Dear Psycho-History Forum Participants,


Background. The following nine axioms are extracted from a number of texts posted at the following site --

www.dialectics.org

-- none of which assert all of these axioms together.

However, the axiom set provided below seems to be a fair representation of those necessary to the "calculus of discovery" described on that site, which seems -- judging from the models and calculations presented in the texts posted there, as well as a number of test models that I have derived on my own [two of which are presented below] -- to be a [somewhat unexpected] fulfillment of the Leibniz quest for a "Characteristica Universalis", a "universal character-language" [universal ideography] for philosophical and scientific inquiry.

The calculus specified by these axioms is a "purely-qualitative", "heuristic", and "intensional" one, rather than a "purely-quantitative" and "extensional" one.

That is, this calculus, when its generic symbols are "interpreted", or "assigned", to a given "ontological category", or "kind of being" category, in order to model some aspect of conceptual and/or experiential reality, are purely connotative, even though the operations of the undergirding, generic calculus are purely deterministic and algorithmic.

The underlying algorithm generates a progression of new "ontological categories" in a "mechanical" way.

However, for a model written in this language to work, the connotations of the interpreted symbols must "make sense of" the algorithmic progression in a way which recapitulates the "psycho-history" of the domain of experience being modeled.

This process of "connotative decipherment" of the new symbols generated by the generic algorithm of this calculus often leads to startling new discoveries and hypotheses.

Below, I provide links to some of the models/discoveries posted on the site, as well as a couple of [more minor] model discoveries of my own, using the new calculus.

The most rudimentary version of this calculus presented involves a new kind of number, which the site names "meta-Natural" number, and which is built up upon the set of standard Natural Numbers, usually denoted by N, where N = {1, 2, 3, ...}, but which appears to be the exact opposite of Standard Natural Number.

That is, if the Standard Natural Numbers can be characterized as forming an arithmetic of "ontologically unqualified pure quantifiers", then the "meta-Naturals", which the site denotes by N\Q, where N\Q = {q|1, q|2, q|3, ... }, forms an arithmetic which the site characterizes as being one of "pure, unquantifiable ontological Qualifiers".



The Axioms.


0. If n is a Natural Number, q|n is a meta-Natural number.

Ideographical shorthand version [using 'e' to denote the phrase "is an element of the set...", and '=>' to denote "implies"]:

n e N => q|n e N\Q.


1. q|1 is a meta-Natural Number.

[N\Q version of Axiom 1 of the Peano Postulates for the Standard Natural Numbers]

Ideographical shorthand version: q|1 e N\Q.


2. The successor of any meta-Natural Number is also a meta-Natural Number.

[N\Q version of Axiom 2 of the Peano Postulates for the Standard Natural Numbers]

Ideographical short hand version [where s denotes the Standard Natural Numbers' Peano "successor function", s(n) = n+1, and where s denotes the meta-Natural Numbers "successor function", S[q|n] = q|n+1]:

[n e N => q|n e N\Q] => S[q|n] = q|s(n) = q|n+1 e N\Q.


3. No two meta-Natural Numbers have the same successor.

[N\Q version of Axiom 3 of the Peano Postulates for the Standard Natural Numbers]

Ideographical short hand version:

[[j, k, e N & j NOT= k =>] qj, qk e NQ] => S[q|j] is qualitatively unequal to S[q|k].


4. q|1 is not the successor of any meta-Natural Number.

[N\Q version of Axiom 4 of the Peano Postulates for the Standard Natural Numbers]

Ideographical short hand version [using 'E(x)' to denote "there Exists at least one value x"]:

NOT[E(x) such that x e N & q|x e N\Q ] such that S[q|x] = q|1.


5. Any two meta-Natural Numbers differ qualitatively, if their subscripts differ quantitatively.

Ideographical shorthand version [using '>=