Dear Reader,
These comments are intended to help you to "get a handle" on "F.E.D. Dialectics", by sharing some things that, over a long period of study, I've come to notice about the "F.E.D. Dialectical Method" -- their "Organonic Algebraic Method".
Note: This exposition will be continued in the immediately-subsequent blog entry.
The F.E.D. First Dialectical Algebra's "Organonic Algebraic Method" is best understood, I think, as a heuristic -- as a heuristic, intensional-intuitional tool [<<organon>>] for "re-constructing" the mysteries of the deep-past-to-present, of natural systems" [including of "human-nature-al" systems, external-physical and/or "ideative"], and of "pre-constructing" ["pre-dict-ing"] their futures.
This Method is "intensional", because it deals with connotations.
It deals with deciphering the algorithmically-generated [ideographical] symbols of an NQ dialectical meta-model for the new qualities of new ontology -- of both new "ideo-ontology" and of new "physio-ontology" -- that they represent; in short, for the "meanings" -- or "in-tensions" -- of those symbols.
Therefore, this Method is not an "exact science".
Its being "intensional" is also what makes it "heuristic" and "intuitional".
What precisely does "intension" mean in this -- "symbolic logics" -- context?
It means the opposite of "extension".
[One will also find that, for F.E.D., "in-tension" has connotations of "internal tension", i.e., of "dialectical internal contradiction", "intra-duality", or "self-duality", while "ex-tension" has connotations of "external tension", i.e., of external differences / conflicts / "self-bifurcations" / "singularities", e.g., the stellar "self-implosion # self-explosion" internal tension, or physio-self-duality, that "extends" stars' existential [contribution to] time as the [diverse stages of the] life-histories of stars, or the ideo-self-duality of the "cardinality # ordinality" [in]tension "inside" the concept of the "Natural Numbers" system of arithmetic, that "extends" as the NQ "meta-Natural meta-Numbers" system of arithmetic, and in the forms of all of the other, subsequent systems of dialectical arithmetic in the systems-progression of the F.E.D. "Meta-System-atic Dialectic of the Dialectical Arithmetics".]
For example, the "extension" of the concept named "celestial [spheroidal] orbs of the stellar/planetary system of the star Sol" is and explicit list, by name, of all of those objects which fit this concept, e.g., the "set" containing the names of all of those objects --
{Sol, Earth, Moon, Mercury, Venus, Mars, Jupiter, Europa, Ganymede, Callisto, . . ., Saturn, Enceladus, Titan, . . ., Neptune, Triton, . . ., Pluto, Charon, . . .}
-- whereas the "intension" of that "extension" is just the concept that encompasses precisely that "set" -- no more and no less -- as re-called to mind by its name, or by its words/phrase-definition: "celestial orbs of the stellar/planetary system of the star Sol" -- i.e., is the meaning, or "unified connotations-cluster", of that "phrasal predicate".
Here's what Leibniz had to say about what F.E.D. characterizes as the "intension # extension" internal duality dialectic within [symbolic] logics --
"The common manner of statement concerns individuals, whereas Aristotle's refers rather to ideas or universals [Aristotle's version of Plato's <<Eide>> -- M.D.].
For when I say Every man is an animal I mean that all men are included amongst all the animals; but at the same time I mean that the idea of animal is included in the idea of man.
[cf. George Boole's "deductive-logical ideography" equation --
h = r x a,
-- or --
"humans are rational animals",
-- and --
Karl Seldon's "dialectical-logical ideography" equation --
l(l) = l + h,
-- or --
'''the self-intra-action of the populations of proto-language-based animal-societies quant[o-qual]itatively expandedly-reproduces those populations of proto-language-based animal-societies, and also[, at & after they achieve their critical density, ] qual[o-quant]itatively expandedly-reproduces those populations by each irrupting a new, unprecedented population of human[oid] meta-societies'''].
'Animal' comprises more individuals than 'man' does, but 'man' comprises more ideas or attributes [more predicates, more "speci-fications", i.e., more "determinations" [key term used by both Hegel and Marx] -- M.D.]:
one has more instances, the other more degrees of reality;
one has the greater extension, the other the greater intension."
[Wolfgang Lenzen, "Leibniz's Logic", in Dov M. Gabbay and John Woods, eds., Handbook of the History of Logic, volume 3, The Rise of Modern Logic: From Leibniz to Frege, Elsevier B. V., [San Francisco: 2004], p. 11, emphases added].
Now, other than N itself, the axiomatic system of arithmetic of the "Natural" Numbers ['=' signifies "is equal to by definition"] --
N = { 1, 2, 3, 4, . . .}
-- itself, the first "dialectical" arithmetic in the F.E.D. "meta-system", or "dialectical systems-progression", of dialectical arithmetics, which can only "quantify" or "cardinize" [and quantitatively order or "ordin[al]ize"], the second axiomatic system of "dialectical arithmetic" in that "meta-system", or systems-progression", that of the NQ "meta-Natural meta-Numbers" --
NQ = { q/1, q/2, q/3, q/4, . . .}
-- is the most "intensional" -- the least "extensional" -- of all of the F.E.D. axiomatic systems of dialectical arithmetic in that dialectical systems-progression.
This is because the NQ system of dialectical arithmetic is less rich -- less complex and less "thought-concrete"; less "determinate" -- in its syntax, and in its semantics -- than are all of its subsequent, successor systems of dialectical arithmetic in that F.E.D. dialectical progression of dialectical systems of arithmetic, and, therefore, the NQ system has less descriptive power as an [ideographical] language for systems-modeling than do all of its subsequent, successor systems in that dialectical systems-progression.
Such gradients of growing "determinateness" [Hegel, Marx], complexity, and "thought-concreteness" are of the essence of [e.g., Systematic] Dialectical ontological-categorial progressions in general.
Thus, dialectical "meta-models" [ideographical representations which encompass both the other-induced / self-induced "evolution" [mere "model" aspect], and the "self-induced revolution" [meta-model aspect], of the systems they re-construct/pre-construct], if written in those NQ-subsequent systems of dialectical ideography, can be expected to be no longer so "heuristic", so "intuitional", so "intensional" as the versions of those "meta-models" expressed in N or in NQ.
Many of those NQ-subsequent systems of dialectical ideography, i.e., of dialectical arithmetic, are capable of being used to represent the detailed "determinations", or "speci[es]-fications", of the dialectical, [meta-]evolutionary, [meta-]dynamical [meta-super^z-]systems of "dialectical systems theory", in exact, precise, "quanto-qualitative", state-space trajectory / control-parameter-space path detail [in the cases of the axioms-systems of dialectical arithmetic that correspond to generic NQ "meta-numerals" with subscripts of the form 2^n - 1, for n > 3, i.e., that correspond to NQ "generic meta-numerals" of the form q/(2^n - 1), , for n > 3, in the NQ "meta-model" of that "meta-system-atic" dialectical progression of axioms-systems of dialectical arithmetic], or in the form of explicit, trans-Platonian, Human-Phenomic <<arithmoi eidetikoi>> content-structures of, e.g., Super-<<Genos>> / <<Gene>> / <<Species>> / Sub-<<Species>> / . . . "qualo-fractals" [in the cases of the axioms-systems of dialectical arithmetic that correspond to generic NQ "meta-numerals" with subscripts of the form 8xn, for n > 2, i.e., that correspond to NQ "generic meta-numerals" of the form q/(8xn), for n > 2, in the NQ "meta-model" of that "meta-system-atic" dialectical progression of axioms-systems of dialectical arithmetic].
The N-based "Natural Numbers" axioms-system is the <<arche'>> axioms-system in the F.E.D. 'ideo-meta-system' generated by the Seldon-Function formula for the "Meta-Systematic Dialectic" -- for the dialectical systems-progression -- of the F.E.D. axioms-systems of dialectical arithmetic.
That N-based arithmetic is "dialectical" only in the "pre-vestigial" form of the purely-quantitative, vestigial <<aufheben>> content of the Peano successor function, s, per the second axiom of that axiomatic system of arithmetic, "Peano Postulate 2", or PP2 for short:
PP2. If n is in set N, then s(n) = n + 1 is also in set N.
The successor function, or succession operator, s, operating upon any "Natural" Number, n, concurrently (a.) <<aufheben>>-conserves n, in the sense that n reappears in and is "contained" in n + 1, (b.) <<aufheben>>-elevates n, by adding to it a single unit of quantitative increment, + 1, and also (c.) <<aufheben>>-determinately negates n, by changing n into its "next", into "not-n", in the specific, determinate form of n + 1 --
n ~= n + 1; n < n + 1; n + 1 is [a specific case of] not-n.
These comments are intended to help you to "get a handle" on "F.E.D. Dialectics", by sharing some things that, over a long period of study, I've come to notice about the "F.E.D. Dialectical Method" -- their "Organonic Algebraic Method".
Note: This exposition will be continued in the immediately-subsequent blog entry.
The F.E.D. First Dialectical Algebra's "Organonic Algebraic Method" is best understood, I think, as a heuristic -- as a heuristic, intensional-intuitional tool [<<organon>>] for "re-constructing" the mysteries of the deep-past-to-present, of natural systems" [including of "human-nature-al" systems, external-physical and/or "ideative"], and of "pre-constructing" ["pre-dict-ing"] their futures.
This Method is "intensional", because it deals with connotations.
It deals with deciphering the algorithmically-generated [ideographical] symbols of an NQ dialectical meta-model for the new qualities of new ontology -- of both new "ideo-ontology" and of new "physio-ontology" -- that they represent; in short, for the "meanings" -- or "in-tensions" -- of those symbols.
Therefore, this Method is not an "exact science".
Its being "intensional" is also what makes it "heuristic" and "intuitional".
What precisely does "intension" mean in this -- "symbolic logics" -- context?
It means the opposite of "extension".
[One will also find that, for F.E.D., "in-tension" has connotations of "internal tension", i.e., of "dialectical internal contradiction", "intra-duality", or "self-duality", while "ex-tension" has connotations of "external tension", i.e., of external differences / conflicts / "self-bifurcations" / "singularities", e.g., the stellar "self-implosion # self-explosion" internal tension, or physio-self-duality, that "extends" stars' existential [contribution to] time as the [diverse stages of the] life-histories of stars, or the ideo-self-duality of the "cardinality # ordinality" [in]tension "inside" the concept of the "Natural Numbers" system of arithmetic, that "extends" as the NQ "meta-Natural meta-Numbers" system of arithmetic, and in the forms of all of the other, subsequent systems of dialectical arithmetic in the systems-progression of the F.E.D. "Meta-System-atic Dialectic of the Dialectical Arithmetics".]
For example, the "extension" of the concept named "celestial [spheroidal] orbs of the stellar/planetary system of the star Sol" is and explicit list, by name, of all of those objects which fit this concept, e.g., the "set" containing the names of all of those objects --
{Sol, Earth, Moon, Mercury, Venus, Mars, Jupiter, Europa, Ganymede, Callisto, . . ., Saturn, Enceladus, Titan, . . ., Neptune, Triton, . . ., Pluto, Charon, . . .}
-- whereas the "intension" of that "extension" is just the concept that encompasses precisely that "set" -- no more and no less -- as re-called to mind by its name, or by its words/phrase-definition: "celestial orbs of the stellar/planetary system of the star Sol" -- i.e., is the meaning, or "unified connotations-cluster", of that "phrasal predicate".
Here's what Leibniz had to say about what F.E.D. characterizes as the "intension # extension" internal duality dialectic within [symbolic] logics --
"The common manner of statement concerns individuals, whereas Aristotle's refers rather to ideas or universals [Aristotle's version of Plato's <<Eide>> -- M.D.].
For when I say Every man is an animal I mean that all men are included amongst all the animals; but at the same time I mean that the idea of animal is included in the idea of man.
[cf. George Boole's "deductive-logical ideography" equation --
h = r x a,
-- or --
"humans are rational animals",
-- and --
Karl Seldon's "dialectical-logical ideography" equation --
l(l) = l + h,
-- or --
'''the self-intra-action of the populations of proto-language-based animal-societies quant[o-qual]itatively expandedly-reproduces those populations of proto-language-based animal-societies, and also[, at & after they achieve their critical density, ] qual[o-quant]itatively expandedly-reproduces those populations by each irrupting a new, unprecedented population of human[oid] meta-societies'''].
'Animal' comprises more individuals than 'man' does, but 'man' comprises more ideas or attributes [more predicates, more "speci-fications", i.e., more "determinations" [key term used by both Hegel and Marx] -- M.D.]:
one has more instances, the other more degrees of reality;
one has the greater extension, the other the greater intension."
[Wolfgang Lenzen, "Leibniz's Logic", in Dov M. Gabbay and John Woods, eds., Handbook of the History of Logic, volume 3, The Rise of Modern Logic: From Leibniz to Frege, Elsevier B. V., [San Francisco: 2004], p. 11, emphases added].
Now, other than N itself, the axiomatic system of arithmetic of the "Natural" Numbers ['=' signifies "is equal to by definition"] --
N = { 1, 2, 3, 4, . . .}
-- itself, the first "dialectical" arithmetic in the F.E.D. "meta-system", or "dialectical systems-progression", of dialectical arithmetics, which can only "quantify" or "cardinize" [and quantitatively order or "ordin[al]ize"], the second axiomatic system of "dialectical arithmetic" in that "meta-system", or systems-progression", that of the NQ "meta-Natural meta-Numbers" --
NQ = { q/1, q/2, q/3, q/4, . . .}
-- is the most "intensional" -- the least "extensional" -- of all of the F.E.D. axiomatic systems of dialectical arithmetic in that dialectical systems-progression.
This is because the NQ system of dialectical arithmetic is less rich -- less complex and less "thought-concrete"; less "determinate" -- in its syntax, and in its semantics -- than are all of its subsequent, successor systems of dialectical arithmetic in that F.E.D. dialectical progression of dialectical systems of arithmetic, and, therefore, the NQ system has less descriptive power as an [ideographical] language for systems-modeling than do all of its subsequent, successor systems in that dialectical systems-progression.
Such gradients of growing "determinateness" [Hegel, Marx], complexity, and "thought-concreteness" are of the essence of [e.g., Systematic] Dialectical ontological-categorial progressions in general.
Thus, dialectical "meta-models" [ideographical representations which encompass both the other-induced / self-induced "evolution" [mere "model" aspect], and the "self-induced revolution" [meta-model aspect], of the systems they re-construct/pre-construct], if written in those NQ-subsequent systems of dialectical ideography, can be expected to be no longer so "heuristic", so "intuitional", so "intensional" as the versions of those "meta-models" expressed in N or in NQ.
Many of those NQ-subsequent systems of dialectical ideography, i.e., of dialectical arithmetic, are capable of being used to represent the detailed "determinations", or "speci[es]-fications", of the dialectical, [meta-]evolutionary, [meta-]dynamical [meta-super^z-]systems of "dialectical systems theory", in exact, precise, "quanto-qualitative", state-space trajectory / control-parameter-space path detail [in the cases of the axioms-systems of dialectical arithmetic that correspond to generic NQ "meta-numerals" with subscripts of the form 2^n - 1, for n > 3, i.e., that correspond to NQ "generic meta-numerals" of the form q/(2^n - 1), , for n > 3, in the NQ "meta-model" of that "meta-system-atic" dialectical progression of axioms-systems of dialectical arithmetic], or in the form of explicit, trans-Platonian, Human-Phenomic <<arithmoi eidetikoi>> content-structures of, e.g., Super-<<Genos>> / <<Gene>> / <<Species>> / Sub-<<Species>> / . . . "qualo-fractals" [in the cases of the axioms-systems of dialectical arithmetic that correspond to generic NQ "meta-numerals" with subscripts of the form 8xn, for n > 2, i.e., that correspond to NQ "generic meta-numerals" of the form q/(8xn), for n > 2, in the NQ "meta-model" of that "meta-system-atic" dialectical progression of axioms-systems of dialectical arithmetic].
The N-based "Natural Numbers" axioms-system is the <<arche'>> axioms-system in the F.E.D. 'ideo-meta-system' generated by the Seldon-Function formula for the "Meta-Systematic Dialectic" -- for the dialectical systems-progression -- of the F.E.D. axioms-systems of dialectical arithmetic.
That N-based arithmetic is "dialectical" only in the "pre-vestigial" form of the purely-quantitative, vestigial <<aufheben>> content of the Peano successor function, s, per the second axiom of that axiomatic system of arithmetic, "Peano Postulate 2", or PP2 for short:
PP2. If n is in set N, then s(n) = n + 1 is also in set N.
The successor function, or succession operator, s, operating upon any "Natural" Number, n, concurrently (a.) <<aufheben>>-conserves n, in the sense that n reappears in and is "contained" in n + 1, (b.) <<aufheben>>-elevates n, by adding to it a single unit of quantitative increment, + 1, and also (c.) <<aufheben>>-determinately negates n, by changing n into its "next", into "not-n", in the specific, determinate form of n + 1 --
n ~= n + 1; n < n + 1; n + 1 is [a specific case of] not-n.
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