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

1.Mathematics '''metaevolves metaaxiomatically'''. Mathematical, 'theoremetical' '''evolution''' may continue to progress within each «aufheben»conserved axiomatic system of arithmetic+, but 'metaaxiomatic metaevolution' leaps from within the selfinadequacies of each given, predecessor axiomatic system, to outside and beyond that system — to the "revolutionary", 'ideaontology'expanding construction of its successor axiomatic system. These 'intersystemic', 'transsystemic', «aufheben» '''metaevolutions''' — from each predecessor axiomssystem to its «aufheben»expanded successor axiomssystem — 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 'metastate' — of the selfdevelopment of the 'Human Phenome' as a whole at any given moment in human history.

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.Each such crisis resolves by expansion of the number concept, of number 'ideoontology', to admit new kinds of numbers.

4.These new kinds of numbers, with their new rules / axioms, enable solution of those previously unsolvable [in]equations. Such cognitive gains in solutioncapabilities tie to qualitative, physicalontological expansions in humansocial selfreproductive praxis — in the growth of the humansocial forces of humansocial selfproductivity — and, thus, in the growth of the real and necessary wealth of humansocial life.

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 333yearprotracted, '''psychohistorical''' 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 humansocial forces of humansocietyexpanding — of humanityexpanding — 'PhenomoGenomic' selfreproduction of global human society as a whole, i.e., to that 'qualoquantitative', 'socioontological growth' of global humanity, which, alone, can overcome the burgeoning forces toward a speciessuicidal New and Final "Dark Age". Those New Dark Ages forces are the forces of a selfdegenerating, '''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 humansocial [re]productivity, and of educated, middleworkingclass 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 multigenocide. All of this arises from a ruling plutocracy, hypermotivated 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 'intraduality', or 'selfduality', of accumulating capitalvalues, namely, that such values form an in[ternal]tension, a "dynamical" — and, ultimately, a 'metadynamical' — '''complex unity''' of "selfexpanding value" visávis 'selfcontracting value', i.e., which is posed by the '''nonlinear law''', immanent to the system of humansocial relations founded upon the Capitalvaluerelation, of the '''selfreflexive''', 'selfrefluxive', competitionenforced 'selfdevalueation' of accumulating [fixed]capitalvalue, i.e., which is posed by the profitincentivesaccelerated scientificotechnological obsolescence depreciation of that plutocracy's ownershipconcentrated, accumulated fixed capital, in the context of the rising '''technological composition''' / productivity / "productive force" of that accumulated capital, as of its corresponding laborpower. The source of this plutocratic motivation is exemplified in the threat that inherently NonlinearProcessesBased FUSION ATOMIC/SUBATOMIC POWER poses to the value of the vast accumulated fixedcapital of the core plutocracy's obsolescent MOLECULAR POWER Industry, e.g., to the capitalvalue of their global Petroleum Industry, and thus to the very foundations of the power of their 'econosociopolitical' 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 PoliticalEconomic 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.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 'ideoontology' of arithmetic; by a further expansion of the 'ideoontology' of number.

7.This resolution requires the discernment of yet new kinds of [meta]numbers, new '''qualities''' of "quantity", beyond those of the "hyperreal", "Complex", Quaternion, Octonion, Clifford, Grassmannian, Boolean, and Cantorian arithmetics, for example. Each successor kind of number, belonging to the "Standard" kinds of new numberontology generated by this 'Gödelian Dialectic', todate, can be modeled by sets of a fixed RussellianGödelian "logical type" [or depth of sets selfmembership], escalated by one unit above the logical type of the sets modeling its immediate predecessor kind of number. The first of the '''NonStandard''' 'dialectical metanumbers' 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 metanumbers' 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 'intratointersystemic' movement, of cumulative axiomssystem expansion/progression, mediated by "unsolvable" [in]equations, rising from lower axiomssystem to the nexthigher "conservative extension" axiomssystem — to a moreinclusive axiomssystem — as an axiomssysteminadequacydriven, [Gödel]incompletenessdriven, selfinduced movement, of the immanent critique, or selfcritique, of arithmetics. More specifically, these 'dialectical metanumbers' model, and are also modeled by, in particular, the selfescalation 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 settheoretical definition, or extensional definition, of the set concept itself, namely, the finitary, 'contraBoolean', 'ideoautokinesic' '''Set of All Sets''', { Sτ  τ ∈ W }, wherein τ denotes a discrete time variable ['epochcount'], 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 'ideoontology', to be added to Sτ. The solution of the foregoing nonlinear setequation 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 [nonset] [idea]objects / "Urelements" / "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 'ideaontology', of the distinct ideaobjects / elements / '''extensional predicates''', that this set of subsets "contains" — is denoted by
2^U = 2^U, wherein U denotes the number of distinct ideaobjects /
elements "contained" in the set U. Thus, the '''Set of All Sets''', { Sτ = S02^τ =
— by immanent critique [i.e., via self«aufheben», 'selfmetaelementizing', 'selfmeta«monad»izing', selfcritique]. It does so because every τmomentary 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 'PredicoDynamasis', in contrast to Boolean 'PredicoStasis', the former being the selfincreasing '''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 'physio«autokinesis»' of the «physis»; of the physical cosmos — the 'selfmetaunitizing', 'selfmeta«monad»izing' '''Dialectic of Nature''' itself. The 'ideo«autokinesis»' inherent in this defining object of Set Theory, '''The Set of All Sets''', also constitutes an immanent critique — indeed, a «reductio ad absurdum» selfrefutation — of all Set Theories which harbor the — usually unstated, but typically tacitly presumed — 'Parmenidean Postulate', of 'ideo«ontostasis»'. This assumption is native, in particular, to socalled "Mathematical Platonism", the proposition that all valid settheoretical ideaobjects 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 truthvalue '''selfoscillating''' 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 'NonStandard Paradox' of this qualitatively, ontologically selfexpanding '''Set of All Sets'''. The assertion of Parmenidean 'ideostasis' for all setobjects — the assertion that "Set ideaobjects do not change." — together with the assertion of the other axioms of "naive" Set Theory, deductively yields the construction of these "paradoxical" sets as counterexamples, 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 counterproposition that "Set ideaobjects do change [and even selfchange]", given the assumed truth of those axioms. This result, by the rules of formal logic itself, establishes — thus, immanently — the axiomsrelative 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 'ideaontology' — of 'ideoontodynamasis' — in place of Parmenidean 'ideoontostasis'. This [re]affirmation, in the case of the fundamental object of Set Theory, '''The Set of All Sets''' — the very settheoretical definition of the set concept itself — turns out to provide a settheoretical model of the «physis»inherent generation of time itself, as the unisonance of the concerted «autokinesis», and 'inter«kinesis»'; of the selfaction«cum»uponother(s)action — the «karma» — of all [ev]entities, and, thus, not as abstract, universeexternal, exogenous, formal "time", but as concrete, 'contental', endogenous, immanent time, ultimately produced and continually reproduced out of the 'dynamasis'driving subject/object, or actioninitiator/actionrecipient, 'selfduality', 'intraduality', or 'indivi[sible]duality', of each [ev]entity within the cosmological totality of subjectverbobjectidentical [ev]entities. This '''Set of All Sets''' turns out to be, therefore, also a settheoretical model of none other than the dialectic itself.

8.The higher degree terms which render the unsolved dynamical equations "nonlinear" essentially signify the "selfreflexivity" and 'selfrefluxivity' of the 'externalworld' processes / agents that those equations describe, reflecting modes of «autokinesis», of selfaction, of selfmovement, and of selfchange, rooted in selfsame subject/object 'intraduality'. The homologous process, among 'internalworld' / mental processobjects, forms the paradoxes — both the "Standard" paradoxes, and the 'NonStandard' paradoxes — the "insolubilia" of formal logic and set theory.

9.Selfreflexivity, selfrefluxivity, 'selfdialogue', selfactivity; the subject/verb/objectidentical 'metafinite metadynamic' of selfchanging, selfdeveloping, selfrevolutionizing, 'viasingularityselfbifurcating diachronic metasuper^nsystems', constitutes the essence of historicaldialectical process.

10.The linearizing "Fundamental Law of Thought" / "Law of Duality" of Boole's original logicalgebra, expressed by Boole via the logicequation 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 dimensionalitydenying, reductionist, pointatomistic, ontostatical, fixedpointsonly, [≤]1attractor, monolithic / nicheless logic, a logic of equilibrium [linear] 'antidynamics', or 'pseudodynamics'.

11.The partial contradiction of reality by ParmenideanBoolean logicomathematical idealizations, making them unfit to decode the nonlinear — i.e., selfreflexive / selfrefluxive, 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 Gödelundecidability, visàvis any other axioms. This points to new, "NonStandard", 'NonParmenidean', 'ContraBoolean' — nonlinear, that is dialectical — logics / totality theories, and to new, 'metafinite' arithmetics, analogous to the NonEuclidean geometries arising from various negations of Euclid's parallels postulate. Adding these 'ideoincrements' to the 'multimetaontic', 'metafractal ideocumulum', of number 'ideoontology' 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.

12.The strong negation of Boole's linearizing axiom of 'exoduality', i.e., the strong negation of the logicequation x2 = x1, or, of the equivalent logicequation 
x(1 − x) = 0
— namely, the inequation
inequality — is solvable within certain new, 'contraBoolean', 'contraCantorian', full
unitinterval [onto]logical arithmetics and totality theories — with the latter
inequation serving as an algebraically nonlinear logicequation, and one which
founds and grounds an algebraic logic of / for nonlinearityingeneral. Its logic of
'metanumber unitqualifiers' — of 'dialector', «aufheben» operators — provides
a unified algorithmic «mimesis» for all of the key characters of dialectical,
'metafinitecumulum metadynamics', and an alternative, nonBoolean
architectonic for computer design. Its extension from the unitinterval realms of
Boolean [onto]logical '''qualification''' and quantification to fullmultiplicity realms of
dimensionallyqualified [metricallyqualified, or qualitative unitsofmeasure
qualified], as well as of ontologicallyqualified, '''arithmetical''', and '''analytical''',
quantification proper [as with Boole's logic, for linear dynamics, which Boole himself
characterized as the logic of linear partialdifferential equations], may lead