Dialectical Ideography

Volume One: Prolegomena

by Karl Seldon and Sophya St. Germain for F.E.D.

Original draft first distributed on 25 November 1999; last HTML update on 27 December 2009

Section A: Epitome

Kernel

The heart of ideographic dialectics as a set forth herein is a "conceptual meta-fractal". It can be expanded and explored on many different scales of exposition, as well as within many distinct contexts, points of view, or applications. A selection of applications are investigated in the main body of this treatise. Several scales of exposition are explored by sections located at its various extremities, viz. — this opening section [micro-scale], the concluding Condensations section [small-scale], and the middle-most sub-section [medium scale]. Potentially larger-scales of exposition, as well as of application, yet to be completed, loom beyond this text. We hope readers who find compelling either this conception of dialectical ideography, or various of its contraries, several of which are also reconnoitered herein, will contribute to that latter scale of exposition and application. This inaugural section lays out the primary hypotheses of Dialectical Ideography in the briefest form offered herein. Meanings that the propositions arrayed below initially hold for you may mutate markedly, as you read the main sections of this treatise. Even so, we believe that a capsule summary of the whole may prove useful to you, now, and later. Here it is. It outlines 'The Gödelian Dialectic' — the dialectic of inherent axiomatic incompleteness; the syntactico-semantic dialectic of the solution of "unsolvable" equations — as mapped into human history, as a project of the '''meta-science''' that we call '''Cognitive Psychohistory''':
  1. 1.Mathematics '''meta-evolves meta-axiomatically'''. Mathematical, 'theorem-etical' '''evolution''' may continue to progress within each «aufheben»-conserved axiomatic system of arithmetic+, but 'meta-axiomatic meta-evolution' leaps from within the self-inadequacies of each given, predecessor axiomatic system, to outside and beyond that system — to the "revolutionary", 'idea-ontology'-expanding construction of its successor axiomatic system. These 'inter-system-ic', 'trans-system-ic', «aufheben» '''meta-evolutions''' — from each predecessor axioms-system to its «aufheben»-expanded successor axioms-system — cumulatively accrue new axioms, by «aufheben» / conservative extensions. These "mathematical revolutions" punctuate and mediate a progression of '''psychohistorical''' cognitive crises. These revolutions serve as an index, i.e., as a barometer, of the attained level — of the 'meta-state' — of the self-development of the 'Human Phenome' as a whole at any given moment in human history.

  2. 2.Each such '''psychohistorical''' crisis involves — either explicitly, or merely "in effect" — the discovery of "unsolvable" [in]equations — i.e., the discovery of "unsolvable" equalities, and/or the discovery of "unsolvable" inequalities.

  3. 3.Each such crisis resolves by expansion of the number concept, of number 'ideo-ontology', to admit new kinds of numbers.

  4. 4.These new kinds of numbers, with their new rules / axioms, enable solution of those previously unsolvable [in]equations. Such cognitive gains in solution-capabilities tie to qualitative, physical-ontological expansions in human-social self-reproductive praxis — in the growth of the human-social forces of human-social self-productivity — and, thus, in the growth of the real and necessary wealth of human-social life.

  5. 5.'''The Nonlinearity Barrier''', the incapacity of modern mathematics to solve, in general, the sciences' nonlinear total and partial integrodifferential equations, especially those which embody its most advanced conceptions of the "laws", or "habits", of nature, constitutes the latest, and 333-year-protracted, '''psycho-historical''' crisis of unsolvability. The overcoming of this '''Nonlinearity Barrier''' within the current period of Terran human history — the overcoming which we term 'The Nonlinearity Breakthrough' — is crucial for our times — for our very lives, as for the very existence of our posterity. 'The Nonlinearity Breakthrough' is crucial to that renewed upsurge in the development of the human-social forces of human-society-expanding — of humanity-expanding — 'Phenomo-Genomic' self-reproduction of global human society as a whole, i.e., to that 'qualo-quantitative', 'socio-ontological growth' of global humanity, which, alone, can overcome the burgeoning forces toward a species-suicidal New and Final "Dark Age". Those New Dark Ages forces are the forces of a self-degenerating, '''decadent''' later capitalism. That capitalism is one in which the ruling plutocracy has organized itself to block the further development of science, of technology, of real/healthy, necessary wealth production, of human-social [re-]productivity, and of educated, middle-working-class expansion, so that this plutocracy seeks to preserve their power, and their lives of "privilege", against the threat of global revolt due to the consequent global growth of ever increasing populations of ever more impoverished people, by attempting to impose global multi-genocide. All of this arises from a ruling plutocracy, hyper-motivated to counter the threat to the concentrated capital base of their power — and, therefore, to the perks of their power — which is posed by the 'intra-duality', or 'self-duality', of accumulating capital-values, namely, that such values form an in[ternal-]tension, a "dynamical" — and, ultimately, a 'meta-dynamical' — '''complex unity''' of "self-expanding value" vis-á-vis 'self-contracting value', i.e., which is posed by the '''nonlinear law''', immanent to the system of human-social relations founded upon the Capital-value-relation, of the '''self-reflexive''', 'self-refluxive', competition-enforced 'self-de-value-ation' of accumulating [fixed-]capital-value, i.e., which is posed by the profit-incentives-accelerated scientifico-technological obsolescence depreciation of that plutocracy's ownership-concentrated, accumulated fixed capital, in the context of the rising '''technological composition''' / productivity / "productive force" of that accumulated capital, as of its corresponding labor-power. The source of this plutocratic motivation is exemplified in the threat that inherently Nonlinear-Processes-Based FUSION ATOMIC/SUB-ATOMIC POWER poses to the value of the vast accumulated fixed-capital of the core plutocracy's obsolescent MOLECULAR POWER Industry, e.g., to the capital-value of their global Petroleum Industry, and thus to the very foundations of the power of their 'econo-socio-political' global dictatorship. 'The Nonlinearity Breakthrough' will enable the rest of humanity to create — as an alternative to the plutocracy's New/Final Dark Age — the material and spiritual foundations for global Political-Economic Democracy, and for the first Global Renaissance of humanity in human history. 'The Nonlinearity Breakthrough' is, in its totality, and in its unity, none other than 'The Dialectics Breakthrough'.

  6. 6.As in the past, so presently, the present 'crisis of unsolvability' — that of '''The Nonlinearity Barrier''' — can be resolved by a further expansion of the 'ideo-ontology' of arithmetic; by a further expansion of the 'ideo-ontology' of number.

  7. 7.This resolution requires the discernment of yet new kinds of [meta-]numbers, new '''qualities''' of "quantity", beyond those of the "hyper-real", "Complex", Quaternion, Octonion, Clifford, Grassmannian, Boolean, and Cantorian arithmetics, for example. Each successor kind of number, belonging to the "Standard" kinds of new number-ontology generated by this 'Gödelian Dialectic', to-date, can be modeled by sets of a fixed Russellian-Gödelian "logical type" [or depth of sets self-membership], escalated by one unit above the logical type of the sets modeling its immediate predecessor kind of number. The first of the '''Non-Standard''' 'dialectical meta-numbers' arithmetics, exposited herein, constitutes a break in this pattern of the dialectic of the "Standard" arithmetics. Rather than being modelable by a domain of sets [of ordered pairs] of a single, fixed logical type, the 'dialectical meta-numbers' of the first in the progression of 'dialectical arithmetics' exposited herein are modelable by, and also model, this very movement of continual logical type escalation — the movement of the 'Gödelian Dialectic' itself, the 'intra-to-inter-systemic' movement, of cumulative axioms-system expansion/progression, mediated by "unsolvable" [in]equations, rising from lower axioms-system to the next-higher "conservative extension" axioms-system — to a more-inclusive axioms-system — as an axioms-system-inadequacy-driven, [Gödel-]incompleteness-driven, self-induced movement, of the immanent critique, or self-critique, of arithmetics. More specifically, these 'dialectical meta-numbers' model, and are also modeled by, in particular, the self-escalation of logical type exhibited by none other than the fundamental [idea-]object of set theory [although an object suppressed by "Standard" set theory], one which constitutes the very set-theoretical definition, or extensional definition, of the set concept itself, namely, the finitary, 'contra-Boolean', 'ideo-auto-kinesic' '''Set of All Sets''', { Sτ | τW }, wherein τ denotes a discrete time variable ['epoch-count'], wherein W = { 0, 1, 2, 3, ... }, and such that, Sτ+1 = Sτ2 = Sτ∆Sτ = Sτ ∪ 2^Sτ, wherein ∆Sτ denotes a qualitative, ontological increment of new 'ideo-ontology', to be added to Sτ. The solution of the foregoing nonlinear set-equation is Sτ = S02^τ, wherein S0 = 2^U = the set of all subsets of U, wherein U denotes the "Universal Set", the [finite, '''constructible'''] «arché»-set, composed of all of the [non-set] [idea-]objects / "Ur-elements" / "logical individuals" needed to define the given "universe of discourse" in question. The "population size" of the set of all subsets of U — the count of the 'idea-ontology', of the distinct idea-objects / elements / '''extensional predicates''', that this set of subsets "contains" — is denoted by

        |2^U| = 2^|U|, wherein |U| denotes the number of distinct idea-objects /

        elements "contained" in the set U. Thus, the '''Set of All Sets''', { Sτ = S02^τ =     

         [ 2^U ]2^τ | τW }, self-develops immanently — as,



— by immanent critique [i.e., via selfaufheben», 'self-meta-element-izing', 'self-meta-«monad»-izing', self-critique]. It does so because every τ-momentary existence of this Set always contradicts its own essence, because this Set is defined to be the Set containing ALL sets, but, since every set has a unique set of subsets, qualitatively different from its own content, and qualitatively different from the set of subsets of other sets, this Set is always missing precisely those sets which are its own subsets, including its own "improper" subset: itself as a whole! When we, therefore, expand this Set, to incorporate its [former] subsets, it thereby becomes a different, cumulatively larger version of this Set, with new, qualitatively different, ontologically different subsets, which it therefore also lacks as members of itself, so that it must be expanded again, to include all of those new subsets, which, once again, changes it into a qualitatively different Set, once again lacking its own subsets, .... This self-[driven ]progression of '''The Set of All Sets''' in fact constitutes a schematic, rudimentary 'Psychohistorical Model' — a real, temporal, diachronic '''phenomenology''' — of the progress of human cognition within a given "Universe of discourse" domain. It models the process of 'Predico-Dynamasis', in contrast to Boolean 'Predico-Stasis', the former being the self-increasing '''sensitization''' to, and 'explicitization' of, ever subtler '''qualities''', or "predicates", by which the cognitive function of that fundamental human "complex unity" — the 'Human Phenome / Human Genome' — advances human knowledge for a given domain of human experience, in the context of an accumulating and deepening human exploration thereof. This '''Set of All Sets''' can also be used to aptly model the 'physioauto-kinesis»' of the «physis»; of the physical cosmos — the 'self-meta-unit-izing', 'self-meta-«monad»-izing' '''Dialectic of Nature''' itself. The 'ideoauto-kinesis»' inherent in this defining object of Set Theory, '''The Set of All Sets''', also constitutes an immanent critique — indeed, a «reductio ad absurdum» self-refutation — of all Set Theories which harbor the — usually unstated, but typically tacitly presumed — 'Parmenidean Postulate', of 'ideoonto-stasis»'. This assumption is native, in particular, to so-called "Mathematical Platonism", the proposition that all valid set-theoretical idea-objects must be timeless, eternal, immutable, and unchangeable by any external agency — let alone being internally-, or self-, changing. The axioms of intuitive, "naive", unguarded set construction, together with this 'Parmenidean Postulate', deduce to a propositional negation of that 'Parmenidean Proposition', not only via the construction of the 'Standard Paradoxes', e.g., that of the truth-value '''self-oscillating''' Russell Set proposition — '''The set of all sets that are not members of themselves is [therefore is not] a member of itself.''' — or that of the proposition '''This proposition is false.''', but also via the construction of the 'Non-Standard Paradox' of this qualitatively, ontologically self-expanding '''Set of All Sets'''. The assertion of Parmenidean 'ideo-stasis' for all set-objects — the assertion that "Set idea-objects do not change." — together with the assertion of the other axioms of "naive" Set Theory, deductively yields the construction of these "paradoxical" sets as counter-examples, demonstrating the truth, within '''Natural Set Theory''', of the negation of that Parmenidean assertion; the truth, relative to the axioms of '''Natural Set Theory''', of the counter-proposition that "Set idea-objects do change [and even self-change]", given the assumed truth of those axioms. This result, by the rules of formal logic itself, establishes — thus, immanently — the axioms-relative falsity of the 'Parmenidean Premise' within the axioms of ''Natural Set Theory", and requires the assertion, in its place, of the negation of that [thus deductively self-]contradicted assertion: the affirmation of the possibility, and, indeed, of the ubiquity, of 'ideo-«[auto-]kinesis»', of a dynamical 'idea-ontology' — of 'ideo-onto-dynamasis' — in place of Parmenidean 'ideo-onto-stasis'. This [re-]affirmation, in the case of the fundamental object of Set Theory, '''The Set of All Sets''' — the very set-theoretical definition of the set concept itself — turns out to provide a set-theoretical model of the «physis»-inherent generation of time itself, as the unisonance of the concerted «auto-kinesis», and 'interkinesis»'; of the self-actioncum»-upon-other(s)-action — the «karma» — of all [ev]entities, and, thus, not as abstract, universe-external, exogenous, formal "time", but as concrete, 'contental', endogenous, immanent time, ultimately produced and continually reproduced out of the 'dynamasis'-driving subject/object, or action-initiator/action-recipient, 'self-duality', 'intra-duality', or 'indivi[sible]-duality', of each [ev]entity within the cosmological totality of subject-verb-object-identical [ev]entities. This '''Set of All Sets''' turns out to be, therefore, also a set-theoretical model of none other than the dialectic itself.

  1. 8.The higher degree terms which render the unsolved dynamical equations "nonlinear" essentially signify the "self-reflexivity" and 'self-refluxivity' of the 'external-world' processes / agents that those equations describe, reflecting modes of «autokinesis», of self-action, of self-movement, and of self-change, rooted in self-same subject/object 'intra-duality'. The homologous process, among 'internal-world' / mental process-objects, forms the paradoxes — both the "Standard" paradoxes, and the 'Non-Standard' paradoxes — the "insolubilia" of formal logic and set theory.

  2. 9.Self-reflexivity, self-refluxivity, 'self-dialogue', self-activity; the subject-/verb-/object-identical 'metafinite meta-dynamic' of self-changing, self-developing, self-revolutionizing, 'via-singularity-self-bifurcating diachronic meta-super^n-systems', constitutes the essence of historical-dialectical process.

  3. 10.The linearizing "Fundamental Law of Thought" / "Law of Duality" of Boole's original logic-algebra, expressed by Boole via the logic-equation x2 = x1 — akin to Cantor's |Rn>2| = |R2| = |R1| = c = Aleph1, despite Rn>2
    ...
    R2
    R1 — works as unitary axiom for the mathematics which inherits '''The Nonlinearity Barrier''', positing a dimensionality-denying, reductionist, point-atomistic, onto-statical, fixed-points-only, [≤]1-attractor, monolithic / niche-less logic, a logic of equilibrium [linear] 'anti-dynamics', or 'pseudo-dynamics'.

  4. 11.The partial contradiction of reality by Parmenidean-Boolean logico-mathematical idealizations, making them unfit to decode the nonlinear — i.e., self-reflexive / self-refluxive, i.e., dialectical — "laws", or "habits", of nature, may imply: (1) a kind of «reductio ad absurdum» / empirical refutation of the premise x2 = x1, or (2) evidence of its "independence" or Gödel-undecidability, vis-à-vis any other axioms. This points to new, "Non-Standard", 'Non-Parmenidean', 'Contra-Boolean' — nonlinear, that is dialectical — logics / totality theories, and to new, 'metafinite' arithmetics, analogous to the Non-Euclidean geometries arising from various negations of Euclid's parallels postulate. Adding these 'ideo-increments' to the 'multi-meta-ontic', 'meta-fractal ideo-cumulum', of number 'ideo-ontology' may render solvable these presently unsolvable, because nonlinear, integrodifferential equations, especially those which embody the most advanced conceptions of the "laws", or "habits", of nature, so far offered up by modern science.

  5. 12.The strong negation of Boole's linearizing axiom of 'exo-duality', i.e., the strong negation of the logic-equation x2 = x1, or, of the equivalent logic-equation --

   

                                                   x(1 − x) = 0

        — namely, the inequation

                                                    x2
x1


     -- wherein the ideogram '
' signifies non-quantitative, '''ontological''', qualitative    

     inequality — is solvable within certain new, 'contra-Boolean', 'contra-Cantorian', full-

     unit-interval [onto-]logical arithmetics and totality theories — with the latter

     inequation serving as an algebraically nonlinear logic-equation, and one which

     founds and grounds an algebraic logic of / for nonlinearity-in-general. Its logic of

     'meta-number unit-qualifiers' — of 'dialector', «aufheben» operators — provides

     a unified algorithmic «mimesis» for all of the key characters of dialectical,

    'metafinite-cumulum meta-dynamics', and an alternative, non-Boolean

    architectonic for computer design. Its extension from the unit-interval realms of

    Boolean [onto-]logical '''qualification''' and quantification to full-multiplicity realms of

    dimensionally-qualified [metrically-qualified, or qualitative units-of-measure-

    qualified], as well as of ontologically-qualified, '''arithmetical''', and '''analytical''',

    quantification proper [as with Boole's logic, for linear dynamics, which Boole himself

    characterized as the logic of linear partial-differential equations], may lead

    you to '''The Nonlinearity Breakthrough'''.