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.



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