Part 1 of 2 -- A Key Difference Between Ordinary Algebra and F.E.D.'s "First 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, Part 1.


Note:  Throughout his blog entry, I will be using visible light spectrum/rainbow 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 Reader,


I often encounter, from discussants, questions such as that below, which I have "genericized" for the purposes of this blog entry --

"The problem I see with the F.E.D. dialectic at present is that it exists as an "elite" tool of the very few."

"How might we bring it to earth in popular language that would facilitate a popular global revolutionary movement by which humanity can achieve "escape velocity" from its present sand-pit: its "capitalism attractor"?"


INTRODUCTION. Every innovation begins as an esoteric, "elite" tool.

Even the supposedly "ultra simple" "Natural Numbers" arithmetic began that way.

Only the work of communication, exemplification, and familiarization can change that.


The "Natural" Arithmetic that is "second nature" no doubt, to all of my readers, was once profoundly esoteric and sequestered, in its use, among a tiny minority of "elite" Indian and, later, of Arabic, arithmeticians.



Below is an instance of what I understand to be "communication, exemplification, and familiarization" that I hope will be found helpful.





Vignette on Dialectical, Pure-Qualitative Algebra versus Pure-Quantitative Algebra.

Quantity Unknowns: Solving equations in the "purely-quantitative algebras" is about finding (a) number(s) that fit(s) for the unknown-value symbol(s), or algebraic variable(s), in such an equation.

Quality Unknowns: Solving equations in the "purely-qualitative algebra" of the F.E.D. "First Dialectical Arithmetic" is about finding (a) categorial meaning(s) that fit(s) the unknown-value symbol(s), or variable(s), in such an equation.





SUBSTANTIATION: THE CONCRETE CONCEPTUAL CONTENT STANDING BEHIND THE VIGNETTE STATED ABOVE, AND CORROBORATING IT IN DETAIL.


Example(s):  General Method [of Solution of the F.E.D. First Dialectical Arithmetic's Algebraic Equations:  The F.E.D. "Organonic Algebraic Method"].


Background. The "meta-numerals" of the NQ arithmetic each stand for -- or can be assigned, in constructing a "dialectical [meta-]model", or "[meta-]model of a [specific] dialectic", to -- an ontological category; a kind-of-being, or <<genos>>-of-being, category, in the form of an "<<arithmos>>", or "number", of "<<monads>>", or "units", of a given kind, or <<genos>>.

A generic assigned dialectical "meta-numeral", such as q/X, denotes a non-amalgamative, ir-reducible "fraction" -- that is, a two-level, two-scale [qualo-]fractal -- consisting of a "higher" [more abstract; more general] scale, denoted by q, the generic quality symbol, or generic categorial qualifier symbol, and a "lower" [less abstract, more specific] scale, consisting of a specifier-qualifier, here X, signifying one single <<species>> that is implicitly "contained" in the [typically multi-<<species>>] <<genos>> denoted generically by q.

This "content-structure" of the NQ dialectical "meta-numerals" reflects that of the Platonian <<arche'>> of all <<dialektike'>>, namely, Plato's <<arithmoi>> [plural of <<arithmos>>] of <<eide>>-<<monads>>, his <<arithmoi eide-tikoi>>, a phrase which translates into English as "assemblages ["numbers"] of idea-units".



The "division" signified by the "separatrix", "separator-bar", or "division-bar" -- '/' -- in the "qualo-fractal fraction" -- 'q/X' -- does not signify the "division" operation native to pure-quantitative arithmetic.

On the contrary, it signifies the "division" operation native to pure-qualitative, dialectical "<<arithmos>>-etic", a kind of "division" which Plato called <<diairesis>>.

This "division-bar", or "<<diairesis>>-bar", functions as a "separator bar", or "delimiter bar", to distinguish, or mark-off the distinction between, two qualitatively different, qualo-fractal scales, or levels, of generalization -- of categorization -- that should not be indiscriminately mixed-up, e.g., the <<genos>> level/scale, and/from the <<species>> level/scale.

This "division" [of] X "into" q , or "of" q "by" X, does not "amalgamate", or "reduce", e.g., to any single value, because q/X signifies a qualitatively "inhomogeneous", or "heterogeneous", value:  q differs in quality, in categorial level/scale, from X; q and X are "qualitatively different" from one another, whereas 4 and 2 in the purely-quantitative fraction 4/2 signify that they are only quantitatively different from one another; that they are mutually homogeneous, mutually non-heterogeneous; of identical scale / level / kind, so that 4/2 does "reduce", or "amalgamate", to a single value -- in this case, to just 2.

The "meta-numeral" q/X thus signifies a minimal, two-<<eide>> Platonian-dialectical <<arithmoi  eide-tikoi>, which can be identified ['='] in the following ways --

q "over" X   =   <<genos>> quality "over" just one of its <<species>> [ordinal] qualities [in this unusual, <<arche'>>, case, "over" the ordinal "quantifier", denoted by X]   =   <<genos>>/<<species>>.

Just X could be substituted for q/X, but, in that case, only the "<<species>>-fic" ontological category would be explicitly asserted, and the <<genos>> category to / within which it "belongs" would be left implicit, elided.



A symbols-expression of the form [q/X]^2 means "q/X raised to the power 2", or "q/X in the 2nd degree", or "q/X squared", i.e., q/X "times itself", or "the [<<aufheben>>] operation denoted by q/X applied to itself".



Procedure. Once you determine what you think should be the <<arche'>>, or first/beginning, ontological category, call it q/A or A, in the dialectical progression of ontological categories that you are trying to model, the next step is to "square", or self-apply, that category, since its category-symbol also, algorithmically, denotes a specific <<aufheben>> operator/function:

A of A   =   A plus q/AA   =   A plus B.