## Wednesday, September 14, 2011

### Most General Principles of Organization per the F.E.D. Dialectic

The Most Abstract/Generic Rules of Order/Rules of Organization Shared by Dialectical Progressions in General, per F.E.D.

The most abstract/generic, universal principles of organization, per F.E.D.'s "Dialectical Theory of Everything", are the following --

0. Precursor to/"Pre-Vestige" of Dialectical Order/Ordinality: [Quanto-]Peanic Succession. Per the first-order "Peano Postulates", the first four of the modern axioms for the "Natural" arithmetic of the "Natural" Numbers --

N = { 1, 2, 3, 4, . . . }

-- are characterized by what F.E.D. terms "Peanic Succession", "Peanic Ordinality", or just
"[quantitative ]Peanicity", based upon the succession-function, s, such that --

s(n)  =  n + 1 --

1..--->.........2.....--->........3......--->.........4....---> . . .
1..--->..s(1)..--->..s(2)...--->..s(3)..---> . . .

First order "Peano Axioms" of "Peanic ordinality" [original version] --

P11 is a "Natural" Number.
P2.  The successor of any "Natural" Number is also a "Natural" Number.
P3.  No two "Natural" Numbers have the same successor.
P41 is not the successor of any "Natural" Number.

1. "Qualo-Peanic Succession". Per F.E.D.'s first-order axioms for their "First Dialectical Arithmetic", the arithmetic of the "Natural Dialectors" --

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

-- is characterized by what F.E.D. terms "Qualo-Peanic Succession", "Qualo-Peanic Ordinality", or just "Qualo-Peanicity", based upon the ontological categories qualifiers succession-function, s, such that --

s[ q/n ] = q/( s(n) ) = q/( n + 1 ) --

q/1..--->..........q/2....--->.........q/3.....--->.........q/4.....---> . . .
q/1..--->....s(q/1)..--->...s(q/2)..--->...s(q/3)..---> . . .

Examples --

a.  Marx's Dialectical Presentation of his theory of the human-societal [sub-]totality known as "capitalism" --

Commodities ---> Monies ---> MMCC ---> Kapitals ---> . . .

-- wherein "MMCC" stands for the "dialectical synthesis", or "complex unity", of "Monies" and/with "Commodities", i.e., the "real subsumption" of "Commodities" by "Monies", or the continuing "conversions" of Commodities into Monies, and vice versa --

C -- M -- C' -- M' -- C'' -- M'' . . .

-- connoted by --

q/MC, or "Monies Mediated Commodities Circulations".

b.  F.E.D.'s "Dialectical Theory of Everything" Equation Meta-Model of the Cosmos / of the universal process of "Cosmo-Genesis" / of "The Dialectic of Nature" --

sub-nuclear "particles" ---> sub-atomic "particles" --->

sub-nuclears-to-sub-atomics-conversions ---> atoms ---> . . .

-- wherein "sub-nuclears-to-sub-atomics-conversions" connotes the "dialectical synthesis", or "complex unity", of "sub-atomic "particles"" and/with "sub-nuclear "particles"", i.e., the "real subsumption" of "sub-nuclear "particles"" by "sub-atomic "particles"", q/sn, or the reproductive accumulation of [more] "sub-atomic "particles"" through the "ontological conversion" of some of the "sub-nuclear "particles"" into "sub-atomic "particles"".

First order "Seldon Axioms" of "Qualo-Peanic ordinality" [for the F.E.D. NQ "First Dialectical Arithmetic"] --

Q1q/1 is a "Natural Dialector".
Q2.  The successor of any "Natural Dialector" is also a "Natural Dialector".
Q3.  No two "Natural Dialectors" have the same successor.
Q4q/1 is not the successor of any "Natural Dialector".

2.  Seldonian, Cumulative [Evolute] Successions of Series / of Sums.

Dyadic Seldon Function, |-|-|k   =   [ q/1 ]^(2^k) --

...............................................................q/1 ---> q/1 + q/2 ---> q/1 + q/2 + q/3 + q/4 --->. . .
count of qualifier terms:......2^0 = 1...................2^1 = 2.........................................2^2 = 4.......................

Triadic Seldon Function, |-|-|k  =   [ q/1 ]^(3^k) --

q/1 ---> q/1 + q/2 + q/3 ---> q/1+ q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8 + q/9 ---> . . .
count of qualifier terms:
3^0 = 1..........3^1 = 3...........................................................................3^2 = 9..............................................

3. F.E.D. Axiom Q9, the Double-<<Aufheben>> Evolute Product Rule.

Rules 1. and 2., given above, concern that aspect of the generic organization of dialectical categorial progressions that is external to the <<monads>> which constitute the <<arithmoi>> of <<monads>>, or assemblages of units, that are the ontological categories, represented by the generic categorial "qualifiers", q/n, that are the terms of the expressions above.

Rule 3. concerns that aspect of the generic organization of dialectical categorial progressions that is internal to each post-<<arche'>> <<monad>>.

Still very abstract, in order to maintain its generality -- its applicability to all such <<monads>>, despite the vast diversity in detail and specificity of dialectical category / <<arithmos>> progressions, Rule 3. nevertheless asserts something definite about the constitution of the <<monads>> of predecessor relative to those of successor "self-hybrid" <<arithmoi>>:  that "[self-]meta-<<monad>>-ization" of the <<monads>> of the predecessor "self-hybrid" <<arithmos>> is the process of genesis of the content of its successor "self-hybrid" <<arithmos>>.

Q9. Axiom 9: For every j and k in N, hence for every q/j and q/k in NQ,

q/j x q/k   =   q/k + q/(j + k) [for generic ontological-categorial qualifiers].

For "interpreted", or "assigned", ontological-categorial qualifiers, e.g., given "X" as representing the first letter of the name of a specific dialectical ontological category, this rule becomes --

X x X   =   X + q/XX   =   q/X x q/X   =   q/X + q/XX

-- wherein q/XX denotes an <<arithmos>> "meta-Xs, each <<monad>> being a "meta-<<monad>>" of the q/X <<monads>>, such that each <<monad>>" of the q/XX <<arithmos>> is made up out of a [usually heterogeneous] multiplicity of <<monads>> of its predecessor self-hybrid, q/X or "X", <<arithmos>>.

If "Y" represents the first letter of the name of this "meta-" category, then --

X x X   =   X + delta-X   =   X + Y

-- wherein delta-X connotes a purely-qualitative, ontological category "incremental" to the ontological category connoted by X, i.e., wherein delta-X connotes Y in terms of its
"meta-genealogical" ancestry, such that ontological category/<<arithmos>> Y is made up out of <<monads>> which are "meta-<<monads>>" of the <<monads>> of ontological category / <<arithmos>> X.

For example, if X = atoms, then --

atoms x atoms   =   atoms + meta-atoms   =   atoms + delta-atoms   =

atoms + molecules

-- or --

a x a   =   a + meta-a   =   a + delta-a   =   a + m

-- wherein each molecule is a "meta-atom", each one made up out of a [usually heterogenous] multiplicity of atoms, e.g. --

Carbon Dioxide molecule unit/<<monad>> = CO2

-- etc.

For another example, if X = Commodities, then --

Commodities x Commodities = Commodities + meta-Commodities =

Commodities + delta-Commodities = Commodities + Monies

-- or --

C x C   =   C + meta-C   =   C + delta-C   =   C + M

-- wherein each unit of Money is a "meta-Commodity" unit, each one memetically made up out of a [usually heterogenous] multiplicity of Commodities, e.g., of the mentalized "prices-list" in the expectation each human agent of the Mon(ey)(ies)-for-Commodit(y)(ies) exchange <<praxis>>, listing the number of units of each [non-money] commodity that will, by long-established convention, exchange for [<-->] what specific number of units of the money-commodity, e.g., for what specific number of units of gold, constituting the gold-"price" of each such commodity --

20 yards of linen <--> 2 ounces of gold

1 coat <--> 2 ounces of gold

10 lbs. of tea <--> 2 ounces of gold

40 lbs. of coffee <--> 2 ounces of gold

1 qr. of corn <--> 2 ounces of gold

1/2 ton of iron <--> 2 ounces of gold

-- etc.

This "product-rule" for the "product" of the interaction of two ontological <<arithmoi>>/categories -- or, in the "self-hybrid" case, for the self-interaction , or "intra-action", within a single such <<arithmos>>/category -- is characterized as a "double-<<aufheben>>" "product-rule".

This is because the "operand", "argument", or "multiplicand" <<arithmos>> -- connoted by the qualifier that is right-most in the product expression, symbolizing the <<arithmos>> that is being acted upon by the qualifier, or <<arithmos>>-symbol, to its left, the "operator", "function", or "multiplier" -- is <<aufheben>>-conserved twice per this "product-rule".

It is <<aufheben>>-conserved in the first instance, in the form of the unchanged "evolute", "Boolean" simple reproduction, without any <<aufheben>>-elevation or <<aufheben>>-transformation, of that <<arithmos>>'s symbol in the left-hand term of the product-expression's result-expression.

It is <<aufheben>>-conserved in the second instance, in the form of its qualitatively, ontologically changed/ expanded reproduction, this time with both <<aufheben>>-elevation and <<aufheben>>-transformation as well, of that <<arithmos>>'s symbol in the right-hand term of the product-expression's result-expression.

For example --

q/1 x q/n   =   q/n + q/(n + 1)
..........................................^...............^
...........................................|................|
..........................................
1st..........2nd

-- or, in general --

q/j x q/k   =   q/k + q/(k + j)
........................................^...............^
.........................................|................|
.......................................
1st...........2nd

-- and, in terms of "interpreted", or "assigned" qualifiers --

q/X x q/X   =   q/X + q/XX   =   q/X + delta-q/X   =   q/X + q/Y
............................................^................^
............................................|.................|
..........................................
1st............2nd

-- which might tempt one to call this product rule the "triple"-<<aufheben>> evolute product rule", except that the double-X in q/XX here signifies the "meta-<<monad>>-ization" of the <<monads>> of X to form the "meta-<<monads>> which are the <<monads>> of Y.

For examples --

atoms x atoms = atoms + meta-atoms = atoms + delta-atoms =

atoms + molecules[ physically made of atoms ]

-- which, in the three result-expressions, exhibits the double-occurrence of the operand / argument / multiplicand category, atoms [here we have atoms as also the operator / function / multiplier category, as this is a "squared", or "self-reflexive function", expression:  an "operator/operand-identical" expression, a "function/argument-identical" expression, or a "multiplier/multiplicand-identical" expression, i.e., a "subject-[verb-]/object-identical" expression], and --

Commodities x Commodities   =

Commodities + meta-Commodities   =

Commodities + delta-Commodities   =

Commodities + Monies[ memetically made of Commodities ]

-- which, in the three result-expressions, exhibits the double-occurrence of the operand / argument / multiplicand category, Commodities [here, again, we have Commodities as also the operator / function / multiplier category, as this is a "squared", or "self-reflexive function", expression:  an "operator/operand-identical" expression, a "function/argument-identical" expression, or a "multiplier/multiplicand-identical" expression, i.e., a
"subject-[verb-]/object-identical" expression].

Regards,

Miguel