Dear Hegel-Hegelians Group Members,

Part III.

[CONTINUED FROM PREVIOUS POST, Part II.]



[NOTE: For more on this Dialectical-Algebraic Model of the dialectical
progression of the systems of Dialectical Algebra, formulated in the ideography
of the F.E.D. First Dialectical Algebra, see, for example --

http://www.dialectics.org/archives/pdf/Fract1-1.pdf

-- and the other texts in that series].


2. On the Generic Meanings of "Non-Cross-Product", "Self-Crossed", Terms in the
Dilectical-Algebraic Model of the Dialectic of Nature.


A. Non-Hybrids as "Self-Hybrids": Phenomenology of "Self-Meta-Monad-ization" as
Self-<<Aufheben>> Process.

Terms of the from q|XX, in models built using the N\Q "First Dialectical
Arithmetic", where q|X, is a mnemonic abbreviation for the name of a
previously-possible ontological category, denote the emergence, into possible
existence, of a qualitatively new ontological category, born out of the
'intra-duality', or 'self-duality' of the predecessor ontological category, q|X,
itself. It is for that reason that q|XX is also standardly denoted by
Delta-q|X, wherein the underscored "Delta" connotes a Non-Quantitative,
QUALITATIVE ontological increment, born from out of category X, or,
equivalently, out of category q|X, itself.

For example, consider the primary foucus of this exposition, the "New Model of
the Universe" for the F.E.D. universal taxonomy, level one, thoroughly narrated,
epoch-by-epoch, in Part IV.

The interaction of the ontological category of atomic nuclei, mnemonically
connoted by (^1)q|a, WITH ITSELF, the interaction of such nuclei, as the
<<monads>>, or units, of the (^1)q|a <<arithmos>>, or assemblage of units,
taking place while the population of these atomic nuclei <<monads>>, or units,
increases at an accelerated rate, and as the physical spatial concentration, or
density, of these atomic nuclei <<monads>>, increases apace, eventually crosses
a threshold where formerly overshadowed forces come into play with sufficient
intensity to precipitate the manifestation of a new kind of unit, or <<monad>>,
of a new kind of <<arithmos>>, or assemblage of units: the molecules,
mnemonically connoted by (^1)q|m.

A molecule, as a unit, or <<monad>>, is a 'meta-unit', or 'meta-<<monad>>',
relative to an atom, taken as a "mere" <<monad>>, or "mere" unit.

The <<arithmos>> of molecules, as a ontological category, is therefore also a
'meta-<<arithmos>>', and an ontological 'meta-category', relative to the
ontological category of atoms, taken as a "mere" <<arithmos>>.

Ideographically, this process of the irruption of the molecular ontology from
out of the heart of the physical spatial 'self-concentration', or
'self-densification', of the atomic ontology, is symbolized as follows, via the
N\Q "First Dialectical Ideography", per the definition of the "<<aufheben>>
evolute product rule" given earlier, above, in these Preliminaries to the
present exposition, and in the earlier posting of the Axioms for a Dialectical
Logic --

(^1)q|a x (^1)q|a = (^1)q|a of (^1)q|a = (^1)q|a<(^1)q|a>

= <(^1)q|a>^2 = (^1)q|a + Delta-(^1)q|a

= (^1)q|a + (^1)q|aa = (^1)q|a + (^1)q|m.

The expression above is simply the '<<species>>-ific', specific, special case
expression -- for the F.E.D. level 1 taxonomy, and for the transition from the
epoch of cosmological "meta-evolution" without molecular ontology as a
"metastable" possibility, to the succeeding epoch, with molecular ontology as a
"metastable" possibility -- of the '<<genos>>-ific', "generic", and general case
dialectical-algebraic expression, alluded to in the paragraph above, i.e. --

q|X x q|X = q|X of q|X = q|X<q|X> = <q|X>^2

= q|X + Delta-q|X = q|X + q|XX = q|X + q|Y.

[Note that, because Delta-q|X, or q|XX, connotes an ontological category that is
"new", and qualitatively different than, the category connoted by q|X, and which
therefore has, or deserves, a new, different name, with a different abbreviation
than "X", we give Delta-q|X, or q|XX, a new, single-letter abbreviation in the
last equation above, namely, q|Y, wherein the letter Y stands for a mnemonic,
one-letter [e.g., first-letter of the name] abbreviation of the name of the new,
emergent ontological category, connoted by q|XX, or Delta-q|X, which is built
upon / out of / from q|X, and which is therefore "higher" than -- "elevated"
with respect to -- q|X, but also not of the same primary quality as q|X, which
means that it exhibits a DETERMNATE negation of the primary quality of q|X, and
which is also '''op-posit[e]''', and '''op-posed to''', q|X, in the
["qualitative-Peanic"] sequence, or succession, or progression, of ontological
categories].

The last equation above illustrates most clearly of all the "contra-Boolean"
character of the N\Q "Dialectical Symbolic Logic", or "Symbolic Dialectics"
["Ideographical Dialectics"].

Boole expressed, via the equation --

X^2 = X

-- an equation that asserts the reduction of algebraic nonlinearity to algebraic
linearity -- his "Fundamental Law of Thought", by which he meant that '''the
selection OF all of the Xs out of the class of all Xs itself, BY the class of
all Xs, acting as an operation[, a self-[s]election operator], UPON itself, can
yield only all Xs again, nothing more, thing less'''.

This fundamental Boolean formula expresses a principle of "the simple
reproduction of ideas" -- that '''X, acting upon X, "reproduces" exactly X once
again, nothing more, nothing less'''.

Note that the "<<aufheben>> evolute product rule" Axiom of the N\Q
contra-Boolean arithmetic still contains this "Boolean" moment, viz. --

X^2 = X + Delta-X

or,

<q|X>^2 = q|X + q|XX

-- as that aspect of the [self-]<<aufheben>> product rule which expresses the
"conservation" aspect of the meaning of the German-Hegelian, core-dialectical
concept of self-<<aufheben>> self-supercession.

However, the kind of "ontological multiplication" stipulated by the
"<<aufheben>> evolute PRODUCT rule" Axiom also supersedes the Boolean
[simple-re-]PRODUCT[ion] rule Axiom, by adding a further term, a qualitative
increment, an increment of new ontology, denoted Delta-q|X, or q|XX,
qualitatively different from the source ontology, denoted q|X, and "meta-" to
it, just as the "physio-ontological category" of molecules is "meta-" to the
"physio-ontological category" of atoms.

The latter term, Delta-q|X, or q|XX, also represents a "conservation" term,
e.g., "atoms" are "conserved" within "molecules", in that each molecule-unit
contains a [usually heterogeneous] multiplicity of atom-units inside itself.

However, this latter form of "conservation" also involves, and thus satisfies,
the "elevation" and "determinate-negation/transformation" aspects of the meaning
of the German-Hegelian, core-dialectical term <<aufheben>>.

Thus, in contrast to the Boolean "Fundamental Law of Thought" --

X^2 = X

-- which asserts a "Simple Reproduction of Ideas" for a fixed and unchanging
universe[ of discourse], an "eternal" and "immutable", "onto-static" and
unevolving, "ideo-ontology", the contra-Boolean "Fundamental Law of Dialectical
Thought" --

X^2 = X + Delta-X,

or

<q|X>^2 = q|X + q|XX

asserts an ontologically-dynamical ["onto-dynamic"] "Expanding Reproduction of
Ideas".

If the ideas being modeled are about the external, extra-mental, sensuous world,
then the model directly asserts a "Self-Expanding Reproduction of
PHYSIO-ontology".

If the ideas being modeled are about the internal, endo-mental, intuitive world
of ideas not directly referred to any external, sensuous objects or experiences,
e.g., about a highly-abstract mathematical system of idea-objects, then the
model directly asserts an "Expanding Reproduction of IDEO-ontology".

The concrete mechanism of the contra-Boolean, dialectical, <<aufheben>> process
that works in a wide variety, but perhaps not in all, of the extant F.E.D.
dialectical models, using the "First Dialectical Ideography", of N\Q, can be
expressed by the following equation:

self-<<aufheben>> self-convervation/self-elevation/self-negation

= "self-meta-<<monad>>-ization"

= "self-meta-unit-ization" =

the self-formation of a "meta-<<arithmos>>" from out of that
self-interacting predecessor <<arithmos>> itself.

That is, each of the units, or <<monads>>, of a successor <<arithmos>>, or
assemblage of <<monads>>/units, relative to the <<monads>>/units of its
predecessor <<arithmos>>, is a "meta-unit", or "meta-<<monad>>", of its
predecessor units, or <<monads>>.

Each unit of that successor, "meta-", <<arithmos>>, is a "meta-unit", made up
out of a [generally] heterogeneous multiplicity of the predecessor units of its
predecessor <<arithmos>>.

Each <<monad>> of that successor, "meta-", assemblage of <<monads>>, is a
"meta-<<monad>>", made up out of a [usually] heterogeneous multiplicity of the
predecessor <<monads>> of its predecessor "assemblage of <<monads>>".

Thus, for example, each molecule is a "meta-atom", made up out of a [typically]
heterogeneous multiplicity of atoms, created through the critically
self-concentrated mutual interaction of atoms, consequent upon the accelerated
quantitative expanded reproduction of their populations.

Likewise, each atomic nucleus is a "meta-sub-atomic particle", made up out of a
[typically] heterogeneous multiplicity of "nuclear sub-atomic particles",
created out of the critically self-concentrated mutual interaction of "nuclear
sub-atomic particles", consequent upon the accelerated quantitative expanded
reproduction of their populations.



B. Emergence of New ANTITheses -- of "Neo-Contra-Thesis" Neo-Ontology.

"Hybrid" terms, of the form q|YX, in F.E.D. dialectical models, using the N\Q
Dialectical Arithmetic and its Dialectical Algebra, are typically interpreted as
connoting "synthesis", or "uni-thesis", ontological categories -- "complex
unity" and "hybrid" ontological categories -- as we have seen, in Part II.

"Self-Hybrid" terms, of the form q|XX, on the contrary, are typically
interpreted as connoting "new contra-thesis", or "new antithesis", ontological
categories.

The F.E.D. expositional model, or "[Meta-]Systematic Dialectic"-al model, of the
progression of the systems of Dialectical Ideography itself, can serve to
illustrate such interpretations.

The "meta-system", and <<arithmos>>, of the systems of Dialectical Arithmetic
[here, an <<arithmos>> having "ideo-systems" as its "ideo-<<monads>>", or
"idea-object-units"], is made up out of a heterogeneous multiplicity -- in the
form of a heterogeneous dialectical PROGRESSION -- of the individual systems of
Dialectical Arithmetic.

The original, or <<arche'>>, thesis "ideo-ontological category / system" of that
"meta-system", and <<arithmos>>, of systems of Dialectical Arithmetics, is the
Standard, "first-order" axiomatic system of the so-called "Natural" Numbers,
denoted by N --

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

That system is characterized, by its "primary quality", as an arithmetic of
numbers which are "unqualified quantifiers".

I.e., "three apples" is a "qualified quantifier" -- the "quantifier", here
three, "qualified" by the "ontological qualifier", apples. Likewise, "three
hours" is a "qualified quantifier"[, and, symmetrically, a "quantified
qualifier" as well] -- the "quantifier", three, "qualified" by the "metrical
unit", or "metrical qualifier", hours. The word "three" by itself is an
"UNqualified quantifier".

The successor system of arithmetic to that of N, in this F.E.D. "Dialectic of
the Systems of Dialectical Arithmetic", is the N\Q "First Dialectical
Arithmetic", the arithmetic of the
"Meta-Natural Meta-Numbers", N\Q = { q|1, q|2, q|3, . . .}.

That second system of arithmetic, N\Q, is the "first contra-thesis" system --
"contra" to the primary quality of the first, "thesis" system, N.

The axiomatic system of the N\Q "meta-numbers" is characterized, by its primary
quality, as an arithmetic of "meta-numbers" which are interpretable as
"unquantifiable ontological qualifiers".

They are "unquantifiable" because of their "idempotent addition rule": for any
n in N, q|n + q|n = q|n.

You can never stably get to even 2q|n, because 2q|n reduces immediately to 1q|n
= q|n.

So, cooting the system of arithmetic of the N ["Natural"] numbers by the symbol
q|N, we then have --

q|N x q|N = q|N + q|NN = q|N + q|Q.

Next, q|Q x q|N = q|N + q|QN.

The term q|QN connotes the "complex unity", or "uni-thesis", of the primary
qualities of the system of N and of the system of N\Q, which F.E.D. calls N\U,
where --

N\U = { u|N } = { u|1, u|2, u|3, . . . }.

The resulting, third system of Dialectical Arithmetic, N\U, is the "first
uni-thesis" system -- "unit-ing" the primary qualities of the first, "thesis"
system, N, and the second, "contra-thesis" system, N\Q.

That is, the third system in this progression of ideo-systems -- q|QN, or q|U,
or N\U -- is characterized, by its primary qualities, as an arithmetic of
"meta-numbers" which are interpretable as "quantifiable ontological qualifiers",
e.g.: 2u|n + 3u|n = 5u|n, for any n in N.

The fourth "ideo-system", in this progression of ideo-systems of Dialectical
Arithmetic, is the "second contra-thesis system / category of ideo-ontology" --
"contra" to ALL of the foregoing systems -- N, N\Q, and N\U.

It is a system of "unquantifiable METRICAL qualifiers", the primitive prototype
for the later-to-emerge fully-algorithmic arithmetic for the "dimensional
units', or "dimensional <<monads>>, of "dimensional analysis" full-blown.

As of the seventh "ideo-ontological category", and system, of Dialectical
Arithmetic, q|MQN, there has emerged an arithmetic for the explicit,
fully-algorithmic "Ontological and Metrical Qualification of Quantifiers", or,
symmetrically, for the Quantification of Metrically Qualified Ontological
Qualifiers, suitable for the formulation of the State Variables and Control
Parameters of an advanced formulation of Dynamical Systems Theory.

For more on this Dialectical-Algebraic Model of the dialectical progression of
the systems of Dialectical Algebra, formulated in the ideography of the First
Dialectical Algebra, see --

http://www.dialectics.org/archives/pdf/Fract1-1.pdf

[TO BE CONTINUED]

Regards,

Miguel