**A Key Difference Between Ordinary,**"

**Purely-**"

__itative__*Quant***Algebra**,

**and F.**

*E***.**'

*D*.**s**"

**Purely-**"

__itative__*Qual***First**

*, that of the*__Dialectical__Algebra**N**

__Q__*, Part*

__Dialectical__Arithmetic**1**.

__: Throughout his blog entry, I will be using visible light spectrum color-order color-coding to call attention to__

**Note**__,__

**dialectical**__itative ordinalities: "__

**qual****R**

**O**

**Y**

**G**.

**BIV**", connoting the

__ities of__

**qual****First**

*-*,

**ness****Second**

*-*,

**ness****Third**

*-*,

**ness****Fourth**

*-*,

**ness****Fifth**

*-*,

**ness****Sixth**

*-*, and

**ness****Seventh**

*-*, respectively.

**ness**Dear Readers,

I often encounter, from discussants, questions such as that below, which I have

*"*ricized" for the purposes of this blog entry --

**gene**"The problem I see with the

**F**.

__.__

*E*__. dialectic at present is that it exists as an "elite" tool of the very few."__

*D*"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"?"

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

**INTRODUCTION**Even the supposedly "ultra simple" "

__atural Numbers" arithmetic began that way.__

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

The "

__atural" 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.__

**N**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****: Solving equations in the "purely-**

__Quantity Unknowns__**algebras" is about finding (a) number(s) that fit(s) for the**

__itative__*quant***known-value symbol(s), or algebraic variable(s), in such an equation.**

*un*__: Solving equations in the "purely-__

**Quality Unknowns****algebra" of the**

__itative__*qual***F**.

__.__

*E*__. "__

*D***First**

**Dialectical***is about finding (a) meaning(s) that fit(s) the*

**Arithmetic**"__known-value symbol(s), or variable(s), in such an equation.__

**un**__:__

**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**'*

**Algebraic Equations: The**

*s***F**.

__.__

*E*__.__

*D**"*].

**Organonic Algebraic Method**"**Background**. The "meta-numerals" of the

**N**

__arithmetic each stand for -- or can be assigned, in constructing a__

**Q***"*[

__dialectical__*]*

**-**__meta__*, or "[*

**model**"*]*

**-**__meta__*of a [*

**model**__fic]__

**speci***, to -- an*

**"**__dialectic__*; a*

**ontological category***of-being, or <<*

**kind**-*>>*

**genos***, in the form of an "<<*

**category***>*

**arithmos***"*, of "<<

**number**"*>*

**monads***"*, of a given

__"__**units***, or <<*

**kind***>*

**genos**A

*ric assigned*

**gene**

**dialectical***"*, such as

**-**__meta__**numeral**"**, denotes a**

__q__/X*amalgamative,*

__-__**non***reducible "fraction" -- that is, a two-level, two-scale [*

**-**__ir__*]fractal -- consisting of a "higher" [more abstract; more*

__o-__**qual***ral] scale, denoted by*

**gene**__, the__

**q***ric*

**gene**__ity symbol, or__

**qual***ric categorial*

**gene**__ifier symbol, and a "lower" [less abstract, more__

**qual**__fic] scale, consisting of a__

**speci**__fier-__

**speci**__ifier, here__

**qual****X**, signifying one single <

*>> that is implicitly "contained" in the [typically multi-<<*

**species**

**species***>> denoted generically by*

**genos****.**

__q__This "content-structure" of the

**N**

**Q***reflects that of the Platonian <*

**"**__dialectical__**meta**-**numerals**"*>> of all <<*

**arche'**__>__

**dialektike'***>> [plural of <<*

**arithmoi***>>] of <*

**arithmos**__>>-<<__

**eide***>*

**monads**

**arithmoi***>>, a phrase which translates into English as*

__-__**eide****tikoi***"*[

**assemblage**__s__*"*] of

**numbers**"*.*

__-__**idea****units**"The "division" signified by the "separatrix", "separator-bar", or "division-bar" -- '

**/**' -- in the

*"*o-fractal fraction" -- '

__qual__

__q__/X__signify the "division" operation native to pure-__

**not**__itative arithmetic.__

**quant**On the contrary, it signifies the "division" operation native to pure-

__itative,__

**qual**__"<<__

**dialectical**__os>__

**arithm***-*, a kind of "division" which Plato called <

**"**__etic__*>>.*

**diairesis**This "division-bar", or "<<

*>*

**diairesis**__itatively different,__

**qual**__o-fractal scales, or levels, of__

**qual***ralization -- of categorization -- that should*

**gene**__be indiscriminately mixed-up, e.g., the <<__

**not***>*

**genos***>*

**species**This "division" [of]

**X**"into"

**, or "of"**

__q__**"by"**

__q__**X**, does

__"amalgamate", or "reduce", e.g., to any single value, because__

**not****signifies a**

__q__/X__itatively__

*qual**"*homogeneous", or

**in***"hetero*geneous", value:

**differs in**

__q____ity, in categorial level/scale, from__

**qual****X**;

__and__

**q****X**are

*"*itatively different" from one another, whereas

__qual__**4**and

**2**in the purely-

__itative fraction__

**quant****4/2**signify that they are only

*itatively different from one another; that they are mutually*

**quant***homo*geneous, mutually

*geneous; of identical scale / level /*

**-hetero**__non__*, so that*

**kind****4/2**

__"reduce", or "amalgamate", to a single value -- in this case, to just__

**does****2**.

The

*"*

__-__**meta****numeral**"**thus signifies a minimal, two-<<**

__q__/X__>__

**eide***<<*

**Platonian**-**dialectical**

**arithmoi**

__-__**eide****tikoi**__'] in the following ways --__

**=**__"over"__

**q****X**

__<<__

**=****>**

*genos*__ity "over" just one of its <__

**qual***>> [ordinal]*

**species**__ities [in this unusual, <<__

**qual***>>, case, "over" the ordinal*

**arche'***"*ifier", denoted by

__quant__**X**]

__=__**>>**

*genos**>>*

**species**Just

**X**

__be substituted for__

**could****, but, in that case, only the "<**

__q__/X__es>>-__

**speci***ontological category would be*

**fic**"__plicitly asserted, and the <<__

**ex***>*

**genos**__plicit, elided.__

**im**A symbols-expression of the form

**[**means "

__q__/X]^2**raised to the power**

__q__/X**2**", or "

**in the**

__q__/X**2**nd degree", or "

**squared", i.e.,**

__q__/X**"times itself", or "the [<<**

__q__/X*>>]*

**aufheben**__denoted by__

*operation*

__q__/X*.*

__itself__"**Procedure**. Once you determine what you think should be the <

*>>, or*

**arche'****first**/

**beginning**, ontological category, call it

**or**

__q__/A__, in the__

**A***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*

__dialectical__progression of ontological categories__fic <__

**speci***>> operator/function:*

**aufheben****.**

__A__of__A__=__A__plus__q__/AA =__A__plus__B__
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