Aoristos’s Blog

The F.E.D. ‘‘‘Psychohistorical Equations’’’

Sat, 19 May 2012 20:20:04 -0700

The Foundation Encyclopedia Dialectica [F.E.D.] ‘‘‘Psychohistorical Equations’’’ were designed, by the General Council of F.E.D., to create a ‘Time-Vault of the Future’, by means of a ‘meta-planetarium projector’, -- named the ‘‘‘F.E.D. Prime Radiant’’’, to project 3-D holographic representations of the solution-scenarios of this system of “simultaneous” equations and their scenario-valued variables, for future values of their time-variables, i.e., to algorithmically ‘pre-construct’ dialectical-mathematically-generated richly-determinate imagery for the future history expected per these equations.
All but one of these seven equations utilize the ‘Dyadic Seldon Function’ formulation of dialectical ontological progressions [ = of [revolutionary new] kinds-of-being progressions], a function which was defined earlier in this series, and which is also defined via the following link –
http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2028OCT2011,%20Definition,%20SELDON%20FUNCTIONS,%20JPEG,%2029OCT2011_1.jpg

The F.E.D. General Council is thereby enabled to anticipate – in timing, and in event-content – approaching “Seldon Crises”, for more about which, in terms of their [science-]fictional instantiation by Isaac Asimov, see –
http://en.wikipedia.org/wiki/Seldon_Crisis

I have listed, below, this system of seven dialectical equations, which are to be solved ‘‘‘simultaneously’’’ for the new ontology possibly emergent during the time period, tx [denoting the generic ‘‘‘epoch’’’ of the dialectical progression denoted by x] of the future value of that time-period variable which is to be solved-for.
The versions of the F.E.D. ‘‘‘Psychohistorical Equations’’’ rendered and partially explicated further below, are all rendered in the rule-based ideographical algebraic language-system of the F.E.D. “First Dialectical Arithmetic / Algebra”, denoted by NQ_.
The nine core algorithms, rules, or “axioms” of this language, presented earlier in this series, are also available via the following links –
http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf
http://www.dialectics.org/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
[pp. I-144 through I-146].

The versions of these equations that are used by the ‘dialectical psychohistorians’ of the F.E.D. General Council, and that would be capable of driving the ‘‘‘F.E.D. Prime Radiant’’’, are formulated in a higher, richer dialectical language than that of the NQ, purely-qualitative algebra used here – a language that arises later in the dialectical, ‘ideo-ontological’ progression-presentation of the F.E.D. systems of dialectical language. Needless to say, this dialectical language is one of the ‘quanto-qualitative’, rather than one of the ‘purely-qualitative’, dialectical languages that arises in that progression.
We do not [yet] publicly disclose the rules-system of that dialectical language. The optimal historical timing of its disclosure is itself a matter for the most delicate psychohistorical-dialectical calculation. However, immediately below, I give a summary description of the early dialectical language systems in the order that they are covered in that dialectical-systematic presentation, a presentation which we can summarily model by a ‘Dyadic Seldon Function’-based dialectical equation, also formulated in the NQ dialectical language.
The generic [unspecified; unsolved] ontology symbols of the NQ dialectic are NQ = {q1, q2, q3, …}. The axiomatic system that governs the ‘algorithmics’ of these NQ ‘meta-numbers’ is denoted by NQ_.
The notational conventions adhered to herein, to the extent that the typographical limitations of this medium allow, are summarized here --
http://www.dialectics.org/dialectics/Welcome_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2029SEP2011,%20Prefatories,%20ENCYCLOPEDIA%20DIALECTICA%20NOTATION,%201st%20Module,%20JPEG,%20for%2030SEP2011.jpg
http://www.dialectics.org/dialectics/Welcome_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2029SEP2011,%20Prefatories,%20ENCYCLOPEDIA%20DIALECTICA%20NOTATION,%202nd%20Module,%20JPEG,%20for%2030SEP2011.jpg

In the sequel, we use the symbol ‘=’ to stand in place of the phrase “is equal to, by definition”.
The underscored mnemonic symbols stand for ‘dialectical meta-numbers’ – i.e., for purely-qualitative values that operate, arithmetically and algebraically, according to the F.E.D. ‘Fundamental Rule of Dialectical Logic’, a rule that was proven deductively here, from the NQ_ axioms, previously in this series.
Since all of the ‘F.E.D. Psychohistorical Equations’ are equations of ‘psychohistorical dialectics’, we use the assignment symbol ‘[--->’ to indicate an association of an unassigned, unsolved, or “generic” NQ ‘meta-natural meta-number’ to a specific, mnemonic [psycho]historical-dialectical ontological category symbol [“kind of being” symbol], within a given [psycho]historical-dialectical ‘[psycho]physio-ontological’ progression:
[interpreted/solved/specific historical category symbol] [---> [generic NQ ‘meta-number’ symbol], e.g. --
The <<Kapital>>-relation as a human[oid]-social relation of production <---> K [---> q16

However, for ‘pure idea-systems’, and for their presentational ‘ideo-ontological’ dialectical categorial progressions, we use the assignment symbol, or solution symbol, ‘[---)’, to indicate such associations, e.g. --

“Thesis” axioms-system; standard “Natural” numbers in 1st order specification; ‘pure-quantifier’ algebra (---) N_ [---) q1

“Antithesis” axioms-system; 1st order, non-standard, dialectical ‘meta-numbers’; ‘pure-qualifier’ algebra (---) NQ_ [---) q2

The beginning – or «arche’» -- category and system of ‘ideas-ontology’ in the F.E.D. presentational-dialectical progression for the F.E.D. axioms-systems of dialectical arithmetic, is the system of the arithmetic of the “Natural” Numbers, N = {1, 2, 3, …}, as formulated by only the four Peano postulates that are expressible in “first order” formal symbolic logic [“first order” logic here means a language of formal logic that makes assertions about individual “Natural Numbers” only, but not about qualities shared by multiple “Natural Numbers”, such as “Even-ness”, “Odd-ness”, “Prime-ness”, “Composite-ness”, etc.]. We denote that “first-order” axioms-system of arithmetic by N_. It is a dialectical arithmetic only in a ‘pre-vestigial’, implicit, quantitative-only sense, in that the Peano successor function, s, is a degenerate, purely-quantitative «aufheben» operation: s(n) = n + 1, given an n in N.
The algorithm that generates this systems-progression is termed, by us, a ‘meta-model’, because a “model” would cover only a single system, whereas this progression contains a potentially infinite multitude of successive systems.
We use the term ‘‘‘evolution’’’ to describe the [self-]developmental dynamics within a single system, whether it be a natural, physical system [‘physio-system’], a human ideas-system [‘ideo-system’] such as a mathematical axiomatic system, or a human psychohistorical system – i.e., an ‘ideo-physio-system’ – such as a human socio-economic system. ‘‘‘Evolutions’’’ are transitions between historically successive states within a single system.
We therefore use the term ‘meta-evolution’ to describe the [revolutionary] transitions between systems – from historical predecessor system to historical successor system, which involve irruptions of new ontology.
The ‘Dyadic Seldon Function Meta-Model’ of the F.E.D. Method of Presentation [ = ‘Meta-Systematic Dialectic’] for this progression of dialectical-mathematical systems, is therefore –
(N_)^(2^s)
wherein s denotes a Whole Number variable, W = {0, 1, 2, 3, … }, representing/counting the cumulative number of steps into the exposition. Four steps in, i.e., for step s = 3, we solve this ‘meta-model’ as follows:

(N_)^(2^3) = (N_)^(8) = a non-amalgamative sum of 8 ‘idea-ontological’ categories, each connoting a qualitatively different, ‘ideo-ontologically’ different dialectical-mathematical axioms-system --
N_ + NQ_ + NU_ + NM_ + NqMN_ + NqMQ_+ NqMQN_+ NA_ [---)
q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8.

These first eight axioms-systems of dialectical arithmetic can be summarily described as follows, using the ‘pure ideo-systems’ assignment symbol, ‘(---)’—
N_ (---) Thesis system of standard natural numbers arithmetic in 1st order specification, as purely quantitative algebra.

NQ_ (---) 1st ‘Anti-thesis’ 1st order axioms-system, non-standard model of N_, 1st dialectical ‘meta-numbers’; ‘pure-qualifier algebra’.

NU_ (---) 1st [full] ‘Synthesis’ system; 1st order, 2nd dialectical ‘meta-numbers’, ‘quantifiable <<monad>>-qualifier’ algebra.

NM_ (---) 2nd ‘Anti-thesis’ system of 1st order, 3rd dialectical ‘metrical qualifier meta-numbers’; purely qualitative algebra.

NqMN_ (---) Partial ‘synthesis’ system of 1st order, 4th dialectical ‘meta-numbers’; ‘quantifiable metrical qualifiers’ algebra.

NqMQ_ (---) Partial ‘synthesis’, 1st order, 5th dialectical ‘meta-numbers’; ‘compound unquantifiable metrical qualifiers’.

NqMQN_ (---) 2nd full ‘Synthesis’ system; fully-algorithmic “dimensional analysis” via quantifiable state-variable qualifiers algebra.

NA_ (---) 3rd ‘Anti-thesis’ system of 1st order, 7th dialectical ‘dynamical system-qualifiers’; purely qualitative algebra
[implicitly, but not yet explicitly, an ideography for dynamical [& ‘‘‘meta-dynamical’’’, super^0-meta-]systems].

The system of seven dialectical equations which constitute ‘the F.E.D. Psychohistorical Equations’ are:
1. The Equation of Human Ideology/Knowledge Meta-Evolution.
2. The Equation of Human-Social Forces of Production Meta-Evolution.
3. The Equation of Human-Social Relations of Production Meta-Evolution.
4. The Equation of Human-Social Formation(s) Meta-Evolution.
5. The Equation of Human Genome / Human ‘Phenome’ Meta-Evolution.
6. The Equation of the Meta-Evolution of Planetary Human[oid]ities.
7. The Equation of The Psychohistorical Dialectic of the Dialectic Itself.

 The NQ_ and ‘Dyadic Seldon Function’ formulation of these psychohistorical dialectical equations can be rendered, and briefly described, as given below.

I. The Equation of Human Ideology/Knowledge Meta-Evolution

a. <<Arche’>>: Mythopoeias, M [---> q1.

b. ‘Dyadic Seldon Function’ Formula: <M>^(2^tk).
c. Solution to epoch tk = 4:

<M>^(2^4) = <M>^(16) =

M + R + qRM + P + qPM + qPR + qPRM + K +

qKM + qKR + qKMR + qKP + qKPM + qKPR + qKPRM + Y [--->

q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8 + q9 + q10 + q11 + q12 + q13 + q14 + q15 + q16.

d. Key categories from the first sixteen categories of the fields/forms of human ideology/knowledge can be summarily described using the ‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

M <---> Mythopoeias, “Myth-Makings”; “Mythologies” [---> q1.

R <---> Religions, codified / standardized / dogmatized, i.e., ‘Meta-Mythopoeias’ [---> q2.

qRM <---> Reconciliations of R & M / conversions from M into R, e.g., Ancient Rome’s Pantheon [---> q3.

P <---> Philosophies, i.e., ‘Meta-Religions’ [---> q4.
.
.
.
qPR <---> R|P reconciliations, even qRP retrograde P to R conversion, e.g., Aquinas’ Summa Theologica [---> q6.
.
.
.
K <---> Scientific Knowledges, i.e., ‘Meta-Philosophies’ [---> q8.
.
.
.
qKPR <---> Includes qPRK partially retrograde K|R to P conversions, e.g., Hegel’s Encyclopedia of the Philosophical Sciences [---> q14.

qKPRM <---> Advanced stage science, K, subsuming/explaining past Philosophy, Religion, & Mythology [possible/not yet actual on Earth] [---> q15.

Y <---> ‘‘‘PsYchohistories’’’, i.e., ‘Meta-Sciences’, e.g., Marx’s theory [---> q16.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
[pp. B-09 through B-17]

II. The Equation of Human-Social Forces of Production Meta-Evolution

a. <<Arche’>>: Whole [proto-]human communities [e.g., hunting bands] as Prime Energy Resources for
their own Human Societal Self-[Re-]Production, Rh [---> q1. This ‘meta-model’ uses somewhat different
notation than the others, because it is not ‘meta-monadic’ in the usual sense, but, on the contrary, recapitulates the ‘meta-monadology’ of the ‘Dialectical Theory of Everything’ self-construction of the cosmos in reverse. The deeper/earlier reaches of the ontology of the cosmos become accessible, for human Appropriation, as energy Resources, in the reverse order of their formation, and, as such energy Resources, form the core of the Marxian “social forces of production”. We are using the term “energy Resources” here in a more concretely determinate, more specific, less ahistorically abstract, less reductionist sense of the term ‘‘‘energy’’’ than is usually the case in contemporary discourse.

b. ‘Dyadic Seldon Function’ Formula: <Rh>^(2^tf).
c. Solution to tf = 4:

<Rh>^(2^4) = <Rh>^(16) =

Rh + Rl + Rlh + Rb + Rbh + Rbl + qblh + Re +

Reh + Rel + Relh + Reb + Rebh + Rebl + Reblh + Rp [--->

q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8 + q9 + q10 + q11 + q12 + q13 + q14 + q15 + q16.

d. Key categories from the first sixteen categories of ‘the existential self-force of human society’ -- of
‘human-societal self-reproductive self-force’ -- can be summarily described as follows, using the
‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

Rh <---> Entire [proto-]human communities [e.g., hunting bands] as prime energy Resource for predation-based social self-Re-production [---> q1.

Rl <---> Other proto-language-based animal societies/ plant communities, as human energy resources [---> q2.

Rλh <---> Reconciliations of Rl & Rh / conversions from Rh into Rl [---> q3, e.g., herding & horticulture.

Rb <---> asocial animals & multi-cellular/’meta-biotan’ animal bodies, including human slaves’ bodies, as exploited
energy Resources for human societal self-Reproduction, e.g., draft animals; slave rowers, etc. [---> q4.
.
.
.
Re <---> eukaryotic cells as energy resources for human societal self-[re-]production [---> q8, e.g., breads, beers,
wines/vinegars/cheeses/jerkies/yogurts/kefirs/pickles/chutneys, other “spoiled food”/fermented products, as food preservation tech.
.
.
.
Rp <---> prokaryotic, e.g., anaerobic, cells as energy sources for social Reproduction [e.g., methane-generating “digesters”] [---> q16.

e. Categories of interest irrupting into possibility in later epochs, tf > 4 --

Rmb comprises much ancient Roman technology, up to, e.g., U.S. Civil War technology [---> q36:
• Ancient Roman temples -- using stone [m]; “drum cranes” using wood [meta-biota, b], rope [b], & sinew [b].
• Ancient Roman waterworks flour mills -- using stone [m]; water [m], grain [b], wood [b], rope [b], etc.
• Ancient Roman aqueducts -- using stone [m]; “drum cranes” using wood [meta-biota, b], rope [b], & sinew [b].
• Ancient Roman glassworks -- using stone [m]; fire [m], metal [m] tools, wood [b] tools & fuels, glass-sand [m], diamond [m] cutting tools, etc., including to produce unsurpassed “Portland vases” and “caged glass”.
• circa 200 B.C.E.+ Chinese / Japanese “Magic Mirrors” -- using metals [m], fire [m], intensive hand polishing [b].
• 17th Century Japanese <<karakuri>> doll automatons [mechanical robots] -- using wood [b], metal [m] springs, etc.
• Ancient Roman <<ballista>> catapults -- using stone missiles [m]; wood [b], plant fiber/hair rope [b], & sinew [b].
• Ancient Roman <<hypocaust>> public baths -- using stone [m]; “drum cranes” [b], wood [b] parts & fuels, & fire [m], water [m], air [m], clay [m] pipes, socializing human beings [h], etc.: <---> Rmbh [---> q37.
• Bushnell’s “Turtle” single-occupant proto-submarine, U. S. First Revolutionary War vintage -- tar [b], wood [b], cork [b], flint [m], glass [m], lead [m], steel [m] bands, trapped water[m], trapped air [m], black powder [m], human-muscle-powered [hand-cranked] propeller [b].
• Confederate H. L. Hunley eight-person crew “fish boat” “torpedo boat”, or “torpedo fish” proto-submarine, vintage U. S. Second Revolutionary [“Civil”] War -- wood [b], rope [b], rubber [b], glass [m], iron [m], copper [m] wire, “blue light” pyrotechnic signal flare [m], trapped water[m], trapped air [m], black powder [m], eight-persons “social” crew, with a seven-persons-hand-cranked propeller [h]: <---> Rmbh [---> q37; both steam-powered [m] and electrical motored [s] propulsion was tried, but failed; there are possible recently-discovered archaeological indications of chemical/molecular battery-powered electrical [s] torpedo detonation: <---> Rsmbh [---> q165.

Rsmh encompasses much recent ‘‘‘android robotics’’’ technology [---> q161:
• Tokyo University of Science, Dr. Hiroshi Kobayashi’s <<Saya>> android proto-robot, using electronics [s], metals [m], human behaviors [h], etc.

Rsmeb encompasses much recent ‘‘‘Genetically-Modified Organisms’’’ [GMO] technology [---> q172:
• Japanese interferon-producing GMO silk worms, created using electronics [s], metals [m], silk worm eukaryotic cell nucleus [e] DNA [m], silkworms [b], etc.

Rsam encompasses many recent designs for subluminal interstellar drive engines [---> q224:
• Ion engines, using electronics [s], [ionized] atoms [a], metals parts [m], etc.

Rnsam encompasses much now-emergent or speculative technology [---> q480:
• Dr. Robert Bussard’s “polywell” fusion power reactor, involving photons [n], atomic [sub-]orbitals’ electrons [s], ionized [plasma] atoms [a], and metal parts [m], etc.
• “Stellar Wind Sail” starship designs, using solar/stellar photon [n] winds, &/or “particle” winds [s], “atom” [ion] plasma winds [a], metal parts [m], etc.
• F.E.D. ‘tachyonic meta-phase’ superluminal interstellar-drive design-hypothesis described earlier in this series.

III. The Equation of Human-Social Relations of Production Meta-Evolution
a. <<Arche’>>: Appropriations [of products of pre-/extra-human Nature, in “raw” form], A [---> q1.

b. ‘Dyadic Seldon Function’ Formula: <A>^(2^tr).
c. Solution to epoch tr = 4:

<A>^(2^4) = <A>^(16) =

A + G + qGA + C + qCA + qCG + qCGA + M +

qMA + qMG + qMGA + qMC + qMCA + qMCG + qMCGA + K [--->

q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8 + q9 + q10 + q11 + q12 + q13 + q14 + q15 + q16.

d. Key categories from the first sixteen categories of human ideology/knowledge can be summarily described as follows, using the ‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

A <---> “Raw” Appropriations, “Predations” [---> q1.

G <---> Goods Production, ‘multi-appropriations’ [e.g., burins from glassy stones]; ‘Meta-Appropriations’ [---> q2.

qGA <---> Reconciliations of G & A / conversions from A into G [---> q3, e.g., spears applied in hunting.

C <---> Bartered Commodities, i.e., ‘Meta-Goods’ [---> q4.
.
.
.
M <---> Monies, i.e., ‘Meta-Commodities’ [---> q8.
.
.
.
qMC <---> Subsumption of Commodities by Monies, i.e., Monies-mediated circulations of Commodities [---> q12.
.
.
.
K <---> <<Kapitals>>, i.e., ‘Meta-Monies’ [---> q16.

e. Categories of interest, observed as having, or predicted to, irrupt into possibility in later epochs, tr > 4 --

qKG <---> Real subsumption/reshaping of Goods-production by the <<Kapital>>-relation; & including <<Kapital>> Goods [---> q18.

qKC <---> Real subsumption of the Commodity-relation by the <<Kapital>>-relation; includes “Commodity <<Kapital>>” [---> q20.

qKM <---> Real subsumption of the Money-relation by the <<Kapital>>-relation; includes “Money <<Kapital>>” [---> q24.

qKK <---> E <---> Self-subsumption of the <<Kapital>>-relation; The Generalized ‘Equity-arian’ ‘Political-ECONOMIC DEMOCRACY’ of “the associated producers” [Marx], i.e., ‘Democratic Communism’ [---> q32.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
[pp. B-24 through B-38]

IV. The Equation of Human-Social Formation(s) Meta-Evolution

a. <<Arche’>>: bands/extended families of hunter-gathers/scavengers/foragers/predators, b [---> q1.

b. ‘Dyadic Seldon Function’ Formula: <b>^(2^tm).
c. Solution to epoch tm = 4:

<b>^(2^4) = <b>^(16) =

b + c + qcb + v + qvb + qvc + qvcb + f +

qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s [--->

q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8 + q9 + q10 + q11 + q12 + q13 + q14 + q15 + q16.

d. Key categories from the first sixteen categories of human social/‘meta-geological’ formation(s) can be summarily described as follows, using the ‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

b <---> bands/extended families of hunter-gathers/scavengers/foragers/predators [---> q1.

c <---> camps, ‘multi-band’ semi-permanent home bases; ‘meta-bands’ [---> q2.

qcb <---> Reconciliations/hybridizations of c & b / conversions from b into c [---> q3, e.g., advantaged, expanding camps.

v <---> villages, ‘multi-camp’, long-duration settlements, ‘meta-camps’ [---> q4.
.
.
.
f <---> chiefdoms, multi-village, “tribal” domains; ‘meta-villages’ [---> q8.
.
.
.
s <---> city-states, i.e., ‘meta-chiefdom’, or ‘meta-tribal’, social formations [---> q16.

e. Categories of interest, observed as having, or predicted to, irrupt into possibility in later epochs, tm > 4 --

e <---> empires, multi-city-state ‘meta-city-states’ [---> q32.

n <---> nation-states, multi-empire-remnant ‘meta-empires’ [---> q64.

p <---> planetary-poli, multi-nation-state ‘meta-nation-states’ [not yet known extant] [---> q128.

m <---> multi-planetary federations; ‘meta-planetary-poli’ [not yet known extant] [---> q256.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

Robert Wright, NONZERO: The Logic of Human Destiny, Pantheon [New York: 2000],
pp. 20, 78-92, 102, 109, 110-111, 12, 113, 125, 165, 353, 355, 357, 363, 365, 368, 376, 402, etc.

http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
[p. B-23]

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DiaRith/Intro/Dialectical-Ideography_An-Introductory-Letter.htm#Example_5

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DiaRith/Intro/Dialectical-Ideography_An-Introductory-Letter.htm#An_Ideography_for_Example_5

http://www.dialectics.org/dialectics/Briefs_files/F.E.D.-Brief2-part3-07DEC2008_OCR.pdf

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/Dialectical_Pictography/Dialectical_Pictography.htm
[slides 54-59]

V. The Equation of Human Genome / Human ‘Phenome’ Meta-Evolution

a. «Arche’»: The human Genome, as a mediately[ ‘Darwinianly’]-developing meta-system, from its proto-human origin , G [---> q1.

b. ‘Triadic Seldon Function’ Formula: <G>^(3^1).
c. Solution to epoch tG = 1:

<G>^(3^1) = <G>^(3) =

G + P + qPG [---> q1 + q2 + q3.

d. The categories of human genomic/human phenomic ‘co-meta-evolution’ can be summarily described
as follows, using the ‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --
G <---> The ‘meta-evolving’ “human Genome” [---> q1.

P <---> The ‘meta- evolving’ ‘human Phenome’ of non-chromosomal, non-genotypic, phenotypic, cultural, “acquired characteristics” [---> q2.

qPG <---> Their ‘‘‘co-meta-evolving’’’ hybrid; the ‘‘‘complex unity’’’ of the human Phenome/human Genome [---> q3.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/PsychoHistory/PsychoHistory.htm

VI. The Equation of the Meta-Evolution of Planetary Human[oid]ities
a. <<Arche’>>: bands/extended families of hunter-gathers/scavengers/foragers/predators, h [---> q1.

b. ‘Dyadic Seldon Function’ Formula: <h>^(2^th).
c. Solution to th = 1:

<h>^(2^1) = <h>^(2) = h + Δh =

h + y [---> q1 + q2.

d. The two ‘[psycho]physio-ontological categories’ for this meta-evolution of planetary human[oid]ities equation, and the three sub-categories of the second-to-emerge into possibility of those categories, can be summarily described as follows, using the ‘historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

h <---> ‘planetized’ ‘humanitys’ [or ‘planetized human[oid]ities’] [---> q1.

Δh <---> y <---> ‘meta-humanitys’ [---> q2; a ‘taxonomy level 1’ «genos» with 3 «species» --

<g>^(3^tg) = <g>^(3^1) = <g>^(3) =

g + r + qrg = g + r + c [---> q1 + q2 + q3 --

g <---> ‘Thesis’ <<species>>: ‘meta-humanity’ via human-genomic self-re-engineering [---> q1.

r <---> ‘Antithesis’ <<species>>: ‘meta-humanity’ via android robotics [---> q2.

qrg <---> c <---> ‘Synthesis’ <<species>>: ‘‘‘complex unity’’’ of r & g;
‘meta-humanity’ via cyborg prosthetics/bionics [---> q3.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

http://www.dialectics.org/dialectics/Dialectic_Ideography_files/DI_Prolegomena-Epitome-24OCT2009.pdf
[p. I-19]

VII. The Equation of The Psychohistorical Dialectic of the Dialectic Itself

a. <<Arche’>>: Systematic / Synchronic Dialectics, S [---> q1.

b. ‘Dyadic Seldon Function’ Formula: <S>^(2^t#).
c. Solution to epoch t# = 2:

<S>^(2^2) = <S>^(4) =

S + qSS + qSSS + qSSSS =

qS + qH + qHS + qHH = qS + qD + qDS + qDD =

S + H + M + Y [---> q1 + q2 + q3 + q4.

d. These four ‘human-historical ideo-ontological’ categories of Dialectics can be summarily described as follows, using the ‘[psycho]historical-systems’ assignment symbols, ‘<--->’ and ‘[--->’ --

S <---> Systematic / Synchronic Dialectics, S [---> q1.

H <---> Historical / Diachronic Dialectics, H [---> q2.

M <---> Meta-Systematic / Diachronico-Synchronic Dialectics, M [---> q3.

Y <---> PsYchohistorical Dialectics, Y [---> q4.

For more amplitude about this particular ‘dialectical meta-model’ and ‘‘‘psychohistorical equation’’’, see –

http://www.dialectics.org/dialectics/Correspondence_files/F.E.D.,Letter%2007,Postludes%20Series,Postlude%20VII,The%20Dialectic%20of%20the%20Dialectic%20Itself,F.E.D.%20to%20Webmaster,www.dialectics.org,18AUG2011.pdf

http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2014SEP2011,%20Definition,%20SYSTEMATIC%20DIALECTICS,%20JPEG,%20for%2015SEP2011.jpg

http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2015SEP2011,%20Definition,%20HISTORICAL%20DIALECTICS,%20JPEG,%20for%2016SEP2011.jpg

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/MetaSystematicDialectics/MetaSystematicDialectics.htm

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/PsychohistoricalDialectics/PsychohistoricalDialectics.htm

Key challenges for the “simultaneous” and ‘quanto-qualitative’ solution of these seven equations –

1. Coordinating / correlating the different ‘‘‘tempos’’’ / temporalities / ‘epochalities’ of these equations, since,
in general, tk tf tr tm tG th tg t#;

2. Modeling the thus coordinated quantitative ‘epochality’ using “continuous” rather than “discrete”
arithmetical models of time, e.g., using tR not in W; i.e., using tR in R = the set of the “Real” numbers;

3. Re-expressing the ‘quality meta-dynamics’, or ‘ontological meta-dynamics’, of the qualifier-sums in these
purely-qualitative, purely-ontological equations, more determinately, as ‘‘‘[space-]time-varying’’’ evolution
trajectory/path dynamics, with [‘metafinite’, ontologically-revolutionary] ‘‘‘singularities’’’, of time-function
quantified state-variables & control-parameters, using the higher dialectical-ideographical languages of
dynamical & ‘’‘meta-dynamical’’’ [super^z meta-]systems, via ‘singularity semantification’ [division-by-
zero ‘semantification’ and ‘metafinitization’];
4. Facilitated by 1. and 2., solving this system of seven equations “simultaneously”;
5. Modeling the ‘inter-mutual’ interactions of the co-existing actualities modeled by these seven equations;
6. Deriving a single equation which unifies / integrates all seven equations into a single ‘meta-equation’.

Dialogically yours,

Aoristos Dyosphainthos

A Breakthough in the Ideography of Dialectical Negation

Sat, 5 May 2012 21:49:47 -0700

Dear Site Users,

I have reproduced, below, the 90th blog entry from Miguel Detonacciones’ blog “F.E.D. Dialectics”, with the permission of the author, and with F.E.D. edits, also approved by the author.

Dialogically Yours,

Aoristos Dyosphainthos
Voting Member of, and Public Liaison Officer for,
the F.E.D. General Council

Dear Readers,

The F.E.D. generic model of dialectical progressions in general, holds as follows --

Suppose that we have a "sum" of mutually qualitatively different qualitative determinations, collectively denoted by Hh, characterizing a given system, as of step h in the dialectical presentation / systematic dialectic of that theoretical system, or as of epoch h in the self-presentation / historical dialectic of that natural system, where h is a discrete variable ranging over the "Whole" Numbers, { 0, 1, 2, 3, . . . }.

If that "sum" of characteristics is a qualitative, heterogeneous [or inhomogeneous], irreducible, and non-amalgamating, or <<asumbletoi>> [Plato: unaddible] sum, then stage h+1 will exhibit the following, expanded non-amalgamative sum, with additional possible determinations/characteristics describable as follows --

Hh   ---> Hh+1    = Hh "of" Hh    = Hh(Hh)    = ~(Hh)    =

Hh x Hh    =    HhHh    =
Hh "squared"   = Hh^2    =    Hh + Delta(Hh)

-- such that the term Delta(Hh) "contains" all of the revolutionary, new "ontology", the “qualitative changes/growth”, that inaugurated the new, h+1 stage, and that eventually emerged from the “quantitative changes” within stage h -- all of the newest possible determinations / characteristics -- all computed from the old determinations / characteristics of stage h.

For example, if h = 1, and if the possible features of the maximal natural system known as the cosmos, as of that epoch, epoch 1, are --

( n + s )   =   ( sub-nuclears + sub-atomics )   =
( bosons & non-composite fermions + hadrons )

-- such that the underscores indicate the "contra-Boolean" computational logic exhibited by these terms as [dialectical] logic elements, or as [dialectical] logic symbols [i.e., x^2   ~= x], then, as of the next discretely representable epoch, epoch h + 1 = 1 + 1 = 2, the possible features of the cosmos are "predicted" by the F.E.D. "Dialectical "Theory of Everything" Equation" to be --

H1 = ( n + s )   ---> H2 = ( qn + qs + qsn + qss )   = H1 "of" H1   =

( n + s ) "of" ( n + s ) = Hh(Hh)    = ( n + s )(( n + s )) =

~(H1) = ~( n + s )    =     H1 x H1    =     ( n + s ) x ( n + s )   =

H1H1    =  ( n + s )( n + s )   =    H1 "squared"   =

( n + s ) "squared"   =

H1^2    =    ( n + s )^2    =    H1 + Delta(H1)    =

( n + s ) + Delta( n + s ).

= ( n + s) + (qsn + a ) = ( n + s + qsn + a ) = H2

= ( sub-nuclears + sub-atomics +
real subsumption of sub-nuclears by sub-atomics + atomics )

= ( sub-nuclears + sub-atomics +
sub-atomic / sub-nuclear hybrids + atomics ).

Note especially the "sub-equation" extracted from the original serial equation above --

Hh(Hh) = ~(Hh).

Its Left-Hand-Side [LHS] expression, read off as "Hh of Hh", denotes the "self-function[ing]" of Hh, that is, the "self-movement" of Hh -- the "autokinesis" [Plato] of Hh.

Its Right-Hand-Side [RHS] expression, read off as "[dialectical] negation of Hh", or as "[determinate] not Hh", denotes the dialectical, determinate negation of Hh, that is, the <<aufheben>> negation of Hh.

So, the sub-equation --

Hh(Hh) = ~(Hh)

-- asserts that the dialectical, determinate, <<aufheben>> negation operation symbol, for a given system-characterization symbol, generically denoted by Hh, itself formulated as a function / operation / action / activity, is, precisely, itself --

~ =     Hh [for Hh]. What more determinate, specific, negator could there be, for Hh, than Hh itself?

That is, the effect of the inherent, immanent, "intra-duality", or "self-duality", of such a system, as of stage h in its [self-]development, denoted Hh -- the "internal dialectical contradiction" or "dialectical self-contradiction" of / immanent to that system -- can be expressed as a dialectical, determinate, <<aufheben>> self-negation of that system-stage --

Hh ---> ~(Hh) = Hh(Hh) = Hh x Hh = HhHh = Hh^2 =

Hh + Delta(Hh) = Hh+1.

[Note that ~(Hh) = Hh(Hh), so ~(~(Hh)) = ~(Hh+1) =

Hh+1(Hh+1) = Hh^2(Hh^2) = Hh^4,

not ~(~(Hh)) = Hh(Hh(Hh)) =

Hh(Hh^2) = Hh^3.

That is, the "tilde" symbol, '~', acts like a pair of "horizontalized" ditto marks].

The series of equations --

Hh ---> ~(Hh) = Hh(Hh) = Hh x Hh = HhHh = Hh^2 =

Hh + Delta(Hh) = Hh+1

-- expresses the core breakthrough discovery of F.E.D. in achieving the first algebraic representation of dialectical processes in general, the first algebraic dialectical logic --

http://www.dialectics.org/dialectics/Dialectic_Ideography_files/5_Dialectics-Part1b-WhyDI_OCR.pdf

-- [F.E.D., Dialectical Ideography, Prolegomena, Part I.b., pages I-73 through I-75].

F.E.D. thus achieved, in 1999 [or even as early as 1996, judging from the day-counter in their document version-numbers], for algebraic dialectical logic, the counterpart and <<surpassement>> of what Leibniz, from 1666 [though he did not publish his algebra of logic], and Boole, from 1847 [who published, in that year, his booklet entitled The Mathematical Analysis of Logic: Being an Essay towards a Calculus of Deductive Reasoning], achieved for algebraic formal logic --

Leibniz / Boole: x^2 = x, i.e., 0^2 = 0, and 1^2 = 1.

F.E.D.: x ---> ~x   = xx   =   x^2   =   x + delta_x, i.e.,

for every n in N = { 1, 2, 3, ... },
qn ---> ~qn =   qn x qn =   qn^2    =    qn + delta_qn    =

qn + qn+n    =    qn + q2n.

The iteration of such ideographically expressed [algebraically expressed] dialectical negation --

Hh   ---> ~(Hh) ---> ~(~(Hh)) --->

~(~(~(Hh))) ---> . . .

-- which is equivalent to the following self-iteration of dialectical self-negation --

Hh^1 ---> Hh^1(Hh^1) = Hh^2 ---> Hh^2(Hh^2) = Hh^4 --->
Hh^4(Hh^4) = Hh^8 ---> . . .

-- is the process that F.E.D. names "The Dyadic Seldon Function" , which can be succinctly summarized as --

Hh   =   H0^(2^h).

Regards,

Miguel

Sartre versus Ollman on the “dialectics of nature” controversy

Sat, 14 Apr 2012 20:33:28 -0700

Dear Site Users,

I have reproduced, below, the 87th blog entry from Miguel Detonacciones’ blog “F.E.D. Dialectics”, with the permission of the author, and with F.E.D. edits, also approved by the author.

Dialogically Yours,

Aoristos Dyosphainthos
Voting Member of, and Public Liaison Officer for,
the F.E.D. General Council

Dear Blog Participants,
Below are two salient quotes on the "dialectics of nature" controversy, the first from Jean-Paul Sartre, the second from Bertell Ollman, plus a record of a recent dialogue about them in which I took part --

The Quotes.

1. [from Jean-Paul Sartre, Search For a Method, translated by Hazel E. Barnes, Vintage Books [New York: 1968], p. xii] --

"Ought we then to deny the existence of dialectical connections at the center of inanimate Nature? Not at all. To tell the truth, I do not see that we are, at the present stage of our knowledge, in a position either to affirm or to deny. Each one is free to believe that physico-chemical laws express a dialectical reason or not to believe it."

2. [from Bertell Ollman, The Dance of the Dialectic: Steps in Marx's Method, University of Illinois Press [Chicago: 2003], pp. 96-97] --

"What are called the "laws of dialectic" are those movements that can be found in one or another recognizable form on every level of generality, that is, in the relations between qualities that fall on each of these levels, including that of inanimate nature. …

Naturally, the particular form taken by a dialectical law will vary somewhat depending on its subject and on the level of generality on which this subject falls.

The mutually supporting and undermining movements that lie at the core of contradiction, for example, appear very different when applied to the forces of inanimate nature than they do when applied to specifically capitalist phenomena.

Striking differences such as these have led a growing band of critics and some followers of Marx to restrict the laws of dialectic to social phenomena and to reject as "un-Marxist" what they label "Engels's dialectics of nature".

Their error, however, is to confuse a particular statement of the laws, usually one appropriate to levels of generality where human consciousness is present, for all possible statements.

…To be adequately apprehended, such movements on each level of generality must be seen as expressions of laws that are specific to that level as well as versions of more general laws.

Most of the work of drafting such multilevel statements of the laws of the dialectic remains to be done."

The Dialogue.

In response to my citing the quotes rendered above, an interlocutor wrote back with the following commentary on them [edited slightly syntactically].

Reply --
"... I agree with Ollman against Sartre: of course, dialectics is inherent to inanimate nature. If we research really complete systems in their development, we can see it: Ilya Prigogine's synergetics (from chaos to order) demonstrates the three basic dialectical laws within inanimate nature clearly."

My Reply to That Reply [edited to add further clarifying content] --
I could not agree more with your views, including your pointing out how Ilya Prigogine's Nobel Prize-winning work on Nonlinear / Nonequilibrium Thermodynamics, and later work on Nonlinear Dynamical Systems Theory, corroborate the dialectical character of "inanimate" -- actually, of decidedly self-animate, self-reflexively self-developing ["nonlinear"] -- Nature.

The dialectic of nature in its pre-human[oid] phases is not a "dialectic without subjects", but is one utterly replete with "subjects" and with "subject-ivity", in the general meaning of the terms "subject" and "subject-ivity", although not, of course, yet in terms of human[oid], self-aware "subjectivity".

But, starting at least with the swarms of bosons, quarks, and leptons believed to have populated our cosmos early-on, the cosmos is populated with entities that fulfill the "subject" role, of action initiation, in Indo-European sentences describing the subjects' actions upon the objects of Nature, then, and ever since.

This 'sentence-iality' includes both those cases wherein the "natural objects" acted upon by these "natural subjects" are other than those "natural objects" -- the merely "flexive" and "fluxive" cases -- but also those cases wherein the "natural objects" acted upon by these "natural subjects" are the self-same populations of [ev]entities -- the "self-reflexive", "self-refluxive", "nonlinear" ["higher degree"] cases, the very cases which constitute the essential core of the most dialectical, "self-aufheben" processes of nature.

That is, Nature -- Natural History -- exhibits a "qualo-quantitatively" -- an 'ONTO-quantitatively" -- growing 'meta-population' of populations of natural "subject-objects", or, better, of "subject-verb-objects"; of "eventities" -- of "eventities which are both each-other-transforming, and self-transforming.

The nonlinear "partial" and "total" differential equations-systems that formulate nonlinear thermodynamics and nonlinear dynamical systems theory also express the dialectics of nature especially when the "singularities" of these equations -- moments at finite values of the t [time] parameter when a division by zero arises, typically marking the transition from system "self-evolution" to system "self-meta-evolution" -- to new-ontology-adding system self-REVOLUTION -- are taken into account.
 

Moreover, your point goes precisely to why I am so excited about the F.E.D. "Dialectical Theory of Everything"!
 

What F.E.D.'s Marxian immanent critique of "natural" arithmetic, in general, and what Karl Seldon's work, in particular, have achieved, is to discover a "non-standard model" of the "natural" numbers, one which is an extreme opposite of the standard model of the "natural" numbers, and one whose "elements" "behave" DIALECTICALLY.

Those same "elements" -- new KINDS of [meta-]numbers ["aufheben operators"] -- also provide solutions to an equation which is the extreme "aufheben-negation" of Boole's "Fundamental Law of Thought" equation of the original Boolean arithmetic/Boolean algebra.

The F.E.D. "first axiomatic system" thus constitutes also a "CONTRA-Boolean Algebra" [one which, of course, aufheben-conserves/contains the Boolean Algebra "fundamental law of thought" equation, but which also supersedes it]; an algebra interpretable as a Dialectical Logic.

This new, dialectical arithmetical/algebraic language is powerful enough to express a univocal "Dialectic of Nature" as a whole — i.e., to dialectically, via concretized versions of the "aufheben" operation alone, generate representations of all known categories of cosmological ontology, in the same order as the historical order in which each such ontological category arose [i.e., was first concretely instantiated] in the natural history of the known cosmos — by means of an equation involving just seventeen symbols [if those symbols are counted in the most maximal way; less otherwise].
 

F.E.D. has also applied that language to generate compact and startlingly-revealing systematic-dialectical models of the contents of Marx's masterpiece, Capital, as well as historical-dialectical models of the historical "meta-evolution" of the social relations of production, of the historical "meta-evolution" of the social forces of production, of the historical "meta-evolution" of human social formations' "geo-morphologies" / "human geographies", etc., etc.
 

F.E.D. got to this new, dialectical-ideographic language by, in effect, in my view, by fulfilling the joint implication of the Goedel completeness theorem and incompleteness theorems at the level of first order Peano "natural numbers" arithmetic.
 
 Mathematical logician John W. Dawson, Jr., also a biographer of Goedel, described this joint implication as follows, in his biography of Goedel ["Logical Dilemmas: The Life and Work of Kurt Goedel", A.K. Peters, Ltd., Wellesley, MA., 1997], pp. 67-68 [square-bracketed parenthetical comments mine] –  "Most discussions of Goedel's ["first incompleteness"] proof … focus on its quasi-paradoxical nature.  It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves.  It is particularly enlightening to consider together both the [Goedel] completeness and incompleteness theorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility.  The confusion arises from the two different senses in which the term "complete" is used within logic.  In the semantic sense, "complete" means "capable of proving whatever is [generally] valid," whereas in the syntactic sense it means "capable of proving or refuting each sentence of the theory."  Goedel's completeness theorem states that every (countable) first-order theory, whatever its non-logical axioms may be, is complete in the former sense: its theorems coincide with the statements true in all models of its axioms.  The incompleteness theorems hold also for higher-order formalizations of number theory.  If only first-order formalizations are considered, then the [Goedel] completeness theorem applies as well, and together they yield not a contradiction, but an interesting conclusion:  Any sentence of ["natural" numbers] arithmetic that is undecidable must be true in some models of Peano's axioms [for the "natural" numbers] (lest it be formally refutable) and false in others (lest it be formally provable). In particular, there must be models of first-order Peano arithmetic whose elements do not "behave" the same as the natural numbers.  Such non-standard models were unforeseen and unintended, but they cannot be ignored, for their existence implies that no first-order axiomatization of ["natural"] number theory can be adequate to the task of deriving as theorems exactly those statements that are true of the [standard] natural numbers."

Regards,

Miguel

Part II. B.: Intuitive Account of the Universality of F.E.D.’s Generic Heuristic Algorithm for Dialectic --The Opening Triad of Marx’s <> -- 86th Blog Entry

Mon, 12 Dec 2011 08:43:18 -0800

Dear Site Users,

I have reproduced, below, the 86th blog entry from Miguel Detonacciones’ blog “F.E.D. Dialectics”, with the permission of the author, and with F.E.D. edits, also approved by the author.

Dialogically Yours,

Aoristos Dyosphainthos
Voting Member of, and Public Liaison Officer for,
the F.E.D. General Council

Part II. B.: Intuitive Account of the F.E.D. Generic Dialectics Algorithm -- 
Systematic-Dialectics Example 2. – The Opening Triad of Marx’s «Kapital».  

 Dear Readers,  Part II. B.: Systematic-Dialectics Example 2. – The Opening Triad of Marx’s «Kapital».  Introduction: [All references to Marx’s Capital, volume I, herein refer to the edition published originally in 1967 by International Publishers, Inc., New York, and, specifically, to their 1973 fifth printing of that edition. Throughout this exposition, it will be helpful to have your copies of the volumes of Marx's Capital handy, and open to the pages being referenced in the various passages below.].  In his Preface to the First German edition of Capital, volume I, Marx wrote as follows regarding the beginning category, or «arché» category, of his method of presentation of his systematic treatise on the human-social historical totality known as “capitalist society” [emphasis added by M.D.] –  “Every beginning is difficult, holds in all sciences.  To understand the first chapter, especially the section that contains the analysis of commodities, will, therefore, present the greatest difficulty.  That which concerns more especially the analysis of the substance of value and the magnitude of value, I have, as much as it was possible, popularised.  The [M.D.: elementary] value-form, whose fully-developed shape is the money-form, is very elementary and simple.  Nevertheless , the human mind has for more than 2,000 years sought in vain to get to the bottom of it, whilst, on the other hand, to the successful analysis of much more composite and complex forms, there has been at least an approximation.  Why?  Because the body, as an organic whole, is more easy of study than are the cells of that body.  But in bourgeois society the commodity-form of the product of labour – or the [M.D.: elementary] value-form of the commodity – is the economic cell form.  To the superficial observer, the analysis of these forms seems to turn upon minutiæ.  It does in fact deal with minutiæ, but they are of the same order as those dealt with in microscopic anatomy.  With the exception of the section on the value-form, therefore, this volume cannot stand accused on the score of difficulty.  I pre-suppose, of course, a reader who is willing to learn something new and therefore to think for himself.” [Karl Marx, Capital, volume I, originally published 1867, International Publishers [New York: 1967], pp. 7-8, emphasis added by M.D.].   I have previously, in this thread, presented dialectical models of the systematic dialectic of the three volumes of Marx’s Capital that take the more superficial of the two «arché» categories that Marx identifies in the “prefatory” passage quoted above – “...the commodity-form of the product of labour...” -- to which the title of the first Chapter of Capital, volume I answers – “CHAPTER I. – Commodities” – and which generate minimally-complete dialectical model reconstructions of Marx’s three-volume argument, reconstructions belonging to the following dialectical categories-sum, dialectical categorial-progression kind --  C + M + qMC + K + qKC + qKM + qKMC + S  -- or, i.e. --  Commodities + Monies +  Monies-Mediated Circulations of Commodities +  «Kapitals» +   Commodities-«Kapitals» + Monies-«Kapitals» +  Circulations of the Total Social «Kapital» +   Social[ist] Revolution   -- covering, connotatively and heuristically, the following portions of the [tables-of-]contents of Marx’s three volumes --  VOLUME I The Productions-Process of Capitals  PART I COMMODITIES AND MONEY CHAPTER I. – Commodities, & CHAPTER III. – Money, or the Circulation of Commodities, &  PART II THE TRANSFORMATION OF MONEY INTO CAPITAL CHAPTER IV. -- The General Formula for Capital CHAPTER V. – Contradictions in the General Formula for Capital  &   VOLUME II The Circulations-Process of Capitals  PART I THE METAMORPHOSES OF CAPITAL AND THEIR CIRCUITS CHAPTER III. – The Circuit of Commodity-Capital, & CHAPTER I. – The Circuit of Money-Capital, &  PART III THE...CIRCULATION OF THE AGGREGATE SOCIAL CAPITAL  &   VOLUME III The Whole-Process of Capitalist Production  PART V ...INTEREST-BEARING CAPITAL CHAPTER XXVII. – The Role of Credit in Capitalist Production /  VOLUME I The Productions-Process of Capitals PART VIII THE SO-CALLED PRIMITIVE ACCUMULATION  CHAPTER XXXII. – Historical Tendency of Capitalist Accumulation.    Now, it has been found – and not unexpectedly so -- in F.E.D.’s research, that dialectical models beginning with this more superficial of the two «arché» categories that Marx identifies in the “prefatory” passage quoted above – “...the commodity-form of the product of labour...” – cannot capture, in such connotative dialectical models, the [table-of-]contents of Marx’s three volumes of «Das Kapital» as deeply as, or in as much in detail as, can dialectical models that begin from the deeper of those two «arché» categories – namely, that begin from the elementary form of commodity value.  We will illustrate such a dialectical model of the contents of Marx’s «Das Kapital», in this post, only for the first three categories in the categorial [self-]expansion, or [self-]progression, of this elementary value-form «arché». [We do plan, in a subsequent post, to present a progression of F.E.D. dialectical models of the contents of the three volumes of Marx’s Capital, with each successor dialectical model beginning from a deeper «arché» than did its predecessor dialectical models, and, thereby, achieving a fuller, more detailed capturing of, or reproduction of, the tables-of-contents of the multiple volumes of Marx’s treatise.].  In terms of Marx’s table of contents for Capital, volume I, this deeper initial triad encompasses the following three sections –   VOLUME I The Productions-Process of Capitals  PART I COMMODITIES AND MONEY  CHAPTER I. – Commodities Section 3. – The Form of Value or Exchange-Value A. Elementary or Accidental Form of Value B. Total or Expanded Form of Value C. The General Form of Value.   Do the detailed content-characteristics of Marx’s Elementary Value-Form, of his Expanded Value-Form, and of his General Value-Form categories exhibit the detailed Generic characteristics claimed by F.E.D. to be universal for the first triads of virtually all «species» of dialectical progressions?  YES !!!   They do so to a far fuller extent than, I think, most would expect, given that Marx could not have had in mind the exact, mathematically-expressed universal algorithm for dialectic that F.E.D. has discovered, and axiomatized, when he organized and wrote the material of «Das Kapital» [although Marx clearly does have some kind of general pattern, procedure, and/or “generic algorithm” for dialectic in mind – albeit a purely-qualitative, non-mathematically-formulated such “generic algorithm”, and his writings often assert to that effect, e.g., in his correspondence, in the «Grundrisse», and in «Das Kapital» itself.].  That is, Marx’s first three “commodity value-form” categories conform, in amazing detail, to F.E.D.’s generic first triad characteristics of –  Thesis + Meta-Thesis + Syn-Thesis  -- or --  Posit + Meta-Posit + Com-Posit  -- or --  «Arché» + Meta-«Arché» + Syn-«Arché».   In particular, each unit, or «monad», or element of the Expanded Form of Value «arithmos», or assemblage of units, is – quite precisely – a concrete-«aufheben» “meta-«monad»-ization, or “meta-unit-ization”, of some or all of the units, or elements, or «monads» of the «arithmos», or assemblage of units, of the Elementary Form of Value, as we shall demonstrate, in detail, below!!!  Moreover, each unit, or «monad», or element of the General Form of Value «arithmos» is a hybrid unit, hybrid «monad», or hybrid element of an Expanded Form of Value «arithmos»’s unit, «monad», or element, together with, or combined with, or “composited with” an Elementary Form of Value «arithmos»’s unit, «monad», or element, such that “half” of each General Form of Value «arithmos»’s unit, «monad», or element comes from one of the Expanded Form of Value «arithmos»’s units, «monads», or elements, and such that the other “half” of each General Form of Value «arithmos»’s unit, «monad», or element comes from one of the Elementary Form of Value «arithmos»’s units, «monads», or elements, as we shall also show below!!!   Finally, note that each hybrid unit, hybrid «monad», or hybrid element of the General Form of Value's «arithmos», or assemblage of units, is a “horizontal inversion negation” of a unit, «monad», or element of the Expanded Form of Value «arithmos», or assemblage of units, as also demonstrated in the illustrations posted below.     Triad:  A & B & C =   Elementary Value-Form &   Expanded Value-Form &   General Value-Form.     Movement:  Elementary Value-Form --->   Elementary Value-Form & Expanded Value-Form --->   Elementary Value-Form & Expanded Value-Form & General Value-Form.    Definitions:  A....Elementary Value-Form. Marx writes of this form: “The whole mystery of the form of value lies hidden in this elementary form. Its analysis, therefore, is our real difficulty. Here two different kinds of commodities (in our example the linen and the coat), evidently play two different parts. The linen expresses its value in the coat; the coat serves as the material in which that value is expressed. The former plays an active, the latter a passive, part. The value of the linen is represented as relative value, or appears in relative form. The coat officiates as equivalent, or appears in equivalent form. The relative form and the equivalent form are two intimately connected, mutually dependent and inseparable elements of the expression of value; but, at the same time, are mutually exclusive, antagonistic extremes – i.e., poles of the same expression.” [ibid., p. 48, emphasis added by M.D.].  The “elementary”, or “accidental”, or “isolated” [Marx] form of commodity [ultimately, commodity-capital-]value is the simplest, or least complex -- and the most abstract, or least concrete, least “determinate”, or least “determinations-rich” -- category, or «arithmos» of «monads» [“assemblage of units”], in which the dialectical polarity -- or “intra-duality”, to use F.E.D.’s term -- of use-value versus exchange-value is still preserved as explicit, and, therefore, in which the historical / diachronic, and also present / synchronic, root of the Capital-relation as predominant, “real-subsuming” social relation of production is still explicitly evident, i.e., such that any further abstraction, or any further simplification, would submerge/repress this "self-dual" root into “implicitude”, so that the elementary form of value «arithmos», or category, is the simplest, most abstract such category that is still suitable to serve as the «arché»-category, or beginning-category, of a systematic-dialectical model -- or theoretical reconstruction/reproduction and explanation -- of the self-reproductive dynamics, and of the social-reproduction process, of the capitals-system [only], including its historical limits as well, on both its past and its future time-sides -- because the [w]hol[e] of the capitals-system, i.e., of its “essence-ial”, core “internal contradiction”, is still “holographically” / holonomically explicitly-enough “contained” in the elementary value-form as arché» category / «arithmos».   Each unit, or «monad», or element of the elementary value-form «arithmos», or assemblage, or category, can be represented as a set of exchange-value-vs.-use-value, or relative-value-pole-vs.-equivalent-value-pole ‘polads’, of ‘polads’ expressed via either the “algebraical”, or “variable” forms, or of the “constant”, or “arithmetical” forms, as given by Marx [Capital, volume I, ibid., p. 48] –   “x commodity A = y commodity B, or  ..x commodity A is worth y commodity B.  20 yards of linen = 1 coat, or  20 yards of linen are worth 1 coat.”  -- or, in F.E.D. "algebraical" shorthand --  xA = yB  -- wherein x and y are quantifier variables standing for counts of the standard units of commodity A and of commodity B, respectively -- of the quantity of units of A -- x -- needed to be able to exchange them directly for, i.e. to have the equivalent value to, a quantity of exactly y units of commodity B – and wherein A and B are what F.E.D. terms “quantifiable ontological-category-as-unit qualifier variables” – i.e., qualitative algebraic variables standing for “quantifiable kind-of-commodity category-qualifiers”[e.g., in the case of B = coat], or for standard, implicltly-combined, "qualitative-multiplication"-combined, metrical-unit-qualifier “of”, or “times”, kind-of-commodity category-as-unit-qualifier variables [e.g., in the case of A = yards of linen = yards “times” linen].   The contents of the category, i.e., of the «arithmos» of «monads», of the elementary form of value, can be rendered explicitly, using the examples of the truncated universe[-of-discourse] of commodities that Marx explicitly exemplifies in Section 3. of volume I of Capital, using them as a set of elements, as follows –  The Elementary Value-Form, as “intension”, corresponds to the following 42-element set of ‘polad’ «monads», or elements, as [a truncated example of] its “extension”, using only Marx’s seven-commodity “exemplificatory” universe of discourse from Ch. I., Sec. 3, of Capital I –  {[20 yds. linen = 1 coat],.............[1 coat = 20 yds. linen],..[10 lbs. tea = 20 yds. linen],...[40 lbs. coffee = 20 yds. linen],.[1 qr. corn = 20 yds. linen], ...[20 yds. linen = 10 lbs. tea],......[1 coat = 10 lbs. tea],......[10 lbs. tea = 1 coat],..............[40 lbs. coffee = 1 coat],............[1 qr. corn = 1 coat], ...[20 yds. linen = 40 lbs. coffee],.[1 coat = 40 lbs. coffee],.[10 lbs. tea = 40 lbs. coffee],.[40 lbs. coffee = 10 lbs. tea],......[1 qr. corn = 10 lbs. tea], ...[20 yds. linen = 1 qr. corn],.......[1 coat = 1 qr. corn],.......[10 lbs. tea = 1 qr. corn],........[40 lbs. coffee = 1 qr. corn],......[1 qr. corn = 40 lbs. coffee], ...[20 yds. linen = 2 oz. gold],.......[1 coat = 2 oz. gold],......[10 lbs. tea = 2 oz. gold],........[40 lbs. coffee = 2 oz. gold],......[1 qr. corn = 2 oz. gold], ...[20 yds. linen = ½ ton iron],......[1 coat = ½ ton iron],......[10 lbs. tea = ½ ton iron],.......[40 lbs. coffee = ½ ton iron],.....[1 qr. corn = ½ ton iron],  ...[2 oz. gold = 20 yds. linen],..[½ ton iron = 20 yds. linen], ...[2 oz. gold = 1 coat],.............[½ ton iron = 1 coat], ...[2 oz. gold = 10 lbs. tea],......[½ ton iron = 10 lbs. tea], ...[2 oz. gold = 40 lbs. coffee],.[½ ton iron = 40 lbs. coffee], ...[2 oz. gold = 1 qr. corn],.......[½ ton iron = 1 qr. corn], ...[2 oz. gold = ½ ton iron],......[½ ton iron = 2 oz. gold] }  -- using Marx’s “arithmetical” or “constant(s)” form, or, using F.E.D.’s abbreviation of Marx’s “algebraical” or “variables(s)” form, to --  {[zA = uB],..[uB = zA],..[vC = zA],..[wD = zA],..[xE = zA],..[yF = zA],..[aG = zA], ...[zA = vC],..[uB = vC],..[vC = uB],..[wD = uB],..[xE = uB],.[yF = uB],..[aG = uB], ...[zA = wD],.[uB = wD],.[vC = wD],.[wD = vC],..[xE = vC],.[yF = vC],..[aG = vC], ...[zA = xE],..[uB = xE],..[vC = xE],...[wD = xE],..[xE = wD],.[yF = wD],.[aG = wD], ...[zA = yF],..[uB = yF],..[vC = yF],...[wD = yF],..[xE = yF],..[yF = xE],...[aG = xE], ...[zA = aG],..[uB = aG],..[vC = aG],..[wD = aG],..[xE = aG],..[yF = aG],..[aG = yF]}    B. Expanded Value-Form. Marx “notates” this value-form as follows –  “z Com. A = u Com. B or = v Com. C or = w Com. D or = x Com. E or = &c.  (20 yards of linen = 1 coat or = 10 lbs. tea or = 40 lbs. coffee or = 1 quarter corn or = 2 ounces gold or = ½ ton iron or = &c.)”   -- or, in F.E.D. shorthand --  zA = uB = vC = wD = xE = yF = aG  -- and describes it thusly [ibid., pp. 62-63] --  “The value of a single commodity, the linen, for example, is now expressed in terms of numberless other elements of the world of commodities. Every other commodity now becomes a mirror of the linen’s value. It is thus that, for the first time, this value shows itself in its true light as a congelation of undifferentiated human labour. For the labour that creates it, now stands expressly revealed, as labour that ranks equally with every other sort of human labour, no matter what its form, whether tailoring, ploughing, mining, &c., and no matter, therefore, whether it is realized in coats, corn, iron, or gold. The linen, by virtue of the form of its value, now stands in a[n exchange-value – M.D.] social relation, no longer with only one [other – M.D.] kind of commodity, but with the whole world of commodities[, itself excepted – M.D.]. As a commodity, it is a citizen of that world. At the same time, the interminable series of value equations implies, that as regards the value of a commodity, it is a matter on indifference under what particular form, or kind, of use-value it appears.”  -- which, note well, also indicates that the units, «monads», or elements of the Expanded Form of Value «arithmos» / category, are, precisely, “meta-units”, “meta-«monads»”, or “meta-elements” of the units, «monads», or elements of the Elementary Form of Value, so that the «arithmos» / category of the “Expanded Value-Form” is a “meta-«arithmos»”, “meta-set”, or “meta-category” of the “Elementary Value-Form”«arithmos» / category / set.   The contents of the “meta-category”, i.e., of the “meta-«arithmos»” of “meta-«monads»”, of the expanded form of value, can be rendered explicitly, using the examples of the truncated universe[-of-discourse] of commodities that Marx explicitly exemplifies in Section 3. of volume I of Capital, as a “meta-set” of “meta-elements”, as follows –  The Expanded Value-Form, as “intension”, corresponds to the following 7-element set of ‘polad’ elements, as [a truncated example of] its “extension”, using only Marx’s seven-commodity “exemplificatory” universe of discourse –  {[20 yds. linen = 1 coat = 10 lbs. tea = 40 lbs. coffee = 1 qr. corn = 2 oz. gold = ½ ton iron], ...[1 coat = 10 lbs. tea = 40 lbs. coffee = 1 qr. corn = 2 oz. gold = ½ ton iron = 20 yds. linen],  ...[10 lbs. tea = 40 lbs. coffee = 1 qr. corn = 2 oz. gold = ½ ton iron = 20 yds. linen = 1 coat], ...[40 lbs. coffee = 1 qr. corn = 2 oz. gold = ½ ton iron = 20 yds. linen = 1 coat = 10 lbs. tea], ...[1 qr. corn = 2 oz. gold = ½ ton iron = 20 yds. linen = 1 coat = 10 lbs. tea = 40 lbs. coffee], ...[2 oz. gold = ½ ton iron = 20 yds. linen = 1 coat = 10 lbs. tea = 40 lbs. coffee = 1 qr. Corn], ...[½ ton iron = 20 yds. linen = 1 coat = 10 lbs. tea = 40 lbs. coffee = 1 qr. corn = 2 oz. gold]}  -- using Marx’s “arithmetical” or “constant(s)” form, or, using F.E.D.’s abbreviation of Marx’s “algebraical” or “variables(s)” form, to --  {[zA = uB = vC = wD = xE = yF = aG], ...[uB = vC = wD = xE = yF = aG = zA],  ...[vC = wD = xE = yF = aG = zA = uB], ...[wD = xE = yF = aG = zA = uB = vC], ...[xE = yF = aG = zA = uB = vC = wD], ...[yF = aG = zA = uB = vC = wD = xE], ...[aG = zA = uB = vC = wD = xE = yF]}  -- which can be equivalently rendered as --  {[zA = uB, or (zA ) = vC, or (zA ) = wD, or (zA ) = xE, or (zA ) = yF, or (zA ) = aG], ...[uB = vC, or (uB) = wD, or (uB) = xE, or (uB) = yF, or (uB) = aG, or (uB) = zA],  ...[vC = wD, or (vC) = xE, or (vC) = yF, or (vC) = aG, or (vC) = zA, or (vC) = uB], ...[wD = xE, or (wD) = yF, or (wD) = aG, or (wD) = zA, or (wD) = uB, or (wD) = vC], ...[xE = yF, or (xE) = aG, or (xE) = zA, or (xE) = uB, or (xE) = vC, or (xE) = wD], ...[yF = aG, or (yF) = zA, or (yF) = uB, or (yF) = vC, or (yF) = wD, or (yF) = xE], ...[aG = zA, or (aG) = uB, or (aG) = vC, or (aG) = wD, or (aG)= xE, or (aG) = yF]}   -- revealing more clearly how each element of the Expanded Form of Commodity Value set is made out of six of the 'polad' elements of the Elementary Form of Commodity Value set.   Note that, in the above, each unit, or «monad», or element of the above-rendered set, or «arithmos», for the Expanded Form of Commodity Value category is a META- Elementary Form of Commodity Value -unit, -«monad», or -element.   That is, each "one" is a META-"one", or META-unit, or META-«monad», or META-element, of the "ones", or units, or «monads», or elements of the set, or «arithmos», of the Elementary Form of Commodity Value category, each "one" being made up out of the heterogeneous multiplicity of the "ones", or units, or «monads», or elements of the set, or «arithmos», of the Elementary Form of Commodity Value category, and hence the Expanded Form of Commodity Value «arithmos» / category is a META-«arithmos», or META-set, of the Elementary Form of Commodity Value category / «arithmos», or set.  Thus, each unit, each «monad», or each element of the Expanded Form of Commodity Value category / «arithmos», is an «aufheben» determinate change ["determinate negation"] of, a conservation of, and a scope-/scale-elevation of, the units, «monads», or elements of the Elementary Form of Commodity Value category / «arithmos».  Thus the generic expectations of the F.E.D. generic algorithm for dialectic are exhaustively fulfilled in this special case -- in this special algorithm for the opening triad of the systematic-dialectical categorial progression of Marx's Capital, in which the whole of the rest of the content of Marx's Capital is implicitly, holographically, holonomically, immanently "contained", for those who at least "chaotically" [Marx], "phenomenologically" experience/empirically know the capitals-system, and from which the rest of the totality of that immanent content of Marx's Capital is systematically, dialectically derived therein.    This “meta-«monad»-ic” character of the Expanded Value-Form «monads», relative to the Elementary Value-Form «monads», is even more visibly apparent in the following rendering, which applies the principle of the rendering that Marx uses for the «monads» of the General Value-Form – Value-Form C. – which are “horizontal inversion negations” of the «monads» of the Expanded Value-Form, Value-Form B., viz. –   .............................................../ 1 coat ............................................./ 10 lbs. of tea .........................................../ 40 lbs. of coffee ........................................./1 quarter of corn 20 yards of linen....=...<  .........................................\2 ounces of gold ..........................................\ ½ ton of iron ...........................................\ x com. A., etc.  -- or, in F.E.D. shorthand --  ......................../ uB ...................../ vC .................../ wD ................./ xE zA..=..< ................\ yF .................\ aG ..................\ bH  -- so that the seven elements of the restricted Expanded Value-Form set are rendered, after this fashion, in Marx’s “algebraical”, or “variables-only”, mode, using the F.E.D. shorthand, as --   {....................../ uB ....................../ vC ..................../ wD ................../ xE [zA..=..< ..................\ yF ...................\ aG],  ........................./ zA ....................../ vC .................../ wD ................./ xE [uB..=..< .................\ yF ..................\ aG],  ......................../ zA ...................../ uB .................../ wD ................./ xE [vC..=..< .................\ yF ..................\ aG],  .........................../ zA ......................../ uB ...................../ vC .................../ xE [wD..=..< ..................\ yF ...................\ aG],  ........................../ zA ......................./ uB ..................../ vC ................../ wD [xE..=..< ..................\ yF ...................\ aG],  ..................../ zA ................./ uB ............../ vC ............/ wD [yF..=..< ............\ xE ..............\ aG],  ........................../ zA ......................./ uB ..................../ vC ................../ wD [aG..=..< ..................\ xE ....................\ yF] }  -- or, after this same fashion, in Marx’s “arithmetical” or “constants-only” mode, as --  {............................................/ 1 coat .............. ............................./ 10 lbs. of tea ............ ............................/ 40 lbs. of coffee ................ ....................../1 quarter of corn [20 yards of linen..=..<  ......... .............................\2 ounces of gold ............... .........................\ ½ ton of iron],  ......................................./ 20 yards of linen ..................................../ 10 lbs. of tea ................................./ 40 lbs. of coffee .............................../1 quarter of corn [1 coat.......=.......<  ...............................\2 ounces of gold ................................\ ½ ton of iron],  ......................................../ 20 yards of linen ...................................../ 1 coat ................................../ 40 lbs. of coffee ................................/1 quarter of corn [10 lbs. of tea..=..<  .................................\2 ounces of gold ...................................\ ½ ton of iron],  ............................................../ 20 yards of linen .........................................../ 1 coat ......................................../ 10 lbs. of tea ....................................../1 quarter of corn [40 lbs. of coffee..=..<  ......................................\2 ounces of gold ........................................\ ½ ton of iron],  ............................................../ 20 yards of linen .........................................../ 1 coat ......................................../ 10 lbs. of tea ....................................../40 lbs. of coffee [1 quarter of corn..=..<  ......................................\2 ounces of gold ........................................\ ½ ton of iron],  ............................................../ 20 yards of linen .........................................../ 1 coat ......................................../ 10 lbs. of tea ....................................../40 lbs. of coffee [2 ounces of gold..=..<  ......................................\1 quarter of corn ........................................\ ½ ton of iron],  ........................................./ 20 yards of linen ....................................../ 1 coat .................................../ 10 lbs. of tea ................................./40 lbs. of coffee [½ ton of iron..=..<  .................................\1 quarter of corn ...................................\ 2 ounces of gold] }  

C. General Value-Form. Marx “notates” this value-form as follows [ibid., p. 65] –

“1 coat.....................\
10 lbs. of tea..........\
40 lbs. of coffee....\
1 quarter of corn..\
.................................................>...=...20 yards of linen
2 ounces of gold.../
½ ton of iron......../
x com. A., etc./”

-- or, in F.E.D. shorthand --

uB........\
vC...........\
wD..........\
xE...............\
...........................>...=...zA
yF................/
aG............/
bH......../

-- and Marx describes the General Value-Form, and the dialectical transition to the General Value-Form from the Expanded Value-Form, and the subsequent dialectical transition from the General Value-Form to the Money Value-Form, in the following, exceedingly brilliant, way [ibid., pp. 64-70, emphasis added by M.D.] --

“3. Defects of the Total or Expanded form of value. In the first place, the relative expression of value is incomplete because the series representing it is interminable. The chain of which each equation of value is a link, is liable at any moment to be lengthened by each new kind of commodity that comes into existence and furnishes [it with – M.D.] the material for a fresh expression of [its – M.D.] value.

In the second place, it is a many-coloured mosaic of disparate and independent expressions of value.

And lastly, if, as must be the case, the relative value of each commodity in turn, becomes expressed in this expanded form, we get for each of them a relative value-form, different in every case, and consisting of an interminable series of expressions of value.

The defects of the expanded relative value form are reflected in the corresponding equivalent form.

Since the bodily form of each single commodity is one particular equivalent form amongst numberless others, we have, on the whole, nothing but fragmentary equivalent forms, each excluding the others.

In the same way, also, the special, concrete, useful kind of labour embodied in each particular equivalent, is presented only as a particular kind of labour, and therefore not as an exhaustive representative of human labour generally.

The latter, indeed, gains adequate manifestation in the totality of its manifold, particular, concrete forms. But, in that case, its expression in an [M.D.: practically-]infinite series is ever incomplete and deficient in unity.

The expanded relative value-form is, however, nothing but the sum of the elementary relative expressions or equations of the first kind, such as

..........................................................20 yards of linen..=..1 coat
..........................................................20 yards of linen..=..10 lbs. of tea, etc.

Each of these implies the corresponding inverted equation,

............................................................1 coat..=..20 yards of linen
..........................................................10 lbs. of tea..=..20 yards of linen, etc.

In fact, when a person exchanges his linen for many other commodities, and thus expresses its value in a series of other commodities, it necessarily follows, that the various owners of the latter exchange them for the linen, and consequently express the value of their various commodities in one and the same third commodity, the linen. If then, we reverse the series, 20 yards of linen = 1 coat or = 10 lbs. of tea, etc., that is to say, if we give expression to the converse relation already implied in the series, we get,

...............................C. The General form of value

..1 coat......................\
10 lbs. of tea...........\
40 lbs. of coffee.....\
.. 1 quarter of corn...\
.................................................>...=...20 yards of linen
...2 ounces of gold.../
...½ ton of iron......../
...x com. A., etc./

1. The altered character of the form of value

All commodities now express their value (1) in an elementary form, because in a single commodity; (2) with unity, because in one and the same commodity. This form of value is elementary and the same for all, therefore general.

. . .

[M.D.: This] universal equivalent form is a form of value in general.

It can, therefore, be assumed by any commodity.

On the other hand, if a commodity be found to have assumed the universal equivalent form (form C), this is only because and in so far as it has been excluded from the rest of all other commodities as their equivalent, and that by their own act.

And from the moment that this exclusion becomes finally restricted to one particular commodity, from that moment only, the general form of relative value of the world of commodities obtains real consistence and general social validity.

The particular commodity, with whose bodily form the equivalent form is thus socially identified, now becomes the money-commodity, or serves as money.

It becomes the special social function of that commodity, and consequently its social monopoly, to play within the world of commodities the part of the universal equivalent.

Amongst the commodities which, in form B, figure as particular equivalents of the linen, and in form C, express in common their relative values in linen, this foremost place has been attained by one in particular -- namely, gold.

If, then, in form C we replace the linen by gold, we get,

..............................................D. The Money-form

..1 coat.........................=.....\
10 lbs. of tea............=.......\
40 lbs. of coffee.....=........\
.. 1 quarter of corn.=..........\
..............................................................>....2 ounces of gold
20 yards of linen....=........./
...½ ton of iron..........= ...../
...x com. A., etc. .....= ./

. . .

The difficulty in forming a concept of the money-form, consists in comprehending the universal equivalent form, and as a necessary corollary, the general form of value, form C.

The latter is deducible from form B, the expanded form of value, the essential component element of which, we saw, is form A, 20 yards of linen = 1 coat or x commodity A = y commodity B.

The simple commodity form is therefore the germ of the money-form.”

The seven elements of the restricted General Value-Form set are rendered, in Marx’s “algebraical”, or “variables-only”, mode, using the F.E.D. shorthand, as --

{

[uB........\
...vC...........\
...wD..........\
...xE...............\
..............................>...=...zA],
...yF.............../
...aG.........../

[zA.........\
...vC...........\
...wD..........\
...xE...............\
..............................>...=...uB],
...yF.............../
...aG.........../

[zA.........\
...uB..........\
...wD..........\
...xE...............\
..............................>...=...vC],
...yF.............../
...aG.........../

[zA.........\
...uB...........\
...vC.............\
...xE...............\
..............................>...=...wD],
...yF.............../
...aG.........../

[zA.........\
...uB...........\
...vC.............\
...wD.............\
..............................>...=...xE],
...yF.............../
...aG.........../

[zA.........\
...uB...........\
...vC..............\
...wD..............\
...............................>...=...yF],
...xE................/
...aG............/

[zA.........\
...uB...........\
...vC..............\
...wD..............\
...............................>...=...aG]
...xE................./
...yF............../
}

-- or, after this same fashion, in Marx’s “arithmetical” or “constants-only” mode, as --

{
[1 coat......................\
10 lbs. of tea...........\
40 lbs. of coffee.....\
.. 1 quarter of corn...\
.................................................>...=...20 yards of linen],
...2 ounces of gold.../
...½ ton of iron......../

[20 yards of linen..\
..10 lbs. of tea...............\
..40 lbs. of coffee.........\
..1 quarter of corn..........\
......................................................>...=...1 coat],
..2 ounces of gold........./
...½ ton of iron............/

[20 yards of linen..\
.....1 coat.............................\
..40 lbs. of coffee..........\
..1 quarter of corn...........\
........................................................>...=...10 lbs. of tea],
..2 ounces of gold........../
...½ ton of iron............./

[20 yards of linen...\
.....1 coat..............................\
..10 lbs. of tea...................\
.....1 quarter of corn.........\
........................................................>...=...40 lbs. of coffee],
..2 ounces of gold........../
...½ ton of iron............./

[20 yards of linen...\
.....1 coat..............................\
..10 lbs. of tea...................\
..40 lbs. of coffee.............\
.........................................................>...=...1 quarter of corn],
..2 ounces of gold.........../
...½ ton of iron............../

[20 yards of linen...\
.....1 coat..............................\
..10 lbs. of tea...................\
..40 lbs. of coffee.............\
.........................................................>...=...2 ounces of gold],
.....1 quarter of corn......./
...½ ton of iron............../

[20 yards of linen...\
.....1 coat..............................\
..10 lbs. of tea...................\
..40 lbs. of coffee.............\
.........................................................>...=...½ ton of iron],
.....1 quarter of corn......./
.....2 ounces of gold.../
}.

Finally, note that we have shown, above, in detail, that each “logical individual”, «monad», unit, or element of the “category of individuals”, “«arithmos» of «monads»”, “assemblage of units”, or “set of elements” known herein as (3) [Marx’s] General Value-Form, or Value Form C., is a “complex unity”, or “dialectical synthesis”, or “hybrid” of (2) “logical individuals”, «monads», units, or elements of the “category of individuals”, “«arithmos» of «monads»”, “assemblage of units”, or “set of elements” known herein as [Marx’s] Expanded Value-Form, or Value Form B., which is the first “meta-«arithmos»”, first “meta-assemblage”, or first “meta-set” of the dialectical progression of categories, which constitutes Marx’s «Kapital», and of, and in intimate inner “unitation” with (1) “logical individuals”, «monads», units, or elements of the “category of individuals”, “«arithmos» of «monads»”, “assemblage of units”, or “set of elements” known herein as [Marx’s] Elementary Value-Form, or Value Form A., which is the “«arché»-«arithmos»”, or beginning category of Marx’s multi-volume «magnum opus» entitled Capital: A Critique of Political Economy, and which is the category from which all subsequent categories in Marx’s «Kapital» “descend” [or “ascend”, in terms of explicit explanatory richness / complexity/ “thought-concreteness” / ”determinateness”].

The “poladic” units, «monads», or elements – the “polads” – of Value Form C. each arise as a hybridization of “polads” of Value Form A. with “polads” of Value Form B..

The generic unit, «monad», or element of Value Form C. , the “General Form of Commodity Value”, requires the following format to express its meaning –

[αΑ......\
...βΒ........\
....................>...=...xX]
.............../
............/

Each “polad” of Value Form C. takes the “solitary” format of its “Equivalent” pole from that of the “Equivalent” pole of the “polads” of Value Form A., the “Elementary Form of Commodity Value”, whose generic unit, «monad», or element is –

[αΑ...=..xX.].

Each “polad” of Value Form C. takes the “array” format of its “Relative” pole from that of the “Equivalent” pole of the “polads” of Value Form B., the “Expanded Form of Commodity Value”, and has that format applied to its “Relative” pole. The generic unit, «monad», or element of Value Form B. is –

...................../αΑ ................../ ...βΒ [xX..=..<......... ..................\.... ....................\......]. 
Thus, each Value Form C. unit is both (i.) a hybridization of Value Form A. with Value Form B., in which the format unique to the “Equivalent” pole of Value Form A. supplies the format for the “Equivalent” pole of Value Form C., and in which the format unique to the “Equivalent” pole of Value Form B. supplies the format for the “Relative” pole of Value Form C., and (ii.) a “horizontal-reversal negation” of each unit of Value Form B., i.e., is a poles[-formats-]reversal of the poles[-formats] of the units of Value Form B..

Commentary: The progression of «arithmoi»[ of «monads»] implicit in the progression of “categorograms” that models the opening dialectical triad of Marx’s Capital, namely, the “categorograms-progression” --

Value Form A. ..--->

Value Form A. + Value Form B. ..--->

Value Form A. + Value Form B. + Value Form C. .. --->...

-- is, using F.E.D.’s shorthand for Marx’s “algebra” --

{[αΑ...=.. xX.]}..--->

................................................................/αΑ ............................................................./ ...βΒ {[αΑ...=... xX]}....&....{[xX..=..<....... .............................................................\..... ...............................................................\..]} ..---> 

...................................................................../αΑ ................................................................../ ...βΒ {[αΑ...=...xX]}....&....{[xX.....=.....<........ ..................................................................\..... ....................................................................\..]}

{[αΑ...............\
βΒ..................\
&........................................>...=...xX]} ..--->...
....................................../
................................./
 -- or --

set of all Form A. “polads” --->

set of all Form A. “polads”..&..set of all Form B. “polads” --->

Form A. set..&.Form B. set..&.Form C. set..--->....

Value Form A. is the ultimate «arché»-category / «arithmos» Marx’s multi-volume «magnum opus», corresponding to the F.E.D. dialectical-algebraic symbol A = qA.

Value Form B. is the «arithmos» formed by the «aufheben» “meta-«monad»-ization” of the «monads» the Value Form A. «arché»«arithmos», as we have seen above, corresponding to the F.E.D. dialectical-algebraic symbol B = qB = qAA.

The format of each «monad» of the Value Form C. «arithmos» is a “complex unity” of formats of the «monads» of Value Form A. with those of Value Form B. , as we have also seen above, corresponding to the F.E.D. dialectical-algebraic symbol C = qC = qBA.

TO BE CONTINUED  
 Next: 
Part II. C.: Intuitive Account of the F.E.D. Generic Dialectics Algorithm --  [Meta-]Systematic Dialectics Example 3. – The Opening Triad of F.E.D.’s Dialectic of the Dialectical Arithmetics.

M.D.: Deductive Proof of the F.E.D. “Fundamental Law of Dialectics” from the Axioms of the NQ_ Dialectical Arithmetic [expanded on 12/11/2011] -- 85th Blog Entry

Fri, 9 Dec 2011 14:45:07 -0800

Dear Site Users,

I have reproduced, below, the 85th blog entry from Miguel Detonacciones’ blog “F.E.D. Dialectics”, with the permission of the author, and with F.E.D. edits, also approved by the author.

Dialogically Yours,

Aoristos Dyosphainthos
Voting Member of, and Public Liaison Officer for,
the F.E.D. General Council

A Simple Deductive Proof of the F.E.D. “Fundamental Law of Dialectical Thought” from the Axioms of the NQ_ Axiomatic System of Dialectical Arithmetic.    Dear Readers,  Introduction. The F.E.D. “first dialectical arithmetic” is [also] a dialectical symbolic logic, whose algebra serves as a first [i.e., as a simplest, most abstract, least determinate in expressive capability] algebra for dialectical logic.   This F.E.D. algebra for dialectical logic [a]mends the original, Boolean algebra for [syllogistic formal] logic, at its very root: it [a]mends the key flaw in Boole’s “Fundamental Law of Thought”, which Boole expressed – in his 1847 work “The Mathematical Analysis of Logic”, and, again, in his 1854 work “The Laws of Thought”, as explicated below.  
 Boole used x as an algebraic variable, or algebraic “unknown”, standing for a generic class or category – standing for any class or category – and wrote his “Fundamental Law of Thought” as a universal assertion of the reduction of “logical nonlinearity” [e.g., second [or higher] degree or power] to “logical linearity” [first degree or first power]: for every category, x –   x^2 equals x^1  -- or --  x squared equals x  -- or simply --  xx = x.  The Boolean versions of the numbers ‘0’ and ‘1’ conform to this “law”: 0 × 0 = 0; 1 × 1 = 1.  Boole’s law thus captures only a purely external version of the conservation “moment” – only the first “moment” of the dialectical «aufheben» operation, which dialectical logic asserts to be the universal operation of nature-process, including of human[oid] mind-process.  The other two “moments” or aspects of the dialectical «aufheben» operation are transformation, meaning qualitative, determinate change or determinate negation, and elevation [in level/scale of existence].  The F.E.D. “Fundamental Law of Dialectical Thought”, “Fundamental Law of Dialectical Logic”, or [just] “Fundamental Law of Dialectical Process”,
[a]mends Boole’s law as explicated below. 
 F.E.D. uses x as an algebraic variable, or “algebraic unknown”, standing for a generic ontological class or category – standing for any primary “kind-of-being” class or category – as an «aufheben» [self- or other-]operation, or operator.  Note Tony Smith’s definition of a “category” in his 1990 work “The Logic of Marx’s Capital”, in which he describes the systematic dialectical logic employed by Marx to construct his «magnum opus»:  “. . . 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”. [p. 5].  That is, the meaning of a category, as standing for an “assemblage” or “set” or “population” of individuals, “contains” both the unity-moment, the aspect of what those individuals have in common, the similarity that they all share, and also, equally, the differences-moment, the aspect of the dis-similarities – the individual differences – that “differentiate”, “distinguish”, and “divide” each individual “in” a category from all others “in” its same class or category.  Given that context, F.E.D. writes their “Fundamental Law of Dialectics” as a universal assertion that “logical nonlinearity” does not reduce to “logical linearity”; that, on the contrary, “logical nonlinearity” makes a qualitative difference: for every primary ontological category, x --  x squared is qualitatively unequal to x  -- or --  xx = x + delta_x  -- or --  xx = x + qxx  -- such that delta_x or qxx represents a category or class that is qualitatively different from that for which x stands, by way of an «aufheben» conservation/transformation/elevation of the individuals for which x stands, and specifically, such that delta_x or qxx stands for a new ontological category, representing for a different category or class -- a different kind of individuals than those for which x stands. 
 F.E.D.’s “Fundamental Law of Dialectical Thought” is a “strong contrary” to Boole’s “Fundamental Law of Thought”.   A “weak contrary” to Boole’s law would be –  “x squared is quantitatively unequal to x”, e.g., “x squared is greater than x”, which works for all “Whole Numbers” other than 0 and 1.  That is, F.E.D.’s “law” requires ‘dialectical meta-numbers’ whose squaring or second power generates a value which is not even within the same “number-line” as its linear or first power; a nonlinear or second power value which resides in a different perpendicular direction – in a different dimension – from its first or linear power. F.E.D. has discovered such “meta-numbers”, and has axiomatized their workings [see Blog-entry # 4 of this Blog].  In general, x ---> xx = x + qxx expresses an abstracted, simplified, purely-qualitative description of what is in reality a “quanto-qualitative” dialectical process of “quantitative change leading to / becoming / turning into qualitative change”.  For example, if we “apply” x to stand for a cosmological population of atoms, a, and qaa to stand for a cosmological population of molecules, also denoted by m, then –  a ---> aa = a + qaa = a + m  -- describes a cosmological new-ontology-generating process in which the expanded self-reproduction of the population-quantity of atoms, together with self-increase of the measure of their quantitative “density”, i.e., of their physical-spatial “concentration”, leads to a growing frequency and intensity of interactions between/among atoms – also presented as an “intra-action” within the population of atoms, or as a “self-interaction” of the category “atoms” – denoted by aa – which reaches a critical threshold, or “metafinite resonance singularity”, whereafter it gives rise, not only, possibly, to further increases in its own population-quantity measure, and in its own quantitative density measure, but possibly also to a new kind of population of a new kind of “individuals” – molecules.  This description implies that qaa = m because molecules are the result of a self-«aufheben» operation of atoms upon atoms: each molecule is “meta-atom”, made up out of, and “internalizing”, a typically heterogeneous multiplicity of atoms. Each molecule thus represents an «aufheben» conservation / transformation / elevation of a sub-population of [“former”] atoms-only self-organizations.  Thus a "qaa" means a "meta-atom", or "atom of the second degree", made up out of a typically heterogeneous multiplicity of "atoms of the first degree".

Therefore, while the Boolean x operator -- x grasped as a [potential] operation upon itself, or upon other Boolean operators/operations -- is only a "preservation" operation, omitting the other two aspects of the full «aufheben» conservation / transformation / elevation operation, on the contrary, the F.E.D. x operation -- x gasped as a [potential] operation upon itself, or upon other dialectical-logic classes-or-categories-as-operators/-operations -- represents a full «aufheben» conservation / transformation / elevation operation, under the interpretation that qxx denotes a "meta-«arithmos»" relative to the qx or x «arithmos», that is, that qxx denotes an «arithmos» each of whose «monads» is a "meta-«monad»-ization", or a "meta-unit-ization", of the «monads» of the qx or x «arithmos».    Motivation: Historical Context and Application. It may assist comprehension and conceptualization of the full import and meaning of F.E.D.’s “Fundamental Law of Dialectical Process” -- a simple deductive demonstration of which is presented below -- if we locate that “Law” within the tide of human history – placing it in context within the development of the human phenome, of the “memes pool”, of the “psychohistorical material[ity]” of human history, and of the historical labor of the self-production of humanity.  In the language of the ancient Mediterranean original awakening of [Occidental] philosophy and science -- in the language of the original philosophy of mathematics, which both predates, and continues beyond, the time of Plato –  An «arithmos» is an assemblage of qualitative units – a “population” of individuals of a given kind: a “number” of individuals / individual units.  A «monad» is a qualitative unit.  If we inwardly form a mental abstraction – an idealized object, or mental object – representing such a qualitative unit, or qualitative «monad», in general, then, by referencing assemblages of such generic unit, or generic «monad», idea-objects -- anciently named, by Plato, the «arithmoi monadikoi», the mental “assemblages”, or groups/populations, of ‘mentalized’, or generic, «monads»: abstract, idealized mental “numbers” of abstract, idealized, mental «monads» / units -- then we can construct an «arithmetiké», or «arithmoi»-«techné» -- a universal “«arithmos»-art” [“art” of “number”], a universal “«arithmos»-craft” [“craft” of “number”], or a universal “«arithmos»-tech” [“technique” or “technology” of “number”]: “theoretical arithmetic”.  Diophantus of Alexandria, the father of algebra, in his «circa» 250 C.E. proto-algebraic manuscript entitled “The Arithmetica” -- or the «Arithmetiké» in transliterated Greek – actually “abbreviatively” symbolized the generic «Monadikos», or abstract qualitative unit, long before the advent of the Hindu’s abstract purely-quantitative unit symbol, 1, by the abbreviation Mo [actually, the Greek capital letter “Mu” with the lower-case letter omicron, o, written atop it], so that he symbolized our modern 1 + 1 = 2 by something more like a’Mo & a’Mo = b’Mo [the ancient Mediterranean Greek/Hellenistic civilization used “primed” letters of the Greek alphabet as numerals: «Gematria» or «Isopsephy»], not just by a’ & a’ = b’ as would later, more abstract notations such as our own, reflecting our modern, far deeper phenomic immersion in the “meme-etic” paradigm of what Marx called “The Elementary Form of [Commodity[-Capital-]]Value”, that is, in the core capital-relation consequence of the [mostly unconscious] “law of [capital-]value”, first fully raised into widespread human consciousness as such by Marx. 
 Our earlier example, in the introductory section above, concerned a cosmological population, or «arithmos», of atom units: a “number” of atoms – e.g., a galactic interstellar “atomic cloud”, soon to transform itself into a proto-stellar “molecular cloud”.  Unlike Boole’s “primary” algebra for [formal] logic -- as his “existential logic”, “existence logic”, or “logic of things” -- in which aa = a, the F.E.D. algebra for dialectic does not get stuck in stasis –  a ---> aa = a + qaa = a + m  -- or –  “atomic cloud” ---> [goes to”, or, in time, “becomes”] “molecular cloud”.   That is, once sufficiently expandedly self-reproduced quantitatively – once “number-ous” or “numerous” enough, or “populous” enough, and “densified” enough -- the atoms in the “atomic cloud” give rise to molecules as well, that is, to a qualitatively-expanded reproduction; to the irruption of the new “cosmo-ontology”; to that new kind of being, that we call molecules.   The atoms no longer conduce to mere quantitative change, within a qualitative “onto-stasis”, but to qualitative change in the form of “onto-dynamasis”: to the irruption of a new, “molecules” ontology, or kind of being, where there once was only the atoms kind of being [ + its predecessor kinds].  The thus self-transformed, proto-stellar “molecular cloud” is an [“evolute”] “cumulum” of [at least] two ontologies; a “cumulum” of both the type a and the type m “kinds of being” --   “atomic cloud” = . . .+ a;  “molecular cloud” = . . .+ a + m.  The Boolean “Fundamental Law”, xx = x, the law of eternal simple reproduction, is insufficient to construct a more scientific science, a more empirically faithful science, a science that more fully “follows nature”.  To achieve the later science – Marxian, dialectical science, the positive fruition of the dialectical, immanent critique of [mostly] unconsciously ideology-vitiated capitalist science – we need a[n a]mended “Fundamental Law”, the “Law” of the qualitatively/ontologically self-expanding self-reproduction of universe ontology –  x ---> xx = x + qxx  -- or --  qx ---> qx × qx = qx + qxx.  

Juxtaposing the two “laws” – the Boolean and the “contra-Boolean” --

xx = x

-- versus –

xx = x + qxx 
-- highlights how the F.E.D. “Fundamental Law” still [externally] «aufheben»-contains and [partially] retains the Boolean “Fundamental Law”, in the first term of the RHS of the F.E.D. law’s equation --

xx = x . . .
 -- but how it also goes beyond that Boolean law, to [internally] «aufheben»-conserve, [internally] «aufheben»-elevate, and to qualitatively-transform, determinately-change, or «aufheben»-negate that “Boolean” term, x, via the second term of the RHS of the F.E.D. law’s equation --

xx = . . . qxx
 -- which “double-contains” x , as xx, at the sub-script level of its generic ideographical symbol, qxx.

Thus, «in toto», the F.E.D. “Fundamental Law of Dialectical Thought” is a full-«aufheben» containment/conservation, qualitative transformation / determinate negation, and elevation [to a new scope/scale/level], of the “Boolean” term, of the x term – the “operand” or “argument” upon which x itself again, but this time as “function” or “operator – operates, or “functions” in –

x(x) = x + delta_x = x + qxx

--- and, indeed, the F.E.D. rule is the positive fruition of a dialectical, immanent critique of the Boolean “law”, of “Boole’s Rule”, as expressed in the first triad of the Dialectic of the Algebras for Logic which we explicated in an earlier blog-entry [wherein WE_ denotes the axioms-system of the Boolean arithmetic for logic, wherein WQ_ denotes the Whole numbers-based version – whose number-space is W = { 0, 1, 2, 3, . . .} -- of the axioms-system of the F.E.D. “First Dialectical Arithmetic”, and wherein WqQE_ denotes the axioms-system of the dialectical synthesis of WQ_ and WE_ ]:

WE_ ---> WE_ + WQ_ ---> WE_ + WQ_ + WqQE_.

F.E.D. has identified four qualitatively-distinct «species» of the «genos» Dialectics in their “[Psycho]Historical Dialectic of the Dialectic Itself [a Dialectic which itself also accords with the F.E.D. “Fundamental Law of Dialectics”].

These four «species» are --

Systematic Dialectics & Historical Dialectics & Meta-Systematic Dialectics & Psychohistorical Dialectics.

For additional background regarding these «species», see blog-entry # 9, and/or the following definitional JPGs in the F.E.D. Glossary [ http://www.dialectics.org/dialectics/Glossary.html ] --

Systematic Dialectics --
http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2014SEP2011,%20Definition,%20SYSTEMATIC%20DIALECTICS,%20JPEG,%20for%2015SEP2011.jpg 
Historical Dialectics --
http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2015SEP2011,%20Definition,%20HISTORICAL%20DIALECTICS,%20JPEG,%20for%2016SEP2011.jpg

Meta-Systematic Dialectics --
http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2015SEP2011,%20Definition,%20%27META-SYSTEMATIC%20DIALECTICS%27,%20JPEG,%20for%2017SEP2011.jpg

Psychohistorical Dialectics --
http://www.dialectics.org/dialectics/Glossary_files/F.E.D.,%20A%20Dialectical%20%27%27Theory%20of%20Everything%27%27,%20Volume%200,%20FOUNDATIONS,%20Edition%201.00,%20last%20updated%2017SEP2011,%20Definition,%20%27PSYCHOHISTORICAL%20DIALECTICS%27,%20JPEG,%20for%2018SEP2011.jpg

In the context of Historical Dialectics, and especially for an x which represents a physical «arithmos», or empirically-given “population” of “individuals”, the formula x^1 ---> x^2 = x + qxx is true primarily because qxx denotes an emergent phenomenon, an emergent property, an emergent quality, and an emergent ontology potential, an addition of new ontology that is already present «qua» potential, [with]in x. This is partially also the case for Psychohistorical Dialectics and for Meta-Systematic Dialectics.

In the context of Systematic Dialectics -- the Dialectics of theoretical systems as generalized, “genericized” idealizations; of “idea-objects” or “mental objects” -- where x(x) = x + qxx is a law of the “expanding reproduction” of ideas, and, especially, for an x which represents a mental-only «arithmos», such as N, or W, i.e., an abstractly-given “population” of “internal”, “mental objects”, most remote from physical/”external” experience, existing as such only for “internal”, “mental experience”, the fundamental formula of Dialectics, x^1 ---> x^2 = x + qxx, is true primarily because the category/concept/idea denoted by qxx is semantically, conceptually implicit [with]in -- is already present as a “presupposition” [with]in – x, but requires mental reflection [“self-dialogue”, or “intra-mind dialogue”], or inter-personal dialogue [“inter-mind dialogue”], to become explicit; to be “lifted up’, or “elevated”, into “explicitude”. This is partially also the case for Meta-Systematic Dialectics and for Psychohistorical Dialectics.

As with a holographic image, or hologram, in which each part of the whole image contains a “sided”, “tendentious”, “distorted”, “deformed”, “fuzzy” replica of that whole image, such that the image as a whole, as “made up out of its parts”, is, in this sense, “made up out of itself”, so it is also with a well-developed, matured theory, or conceptual system. That is, such a theory-system is “holonomic”, meaning that each of its category/concept parts “contains” – entails / involves / presupposes -- all of the other category/concepts of that system implicitly. This “implicitude” can be evoked, or “out-voked”, into “explicitude”, via introspective reflection, or via inter-personal [“Socratic”] dialogue, or via skillful oral, or written, “monologue-ic” presentation.

The pedagogical strategy of Systematic Dialectics , as a dialectical method of totality-theories presentation, is to start with/from the simplest, most abstract of the categories/concepts in this “holonomic” network of mutually-involved, “inter-pre-supposing”, “intergenerate” categories/concepts, as «arché» / beginning, and to proceed from there, evoking a[n ever-lengthening] series of more complex, more “thought-concrete” [more “determinate”; more richly «speci»-fied] categories/concepts, as successor series, from each predecessor series, -- from each less lengthy, less complex, less “thought-concrete”, series -- until the most lengthy series, ending with the most-complex, most-“thought-concrete” category/concept -- one which most richly in-itself represents the total human experience of the [sub-]totality of human experience so theorized -- is reached / presented / evoked.

   Demonstration. A Simple Deductive Proof of the F.E.D. “Fundamental Law of Dialectical Thought” from the Axioms of the NQ_ axioms-system of Dialectical Arithmetic.    Given: n is an element of the set N = { 1, 2, 3, . . .} of "Natural" Numbers.  To Prove: x "times" x is qualitatively unequal to x,   for every x in NQ = { q1, q2, q3, . . .}, the set of the F.E.D. "First Dialectical Meta-Numbers".   Proved:  Assertion #.......Assertion Content_____________________....Assertion
......................................................................................................Justification  1...................................n is an element of N.............................................................Given  2...................................qn is an element of NQ, for every n in N..........Axiom 5  3...................................qn squared = qn + q2n, for every n in N...Axiom 9  4...................................n does not equal 2n, for every n in N..............*  5...................................qn is qualitatively unequal to q2n,  .........................................for every n in N......................................................................Axiom 6  6...................................qn + q2n is qualitatively unequal to qn,  .......................................for every n in N........................................................................Axiom 8  7...................................qn may be represented by x, .......................................for any n in N.............................................................................Equivalent
......................................................................................................Substitution  8...................................x squared is qualitatively unequal to x.............Assertion # 3 &
........................................................................................................................................................Assertion # 6
.....................................................................................................above: ........................................................................................................................................................Q.E.D.   *The proposition "n does not equal 2n, for every n in N." is justifiable, for example, from a corollary of a theorem of the "Natural Numbers" axiomatic system of arithmetic that derives from the "multiplicative identity element" axiom for that system. This theorem asserts that for any element of the set of "Natural Numbers", call it m, that is not the "multiplicative identity element" of that set, and for any element of the set of "Natural Numbers", call it x, x < mx.   Key Upshot: The “contra-Boolean” F.E.D. “Fundamental Law of Dialectical Thought” is not a direct premise, assumption, presumption, postulate, or axiom of the NQ_ axiomatic system of arithmetic for dialectical algebra: it is a proven theorem of that system.  
 The proof above invokes the following Axioms of the NQ_ axiomatic system of arithmetic for dialectical algebra, posted previously to this Blog, in Blog-entry # 4 -- 
   “Dear Readers,  Thought it might be useful to reproduce here the core "axioms" -- the main rules -- of the "rules-system", or "axioms-system" ["axiomatic system"] of the F.E.D. "First Dialectical Arithmetic".   . . .   Axiom 5. For every n in N, qn is in NQ.   [this states the «aufheben» tie between the N and the NQ. The rest of the rules below state "Non-Standard" aspects of the NQ relative to the "Standard" N].   Axiom 6. For any j and k, both in N:   If j is quantitatively unequal to k, then qj is qualitatively unequal to qk.   [This axiom expands the "trichotomy principle" of the "Standard" arithmetics -- the principle that for any arithmetical objects a and b, just one of the following three relations obtains:  a < b, or a = b, or a > b   -- to a "tetrachotomy principle", that adds a fourth possibility, that of qualitative inequality, to the basic possible relations between any pair of arithmetical objects].   . . .    Axiom 8. For all j, and k, both in N:   If j is quantitatively unequal to k, then   qj + qk is qualitatively unequal to qx   for any x in N.
 [This is the rule that makes dialectically interpreted NQ expressions "ontologically anti-reductionist". It means that a "heterogeneous sum" of ontological categories, like C + M in the previous blog-entry, does not reduce or collapse into any single ontological category at the same level of generalization as its summands. This rule too is crucial to the calculations summarized in that previous blog-entry].   Axiom 9. For every j and k, both in N:   [qk] x [qj] = [qj] + [q(k+j)].  [This rule is most crucial of all to the "purely-qualitative, algorithmic calculations" presented in the previous blog-entry. It defines what "multiplication" means among the NQ "dialectical meta-numbers". It is called, by F.E.D., "the double-«aufheben» evolute product rule".]  . . .”     Regards,  Miguel