Tuesday, September 20, 2011

What ARE the F.E.D. "Dialectical Meta-Numbers"?


What are the meta-numbers of the F.E.D. First Dialectical Arithmetic, NQ_ --

My Take on an Intuitive, Dialectical Understanding of the NQ, and of how they emerge from the N.




Dear Readers,


What is an NQ "dialectical meta-number", such as q/1, q/2, q/3, or q/4, and what is so "dialectical" about it?


First of all, let us state clearly that the symbol q in the above does not stand for a "variable quantity", or "quantitative variable", such as, e.g., q -- without the underscore -- would stand for in ordinary algebra.

The q used above is not a "quantifier":  it is a "qualifier".

It stands for a [generic] quality, not for a generic [variable] quantity.

But for what quality?


[Note: We use, in this blog entry, below, our usual textual color-coding, in visible-light-spectrum color-order, to call attention to categorial ordinalities -- e.g.,
A, B, C, D, E, F, ...]


The historical <<arche'>> of all Occidental dialectic is Plato's <<arithmoi eidetikoi>>, that is, Plato's "assemblages of qualitative, ontological <<eide>>-<<monads>>", or "ensembles of qualitative units in the form of idea-units": "numbers" of qualitative units in the shape of idea-units; of ontological categories.

<<Dialektike'>>, for Plato, was essentially a matter of "ideo-taxonomy", "ideo-classification", or "ideo-systematics", that is, of discovering the ramified, genetic -- inverted, "qualo-fractal" -- "tree of fundamental concepts" that, per Plato, undergirded all of human experiential reality, and to which the empirical-physical taxonomy of the external-to-mind world of experience could only provide rough clues.

An [ideo-]<<genos>> means a kind of [idea-]being, for example, the [idea-]category of "animals" might name an [ideo-]<<genos>>.  The term [ideo-]<<gene>> is the plural of the term [ideo-]<<genos>>, for example "animals, vegetables, and minerals are category-names for three qualitatively-distinct [idea-]<<gene>>".  An [ideo-]<<species>>, relative to its [ideo-]<<genos>>, means a more "speci-fic" kind of [idea-]being, one that is implicitly, but not [usually] explicitly, <<aufheben>>-"contained" in its [ideo-]<<genos>>; its "<<gene>>-ralization"; its category of higher "<<gene>>-rality". The term [ideo-]<<species>>, taken as a singular term, is also its own plural term as well.

That is, <<Dialektike'>>, per Plato, is a matter of systematically organizing/dividing the qualitative, non-sensuous idea-objects, perceived by the "minds' eyes" of human minds, into "kinds" [into idea-<<gene>>], and into their <<aufheben>>-connected "sub-kinds" [idea-<<species>>], etc.  Of course, Plato did not use the -- later -- German word <<aufheben>> to characterize his <<arithmoi eidetikoi>> inverted tree content-structure. Nevertheless, the relationship of, e.g., the heterogeneous multiplicity of <<species>> categorial units, or of <<species>> <<idea>>-units, or of <<species>> <<eide>>-units, to their one-level-of-generalization-higher/one-scale-higher, or next-up layer-of-<<eide>>-units/"qualo-fractal scale" <<genos>> categorial unit, <<genos>> <<idea>>-unit, or <<genos>> <<eide>>-unit, is an <<aufheben>>, "meta-<<monad>>-ization" relationship, i.e., each <<genos>> <<eide>>-unit is a "meta-unit" of its <<species>> <<eide>>-units, each <<genos>> <<eide>>-unit being implicitly made up out of the qualitatively heterogeneous multiplicity of its <<species>> <<eide>>-units, so that each <<genos>> <<idea>>-unit is concurrently a conservation, an elevation, and a determinate negation [via the removal, by abstraction, or by "generalization", of all <<species>>-specific determinations of its <<species'>> <<idea>>-units in the formation of that <<genos>> <<idea>>-unit, which explicitly contains only those determinations which are shared alike by all of its <<aufheben>>-"contained" <<species'>> <<idea>>-units.  Moreover, e.g., <<species>> <<idea>>-units, and, indeed, <<idea>>-units at all "qualo-fractal scales" / levels of "generalization", of the <<arithmoi eidetikoi>> "qualo-fractal" content-structure, may come in triads, with a "thesis" <<species>> category-unit, followed by an "anti-thesis" <<species>> category-unit, followed by a "synthesis" <<species>> category-unit, as the three units "coming under", or "feeding-in to", each single <<genos>> category meta-unit, thus resulting in a very regular "qualo-fractal" "self-similarity structure" to the whole inverted tree of the thus dialectical <<arithmoi eidetikoi>> "meta-fractal".], viz. --


"..........STRANGER: Well, now that we have agreed that the kinds [the «gene» -- M.D.] stand toward one another in the same way as regards blending, is not some science needed as a guide on the voyage of discourse, if one is to succeed in pointing out which kinds are consonant, and which are incompatible with one another – also, whether there are certain kinds that pervade them all and connect them so that they can blend, and again, where there are divisions [separations], whether there are certain others that traverse wholes and are responsible for the division?

...........THEAETETUS: Surely some science is needed – perhaps the most important of all.

...........STRANGER: And what name shall we give to this science? Or – good gracious, Theaetetus, have we stumbled unawares upon the free man’s knowledge and, in seeking for the Sophist, chanced to find the philosopher first?

...........THEAETETUS: How do you mean?

...........STRANGER: Dividing according to kinds, not taking the same formeidos»; «idea» -- M.D.] for a different one or a different one for the same – is not that the business of the science of dialectics?

...........THEAETETUS: Yes.

...........STRANGER: And the man who can do that discerns clearly one form everywhere extended throughout many, where each one lies apart, and many forms, different from one another, embraced from without by one form, and again one form connected in a unity through many wholes, and many forms, entirely marked off apart. That means knowing how to distinguish, kind by kind, in what ways the several kinds can or cannot combine.

...........THEAETETUS: Most certainly.

...........STRANGER: And the only person, I imagine, to whom you would allow this mastery of dialectic is the pure and rightful lover [the «philo» -- M.D.] of wisdom [of the «sophia»: of skill; deeper knowledge -- M.D.]."

[Source: E. Hamilton, H. Cairns, editors, Plato: The Collected Dialogues, Princeton University Press [Princeton, New Jersey: 1989], pp. 998-999, Sophist, 253b 254d [emphasis added by M.D.]].



Now, Marxians recognize Plato's view, not only as a philosophy, but as an ideology, in Marx's sense.

Yet also, as with Hegel's capitalist philosophy/ideology -- and as with the bourgeois ideological-science of "classical political economy", especially in the work of David Ricardo -- for Marx, in the Platonian ideology, F.E.D. recognized a "rational kernel", a valuable seed of real science -- in the form of Plato's <<arithmoi eidetikoi>> conception of dialectic -- which can be, first, extricated, and, then, progressed anew, by means of the Marxian method of dialectical, immanent critique.

This is especially so because the Platonian <<mentalite'>> arose in a social milieu not so pervaded by capital-relation-inherent the law of value as is the modern, post-Occidental Dark Ages <<mentalite'>>, and was, hence, able to more easily discern aspects of the concept of <<arithmoi>> -- of "numbers" of [qualitative] things -- which are exceedingly difficult for the presently-prevailing, law-of-value-pervaded human <<mentalite'>> to "see".


The positive fruition of the F.E.D. immanent critique of Plato's <<arithmoi eidetikoi>> ideology of dialectics, is a more explicit version of the Marxian paradigm of "psycho-historical materialism", and of the Marxian "meta-science" of "psycho-historical dialectics", which locate the actuality of Plato's phantasy dialectical <<eide>>, or <<ideas>> -- which Plato "located" in his delusory, mystical, meta-physical, transcendental, eternal, immutable, Parmenidean causal heaven -- in the human-historical realm of the human "memes-pool", 'memenome', or '''phenome''', and, principally, in the collectively, culturally created [both deliberately and unconsciously], and shared, dynamical [psycho-]historical-materiality of human language.

Thus, once the dynamicity, and the phenomic location, of "the <<eide>>" is accepted, F.E.D. has shown that it has no qualms about stating that every instance of dialectic still "contains" the Platonian <<arche'> of all dialectic: that every dialectic is an <<arithmos eidetikos>>: a systematic assemblage of [at least two; of two or more] categorial units.


That is also what the NQ are.

That is, most fundamentally, why they are "dialectical".

Each NQ "dialectical meta-number" is an <<arithmos eidetikos>> in miniature.

Each NQ "dialectical meta-number" is explicitly a minimal <<arithmos eidetikos>>, while each N "Natural Number" is so only implicitly, as we will see.

[Note: herein, '
=' means "equals by definition"].

Using n as a ["purely-quantitative"] variable, to stand for any one of the "Natural" Numbers --

 N...=...{ 1, 2, 3, 4, . . .}

-- then q/n is also a variable, but a qualitative variable, standing for the generic "dialectical meta-number", i.e., standing, in general, for any one of the specific NQ "meta-numbers"--

{ q/1, q/2, q/3, q/4, . . .}...=...NQ...=...{ q/n }.


It is typographically difficult, but is more forthcoming conceptually, to render this generic "meta-number", q/n, and all specific NQ meta-numbers", such as q/2, q/3, q/4, . . ., in "qualitative fraction / qualo-fract-al" form, as --

...................................................................................q
.............................................................................________
...................................................................................n


-- because each of the specific NQ "meta-numbers" is of the <<aufheben>> qualo-fractal / qualo-fraction-al form --

..................................................................................<<genos>>
...............................................................................______________
................................................................................<<species>>


-- that is, the "space", or "set", of the NQ "dialectical meta-numbers" may be decoded as --




..................<<genos>>.........<<genos>>........<<genos>>.........<<genos>>
NQ...=...{...______________,..______________,..______________,.._______________, . . .}
...............<<species>> 1..<<species>> 2..<<species>> 3..<<species>> 4




-- wherein the <<genos>>, in every case above, is that of "ordinal-quality-in-general", or of "qualitative ordinality-in-general"; the <<genos>> of generic ordinality, or of generic discrete, consecutive order.


So, the q/n that constitute the set or space of NQ are all ordinal qualifiers; order qualifiers.

They each signify a specific quality of consecutive, discrete order that can be used for modeling, mathematically, algebraically, a categorial progression, a systematic progression of categories, in which each [successor] category dialectically-logically "follows from" its predecessor categor(y)(ies).

Thus, q/n generically denotes the quality of "nth-ness", and the NQ may thus be re-written as --

NQ...=

{ q/n }...=

{ q/1, q/2, q/3, q/4, . . .}...=

{ first-ness, second-ness, third-ness, fourth-ness, . . .}...=

{ 1st-ness, 2nd-ness, 3rd-ness, 4th-ness, . . .}

-- and --

q/1 means/stands for the [ordinal] quality of/"qualifier" for "first-ness",

q/2 means/stands for the [ordinal] quality of/"qualifier" for "second-ness",

q/3 means/stands for the [ordinal] quality of/"qualifier" for "third-ness",

q/4 means/stands for the [ordinal] quality of/"qualifier" for "fourth-ness", etc.


-- all in the context of some type of categorial progression, such as a systems-progression.


Each NQ meta-number's "meta-numeral" consists of exactly two <<eide>>, two categories, and, thus, qualifies as a minimal <<arithmos eidetikos>>, and, thus, as "dialectical" in that -- <<arche'>>-ic -- sense.


That is, each NQ "meta-numeral" consists of a more general, <<genos>> category, denoted by its "qualitative numerator", q, denoting the more abstract category of "ordinality-in-general", and a more specific, <<species>>, specifier category, denoted by its "quantitative denominator", n, from N, which tells the particular quality of nth-ness that q/n denotes.


The
N numbers are all quantitatively unequal to one another, i.e., 1 < 2 < 3 < 4 < . . ., or 1 is less than 2 is less than 3 is less than 4 is less than . . ..

The NQ "meta-numbers" are all non-quantitatively unequal to one another, i.e., are qualitatively unequal to one another, viz. --

q/1 is neither less than, nor equal to, nor greater than q/2;

q/2 is neither less than, nor equal to, nor greater than q/3;

q/3 is neither less than, nor equal to, nor greater than q/4, etc., e.g. --


the quality of "first-ness" is neither less than, nor equal to, nor greater than the quality of "second-ness";

the quality of "
second-ness" is neither less than, nor equal to, nor greater than the quality of "third-ness";

the quality of "
third-ness" is neither less than, nor equal to, nor greater than the quality of "fourth-ness", etc.


Thus, the NQ "meta-numbers" transcend the "trichotomy principle" of ordinary, "purely-quantitative" arithmetics, and extend that principle into an expanded, "tetrachotomy principle" for its "purely-qualitative arithmetic", whereby NQ_ thus constitutes a qualitatively antithetical contra-system to the N_ system of arithmetic.


True, the "dialectical-ness" of the q/n grows greater as further cumulative layers of generic interpretation are laid down.

That is, the "Seldon Functions" divide/punctuate their generic ontological-categorial progression into "stages" or "epochs", e.g., for the "Dyadic Seldon Function" - -

epoch 0, q/1 only is possible / possibly extant/existent;

epoch 1, q/1 + q/2 are [each][both] possible / possibly extant/existent;

epoch 2, q/1 + q/2 + q/3 + q/4 are [each][all] possible / possibly extant/existent, and so on; . . .

-- and then, next, the following "interpretational" extension of the ordinal meaning of the successive NQ qualifiers adds even more of [traditional] "dialecticality" --

q/1 connotes the 1st <<arche'>> thesis [systematic dialectic], or 1st <<arche'>> <<physis>> [historical dialectic];

q/2 connotes the 1st contra-thesis [systematic dialectic], or 1st meta-<<physis>> [historical dialectic];

q/3 connotes the 1st full uni-thesis [systematic dialectic], or 1st full uni-<<physis>> [historical dialectic];

q/4 connotes the 2nd contra-thesis [systematic dialectic], or 2ndmeta-<<physis>> [historical dialectic], . . .

-- but the categorial-progressions' ordinality meanings of the q/n remains the root of their "dialecticity".


[ Note:  For every w in W = { 0, 1, 2, 3, . . . }, the generic "Dyadic Seldon Function" is --

 2|-|-|w....=....2|-|-|0^(2^w)....=....[ q/1 ]^(2^w)

-- and the generic "Triadic Seldon Function" is --

3|-|-|w....=....3|-|-|0^(3^w)....=....[ q/1 ]^(3^w) ].


The full generic dialectical interpretation of the NQ "meta-numbers", using the "Dyadic Seldon Function", thus becomes, for "stages", or s values, or for "epochs", or t values:  0, 1, 2, and 3 [using '[---)' and '[--->' and '[---]' to denote the relation "interprets", for systematic dialectics, and for historical dialectics, and for generic dialectics, respectively] --


stage s = 0: <<arche'>>-thesis [---) q/1, per the generic "Dyadic Seldon Function" for systematic dialectics, )-|-(s....=....)-|-(0^(2^s);

stage t = 0: <<arche'>>-<<physis>> [---> q/1, per the generic "Dyadic Seldon Function" for historical dialectics, >-|-<t....=....>-|-<0^(2^t);



stage s = 1:  <<arche'>>-thesis & 1st contra-thesis [---) q/1 + q/2;

stage t = 1:  <<arche'>>-<<physis>> & 1st meta-<<physis>> [---> q/1 + q/2;



stage s = 2:  <<arche'>>-thesis & 1st contra-thesis & 1st full uni-thesis & 2nd contra-thesis
[---) q/1 + q/2 + q/3 + q/4;

stage t = 2:  <<arche'>>-<<physis>> & 1st meta-<<physis>> & 1st full uni-<<physis>> &
2nd meta-<<physis>> [---> q/1 + q/2 + q/3 + q/4;



stage s = 3:  <<arche'>>-thesis & 1st contra-thesis & 1st full uni-thesis & 2nd contra-thesis & 1st partial uni-thesis & 2nd partial uni-thesis & 2nd full uni-thesis & 3rd contra-thesis

[---) q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8;


stage t = 3:  <<arche'>>-<<physis>> & 1st meta-<<physis>> & 1st full uni-<<physis>> & 2nd meta-<<physis>> & 1st partial uni-<<physis>> & 2nd partial uni-<<physis>> & 2nd full uni-<<physis>> & 3rd meta-<<physis>>

[---> q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8.


We can get a generic, one-take overview picture of the "dialectical combinatorics" of all of the above, lengthier, expressions, if we use letter-characters, in their alphabetical order, using "A" to stand for the <<arche'>> ontological category, and using subsequent, say nth, letter-characters, to stand for the (n-1)st "contra-thesis", or "meta-<<physis>>", ontological categories, and "subscripted", or "denominated", combinations of these "thesis" and "contra-thesis" letter-characters to stand for the full and the partial "uni-thesis", or "uni-<<physis>>", ontological categories, as follows:

|-|-|3....=....A.....+....B....+....q/BA..+..C....+.....q/CA..+..q/CB..+..q/CBA..+..D

[---].................q/1..+..q/2..+..q/3.....+....q/4..+..q/5....+.....q/6....+.....q/7......+......q/8.



But how may we best conceive the emergence, or "followership", of the NQ from the N?


Some Clues:

For n in N, the q of q/n is a generic "meta-<<monad>>", or "meta-unit", of the <<monads>>/units that make up the <<arithmos>>-of-<<monads>>/"number"-of-units named n.

q/1 is a [self-]<<aufheben>> of 1.

q/2 is a [self-]<<aufheben>> "[self-]meta-<<monad>>-ization" of the <<arithmos>>-of-<<monads>>/units "1+1"  =  2.

q/3 is a [self-]<<aufheben>> "[self-]meta-<<monad>>-ization" of the <<arithmos>>-of-<<monads>>/units "1+1+1"  =  3.

. . .

But how so?



TO BE CONTINUED



Regards,

Miguel

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