F.E.D. Dialectics: October 2022

‘Dialectically Developing a Dialectical SCIENCE for a new ontological Domain’ –

‘A Dialectical Order of Development’ –

Implicit in the Order of Presentation of the Meta-Systematic Dialectic of the Encyclopedia Dialectica Ideographies for Dialectics.

Dear Reader,

We have, in various posts herein, narratively, as well as in various ‘dialectograms’ [e.g., those posted below] outlined the Encyclopedia Dialectica ‘Dialectic of the Seldonian Arithmetics for Dialectics Themselves’.

That dialectic is of the ‘Meta-Systematic Dialectics’ species, the third species of dialectics per the Encyclopedia Dialectica ‘Systematic-Dialectic of the Dialectic Itself’.

That is, its categorial progression presentation is not only a categories-progression presentation, it is also a systems-progression presentation.

The categories, as «arithmoi», in this categorial progression, have, as their units, or «monads», the variant axioms-systems all inhering in the qualitatively, ‘ideo-ontologically’ distinct family of axioms-systems that each category in this categories/systems-progression presentation represents.

It can be noticed that this sequence of dialectical-mathematical systems -- ordered by their increasing ‘thought-complexity’, ‘thought-concreteness’, ‘ideo-determinateness’ and ‘ideo-specificity’, actualizing increasingly-replete mathematical-language and mathematical-model descriptive power and richness -- mentally-embodies yet a different kind of order as well.

That other kind of order ‘implicitizes’ a coherent, systematic, progressive, advantaged order of development for the new, incipient science of a new ontological Domain, perhaps one previously unknown, previously unexplored, and/or as yet unmastered scientifically.

The first eight solved categories/axioms-systems of the Encyclopedia Dialectica dialectical ideographies, in the long-form presentation version, and the steps in science-development that they prescribe, plus a few of their later systems/categories, are characterized below, in order of their ideographical-linguistic system descriptive power and richness.

Dialectical-Mathematical Axioms-Systems Characterizations/Definitions.

q₁(--] N_ = The ‘ideo-ontological’ category of the axioms-systems variants of the 1st 4 first order logic, standardly-interpreted Peano axioms, which, superficially, describe “purely”-quantitative, “ordinal quantities”; “ordinal numbers” only;

q₂(--] _N_{_}Q_ = The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted “purely”-qualitatively, as representing qualitatively, ontologically distinct, unquantifiable ontological categories as ARITHMOI – as ‘purely-qualitative’ ‘‘‘numbers’’’ -- forming a strictly-ordered taxonomy for a given ontological Domain to which they are applied;

q₃(--] _N_{_}U_ = The ‘meta-numerals’ of this category of dialectical- mathematical systems/languages are interpreted ‘quanto-qualitatively’, as definite population quantities of the qualitative units, the qualitatively/ontologically different units, that ontological categories represent;

q₄(--] _N_{_}M_ = The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted “purely”-QUALitatively, as representing qualitatively distinct, ontologically distinct, but UNquantifiable [Plato: «asumbletoi»] elementary metrological units, e.g., “sec.”, “gm.”, and “cm.”, in their current, still primitive – “syncopated” [a la Diophantus, circa 250 C.E.] -- notations;

q₅(--] _N_{_}q__M_N = The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted ‘QUANTo-qualitatively’, as representing qualitatively distinct, ontologically distinct, but quantifiable elementary metrological units, e.g., 5 x sec., or 9 x gm.;

q₆(--] _N_{_}q__M_Q = The ‘meta-numerals’ of this category of dialectical-mathematical-systems/languages are interpreted “purely”-qualitatively, as representing qualitatively distinct, ontologically distinct, UNquantifiable [«asumbletoi»] but COMPOUNDABLE metrological units, e.g., in the current, primitive notation, cm./sec. [velocity]; cm./[sec. x sec.] [acceleration]; gm. x cm./[sec. x sec.] [force], etc.;

q₇(--] _N_{_}q__M_Q_N_¶_|-= _N_{_}q__M_U_|-= _N_{_}m_= The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted ‘QUANTo-QUALitatively’, as representing qualitatively distinct, ontologically distinct, but QUANTifiable and COMPOUNDABLE metrological units, such as 500,000 x cm./sec. [velocity], or 50,000 x cm./[sec. x sec.] [acceleration], or 50 x gm. x cm./[sec. x sec.] [force], etc.;

q₈(--] _N_{_}q__M_M_|-= _N_{_}A_= The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted “purely”-qualitatively, as representing qualitatively distinct, ontologically distinct, but unquantifiable [e.g., dynamical] system units.

q₁₅(--] _N_{_}q__A_M_U_|-= _am_= ‘Meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted ‘quanto-qualitatively’, as dynamical system units, denoted by the DYNAMICAL-system ‘arithmetical qualifier meta-numeral’ a, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a sum of type q__M_U dynamical, quantified state-variables [and dynamical, quantified control parameters].

q₂₄(--] _N_{_}q__B_A= The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted “purely”-qualitatively’, as TAXONOMICAL system units, a ‘minimal [2-scales] ‘qualo-fractal’, consisting of a b, or beta-level, relative-«genos» category ‘arithmetical qualifier’ first/more abstract layer/level/scale, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a -- non-amalgamative -- sum of qualitatively different/heterogeneous a, or alpha-level, relative-«species» category ‘arithmetical qualifier meta-numerals’ second/lower/more-concrete layer/level/scale.

q₃₁(--] _N_{_}q__B_A_M_U_|-= _bm_= ‘Meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted ‘quanto-qualitatively’, as dynamical system units, denoted by a beta level/layer/‘qualo-fractal’ scale DYNAMICAL-system ‘arithmetical qualifier meta-numeral’, b, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a non-amalgamative sum of type q__M_U dynamical, quantified state-variables [and dynamical, quantified control parameters] as top, beta-level. The alpha level/layer/‘qualo-fractal’ scale, “underneath” that beta level, contains a non-amalgamative sum of explicit dynamical SUB-system ‘arithmetical qualifier meta-numerals’, the {a_i}, EACH ONE ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a sum of type q__M_U dynamical, quantified state-variables [and dynamical, quantified control parameters].

q₅₆(--] _N_{_}q__G_B_A= The ‘meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted “purely”-qualitatively’, as TAXONOMICAL system units, forming a 3-scales ‘qualo-fractal meta-numeral’, consisting of a g, or gamma-level, relative-super-«genos» category ‘arithmetical qualifier’ top/more abstract layer/level/scale, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a non-amalgamative sum of qualitatively different/heterogeneous, multiple b, or beta-level, of relative-«gene» categories’ ‘arithmetical qualifier’ second/less abstract layer/level/scale, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a non-amalgamative sum of qualitatively different/heterogeneous a, or alpha-level, relative-«species» categories’ ‘arithmetical qualifier meta-numerals’ as third/lower/more-concrete taxonomic layer/level/scale.

q₆₃(--] _N_{_}q__G_B_A_M_U_|-= _gm_= ‘Meta-numerals’ of this category of dialectical-mathematical systems/languages are interpreted ‘quanto-qualitatively’, as dynamical system units, denoted by the DYNAMICAL-system ‘arithmetical qualifier meta-numeral’ g, ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a non-amalgamative sum of its own type q__M_U dynamical, quantified state-variables [and

dynamical, quantified control parameters] as third, top, gamma-level.

Underneath/‘‘‘contained by’’’ this gamma layer/level/‘qualo-fractal’ scale is a beta layer/level/‘qualo-fractal’ scale, of non-amalgamatively summed {b_i} dynamical SUB-system ‘arithmetical qualifier meta-numerals’, EACH ONE divided by”’/‘‘‘containing’’’ a non-amalgamative sum of its own type q__M_U dynamical, quantified state-variables [and dynamical, quantified control parameters] as beta-level.

Underneath/‘‘‘contained by’’’ this beta layer/level/‘qualo-fractal’ scale is an alpha layer/level/‘qualo-fractal’ scale, consisting of a non-amalgamative sum of explicit dynamical SUB-SUB-system qualifiers, {a_i}, EACH ONE ‘“non-amalgamatively divided by”’/‘‘‘containing’’’ a sum of its own type q__M_U dynamical, quantified state-variables [and dynamical, quantified control parameters].

AND SO ON…

This progression of dialectical-mathematical systems suggests, to us, the following order of development for a science:

1. Taxonomy – develop a system of ontological categories that systematically comprehends the categorial content of the Domain, i.e., in systematic order [not necessarily the same as the historical order of first manifestation], in a gradient from least complex/least determinate ontological category to the most complex/most determinate ontological category of the Domain to-date.

2. Metrology – determine the “physical units”, the metrological “dimensions” needed to describe the present dynamics of this Domain, as described by means of ‘‘‘dimensional analysis’’’.

3. Historical Ordering – The determination of the taxonomic -- systematic -- order of the categories of the Domain in step 1, generates also the desire to determine the historical order-of-appearance of the kinds of units basing each of the ontological categories that constitute and exhaust the Present Domain, thus reconstructing, categorially, the past history of the Domain. On the basis of the Present ontology state, and of the reconstructed Past ontology-states, of the Domain, form hypotheses about its Future. These are thus predictive hypotheses, ‘pre-constructing’, symbolically, the investigators’ expected irruptions of new, unprecedented kinds of units/ontology/«arithmoi» in the future of the Domain, as well as the new ontological categories corresponding to these predicted new, unprecedented kinds of units.

4. Systems Dynamics – determine the families of dynamical systems that inhere Presently in the given Domain, e.g., a family of dynamical systems for each ontological category ‘“contained”’ in that Domain. Build dynamical models that also contain singularities, self-bifurcations, etc., that point to the expected future systems-level dynamics of this Domain.

5. Sub-Systems Dynamics – determine the families of dynamical sub-systems that inhere Presently in the given Domain, e.g., the families of dynamical sub-systems for each family of dynamical systems for each ontological category ‘“contained”’ in that Domain. Build dynamical models that also contain singularities, self-bifurcations, etc., that point to the expected future sub-systems-level dynamics of this Domain.

6. Sub-Sub-Systems Dynamics – determine the families of dynamical sub-sub-systems that inhere Presently in the given Domain, e.g., the families of dynamical sub-sub-systems for each family of dynamical sub-systems, for each family of dynamical systems for each ontological category ‘“contained”’ in that Domain. Build dynamical models that also contain singularities, self-bifurcations, etc., that point to the expected future sub-sub-systems-level dynamics of this Domain.

AND SO ON. . .

For ongoing updates regarding F.E.D. content, please see -- www.dialectics.info .

For F.E.D. books, and for partially pictographical, ‘poster-ized’ visualizations of many of our hypotheses -- specimens of ‘dialectical art’ -- see: