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.
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 "<
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
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 <
[A + B]^2 = q/A + q/AA + q/AAA + q/AAAA =
-- 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 --
The standard interpretation of an initially "unknown" symbol such as q/BA is that it stands for "the [<<aufheben>
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>
So, methodologically-speaking, one asks oneself, right after having "squared"
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
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",
. . . 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'>
Then, we obtain, for the "possible ontology-content" of the tau = 1 epoch , the "non-amalgamative, ir-reducible", two-component,
n^2 = n + q/nn
-- and, suppose, we determine that the "meta-unit-ization" of the pre-nuclear "particles" "<<arithmos>
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 = 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'>>