Monday, December 12, 2011

Part II. B.: Systematic-Dialectics Example 2. – The Opening Triad of Marx’s «Kapital».

Part II. B.:  Intuitive Account of the F.E.D. Generic Dialectics Algorithm --

Systematic-Dialectics Example 2. – The Opening Triad of Marxs «Kapital».




Dear Readers,

Part II. B.:  Systematic-Dialectics Example 2. – The Opening Triad of Marxs «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 blog, 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: "-- ...the value-form of the commodity --".

We will illustrate such a dialectical model of the contents of Marx’s «Das Kapital», in this blog entry, but 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 blog entry, 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 replication of, capturing of, or reproduction of, the tables-of-contents of the multiple volumes of Marx’s master-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 -- which are but three different root, abstract, generic expressions for exchange-relations, i.e., for the human-social relations of production within the "super-«genos»" of exchange-value-based human-social relations of production -- 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 [the Relative "form of value" pole] comes from one of the Expanded Form of Value «arithmos»’s units', «monads»'s, or elements' formats, and such that the other “half” of each General Form of Value «arithmos»’s unit, «monad», or element [the Equivalent "form of value" pole] comes from one of the Elementary Form of Value «arithmos»’s units', «monads»'s, or elements' formats, 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 presented 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.].

Theelementary, 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-polepolads, 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],
...[20 yds. linen = 10 lbs. tea],......[1 coat = 10 lbs. tea],......[10 lbs. tea = 1 coat],..............[40 lbs. coffee = 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],
...[20 yds. linen = 1 qr. corn],.......[1 coat = 1 qr. corn],.......[10 lbs. tea = 1 qr. corn],........[40 lbs. coffee = 1 qr. corn],
...[20 yds. linen = 2 oz. gold],.......[1 coat = 2 oz. gold],......[10 lbs. tea = 2 oz. gold],........[40 lbs. coffee = 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 = 20 yds. linen], 
...[1 qr. corn = 1 coat],
...[1 qr. corn = 10 lbs. tea],
...[1 qr. corn = 40 lbs. coffee],
...[1 qr. corn = 2 oz. gold],
...[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-categoryof 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-FormValue-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,

...........................................................CThe 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 –

[aA......\
..bB.........\
.................>...=...xX]
.............../
...wW./

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 –

[aA...=...wW].

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 –

..................../aA
................./ ...bB
[xX...=...<........
.................\.....
...................\.wW].

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” --

{[aA...=...wW]}..--->

.............................................................../aA
............................................................/....bB
{[aA...=...wW]}....&....{[xX.....=.....<.......
............................................................\....
..............................................................\..wW]} ..--->

................................................................/aA ...........................{[aA.......\
............................................................./ ...bB .................................bB.........\
{[aA...=...wW]}....&....{[xX.....=.....<.......................&..............................>...=...xX]}...
.............................................................\.............................................../
...............................................................\..wW]}...................wW../

-- or --

The set of all Form A. “polads” --->

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

The Form A. “polads” set..&..The Form B. “polads” set..&..The Form C. “polads” 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»-izationof the «monads» the Value Form A. «arch黫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.