Dear Psycho-History Group Members,
Part III. Prelude to the Psycho-Historical Equations -- A New Model of the Universe / of the History of Nature
NOTE: 'q|XX' denotes q, underscored, and with the subscript XX; 'N\Q' denotes Q with the "pre-subscript" N, where N denotes the set of the "Natural" Numbers N = {1, 2, 3,...}, and where 'N\Q' as a whole is understood to have its 'Q' underscored, and, in this context, denotes the N-based set of pure, unquantifiable ["unaddable": q|n + q|n = q|n;
q|n + q|m does not reduce, if n not= m ] ontological qualifier meta-numbers:
N\Q = { q|1, q|2, q|3,...}.
'(^1)q|a' denotes q-underscore-subscript-a with pre-superscript 1.
"Delta-" denotes the delta-symbol underscored; it identifies a purely-qualitative, non-quantitative, ontological increment.
[CONTINUED FROM PRIOR POST, Part II.]
[NOTE: For more on the Dialectical-Algebraic Model of the dialectical progression of the systems of Dialectical Algebra, used as an example below, and formulated in the ideography of the F.E.D. First Dialectical Algebra, see, for example --
www.dialectics.org . . .
-- and the other texts in that series].
2. On the Generic Meanings of "Non-Cross-Product", "Self-Crossed", Terms in the Dialectical-Algebraic Model of the Dialectic of Nature, presented herein.
A. Non-Hybrids as "Self-Hybrids": Phenomenology of "Self-Meta-Monad-ization" as Self-Process.
Terms of the from q|XX, in models built using F.E.D.'s 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 q|X itself.
Consider the primary focus 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 V.
The interaction of the F.E.D. taxonomy level 1 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 "arihmos", or assemblage of units, taking place as 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 an ontological category, is therefore also a "meta-arithmos", and an ontological 'meta-category', relative to the ontological category of atoms, taken as a "mere" category / "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 = [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]^2 =
qX + 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 from, the category connoted by q|X, and which therefore has, or deserves, a new, different name, with a different abbreviation than the abbreviation denoted by "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 abbreviation [e.g., the first-letter of the new ontological category name] of the name of the new, emergent ontological category, connoted by q|XX, or Delta-q|X, which is built upon / built out of / built-up from q|X, and which is therefore "higher" than -- which is "elevated" with respect to -- q|X, but which is also not of the same primary quality as q|X, which means that it exhibits a DETERMINATE 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 ideograms].
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 for Boolean algebra -- his "Fundamental Law of Thought", by which Bole meant that '''the selection OF all Xs out of the class of all Xs itself, BY the class of all Xs itself, acting as an operation[, a self-(s)election operator], UPON itself, can yield only all Xs again, nothing more, nothing 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, one of the Axioms specified in an earlier posting here, staing those axioms, still contains this "Boolean" moment, viz. --
[q|X]^2 = q|X [+ Delta-q|X]
-- as that aspect of the 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 its 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, in a sense, another -- a special kind of -- "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" --
[q|X]^2 = q|X + Delta-q|X
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 Self-Reproduction of PHYSIS-Ontology", i.e., "of PHYSIO-Ontology", and, INDIRECTLY, an "Expanded Reproduction of IDEAs ABOUT that 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., are 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, 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 predecessor "arithmos" itself.
That is, each of the units, or "monads", of a successor "arithmos", or assemblage of "monads"/units, relative to the /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 "monads" of that successor, "meta-" arithmos, 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 such atoms, consequent upon the accelerated quantitative expanded reproduction of that local population of atoms.
Likewise, each atomic nucleus is a "meta-sub-atomic particle", made up out of a [typically] heterogeneous multiplicity of proto-nuclear sub-atomic particles, created out of the critically self-concentrated mutual interaction of such proto-nuclear sub-atomic particles, consequent upon the accelerated quantitative expanded reproduction of the local population of such sub-atomic particles.
B. Emergence of New Anti-Theses -- 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 -- whether of the "full uni-thesis", or of the "partial uni-thesis" "species", 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 systems of Dialectical Arithmetic themselves form a "genos-arithmos", here, an "arithmos" having "ideo-systems" as its "ideo-monads", or "ideo-units". This "cumulum" of axiomatic systems, this assemblage of "ideo-ontology" -- this "ideo-arithmos" of Dialectical Arithmetics -- is made up out of a heterogeneous multiplicity -- in the form of a heterogeneous progression -- of the individual systems of Dialectical Arithmetic as its units, or "monads".
The original, or "arche'", thesis "ideo-ontological category / system" of that "meta-system", 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, each and all, "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. "Meta-System-atic 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, each and all, interpretable as "unquantifiable ontological qualifiers".
They are "unquantifiable" because of their "idempotent addition rule": for any k in N, q|k + q|k = q|k.
You can never stably get to even 2q|k, because 2q|k reduces immediately to 1q|k = q|k.
This axiomatic rule, of "idempotent addition", is like Boolean addition -- 0 + 0 = 0 and 1 + 1 = 1 -- and is posited, as per Plato's "Arithmoi Eidetikoi", and its "asumbletoi" [un-summable] "idea-numbers", because it makes no sense to DISTINGUISH "multiple copies" of "the same" ontological category.
If we connote the axiomatic system of the N numbers, by q|N, and the axiomatic system of the N\Q "meta-numbers" by q|Q, then we have --
q|N x q|N = q|N + q|NN = q|N + q|Q,
-- and, subsequently --
q|Q x q|N = q|N + q|QN = q|N + q|U.
The term q|QN = q|U connotes the "complex unity", or "uni-thesis", of the primary qualities of the system of N, with the primary qualities of the system of N\Q, whose result F.E.D. calls N\U, where --
N\U = { Nu|N } = { Nu|1, Nu|2, Nu|3, . . . }.
This 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 -- N\q|QN, or N\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 to EACH of the three, of the foregoing axiomatic systems -- N, N\Q, and N\U.
It is an axiomatic system of "unquantifiable METRICAL qualifiers", N/M, the primitive prototype for the later-to-emerge fully-algorithmic arithmetic for the "dimensional units', or "dimensional , 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 [Nonlinear] Dynamical Systems Theory.
Thus --
N\q|Q x N\q|Q = N\q|Q + N\q|QQ = N\q|Q + N\q|M
=
the system of arithmetic of the N\Q "meta-numbers"
PLUS
the system of arithmetic of the N\M "meta-numbers"
-- where N\M = { m|N } = { m|1, m|2, m|3, ... }
-- and where the full systems-progression "cumulum", to stage 2 of this "Meta-System-atic Dialectic of the Systems of Dialectical Arithmetics", is --
[q|N]^(2^2) = [q|N]^4 = N + N\Q + N\U + N\M.
For more on this Dialectical-Algebraic Model of the dialectical progression of the systems of Dialectical Algebra, used as an example above, and formulated in the ideography of the F.E.D. First Dialectical Algebra, see, for example --
www.dialectics.org . . .
-- and the other texts in that series.
