Dear Psycho-History Forum Participants,
Part 2.
5. Any two meta-Natural Numbers differ qualitatively, if their subscripts differ quantitatively.
Ideographical shorthand version:
j, k e N & j >< k => q|j ~>=< q|k.
6. The self-addition [or "doubling"] of any meta-Natural Number is redundant. [The principle of "additive idempotency", or of "UNQUANTIFIABILITY"].
Ideographical shorthand version [using '(n)' to denote "for every value "denote-able" by the variable n"]:
[(n) e N][ q|n + q|n = q|n ].
[Note similarity to "Boolean addition": 0 + 0 = 0, and 1 + 1 = 1].
Corrolary. 2q|n = q|n.
Indeed, (m) e N => m "times" q|n = mq|n = q|n.
7. The addition of one ontological qualifier meta-Natural Number to another DOES NOT REDUCE to any SINGLE meta-Natural Number value.
Ideographical shorthand version:
j, k e N & j k => NOT[ E(x) e N such that q|j + q|k = q|x ].
8. Multiplication of ontological qualifiers multiples the ontology explicitly qualified [adds new ontology].
Ideographical shorthand version:
j, k e N & j k => q|j "times" q|k = q|k + q|(j+k).
That is, q|j, "operating upon" or "multiplying" q|k, gives back q|k again, but also together with [added to] a QUALITATIVE, ontological INCREMENT, q|(j+k), which is denoted by a[n additive] combination of [the Natural Number subscripts of] q|j and q|k.
NOTE how this product rule implements a Hegelian-like "aufheben" -- annulment/conservation/elevation -- operation!
Sample Applications.
The texts posted to the site linked-to above set forth a large number of natural-scientific -- and social-scientific, or "psycho-historical" -- models, constructed using the calculus whose rules I have attempted to summarize above,
with many startlingly insightful results.
For example --
a. A model of the "taxonomy level one" evolution, and "meta-evolution", of the cosmos as a whole: Pages B-19 ff., Link --
www.dialectics.org . . .\
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B-1,%20pp.%2012-22,%20v.2.pdf
b. A "psycho-historical" model of human social evolution, and "meta-evolution", in terms of human "social relations [of [social re-]production]": Pages B-24 ff., Link --
www.dialectics.org . . .\
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B-1,%20pp.%2023-33,%20v.2.pdf
c. A "psycho-historical" model of human social evolution, and "meta-evolution", in terms of "human socio-geography": Page B-23, link --
www.dialectics.org . . .\
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B-1,%20pp.%2023-33,%20v.2.pdf
d. A "psycho-historical" model of the progressive emergence of the various "fields" of human inquiry -- Mythopoeia --> Religions --> Philosophies --> Sciences --> to the "meta-Sciences" of "Psycho-History" itself: Pages B-9 ff.,
Link --
www.dialectics.org . . .\
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B-1,%20pp.%2001-11,%20v.2.pdf
e. A "systematic presentation", or "pedagogical" model of the progression of the "Characteristica Universalis" systems of mathematics, starting with the Natural Numbers arithmetic of N, as "thesis", thence provoking the emergence of the arithmetic of N\Q, as "contra-thesis", thence provoking the emergence of the arithmetic of N\U, the arithmetic of "quantifiable ontological qualifiers", or of "ontologically qualifiable quantifiers", thence onwards from there, step by step, into an indefinite, potentially infinite expansion of the horizons of mathematical/ideographical language expressive power: Pages B-6 ff., Link --
www.dialectics.org . . .\
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B-1,%20pp.%2001-11,%20v.2.pdf
Of course, this dialectical calculus cannot, and does not, escape "The Godelian Dialectic" -- 'semantics #syntactics' -- i.e., cannot escape that "reductio ad absurdum" of all formalist, "deductivist" claims to the mechanization of all logical truth that is codified in the Godel Incompleteness Theorems.
At the higher-than-first-order levels of formal logic, this means that there will be well-formed equations in N\Q, and assertions that those equations are unsolvable -- assertion which are true theorems, but which theorems cannot be proven within N\Q, but only within all successor systems to N\Q, in the systems progression denoted, in the N\Q language itself, by [N]^(2^t), where N connotes the first order axiomatic system of the Standard Natural Numbers.
An example of an equation which is unsolvable in N\Q is:
x = [q|1] - [q|1].
The latter equations IS SOLVABLE in W\Q, where
W = { 0, 1, 2, 3, . . . },
with a very interesting expansion of mathematical "ideo-ontology", to include a new "meta-number", with a surprising praxis: q|0.
This means that, at the "first order" logical level, despite its "semantic" completeness, i.e., where the Godel "[semantic] Completeness Theorem" for first order predicate calculus also applies, there will always be "syntactical INcompleteness", that is, there will exist well-formed formulas of N\Q that are true in some models of its first order axioms, but false in others, so that the co-implicitude of Non-Standard Models together with Standard Models cannot be avoided.
I will illustrate two "sample models" of my own, using the N\Q algebra, in a reply post to this one.
Regards,
Miguel
