Monday, October 29, 2012

'Evoluteness' & 'Cumulativity' of Dialectical Progressions

Full Title:

Evolute-nessandCumulativityof Dialectical, Ontological-Categorial Progressions.

Dear Readers,

The text below is an excerpt from the seventh section of Part II. of the essay entitled --

The Goedelian Dialectic of the Standard Arithmetics --

recently enhanced in a new version.

The Evolute-ness and Cumulativity of Dialectical, Ontological-Categorial Progressions.

The dialectical ‘meta-models’ rendered by ‘Dialectical Equations’ formulated using the NQ_#_ dialectical arithmetic, all take the form of categorial progressions -- of a stepwise-advancing, all-predecessors-containing, sequence of ‘“series”’ [of inhomogeneous ‘“sums”’; of ‘qualitative superpositions’], via ‘category-symbolizing ideograms’, or categorograms -- for whatever «species» of the «genos» of Dialectics-in-general, in whatever «species» of ‘the dialectic of the dialectic itself’, be it Systematic Dialectics, Historical Dialectics, ‘Meta-Systematic Dialectics’, or ‘Psychohistorical Dialectics -- that ‘meta-model’ resides.

¿But what is a categorial progression?

It is an advancing, non-amalgamative ‘“sum”’ of ‘‘‘ontological categories’’’ symbols; i.e., of “kind of being” categories’ symbols, wherein the symbol-represented “kinds of being” include, not only tangible, physical beings, but ‘human-mental beings’ -- ‘ideo-beings’; ‘idea-objects’; ‘idea-material’ -- as well.

¿Are there any common, everyday examples we can point to which involve, or instantiate, “progressions of ontological categories”?


Some of the best ‘exhibitors’ of such “advancing category sums” are books, especially books whose contents involve ‘ideo-systematics’, and ‘ideo-taxonomics’, such as Hegel’s «Logik» and Marx’s «Kapital».

Consider the tables of contents of such books. Volume, Part, Section, Chapter, and sub-chapter titles in such books -- such as Quality [Determinateness], Being, Nothing, Becoming, Determinate Being, Being-for-self, Quantity, and Measure, or such as Commodities, Money, and “[The General Formula for ]Capital”, are also the names of the central, principal ‘ideo[-physio]-ontological’ categories that ‘‘‘contain’’’, capture, classify, and comprehend the ‘ideo-meta-anatomy’, and the ‘ideo-meta-physiology’ of the universes of discourse that these books attempt to systematize and to comprehensively theorize.

Such tables of contents thus spell out, as a whole, frozen at completion, the categorial progression that a reader experiences progressively as that reader reads through the contents of such books from start to finish.

Such tables do so by listing those category-names, which also serve as division-titles, in the exact order of their presentation to such a reader.

A reader is not expected to entirely forget the contents of every chapter that the reader has read in such a book as soon as that reader reads it, as if the reader had ingested some [ Transient Global Amnesia ] TGA-inducing statin drug.

Yes, the previous chapters read are expected to fade in the reader’s mind, relative to the vividness of the chapter presently being read, and to merge into a background which frames the foreground of the chapter presently being read.

But the full comprehension of present reading depends upon and presupposes the remembrance of material already read, even given a relative attenuation of that remembrance.

The message of such a book is a cumulative one.

The full communication of its content to the reader depends upon a partially subconscious semantic summation, or qualitative superposition, of the entire material already read, in the reader’s mind, until the end of the reading, when the full cumulum of its meaning should have been conveyed to that reader.

Each new chapter read is supposed to ‘‘‘add’’’ itself to the cumulative effect of all of the previously-read chapters, in the mind of the reader; to ‘cumulate’ in the reader’s mind, and to superimpose itself -- its new content, its incremental meaning -- on the retained content of each of the chapters read before it.

Or, as with a lecture, the import of the whole previous discourse heard is supposed to ‘cumulate’ in the mind of the hearer, and each new enunciation is supposed to operate upon, in the hearer’s memory, and to re-clarify, each element of that previous heard discourse.

Please consider, in the light of the examples given above, the following description of dialectical categorial progressions from the work of Tony Smith.

Prof. Smith wrote as follows regarding “Systematic Dialectics”, the categorial progression method of systematic presentation of theories advanced by Hegel:

“Hegel attempted to provide an immanent ordering of the basic categories ...

To see this we have first to consider what a category is.

It is a principle (a universal) for unifying a manifold of some sort or other (different individuals, or particulars).

A category thus articulates a structure with two poles, a pole of unity and a pole of differences.

In Hegelian language this sort of structure, captured in some category, can be described as a unity of identity in difference, or as a reconciliation of universal and individuals.

From this general notion of a category we can go on to derive three general types of categorial structures.

In one the moment of unity is stressed, with the moment of differences implicit.

In another the moment of difference is emphasized, with the moment of unity now being only implicit.

In a third both unity and differences are made explicit together.

[Tony Smith, The Logic of Marx’s CapitalReplies to Hegelian Criticisms, SUNY Press [NY:    1990], pp. 5-8].

Thusly, Dr. Smith notes what we term a ‘trans-Platonian’ «arithmos eidetikos», a dialectical ‘ideo-systematic cumulum’.

This «arithmos» is an ‘ideo-taxonomy’, in which the «genos» category-«monad» of ‘“categories-in-«gene»-ral”’ is also the «aufheben» meta-«monad»-ization of its 3 «species» category-[sub-]«monads» -- «species» alpha., unity-biased categories; «species» beta., difference(s)-biased categories, and; «species» gamma., unity/difference(s) combined-emphasis categories. This «arithmos eidetikos» can be modeled by a Triadic Seldon Function --

)-|-(1    =    (ac)^(3^1)   =   ( ( ac ~+~ bc ) ~+~ gc )

-- which forms a ‘tri-oppositional synthesis-sum’ for this dialectical theory of the tripartite ‘ideo-ontology’ of categories.

Prof. Smith notes next how these three category-kinds can be sequenced to constitute a dialectical logic of ‘tri-categorial oppositions’:

“Hegel’s next claim is that there is a systematic order immanently connecting these three categorial structures.

A structure of unity in which differences are merely implicit is simpler than one in which differences are explicitly introduced; and one in which both unity and differences are explicit is yet more complex still.

Similarly, the first sort of structure is the most abstract, while the other structures are successively more [F.E.D.:  thought-]concrete.

Yet another way of speaking about the immanent connections here is through the idea of a dialectical contradiction.

Hegel’s views on contradiction have been quite controversial.

But at least in the context of constructing a systematic theory of categories he appears to have meant something fairly straightforward.

If a category is in general a principle that unifies a manifold, then if a specific category only explicates the moment of unity, leaving the moment of difference implicit, then there is a “contradiction” between what it inherently is qua category (a unifier of a manifold) and what it is explicitly (the moment of unity alone).

Overcoming this contradiction requires that the initial category be “negated” in the sense that a second category must be formulated that makes the moment of difference explicit.

But when this is done the moment of difference will be emphasized at the cost of having the moment of unity made merely implicit.

Once again there is a contradiction between what a category inherently is and what it is explicitly.

Overcoming this contradiction demands that the second sort of category also be negated, and replaced with a category in which both poles, unity and difference, are each made explicit simultaneously.

Hegel is well aware that “contradiction” and “negation” are not being used here in the sense given to them in formal logic.

Following a tradition that goes back to Plato, he asserts that in the above usage “contradiction” and “negation” are logical operators for ordering categories systematically, as opposed to logical operators for making formal inferences.

The logic with which we are concerned here is dialectical logic.”

[Tony Smith, The Logic of Marxs Capital, ibid., p. 6].

Thusly, Prof. Smith notes the ‘qualitative gradient’ of increasing complexity and of increasing ‘thought-concreteness’, or ‘thought-determinateness’ [‘determinations-richness’] that runs through these three kinds of categories when arrayed in their native order.

He also so narrates the dialectical logic of ‘‘‘categorial dialectical contradiction’’’ -- one of categorial insufficiency and incompleteness -- and one of the categorial ‘intra-duality’ of unity versus differences, that moves our minds from alpha., to beta., to gamma., and beyond.

Prof. Smith notes next how sequences of these three category-kinds can be combined in layers, [‘qualo-fractal’] scales, or levels, to express a categorial-progression theorization, or comprehension, of a given [sub-]totality of human experience:

“Since a category of unity-in-difference on one level can itself prove to be a category of simple unity from a higher-level perspective, thereby initiating another dialectical progression from unity through difference to unity-in-difference, we can construct a systematic theory of categories by employing the dialectical method.

In this sort of theory we move in a step-by-step fashion from simple and abstract categories to those that are complex and [F.E.D.:  thought-]concrete, with dialectical logic providing the warrant for each transition ...

At the conclusion of the linear progression of categories we once again arrive at the initial starting point. But it has now been apprehended in thought.

If dialectical logic is rigorously adhered to, the move from one category to the next is not ad hoc.

The linear progression from a category of immediate unity to one of difference, and from there to a category of unity-in-difference, is not a mere formal schema imposed by Hegel externally.

It is instead “the absolute method. . .[F.E.D.: which] does not behave like external reflection but takes the determinate element from its own subject matter, since it is itself that subject matter’s immanent principle and soul.”

In this manner the object realm of [F.E.D.:  “chaotic” [cf. Marx, Grundrisse]] experience has been [F.E.D.:   ‘ideo-systematically’ / ‘ideo-taxonomically’/ ‘well-orderedly’] reconstructed in thought [F.E.D.:    i.e., comprehensively and ‘comprendingly’ theorized].

[The Logic of Marx’s Capital, ibid., pp. 7-8].

But why do we bother at all, in Systematic Dialectics, i.e., in the dialectical theory-presentation method, with these known inadequacies of defective category-kinds alpha., to beta?

Why don’t we just jump from the gamma. of one level or scale of the theory to the gamma. of the next such level or scale?

Or, why don’t we just jump once, to the gamma. of the highest scale of the theory, and be done with it?

The reasons are pedagogical.

Intelligibility for the learner is lost with such jumps.

If you are not already a physicist, just try jumping to the penultimate or ‘culminant’ chapter of a modern physics textbook, without having read anything in the earlier chapters.

If you’re like most, you won’t comprehend very much of modern physics from that chapter alone.

New categories -- new concepts, and their definitions -- need to be introduced gradually, and with their introduction motivated, in part, by the need for them that is revealed by revealing the inadequacies of the old, preceding categories/concepts/definitions.

Similar incomprehension would arise if we started kindergartner’s learning of arithmetic with C or H, instead of with N.

Now, to F.E.D.’s reading of Prof. Smith’s account, extracted above, there is an ambiguity about what, from our perspective, is a key issue regarding the categorial progression method of theory formulation/presentation.

The following question poses this issue most directly:

¿Is a categorial progression best described by a ‘sequence of singletons’? --

ax   ----)   bx   ----)   gx

-- or --

¿Is a categorial progression best described by a ‘sequence of series/of non-amalgamative sums’? --

ax   ----)  ax ~+~ bx   ----)  ax ~+~ bx ~+~ gx   =

xqa ~+~ xqaa ~+~ xqba

-- wherein all previously-emerged categories are «aufheben»-“conserved”, ‘‘‘externally’’’ and additively, as well as ‘‘‘internally’’’ [notated ‘subscriptally’] -- implicitly conserved “inside”/as-“contained”-in[-by], later-to-emerge categories.

Let’s consider an early phase of ‘‘‘the dialectic of nature’’’, for example, the self-conversion of the early sub-atomic “particles” ‘physio-ontology’, s ..., into that plus a new, ‘plasmic’, ‘‘‘ionized atoms/naked atomic nuclei”’ ‘physio-ontology’, a  =  qss.

We have therein a sub-phase of the ‘‘‘formal subsumption’’’ of sub-atomic “particles” by atomic nuclei, called [primordial] “cosmological nucleosynthesis”, of proton-«monads» and neutron-«monads» «aufheben»-self-meta-«monad»-izing into, e.g., ionic, Helium-plasma atomic-nuclei. Next, we have a sub-phase of ‘‘‘real subsumption’’’ of sub-atomic “particles” by atomic nuclei, called [primordial] “stellar nucleosynthesis”.

In this sub-phase, those hybrid bodies/processes that we call “first generation stars”, made up out of both atomic nuclei, e.g., Helium ions, and, e.g., proton sub-atomic “particles” [i.e., ‘plasmic’, ionized Hydrogen atoms] convert, e.g., proton and neutronsub-atomic “particles’ into, e.g., even more Helium-ion atomic nuclei, and, later, convert Helium atomic nuclei into atomic nuclei of higher atomic «species», in the cores of these first generation stars.

Note that, even in this ‘‘‘real subsumption’’’ phase, sub-atomic matter does not disappear from the universe. Masses of matter organized only up to the sub-atomic level, but no higher, e.g., unbound protons, continue to coexist with rising, but still much smaller, masses of matter organized up to the ionic-atomic level, but no higher.

Thus, a ‘sequence of series’ description --

s   ---->   s + a   ---->   s + a + qas ...   =   qs + qss + qas ...

-- is more apt to the qualitative data of the successive phases of this growing ‘physio-ontological’ content of the cosmos in general, and of this phase of ‘universal meta-evolution’ in particular, than is a ‘sequence of singletons’ description --

s   ---->   a   ---->   qas.

Likewise, in the kind of example already considered at the outset of this sub-section, of an audience’s reception of a lecture-presentation, and of a reader’s reception of a book-presentation, the content currently being encountered should eclipse, but should not erase, the content encountered earlier.

The whole advancing content should ‘cumulate’ in the reader’s mind.

E.g., Chapter I. of Marx's «Kapital», entitled “Commodities”, should not be erased from the reader’s mind in and by reading Chapter III., entitled “Money...”, and, likewise, the ‘‘‘impressions’’’ left upon the reader’s mind after reading both the “Commodities” and “Money” Chapters should not be lost-to-mind, but, on the contrary, should inform, be ingredient in, the reading of Chapter IV., entitled ‘‘‘...«Kapital»...’’’, like this --

...C«K»...  ----)   ...C«K»... ~+~ ...M«K»...   ...----)...   ...C«K»... ~+~ ...M«K»... ~+~ ...K«K»...

-- not, as if reading under the influence of an ingested substance such as some TGA-inducing statin drug, like this --

...C«K»...   ----)   ...M«K»...   ...----)...   ...K«K»... .

We at F.E.D. term the former an evolute process’, after the name used by, e.g., paleontologists to characterize the shell of a marine organism, in which successive whorls of that shell rise higher with each turn, thus rendering all whorls visible concurrently to a sidewise view of that seashell.

We term the latter a convolute process’, after the corresponding seashell-type, in which succeeding whorls do not rise, and hence cover-up all preceding whorls, hiding them from a sidewise view.

The evolute-tion’ description is codified in the double-conservation «aufheben» evolute product rule, axiom §9 of the NQ_#_ axioms-system [see the sub-section immediately after this one, for a full rendering of the core axioms of NQ_#_ ].

The empirical arguments set forth above for modeling dialectic evolute-ly’ could suffice.

However, it is interesting to note that Hegel is repeatedly unequivocal about the evolute-ness’ of dialectic [although, of course, not using the term ‘evolute’].

For example, in a series of lectures on the «Logik», noted down by his son, and recently published in English translation, Hegel stated:

The first determination [F.E.D.:  first «speci»-fication] is immediate, while the second one constitutes the sphere posited in its differentiation from the first.

Within every simple first determination, [e.g., ground,] what is determinately different from it [, e.g., the consequence of the ground] is at once also present, but it is at first present without being explicitly posited.

In the second determination, finitude [and with it contradiction] again enters.

The third determination is the unity of the first and the second, in which the contradiction is resolved. ...

The progression is as follows.

The beginning is simple
, immediate. ... Every newly emerging concept [F.E.D.:  category] is more concretely determinate than its predecessor.

We are always carrying everything that went before along with ourselves into what is new, but everything prior is, within what is new, put in its determinate place.

, in what preceded, each [momentarily immediate] determination passed as ultimate, it is now demoted into being only a moment.”

[G. W. F. Hegel, Lectures on Logic, Clark Butler, translator, Introduction to the Lectures on Logic, More Exact Concept and Division of the Science of Logic, [I. Being], Indiana U. Press [Indianapolis:  2008], pp. 79-80, bold, italic, underline, shadow, and color emphasis, plus [F.E.D.-labeled square-bracketed commentary] added by F.E.D.].

At the very outset of his book «Wissenshaft der Logik» -- The Science of Logic -- Hegel also asserts what we call ‘the «arché»-onic and ‘‘‘evolute’’’ principle of dialectical categorial progression:

“...the progress from that which forms the beginning is to be regarded as only a further determination of it, hence that which forms the starting point of the development remains at the base of all that follows and does not vanish from it.

progress does not consist merely in the derivation of an other, or in the effected transition into a genuine other; and in so far as this transition does occur it is equally sublated [F.E.D.:  ‘«aufheben»-ated] again.

the beginning ... is the foundation which is present and preserved throughout the entire subsequent development, remaining completely immanent in its further determinations.”

[G. W. F. Hegel, Science of Logic, Translated by A. V. Miller, Humanity Books [NY:  1969 [originally published in 1812]], Volume One, With What Must The Science Begin [placed before Chapter 1], p. 71, bold, italic, underline, shadow, and color emphasis, plus [square-bracketed commentary] added by F.E.D.].

At the conclusion of the same book, «Wissenshaft der Logik», in its ultimate chapter, Hegel reiterates this ‘«arché»-onic and ‘‘‘evolute’’’ principle of dialectical categorial progression once again:

... the determinateness [F.E.D.:  ‘«speci»-ficity] which was a result is itself, by virtue of the form of simplicity into which it has withdrawn, a fresh beginning; as this beginning is distinguished from its predecessor precisely by that determinateness, cognition rolls onward from content to content.

First of all
, this advance is determined as beginning from simple determinatenesses, the succeeding ones becoming ever richer and more [F.E.D.:  thought-]concrete.

For the result contains its beginning and its course has enriched it by a fresh

The universal constitutes the foundation
; the advance is therefore not to be taken as a flowing from one other to the next other.

In the absolute method the Notion
maintains itself in its otherness, the universal in its particularization, in judgment and reality;  at each stage of its further determination [F.E.D.:  further«speci»-fication]  it raises the entire mass of its preceding content, and by its dialectical advance it not only does not lose anything or leave anything behind, but carries along with it all it has gained, and inwardly enriches and consolidates itself.”

[G. W. F. Hegel, Science of Logic, ibid., Volume Two, Section Three, Chapter 3, The Absolute Idea, p. 840, bold-italic, underline, shadow, and color emphasis, plus [square-bracketed commentary] added by F.E.D.].

In our ‘ideo-perception’ of physio-dialectic, as of ideo-[physio-]dialectic, dialectical progression is ‘‘‘evolute’’’.

Evolute-ness’ is cumulative.

Dialectical progression cumulates.

It presents [ideo-][physio-]ontological cumula, which ‘‘‘contain’’’ the past as the pre-sent’, the ‘sent to now’s now from previous’, prior-nows’.



Wednesday, October 17, 2012

The Historical Dialectic of Arithmetics & The Historical Dialectic of the Social Forces / Social Relations of Production

Full Title:

PsychoHistorical Dialectic of Universal Labor” [Marx] -- Correlation of the Collective-Cognitive PsychoHistorical Dialectic of Arithmetics with the Collective PsychoHistorical  Dialectic of the Social Forces and Social Relations of production within the historically self-developing ‘‘‘Human Phenome’’’.

Dear Readers,

The extract below is quoted from the essay entitled The Gödelian Dialectic of the Standard Arithmetics, from the second-to-final Section of its second and final Part.

The following links, if clicked, will jump you to the full text of Part II. --,Part_II_of_II,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,posted_20SEP2012.pdf

[Preliminaries:   Skip to Main Section, Unless You Want to Know Its Deeper Context]

A far deeper dialectical questioning of the concept of “number” requires a far deeper, far richer «arché»-thesis, and, hence, one with a far deeper, far richer first contra-thesis -- or ‘‘‘first counter-example’’’ -- system of ‘‘‘number’’’, with a far wider chasm [‘~+~’] between the two, and, hence, with a much more profound initial «aporia» [denoted by ‘~+~’], than that between counts as numbers, N#, and no[n]-counts as numbers, a#, in --

1.   #)-|-(1   =   ( N# )^(2^1 ( N# )^2  =  N# x N#  =  ~( N#  

N# ~+~ a#

-- as a dialectic within the ‘‘‘meta-system’’’ of higher-than-first-order axiomatic systems of arithmetic [using ‘~’ to denote ‘dialectical [self-]negation’, or ‘‘‘immanent/«aufheben» negation’’’].

That deeper dialectic is given as 2., below:  The Meta-Systematic Dialectic of the Axioms-Systems of the Dialectical Arithmetics, #_, as given by the Encyclopedia Dialectica ‘meta-model’ --

2#_)-|-(s   =   ( _N#_ )^(2^s)

-- such that --

#_)-|-(1 = ( _N#_ )^(2^1) = ( _N#_ )^1 = _N#_( _N#_ ) = ~( _N#_ ) = _N#_ ~+~ NQ_#_

-- whose «aporia» is that between an arithmetic of “pure, unqualified ordinal quantifiers as numbers, _N#_, vs. of pure, unquantifiable ordinal qualifiers as meta-numbers, NQ_#_, for first-order-only axioms-systems.

The ‘‘‘externalized’’’ ‘oppositional sum’ of the apparently ‘mutually external’ terms, _N#_ and NQ_#_, namely --

_N#_ ~+~ NQ_#_

-- is founded, at root, upon what we term the ‘self-duality’ of _N#_, i.e., in its immanent, internal, qualitative ‘self-opposition’, or ‘self-antithesis’[denoted by the ‘ ˾̚     relation-sign]:

quantitative ordinality     ˾̚  qualitative ordinality.

As this essay is intended, in part, to demonstrate/exemplify, it is the case that the NQ_#_  dialectical arithmetic forms a fundamental supplement, and ‘supplementary opposite’, to the N# arithmetic, to which _N#_ is typically reduced.

The NQ_#_  arithmetic instantiates, inculcates, and facilitates powerful, ‘‘‘purely-qualitative’’’ cognitive capabilities that reside beyond the ken of the N#  -- or ‘‘‘purely-quantitative’’’ -- capital-value «mentalité», the «mentalité» that, most of all, characterizes the ‘‘‘human phenome’’’ in general, including its partly consciously, but mostly unconsciously ideology-compromised sciences, during the capitalist epoch of the self-development of the human «species».

Of course, the way forward for humanity also involves transcendence of the «aporia» of  

_N#_ ˾̚  NQ_#_,

to the system of their first dialectical synthesis, NU_#_, and beyond, to the seqq. systems of NU_#_, each one more determinate -- more ‘thought-concrete’ -- in its expressive power than its predecessor.

This deeper dialectic, one that encompasses the dialectic of number as «arithmos» in the ancient Hellenistic(+) sense, also intersects, or interconnects with, five other key Encyclopedia Dialectica dialectics of ‘The Human Phenome’ --

3.  The Axioms-Systems Dialectic of Arithmetical Logics, L, | WEL denotes Boolean Algebra’s ‘logic-arithmetic’;

4.  The [Psycho]Historical Dialectic of the ‘‘‘Meta-Evolution’’’ of Arithmetics, #.

5.  The [Psycho]Historical Dialectic of Human-Social formation(s) ‘‘‘Meta-Evolution’’’, f.

6.  The [Psycho]Historical Dialectic of Human-Social Relations of Production ‘‘‘Meta-Evolution’’’, R.

7.  The [Psycho]Historical Dialectic of Human-Social Forces of Production ‘‘‘Meta-Evolution’’’, F.

You will find, rendered below, multi-directional systems-progression content-structures, which depict the interweaving of the first three of the four ‘Meta-Models’ that, together, form the core of the Encyclopedia Dialectica Immanent Critique Of Arithmetics [ICOA].

The following is a list of all seven of the ‘meta-models’ cited in this section --

1.   #)-|-(s          =      ( N# )^(2^s)

2.   #_)-|-(s         =      ( _N#_ )^(2^s)

3.   L)-|-(s         =       ( EL )^(2^s)

4.    #>-|-<t     =      < _N# >^(2^t)

5.   f>-|-<t       =     < bf >^(2^t)

6.    R>-|-<t     =     < AR >^(2^t)

7.   F>-|-<t       =     < Rh,F >^(2^t)

The «arché» for Dialectical Equation 4., above, _N#, connotes Dr. Denise Schmandt-Besserat’s reconstruction of the practical, informal beginning of written numbers . . . in ancient Babylonia.

The «arché» for Dialectical Equation 5., bf, connotes the primordial, “bands” of predatory/foraging/scavenging/hunting-and-gathering ‘proto-human[oid]s’ who, to the best of our contemporary knowledge, constituted the ultimate ancestor-population of Terran humanity [see, for example, Robert Wright, Non-Zero: The Logic of Human Destiny, Pantheon, New York, 2000].

The «arché» for Dialectical Equation 6., AR, connotes the primordial mode of ‘‘‘production’’’ of humanity, that of non-production’, i.e., of direct Appropriation of the products of Nature in their “Raw” forms, unimproved, for human consumption, by human labor, carried on by those primordial “bands” of predatory/foraging/scavenging hunter-gatherers.

The «arché» for Dialectical Equation 7., Rh,F, connotes the primordial human-society self-reproductive force Resource of, or the primordial form of potential human-social ‘‘‘free energy’’’/human-social ‘‘‘negentropy’’’, harnessed/actualized by humanity, namely, that of the human community of those primordial “bands” of proto-human hunter-gatherers who carried on that “raw” Appropriation of the products of Nature.

The following links access extant F.E.D. texts which address the 2nd, and the remaining 5 related, ‘dialectical meta-models’ --

Meta-Systematic Dialectic of the Dialectical Arithmetics

Example (3), pages 3-01 to 3-12

Meta-Systematic Dialectic of Arithmetical/Algebraic Logic

pages 12to 19

.  Historical Dialectic of the Meta-Evolution of Arithmetics
pages I-122 to I-128

.  Historical Dialectic of the Meta-Evolution of Human-Social Formation(s)

pages III.A-01 to III.A-29


6.  Historical Dialectic of the Social Relations of Production

pages B-24 to B-37,%20Supplement%20B-1,%20v.2_OCR.pdf

.  Historical Dialectic of the Social Forces of Production

section II

Progressions 4., 5., 6., and 7. tie-in to the three-fold interconnexion of the other three progressions at the level of that collective, human-phenomic cognitive ‘‘‘meta-evolution’’’, a ‘‘‘meta-evolution’’’ which is driven by the development of the human-social forces of production/human-social relations of production praxis, which undergirds the historical dialectic of ‘the meta-evolution of arithmetics’, as of so much else in the human phenome.


The “
Natural Numbers”, in their advanced but still pre-“aught numbers” representations, e.g.,  

N = { I, II, III, ... }

are associated with vestiges extending all of the way back into the earliest Terran human[oid] social Relations of production epoch, tR = 0, with the predatory mode of self-reproduction of early hunting, gathering, and scavenging Paleolithic bands, in the form of, e.g., notches carved in bone, to mark time by marking re-occurrences of ~periodically recurring events, such as full moons.

The “
Natural Numbers”, as the upper limit of human arithmetical «mentalité», thus correspond to the AR [Appropriation of products of nature in their “raw” forms, not yet improved, for human use, by human labor] & early GR [ Goods/obligatory-Gifts production] ‘‘‘human-social Relations of production’’’/‘‘‘modes of [human-societal self-re-]production’’’ [cf. Marx].

Something resembling what we would mean today by the phrase “the positive Rational Numbers”, i.e., the positive fractions, Q+ = { (n1/n2) | n1, n2 are in N}, may have emerged as early as, and in connexion with, the emergent Neolithic(+) CR [Commodity Barter] human-social Relation of [human-societal-re-]production, ‘‘‘The Commodity-Relation’’’ [cf. Marx], in response to pre-Money/Money-less exchange of Commodities, often forcing exchange-value-equating of non-Whole-number physical amounts of Commodities in barter exchange, among camp, then village, then chiefdom human-social formations.

Something resembling what we would mean today by the phrase “the positive Real numbers”, R+, i.e., the positive ‘[hexa-]decimalized’ fractions, used by the ‘later-ancient’ Babylonia astronomers, including potentially-infinitely-repeating and potentially-infinitely-non-repeating [hexa-]decimal, or anthyphairesis, approximations, may have begun to emerge in association with late Neolithic(+) agricultural production practices involving “land-measurement” or “earth- measurement” [‘‘‘geo-metry’’’], leading to the discovery of “incommensurable” geometrical magnitudes.

This development belongs to the ancient Occidental classical period, in the Mediterranean venue, which increasingly featured ‘meta-chiefdom’ city-state, and ‘‘‘multi-city-state empire’’’ human-social formations, waxing region-wide and even into proto-global exchanges of Commodities, mediated by Monies, i.e., by MR, “The Money-Relation” [Marx], grasped, per Marx, as a social Relation of production.

Eventually, even “antediluvian”, ‘protoic’, pre-industrial ‘pre-vestiges’ of “The Capital-Relation” [Marx] -- e.g., usurers’ «Kapital» and mercantile «Kapital» non-production forms [‘‘‘circulation-forms’’’], and chattel-slaves-worked, ‘latifundial’-plantation, capitalist agricultural production-forms of «Kapital» [forms of which re-appeared later, and persisted, in the colonial, confederation, pre-Civil-War federal, and Civil-War confederacy plantation-states of southern North America] -- emerged in the ancient Mediterranean world, all three representing historical «species» of «Kapital», KR, as a human-social Relation of production.

The formation, in advanced human-phenomic, collective cognition, of something resembling what we might describe today as the “non-negative Reals”, R>0, including the use [e.g., by Eudoxus ..., and by Archimedes, e.g., in their “Method of Exhaustion” proto-integration algorithm] of informal/heuristic positive numbers of infinitesimal magnitude, verging on 0, and even of the use of 0 itself as a fully-operative number, constituting R>0, seems, in the Mediterranean locus, to have required the context of ancient classical multi-city-state empire social formations, at least, involving advanced Money-/proto-«Kapital»-based, proto-global commercial trading economies.

The acceptance, in the prevailing human-phenome, of a fullReal” numbers based -- R based -- conception of arithmetic, including of the negativeReals”, belongs to the aftermath of the catastrophic contracted human-social reproduction of the last Western EuropeanDark Ages, to the late-Medieval/Renaissance period’s resurrection of city-state republics, as well as of ‘protoic’ nation-state social formations, and of mercantile, banking, and, eventually, of large-scale manufacturing capitalism in that Occidental locus.

That emergence of proto-industrial capitalism is epitomized by the use of oppositely-signed decimal numbers to present “debit” vs. “credit” Monetary-value & «Kapital»-value ‘‘‘quantifiers’’’ in the double-entry bookkeeping practices that also emerged, for the first time in known human history, during that period.

The widespread acceptance and use of the Complex numbers, C = { Rr + Rri } [which were first discovered/invented during Europe’s Renaissance rebirth from its last Dark Ages, by Rafael Bombelli, a civil engineer who worked in Italy, circa 1572 C.E./B.U.E], appears to belong to the zenith of the ascendant phase of the industrial capitalist epoch, within the sub-epoch of the prevalence of the nation-state social formation, and with the industrial development and application of “electro-chemical” productive forces, then of [sinusoidal alternating current] “electrical” productive forces [in whose formulae our i of the C# “Standard Arithmetic” is traditionally notated as j instead, to help avert a potential confusion, given the traditional use of i to denote the electrical current variable], then of “electronic” productive forces, culminating, to-date, in sinusoidal EMR [ElectroMagnetic Radiation]-based, e.g., the radio wave/microwave technologies fundamental in today’s production, transportation, & communication, epitomized by “phasor” sinusoidal dynamics [for an angle quantifier, A] --

e^(i x A)    =    cos(A) x r + sin(A) x i

These stages in the ‘meta-evolution of arithmetic’ correspond, of course, to ‘The [Psycho]Historical, diachronic Dialectic of Arithmetics, not -- not directly at least -- to the essentially synchronic [Meta-]Systematic Dialectical presentation of The Gödelian Dialectic of the Standard Arithmetics of the present historical moment, central to this essay.

Nevertheless, the f>-|-<t, the R>-|-<t, and the F>-|-<t cumula provide some of the historical, diachronic grounding which is still ingredient -- however ‘complexly’ and implicitly so -- in the essentially synchronic view native to the #)-|-(s dialectic, which forms the primary subject-matter of this essay.