sub-atomic "particles" " <<
arithmos>>, or "population" -- e.g., proton units, each one made up out of a heterogeneous multiplicity of quarks as its sub-units].

Thereby, we have solved the equation --

n^2   =   n + q/nn

-- via the equation / valuation / identification --

n^2   =   n + s

-- and have "solved for" the specific meaning, or specific connotation, of the initially "unknown-meaning" symbol,
q/nn, by the equation / identification [the relationship of "identification" being herein symbolized by '='] --

q/nn   =   s.



2.  Suppose we decide, in the context of the Marxian "method of presentation" dialectic, synchronic dialectic, or Systematic Dialectic, presentation of the Marxian theory of the Capitalist System of human-societal self-reproduction as a [sub-]totality of Nature, that the category we name "Commodities", should be the <<arche'>>>, or starting, category of our categorial progression exposition of the self-reproductive, and of the ultimately self-dis-reproductive, workings of that system, thus positing the exposition of C itself as step 0 in our presentation of the auto-catalytic, self-reproductive "meta-anatomy" and "meta-physiology" of the Capital-ist [self-organizing-becoming-self-dis-organizing] System [wherein "Capital" names "the auto-catalytic form of value", in that: "Capital[-value] makes more Capital[-value]", as in the generic Capital-circuit  

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