Introductory Letter

Dialectical Ideography and the Mission of F.E.D.



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Subject: Dialectical, Non-Standard "Natural"-Numbers' Arithmetic & contra-Boolean Algebra [Heuristic, Intensional-Intuitional Calculi for the Catalysis of Conceptual Discovery and Theory-Formation in the Natural and Social Sciences]: Solving for the Successor System.


Dear web-user, viewing the dialectics.org website,

This [revised and expanded] introductory letter and its two Supplements provide an overall introduction to Dialectical  Ideography, and to the mission of Foundation Encyclopedia Dialectica [F.E.D.].

But why do we believe that any of this should be of any interest to you? That question, we can answer. This letter contains our answer. As to whether this material is in fact of interest to you, only you can judge.

Our goal is to help avert the renewed, global, and final Dark Age that threatens Terran humanity, this time, with total extinction: not just with local "genocides", but with a global 'humanocide', culminating the ['''psycho-historically''' foreseeable] historical emergence, convergence, and 'historical singularity', for planet Earth, of 'capitalist anti-capitalism' and 'human anti-humanism', as captured in '''The Psycho-Historical Equations''' of dialectical ideography.

All of our efforts, our very lives, are dedicated to discovering means to overcome the gathering undertow of that new Dark Age, and to applying those means, nonviolently but effectively, in accord with 'The Seldonian Imperative', the ethical imperative to act so as to help avert, if possible, the collective agony of forseeable future catastrophes of contracted social reproduction, or, if too late to completely avert said catastrophes, to act so as to help reduce both their severity and their duration.

Nonviolence is a key to that effectiveness, for violence corrupts, converting its initiators into the very evil which they had intended to overthrow.

The Dialectic According to Plato

As a result, we have embarked upon a project that has bridged, across the abyss of the last Dark Age, the most advanced «problematiques» of the ancient and modern worlds.

This project appropriates the 'meta-fractal' self-similarity of those two different, successive scales of «problematique» so as to resume some final and neglected, zenith breakthroughs within the ancient Alexandrian flowering of humanity's ancient Mediterranean civilization, in relation to what has developed subsequently, and in a way which assimilates, also, the wealth of that subsequent development.

This has led to a rediscovery, in a higher, modernized, and less Parmenidean, more Heraclitean form, of Plato's lost «arithmos eidetikos», his "arithmetic of ideas" or "arithmetic of dialectics".

This «arithmos» is alluded to in his extant writings, but its full exposition is nowhere to be found in those portions of Plato's opus which survived the last Dark Age.

It has been 'psycho-archaeologically' reassembled in a seminal study, by Jacob Klein, as follows:

"While the numbers with which the arithmetician deals, the arithmoi [assemblages of units — F.E.D.] mathematikoi or monadikoi [abstract, generic, idealized, qualitatively homogeneous "monads" or [idea[ized]] unitsF.E.D.] are capable of being counted up, i.e., added, so that, for instance, eight monads [eight abstract idea[ized]-units, unities, or idea-a-tomsF.E.D.] and ten monads make precisely eighteen monads together, the assemblages of eide [of 'mental seeings' or mental visions; of "ideas" — F.E.D.], the "arithmoi eidetikoi" [assemblages, ensembles, '''sets''', or [sub-]totalities of qualitatively heterogeneous ideas or «eide» — F.E.D.], cannot enter into any "community" with one another [i.e., are 'non-reductive', '''nonlinear''', "non-superpositioning", "non-additive", 'non-addable', or "non-amalgamative" — F.E.D.]. Their monads are all of different  kind [i.e., are 'categorially', ontologically, qualitatively unequalF.E.D.] and can be brought "together" only "partially", namely only insofar as they happen to belong to one and the same assemblage, whereas insofar as they are "entirely bounded off" from one another...they are incapable of being thrown together, in-comparable [incapable of being counted as replications of the same unit[y] or monad; incomparable quantitativelyF.E.D.] ... .The monads which constitute an "eidetic number", i.e., an assemblage of ideas, are nothing but a conjunction of eide which belong together. They belong together because they belong to one and the same eidos [singular form of «eide»: one particular 'internal / interior seeing', vision, or «ιδεα» — F.E.D.] of a higher order, namely a "class" or genos [akin to the grouping of multiple species  under a single genus in classical 'taxonomics' — F.E.D.]. But all will together be able to "partake" in this genos (as for instance, "human being", "horse", "dog", etc., partake in "animal") without "partitioning" it among the (finitely) many eide and without losing their indivisible unity only if the genos itself exhibits the mode of being of an arithmos [singular form of «arithmoi»: a single assemblage of units/«monads» — F.E.D.]. Only the arithmos structure with its special koinon [commonality — F.E.D.] character is able to guarantee the essential traits of the community of eide demanded by dialectic; the indivisibility [a-tom-icity or 'un-cut-ability' — F.E.D.] of the single "monads" which form the arithmos assemblage, the limitedness of this assemblage of monads as expressed in the joining of many monads into one assemblage, i.e., into one idea, and the untouchable integrity of this higher idea as well. What the single eide have "in common" is theirs only in their community and is not something which is to be found "beside" and "outside"...them. ...The unity and determinacy of the arithmos assemblage is here rooted in the content of the idea..., that content which the logos [word; rational speech; ratioF.E.D.] reaches in its characteristic activity of uncovering foundations "analytically". A special kind of [all-of-one-kind, generic-units-basedF.E.D.] number of a particular nature is not needed in this realm, as it was among the dianoetic numbers [the «arithmoi monadikoi» — F.E.D.]..., to provide a foundation for this unity. In fact, it is impossible that any kinds of number corresponding to those of the dianoetic realm [the realm of 'dia-noesis' or of '«dianoia»', i.e., of 'pre-/sub-dialectical' thinkingF.E.D.] should exist here, since each eidetic number is, by virtue of its eidetic charactereide»-character or idea-natureF.E.D.], unique in kind [i.e., qualitatively unique/distinct/heterogeneous in comparison to other «eide» — F.E.D.], just as each of its "monads" has not only unity but also uniqueness. For each idea is characterized by being always the same and simply singular [ additively idempotentF.E.D.] in contrast to the unlimitedly many homogeneous monads of the realm of mathematical number, which can be rearranged as often as desired into definite numbers. ...The "pure" mathematical monads are, to be sure, differentiated from the single objects of sense by being outside of change and time, but they are not different in this sense — that they occur in multitudes and are of the same kind (Aristotle, Metaphysics B 6, 1002 b 15 f.: [Mathematical objects] differ not at all in being many and of the same kind...), whereas each eidos is, by contrast, unreproducible [hence modelable by idempotent addition, or 'non-addability'F.E.D.] and truly one (Metaphysics A 6, 987 b 15 ff.: "Mathematical objects differ from objects of sense in being everlasting and unchanged, from the eide, on the other hand, in being many and alike, while an eidos is each by itself one only"...). In consequence, as Aristotle reports (e.g., Metaphysics A 6, 9876 b 14 ff. and N 3, 1090 b 35 f.), there are three kinds of arithmoi: (1) the arithmos eidetikosidea-number, (2) the arithmos aisthetos — sensible number, (3) and "between"...these, the arithmos mathematikos or monadikos — mathematical and monadic number, which shares with the first its "purity" and "changelessness" [here Aristotle reflects only the early, more 'Parmenidean', Plato, not the later, «autokinesis» Plato — F.E.D.] and with the second its manyness and reproducibility. Here the "aisthetic" ["sensible" or sensuousF.E.D.] number represents nothing but the things themselves which happen to be present for aisthesis [sense perceptionF.E.D.] in this number. The mathematical numbers form an independent domain of objects of study which the dianoia [the faculty of 'pre-/sub-dialectical thinking'F.E.D.] reaches by noting that its own activity finds its exemplary fulfillment in "reckoning [i.e., account-giving] and counting".... The eidetic number, finally, indicates the mode of being of the noeton [that which exists "for" thought; that which thought "beholds"; the object of thought; the idea[l]-objectF.E.D.] as such  — it defines the eidos ontologically as a being which has multiple relations to other eide in accordance with their particular nature [i.e., in accord with their contentF.E.D.] and which is nevertheless in itself altogether indivisible. The Platonic theory of the arithmoi eidetikoi is known to us in these terms only from the Aristotelian polemic against it (cf., above all, Metaphysics M 6-9)." [Excerpts from: Jacob Klein; Greek Mathematical Thought and the Origin of Algebra; Dover (New York: 1992); pages 89-91; bold, italic, underline, and color emphasis added by F.E.D.].

aufheben_arithmoi_eidetikoi.

Plato may have already embarked upon an axiomatization of these three arithmetics, circa 380 B.C.E., even prior to Euclid of Alexandria's axiomatization of geometry, circa 300 B.C.E.:

"Plato seems to have realized the gulf between arithmetic and geometry, and it has been conjectured that he may have tried to bridge it by his concept of number and by the establishment of number upon a firm axiomatic basis similar to that which was built up in the nineteenth century independently of geometry; but we cannot be sure, because these thoughts do not occur in his exoteric writings and were not advanced by his successors. If Plato made an attempt to arithmetize mathematics in this sense, he was the last of the ancients to do so, and the problem remained for modern analysis to solve. The thought of Aristotle we shall find diametrically opposed to any such conceptions. It has been suggested that Plato's thought was so opposed by the polemic of Aristotle that it was not even mentioned by Euclid. Certain it is that in Euclid there is no indication of such a view of the relation of arithmetic to geometry; but the evidence is insufficient to warrant the assertion that, in this connection, it was the authority of Aristotle which held back for two thousand years a transformation which the Academy sought to complete." [Carl B. Boyer; The History of the Calculus and its Conceptual Development; Dover (NY: 1949); page 27].

We term 'Peanic' any progression of entities which conforms to the first four (Giuseppe) Peano postulates, circa 1889, which were formulated with the intent to axiomatize just the "standard natural numbers", but which are known to have "non-standard models".

'Dialectic' is a 'logic', or a ['Qualo-Peanic' 'Meta-Peanic'] 'pattern of what follows from what', more general than the "formal logic" of 'propositional followership'.

'Dialectic' generalizes about how natural populations, ensembles, systems, [sub-]totalities — both concrete, physical-'external' «arithmoi», and 'internal', human-conceptual «arithmoi», — change, including especially of how they change themselves.

'Dialectic' is about both '[allo-]flexion' or '[allo-]flexivity' — the 'bending', or '''alteration''', of the 'course of development' of one '[ev]entity' by the actions of others — and about 'self-re-flexivity', 'self-re-fluxivity', 'self-dialogue', 'self-controversion', self-activityself-change, or "self-contra-kinesis" [in summary, about '''self-bending''': the 'self-induced', self-determining '''bending''' of the 'course of development' of an 'eventity' as a result of its own, immanent, '''inertial''', ''ballistic''', 'intra-dual', 'essence-ial', 'self-force'].

'Dialectic' is the name for the fundamental [and ever self-developing] modus operandi of nature, including that of human[ized] nature, but also including that of pre-human and extra-human nature.

'Dialectic' is about the subject/verb/object-identical meta-dynamic of 'quanto-qualitatively', 'quanto-ontologically' [self-]changing, [self-]developing, via-'metafinite'-singularity '[self-]bifurcating' 'meta-systems' or 'process-entities' ['eventities'], manifested in all levels, at all '[meta-]scales', for all '''orders''' of 'natural history', including that part of 'natural history' which we call Terran human history, and, by hypothesis, in the history of humanoid species generally, throughout this cosmos ['human-natur[e-]al history'].

For Plato, «Dialektike», 'dialectical thought-technology', as manifested in his «arithmos eidetikos», names a higher form of human cognition. It is higher than that of «Doxa», mere opinion. It is also higher than that of «Dianoia» or «Dianoesis»; higher than that which Hegel termed «Verstand», "The Understanding" [cf. Plato].

"Dialectical thought" names a higher stage of human cognitive development, a higher "state" of human [self-]awareness, a higher form of human self-identity, and of '''human subject-ivity''', beyond even those associated with the most advanced possible forms of axiomatic, deductive, mathematical logic, still «dianoetic» and partly sub-rational due to the frequent arbitrariness, authoritarianism, and dogmatism of their unjustified axioms and primitives:

"...disputation and debate may be taken as a paradigmatic model for the general process of reasoning in the pursuit of truth, thus making the transition from rational controversy to rational inquiry. There is nothing new about this approach. Already the Socrates of Plato's Theaetetus conceived of inquiring thought as a discussion or dialogue that one carries on with oneself. Charles Saunders Peirce stands prominent among those many subsequent philosophers who held that discursive thought is always dialogical. But Hegel, of course, was the prime exponent of the conception that all genuine knowledge must be developed dialectically. ... These conclusions point in particular towards that aspect of the dialectic which lay at the forefront of Plato's concern. He insisted upon two fundamental ideas: (1) that a starting point for rational argumentation cannot be merely assumed or postulated, but must itself be justified, and (2) that the modus operandi of such a justification can be dialectical. Plato accordingly mooted the prospect of rising above a reliance on unreasoned first principles. He introduced a special device he called "dialectic" to overcome this dependence upon unquestioned axioms. It is worthwhile to see how he puts [this] in his own terms:

"There remain geometry and those other allied studies which, as we have said, do in some measure apprehend reality; but we observe that they cannot yield anything clearer than a dream-like vision of the real so long as they leave the assumptions they employ unquestioned and can give no account of them. If your premise is something you do not really know and your conclusion and the intermediate steps are a tissue of things you do not really know, your reasoning may be consistent with itself, but how can it ever amount to knowledge? ... So... the method of dialectic is the only one which takes this course, doing away with assumptions. ... Dialectic will stand as the coping-stone of the whole structure; there is no other study that deserves to be put above it."

Plato's writings do not detail in explicit terms the exact nature of this crucial enterprise of dialectic. Presumably we are to gain our insight into its nature not so much by way of explanation as by way of example  — the example of Plato's own practice in the dialogues." [Nicholas Rescher; Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge; SUNY Press (Albany, NY: 1977); pages 46-48; bold, italic, underline, and color emphasis added by F.E.D.].

The procedure of formal proof, of deductively deriving theorems from axioms and postulates, is the exercise of «dianoesis» «par excellence». But the process of discovery, formulation, selection, refinement, and optimization of the individual axioms themselves, and of systems of axioms, resides beyond the «dianoetic» realm. Formal and mathematical logic provide it with no algorithmic guidance. That process belongs to the realm of dialectics.

The Enigma of the Platonic Dialectic

The following extracts provide an overview of the difficulties confronting modern scholars of Plato in deciphering the unified meaning of the Platonic dialectic / ‹‹arithmoi eidetikoi››.  Prior to the insights of Jacob Klein, Denise Schmandt-Besserat, and others regarding ancient arithmetic, and the integration of those insights in the work of Karl Seldon and Sophya St. Germain, no such unified meaning had been recovered.

We learn, for instance, in J. O. Urmson’s The Greek Philosophical Vocabulary, in the entry for the Greek word ‹‹arithmos›› which is translated, in this entry, simply as “number of the ‘‘‘psycho-historical’’’ fact that the ancient Greek concept of “number” differed markedly from – and was, in some ways, ‘ideo-ontologically’ shrunken with respect to our own. However, in another way, that ancient conception was ‘ideo-ontologically’ expansive relative to the modern one, in that it included a concept of “nonaddible”, and therefore apparently of qualitative – qualitatively heterogeneous – “numbers”:

“Zero was unknown as a number, and one also was not counted as a number, the first number being the duas two.” [J. O. Urmson, The Greek Philosophical Vocabulary, Duckworth & Co., Ltd. [London: 1990], pp.31-32].

We also learn of a key “obscure” distinction in Plato’s “unwritten doctrines”, between Plato’s concept of ‘dianoiac’ “mathematical numbers", the ‹‹arithmoi monadikoi››, versus his dialectical ‘‘‘idea-numbers’’’, the ‹‹arithmoi eidetikoi››:

“From the Pythagoreans … who consider number to be the first principle number played a great role in metaphysics, especially in Plato’s unwritten doctrines, involving obscure distinctions of e.g. ‹‹sumblêtoi›› and ‹‹asumblêtoi›› addible and non-addible numbers.” [Urmson, ibid., emphasis added by F.E.D.].

Thus it appears that Plato too, with the Pythagoreans, considered ‘“number”’ to be the “first principle”. But Plato ‘‘‘also’’’ considered the “Forms”, the ‹‹eide›› the «ιδεας» to be the “first principle”. However, these ‘‘‘two’’’ considerations, for Plato, constituted no contradiction. The «ιδεας» or ‹‹eide›› were, for Plato, ‘‘‘numbers’’’– i.e., ‹‹arithmoi›› namely, the ‹‹arithmoi eidetikoi››, the very ‹‹arithmoi›› of his ‹‹dialektiké››.

This ‘‘‘idea-number’’’ notion of Plato’s has been replete with all manner of perplexity for modern scholarship:

arithmós:  number (see also arithmos eidetikos and arithmos mathematikos)

   3.  The most perplexing aspect of ancient number theory is Aristotle’s repeated assertions that Plato taught that the eide were numbers (e.g. Meta. 987b), a position that must be distinguished from 1) the existence of the eide of numbers (see arithmos eidetikos) and 2) the existence of the “mathematicals” as an intermediate grade of being (see mathematika, metaxu). But nowhere in the dialogues does Plato seem to identify the eide with number. To meet this difficulty some have postulated a theory of later “esoteric” Platonism known to Aristotle (but see agrapha dogmata), while others have attempted to see the emergence of the eide-arithmos theory described in such passages as Phil. 25a-e, the reduction of physical objects back to geometrical shapes in Tim. 53c-56c (see stoicheion), and the increasing stress on a hierarchy among the Forms (see Soph. 254d and genos, hyperousia), which, according to Theophrastus, Meta. 6b, would suggest the descending series archai (i.e., monas/dyas or peras/apeiron, qq.v.), arithmoi, eide, aistheta. Still others say that Aristotle either deliberately or unknowingly confused the position of Plato with those of Speusippus and Xenocrates (see mathematika).” [F. E. Peters, Greek Philosophical terms:  A Historical Lexicon, NYU Press [NY: 1990], pp. 25-26].

In the entry for the Greek word ‹‹dialektiké››, translated, in this reference, as the English “dialectic”, we learn the following:

On the testimony of Aristotle dialectic was an invention of Zeno the Eliatic, probably to serve as a support for the hypothetical antinomies of Parmenides ... But what was a species of verbal polemic (what Plato would call “eristic” or disputation...) for the Eliatics was transformed by Plato into a high philosophic method. The connecting link was undoubtedly the Socratic technique of question and answer in his search for ethical Definitions…, a technique that Plato explicitly describes as dialectical (Crat. 390c). …With the hypostatization of the Socratic definitions as the Platonic eide … the role of dialectic becomes central and is the crown of the ideal curriculum described in the Republic:  after ten years devoted to mathematics the philosopher-to-be will devote the years between thirty and thirty-five to the study of dialectic. …

What is dialectic?  The question is not an easy one since Plato, as usual, thought about it in a variety of ways. There is the view of the Phaedo and the Republic, which envisions dialectic as a progressively more synoptic ascent, via a series of “positions” (hypotheseis, q.v.; the Theory of Forms is one such in the Phaedo 100b), until an ultimate is reached (Phaedo 101d, Rep. 511e).  In the Republic, where the context of the discussion is confessedly moral, this “unhypothesized principle” is identified with the good-in-itself (auto to agathon; Rep. 532a-b) that subsumes within itself all of the lower hypotheses (ibid., 533c-d) [cf. the Hegelian core concept of dialectic, named by the German word ‹‹aufheben››F.E.D.] … If the dialectic of the Phaedo and the Republic may be described as “synoptic” …, that which emerges from the Phaedrus onward is decidedly “diacritic”… it is introduced in Phaedrus 265c-266b (compare Soph. 253d-e) and consists of two different procedures, “collection” (synagogue, q.v.) and “division” (diairesis, q.v.), the latter process being amply illustrated in subsequent dialogues like the Sophist, Politicus, and Philebus. The earlier dialectic appeared similar to the operations of eros (q.v.) [recall Herbert Marcuse’s comment, in his Reason and Revolution, to the effect that '''eros is the force that binds matter together into ever higher unities'''-- F.E.D.], but here we are transported into an almost Aristotelian world of classification through division; ascent has been replaced by descent. While it is manifest that we are here still dealing with ontological realities, it is likewise clear that a crucial step has been taken along the road to a conceptual logic. The term [i.e., the terminus – F.E.D.] of the diairesis is that eidos which stands immediately above the sensible particulars (Soph. 229d), and, while this is “really real” (ontos on) in the Platonic scheme of things, it is significant that the same process ends, in Aristotle, in the atomon eidos, the infima species in a logical descent (De an. II, 414b)…” [Peters, ibid., pp. 36-37].

Within the kind of ‹‹arithmoi eidetikoi›› structure described in the extract from Jacob Klein’s book, and depicted in the illustrations above, both the ‘‘‘ascending’’’ and ‘‘‘descending’’’ paths are intrinsic. Further clues regarding this supposedly only synchronic dialectical structure may be gleaned from the entry on ‹‹diairesis›› in the above-sited lexicon, by Peters:

diairesis: separation, division, distinction

1. Division, a procedure that did not interest Socrates since the thrust of his enquiry was toward a single eidos (see epagoge), becomes an important feature in the later dialogues where Plato turns his attention to the question of the relationship between eide.  Expressed in terms of Aristotelian logic diairesis is part of the progress from genus to species; but, as is clear from a key passage in the Parmenides, where he first puts the question (129d-e), Plato did not see it as a conceptual exercise. The dialectical search of which diairesis is part has as its object the explication of the ontological realities that are grasped by our reflection (logismos).

2. The pursuit of the interrelated eide begins with an attempt at comprehending a generic form (Phaedrus 265d); this is “collection” (synagoge, q.v.).  It is followed by diairesis, a separation off of the various eide found in the generic eidos, down to the infima species (Soph. 253d-e). Plato is sparing of details in both the theory and the practice of synagoge, and, while the Sophist and the
Politicus are filled with examples of diairesis, there is relatively little instruction on its methodology. We are told, however, that the division is to take place “according to the natural joints” (Phaedrus 265e). What these are becomes clearer from the Politicus:  they are the differences (diaphorai, q.v.) that separate one species from another in the generic form (Pol. 262a-263b, 285b).

3.  The method of division raises certain serious questions, so serious, indeed, that they might very well shake confidence in the existence of the eide. …  Do the species constitute the genus
or are they derived from it? …” [Peters, ibid., pp. 34-35].

Of the meaning of this ‘sub-method’ of the Platonic dialectical method, termed ‹‹diairesis››, the Urmson source provides the following:

diairien (in past tense, dielein), diairesis:  to divide, division, used in many contexts in Greek as in English. In philosophy particularly the logical division of a genus into species. In the Phaedrus and the Sophist Plato speaks of a method of sunagôgê collection – and diairesis division — as the supreme method of philosophy: … and, Phaedrus, I myself am a lover of divisions and collections in order to become able to speak and think (Pl. Phaedrus 266b); … unless one is capable of dividing things and subsuming each thing individually under a single form, one will never become skilled in discussion to the limit of human capacity (Pl. Phaedrus 273d): … a longstanding laziness about dividing genera into species (Pl. Soph. 267d).” [Urmson, ibid., pp. 39-40].

The “mystery” of the first movement, and ‘sub-method’, of the dialectical “method of discovery”, ‹‹synagoge››, is also further addressed in our two sources:

sunagein: to collect; sunagôgê:  the action of collecting. Non-technically: … we shall bring together the brides and the bridegrooms (Pl. Rep. 459e).  Also used as a technical term by Plato, particularly in the Sophist and the Phaedrus, where the contrary of sunagôgê is diairesis, division: … I am myself, Phaedrus, a lover of these divisions and collections (Pl. Phaedrus 266b). Collection appears to be bringing together under a single genus a variety of things which are then to be divided formally into species and sub-species: … to survey under one form things that are scattered in many areas (Pl. Phaedrus 265d).” [Urmson, ibid., pp. 158-159].

synagôgé: collection

The Platonic type of “induction” (for the more normal type of induction, i.e., a collection of individual instances leading to a universal, see epagoge) that must precede a division (diairesis) and that is a survey of specific forms (eide) that might constitute a genus (Phaedrus 265d, Soph. 253d).  An example is Soph. 226a, and the process is also suggested in Rep. 533c-d, and Laws 626d…” [Peters, ibid., p. 188].

Parts of the entries under ‹‹eidos›› in the Peters source can serve as a summary of our findings, above, regarding the enigma of the Platonic dialectic:

eídos:  appearance, constitutive nature, form, type, species, idea

  12. At various points in the dialogues Plato seems to grant preeminence to one or other [sic] of the
eide.  Thus, both the Good (Rep. 504e-509c) and the Beautiful (Symp. 210a-212b) are thrown into relief, to say nothing of the notorious hypotheses of the One in the Parmenides (137c-142; see hen, hyperousia). But the problem of the interrelationship, or, as Plato calls it, “combination” or “communion” (koinonia), and, by implication, of the subordination of the eide is not taken up formally until the Sophist.  It is agreed, again on the basis of predication, that some eide will blend with others and some will not, and that it is the task of dialectic to discern the various groupings, particularly through the diacritic method known as diairesis (q.v.; Soph. 253b-e). …  [Peters, ibid., p. 49, emphasis added by F.E.D.].

  8. Though the eide are the centerpiece of Platonic metaphysics, nowhere does Plato undertake a proof for their existence; they first appear as a hypothesis (see Phaedo 100b-101d) and remain so, even though subjected to a scathing criticism (Parm. 130a-134e).  They are known, in a variety of methods, by the faculty of reason (nous; Rep.532a-b, Tim. 51d). One such early method is that of recollection (anamnesis, q.v.), where the individual soul recalls the eide with which it was in contact before birth (Meno 80d-85b, Phaedo 72c-77d; see palingenesia). Without the attendant religious connotations is the purely philosophical method of dialektike (q.v.; see Rep. 531d-535a; for its difference from mathematical reasoning, ibid., 510b-511a; from eristic, Phil. 15d-16a).  As it is first described the method has to do with the progress from a hypothesis back to an unhypothesized arché (Phaedo 100a, 101d; Rep. 511b), but in the later dialogues dialektiké appears as a fully articulated methodology comprising “collection” (synagoge, q.v.) followed by a “division” (diairesis, q.v.) that moves, via the diaphorai, from a more comprehensive Form down to the atomon eidos.  Finally, one may approach the eide through eros (q.v.), the desiderative parallel to the earlier form of dialectic (see epistrophe).” [Peters, ibid., pp. 47-48].

There is another central Platonic theme more Heraclitean, less Parmenidean; more diachronic, less synchronic that forms a part, in our view, of the enigma of the Platonic dialectic:  that of ‹‹autokinesis››, or of “self-motion of the self-induced motion of a ‘‘‘self’’’, agent, object, or entity. Our re-discovery of Plato's 'dialectical arithmetic' emerged in the context, also, of our study of the most advanced development of Plato's thinking, as embodied in his final dialogues, beginning with The Parmenides. In those later dialogues, Plato advances beyond his earlier, 'Parmenideanic' eternal «stasis» of the "Forms", or «eide», to embrace a theoretical commitment to the fundamentality of “self-change”, or «autokinesis», and to the primacy of this "self-derived motion" over "derived motion", i.e., over other-induced, externally-induced change:

"The dialogues of the Socratic period provide that view of the world usually associated with Plato. The period of transition and criticism, and the final synthesis, are little noted ... The Parmenides can be taken as signaling the change. In this dialogue Socrates is unable to defend his Doctrine of Ideas. ... Where the Republic and Phaedo stressed the unchanging nature of the soul, the emphasis in the Phaedrus is exactly reversed. In this dialogue, the soul is the principle of self-motion, and we are told that the soul is always in motion, and what is always in motion is immortal. The difference now between spirit and matter is not changelessness in contrast with change, but self-motion, the essence of the soul, in contrast with derived motion. The emphasis on self-motion is continued even in the Laws, Plato's final dialogue." [William L. Riese; Dictionary of Religion and Philosophy: Eastern and Western Thought; Humanities Press, Inc. (New Jersey: 1980); pages 442-443, emphasis added by F.E.D.].

Is there a connection between the late-Platonic principles of ‹‹autokinesis››, of self-change and self-movement, and the Platonic concept of ‹‹dialektiké››? Considering the following extracts on ‹‹autokinesis›› from the Platonic dialogues cited in the quote above may help to advance us in our consideration of this question:

[Phaedrus]:  “But that which while imparting motion is itself moved by something else can cease to be in motion, and therefore can cease to live; it is only that which moves itself that never intermits its motion, inasmuch as it cannot abandon its own nature; moreover this self-mover is the source and first principle of motion for all other things that are moved.  Now a first principle cannot come into being, for while anything that comes to be must come to be from a first principle, the latter itself cannot come to be from anything whatsoever; if it did, it would cease any longer to be a first principle. Furthermore, since it did not come into being, it must be imperishable, for assuredly if a first principle were to be destroyed, nothing could come to be out of it, nor could anything bring the principle itself back into existence, seeing that a first principle is needed for anything to come into being.

The self-mover, then, is the first principle of motion, and it is as impossible that it should be destroyed as that it should come into being; were it otherwise, the whole universe, the whole of that which comes to be, would collapse into immobility, and never find another source of motion to bring it back into being.” [Plato, The Collected Dialogues, E. Hamilton, H. Cairns, editors, Princeton U. Press [Princeton:  1989], Phaedrus, 245c-e, pp. 492-493, italic and bold-italic colored text emphasis added by F.E.D.].

[Laws]:  “When we have one thing making a change in a second, the second, in turn, in a third, and so on – will there ever, in such a series, be a first source of change? Why, how can what is set moving by something other than itself ever be the first of the causes of alteration? The thing is an impossibility. But when something which has set itself moving alters a second thing, this second thing still a third, and the motion is thus passed on in course to thousands and tens of thousands of things, will there be any starting point for the whole movement of all, other than the change in the movement which initiated itself?

Suppose all things were to come together and stand still – as most of the party have the hardihood to affirm. Which of the movements we have specified must be the first to arise in things?  Why, of course, that which can move itself; there can be no possible previous origination of change by anything else, since, by hypothesis, change was not previously existent in the system. Consequently, as the source of all motions whatsoever, the first to occur among bodies at rest and the first in rank in moving bodies, the motion which initiates itself we shall pronounce to be necessarily the earliest and mightiest of all changes, while that which is altered by something else and sets something else moving is secondary.” [ibid., Laws, 10.894e-10.895b, pp. 1450, bold-italic, underlined, and colored text emphasis added by
F.E.D.].

The above considerations, then, adumbrate the challenge that Karl Seldon and Sophya St. Germain faced in their project to recover the hypothesized original unity of the Platonic conception of ‹‹dialektiké››, and of its ‹‹arithmoi eidetikoi››, from the enigma of its seemingly disparate doctrines, as portrayed in the extracts above:

1.    of ‘‘‘ideas as ‘unaddable’ numbers’’’, & of ‹‹dialektiké›› as an ‘‘‘arithmetic of ideas’’’; the arithmetic of the ‹‹arithmoi eidetikoi››;

2.   of ‹‹dialektiké›› as the highest philosophic method, one similar in its operation to that of eros; a synoptic method, a method of ascent,  via a series of “positions”, or hypotheses, until an ultimate is reached, that subsumes within itself all of the lower hypotheses;

3.   of ‹‹dialektiké›› as a diacritic method, a method of descent of synchronicideo-systematics’, ‘ideo-taxonomics’, or ‘ideo-meta-genealogy’, for the correct determination of the … «gene», the «species», and the sub-«species» …, etc., of the «eide» a method composed of two distinct, opposite procedures, or movements; first by one of collection”[«synagoge»], into «gene», followed, second, by one of “division” [«diairesis»], into “classes” — into «species», sub-«species» …, etc. of the fundamental «ιδεας» that, per Plato, undergird this «kosmos», and;

4.   of ‹‹arché kinesis›› as ‹‹auto kinesis››.

Moreover, this challenge emerged in the context of the effort of Karl Seldon and Sophya St. Germain to advance the theory of diachronic, historical dialectics, and of its calculus   a theory and a calculus of the auto-kinesic, temporal, ‘‘‘chrono-logical’’’, and, moreover, ‘chronogenicself-speciation of species and self-generation of genera, in a way con-«gene»-al with the more synchronic, ‘‘‘systematic dialectic’’’ that Plato emphasized.

All of these considerations converge in the exposition of the rest of this letter, and of its next section, entitled: The Secret of the Dialectic.

The Secret of the Historical Dialectic

The most fundamental form of dialectical opposition, the most fundamental form of dialectical '''contradiction''' — i.e., of ontological, existential, 'essence-ial', and temporal / 'temporo-genic' contradiction, as distinguished from formal-logical, propositional contradiction — the most fundamental form of thesis versus 'contra-thesis' confrontation — or of «physis» versus 'contra-«physis»' confrontation — is the '''self-reflexive''', 'self-refluxive', '''‹‹karmic»''' self-confrontation of a single 'event-entity' ['''eventity'''], of a single '''[sub-]totality''', of a single ''''[w]hol[on][e]''', of a single '''dynamical system''', of a single '''self''', as [sentential] '''subject''', or '''agent''' of action, versus itself again, as [sentential] '''object''', i.e., as the recipient of the 'essence-ial' action [«karma»]of the subject, through the [sentential] verb-form of that 'subject-object', or 'agent-object'.

Thus, the celestial '''eventities''' whose mature forms we call ["Main Sequence"] "stars" act upon themselves '''self-gravitationally''' — starting during their "proto-stellar" stages — thereby inducing an implosion which, by compressing the Hydrogen atoms / proton plasmas at their hearts beyond a critical threshold, triggers the '''anti-implosive''', explosive "thermonuclear" force of Hydrogen fusion into Helium, that stops their implosive collapse in a [temporary] balance between the [nearly] spherically-symmetrical, [internally-]everywhere-opposing vector-field forces of implosion and explosion, until the complete fusion-conversion of their core Hydrogen into core Helium triggers a further qualitative change.

This fundamental dialectical opposition "between" the '''eventity''' in its aspect as subject versus the self-same '''eventity''' in its aspect as object, is epitomized, in a universal sense, the following sentence, a '''self-reflexive''', 'self-refluxive', '''«karmic»''' sentence, which formulates this universal dialectical principle of '''self-activity''', '''self-change''', '''self-movement''', or «autokinesis», in its generic form:

'subject '''subjects''' subject.' OR 'agent '''agents''' agent.'

In the above sentence, the verb, '''subjects''', is not meant with its usual connotation alone, or mainly, but is meant primarily as the verb form of the noun '''subject''', or '''agent''', connoting the 'essence-ial', and '''essential'''/necessary, ineluctable action of the subject upon all things it encounters, including upon itself as its own object; the 'verb-name', or 'action-name' of the subject in question, whatever its 'noun-name', or 'pronoun-name'.

Of course, since the products of the 'self-production' that these sentences describe multiply the ontology of their domains, the resulting '''subjects'', '''objects''', or '''subject-objects''' engage in true '''INTER-actions''', in addition to the '''self-interactions''', '''self-INTRA-actions''', or '''self-actions''' that the sentence above describes:

'subject '''subjects''' object.'

— wherein '''subject''' is not synonymous with '''object'''. However, the 'self-reflexive', 'self-refluxive', '''«karmic»''' form remains primary and fundamental.

This primary form is 'formulatable' equally as

'object '''objects''' object.'

once it is realized that '''natural objects''' in general — and not just individual humans — are inherently active and 'self-active'; are 'agental'; indeed, that the are 'activities'; that they are 'activity-entities'; that they are [nameable as] "verbs"; that they are "eventities".

For human natur[e-]al history, the primary «specific» forms of the «generic» sentence above are:

'Humanity '''humanifies''' humanity.'

'Humanity produces humanity.'

'Humanity produces itself.'

'Humanity expandedly-reproduces itself.'

and, the fundamental proposition of Marxian theory in this regard is the proposition that 'the accumulation of humanity' — 'phenotypically' as well as 'genotypically'; '''culturally''' as well as "biologically" — within the later, final, '''descendant''' phase of the capital-relation-based epoch of human self-development, comes into conflict with '''the accumulation of capital'''. Using the doubly-negated/slashed equals sign, '#', as the sign for dialectical [self-]opposition / [internal-]contradiction, we then have the claim, for '''descendant-phase''' capitalism that:

'Accumulation of Humanity' # 'Accumulation of Capital'.


The immediate question in this regard, for this section of this Introductory Letter, is: How can we incorporate the '''subject-object identical''' secret of the natural-historical dialectic, as defined just above, into our 'dialectical pictography' within the rest of this Letter?

Here's what we have devised.

The Dialectic of Set Theory

Dialectical 'Meta-Axiomatics' «aufheben»-conserves the full logical rigor of deductive proof-based «dianoesis». But dialectical 'Meta-Axiomatics' also exceeds that «dianoesis» in rigor by virtue of its unified recognition of:

(i) the non-self-evidence of appropriate and optimal axioms generally; the exercise of choice and skillful design required in their development and selection, and the abounding 'alternativity' which that activity confronts;

(ii)
the axioms-dependence or assumptions-relativity of all formal proofs, hence of all formal "truths";

(iii) the logical 'equi-coherence' of non-standard models of "first order" axioms-subsystems with respect to their standard models;

(iv) the independence or Gödel-undecidability of key axioms of "higher-than-first-order" axioms-systems with respect to the rest of the axioms, hence the logical 'equi-coherence' of alternative axioms-systems, built on contraries of those key axioms, and especially;

(v) 'The Gödelian Dialectic'; the psycho-historical, «aufheben»/evolute-cumulative progression of de facto axioms-system within the social and socio-cognitive, '''Phenomic''' progression of a human species.

That is, the controversial, dialogical, dialectical process of discovery, exploration, comparative evaluation, and rational selection of assumptions [of premises, postulates, axioms, definitions, primitives, and rules of inference] is not a final, once-for-all, 'finishable', synchronic activity. Not all possible alternative and/or incremental axioms are known, or even knowable, for humanity, at any given moment in human history. This 'meta-axiomatic' dialectic process is, on the contrary, an ever-renewed, ongoing, and cumulative process, a diachronic activity of expansion of our accessible axiomatic and 'ideo-ontological' foundations. It produces an «aufheben»-progressive ['Qualo-Peanic' or 'Meta-Peanic'] historical sequence of systems of logic and mathematics.

That ['Qualo-Peanic' or 'Meta-Peanic'] «aufheben»-progression reflects the immanent emergence of psycho-cultural 'readiness' for each next epoch of axiomatic and 'ideo-ontological' expansion, borne in the interconnexion between: (1) "technical" or 'technique-al', "technological-ontological" expansion of the activities/practices of a generally acceleratedly-expanding human-societal self-reproduction; of "human species praxis", and (2) maturation in the prevailing level of exo-somatically acquired, trans-genomically transmitted cognitive and affective development of the typical "social individual", hence of the global human culture and "meme pool" [or '''Phenome'''].

The mathematical logic of the '[proto-]«arithmoi» theory', [proto-]'totality theory', "ensembles theory", "manifolds theory", or "set-theory" approach to an axiomatic foundation for all of mathematics created a model, and a kind of metric, for the 'meta-monadizing', 'meta-«monads»' creating, neo-«arithmos» engendering 'self-internalization', 'self-re-entry', 'self-inclusion', 'self-incorporation', or 'self-containment' of sets, i.e., for the becoming "elements" of "sets" themselves; the becoming "elements" of [idea-]objects, of entities which are already sets-of-elements. It is called the theory of logical types.

A set-representation which "contains" only representations of "logical individuals", e.g., of 'fundamental objects', or "ur-objects''', which are not themselves sets, might be assigned to 'logical type 1'. Thus, if a and b denote two such "concrete" or "determinations-rich" 'base-[idea-]objects' [perhaps, at root, idea-representations of physical, sensuous objects], the set denoted {a, b} is then of logical type 1, and represents a more 'determinations-reduced', "abstract" [idea-]object, denoting only those determinations, qualities, or "predicates" which a & b both exhibit/"have in common". A set of logical type 2 would then be a set that includes sets of 'base-objects' among its elements, such as that denoted by:

{ a, b, {a}, {b}, {a, b} }.

The '''logical type''' of a set, per the definition of '''logical type''' given above, can be determined directly by counting the number of '''opening braces''', '{', or of '''closing braces''', '}', to their deepest, or maximal, level within the set whose "logical type" metric is to be evaluated.

Notice that the contents of the set {a, b} are also [«aufheben»] contained/conserved within the contents of the set { a, b, {a}, {b}, {a, b} }, but also that { a, b, {a}, {b}, {a, b} } is a kind of not-{a, b}:

{a, b { a, b, {a}, {b}, {a, b} }.

Indeed, { a, b, {a}, {b}, {a, b} } is qualitatively unequal tonot quantitatively unequal to{a, b}:

{a, b  { a, b, {a}, {b}, {a, b} },

wherein the new, '''non-standard''' relation-symbol, '', enables us to summarize, in a single statement, the 'negated trichotomy' of the conjunction of the three statements '{a, b} is not greater than { a, b, {a}, {b}, {a, b} }', and '{a, b} is not equal to  { a, b, {a}, {b}, {a, b} }', and '{a, bis not less than { a, b, {a}, {b}, {a, b} }'.

What we are saying, in other words, is that mathematics immanently needs to recognize, and distinguish, [at least] two qualitatively distinct «species» of the «genos»
denoted "" of inequality.  One «species» is already recognized, and conventionally denoted by the ideographical symbol "quantitative_inequality_sign ".  The other «species» is currently, in general, unrecognized in conventional mathematics, and is denoted, herein, by the ideographical symbol, and 'neogram', ''.

genus_speciation


Notice also that the 'successor-set',  { a, b, {a}, {b}, {a, b} }, differs, 'contentally', from the 'predecessor-set', {a, b}, in that it contains — together with the 'predecessor-set' itself, {a, b}also [most of] the ["standard"] "sub-sets" of that 'predecessor-set'.  That is, 'the successor-set', { a, b
, {a}, {b}, {a, b} }, contains [most of] the elements of most of the ["standard"] "set of all sub-sets" i.e., the elements of [most of] the so-called "power-set" of the 'predecessor-set', {a, b}, '''plus''' [or "Union", denoted ''] that 'predecessor-set' itself.  The ["standard"] "sub-sets" of {a, b} include the "improper" subset of {a, b} — none other than the whole of {a, b} itself — so that the 'successor-set', { a, b, {a}, {b}, {a, b} }, results from, in part, a 'self-internalization' of the previous whole/entire set, or '''totality''', {a, b}, which '''now''' becomes a '''mere''' [new] part inside the new whole/'''totality'''.

Thus, the 'successor-set', here, is the 'predecessor-set' itself, '''plus''' the elements of [most of] the "power-set" of that 'predecessor' set.

The various parts of the 'successor-set', { a, b, {a}, {b}, {a, b} }, might, for example, be interpreted as follows: 'a' names a concrete, complex, 'full-determinations' '''ur-object''', as does 'b', for a distinct/other such object; '{a}' names a predicate formulated to express, as a univocal, singular quality/'''in-tension''', the total '''nature'''/content unique to 'a'; '{b}' names a predicate formulated to express, as a singular quality/'''in-tension''', the 'total ''nature'''/content unique to 'b', and; '{a, b}' names a predicate formulated to express, as a singular quality/'''in-tension''', just those qualit(y)(ies) shared in common by 'a' and 'b' alone among the totality of '''ur-objects''' constituting the universe[-of-discourse] being modeled. The set-succession — or «aufheben» set-progression — partially depicted here is thus one which models a 'predico-dynamasis', or 'qualo-dynamasis', progressively conceptualizing — or lifting out of "'chaotic'" and '''inchoate''' implicitude; progressively 'explicitizing' — more and more predicates to articulate ever-more distinctly and concretely, '''for-themselves''', the richness of the determinations of that universe's '''ur-objects''', '''in-themselves'''.

Thus, in summary, the 'predecessor-set'/logical-type, above, is «aufheben»-conserved, and also, simultaneously, «aufheben»-elevated [in logical type, as well as being expanded in contents-ontology], and thus also «aufheben»-negated/annulled/canceled/qualitatively-transformed, by this «aufheben» self-product, or 'Power-Set Evolute Self-Product', of sets.

If we denote by T, and by S0, the "universal set", the set of All '''logical individuals''', or the '''Totality''' of '''ur-objects''' that are part(s) of a given universe of discourse, and if s[ T ] denotes the 'successor-set' of the 'predecessor-set', T, and if P[ T ] denotes the "set of all subsets", or "Power-set", of the set T, then the formula for this product-rule can be stated as follows:

s[ T ]    ≡   T × T    ≡   T2      T + Δ[ T ]       T     P[ T ],
or
s[ S0    ≡   S0 × S0    ≡   S02   ≡    S0 ΔS0    ≡    S0    P[ S0 ]      ≡      S1,
or, more generally, for the variable τ  successively taking on the values 0, 1, 2, 3, 4, ..., as:
s[ Sτ ]  Sτ+1 Sτ × Sτ  Sτ2 Sτ   ΔSτ    Sτ    P[ Sτ ],
or
sτ[ S0 ]   =   Sτ   =   S02^τ,

where
 
2^
τ 2τ.

The resulting «aufheben»-progression of sets namely, the set-sequence-containing the set denoted by { Sτ } as τ successively takes on the values 0, 1, 2, 3, 4, ... — i.e., for the "Natural" order of progression of the "Whole" Number value, τ, provides, especially for '''realistic''', finite, '''actually-constructed''' universes of discourse, a propositionally non-self-contradictory, non-paradoxical model of the '''set of all sets'''.

This '''set of all sets''' — since it is set-theory's own, native definition of the "set" itself, the set-theoretical, or "ex-tension-al", definition of the '''in-tension''' of the "set" concept itself is the central idea-object of set-theory, though it is suppressed in "Standard" Set Theory.  Hence, also, the '''set of all sets''' is the central locus of a dialectical, immanent critique of that set theory.

This '''set of all sets''' is a 'contra-Parmenidean' mental 'eventity'; a mental ''self-movement'''; an 'ideo-auto-kinesic', '[ideo-onto-]dynamical', 'ideo-onto-logic-ally' self-expanding '''idea-object''', and one which, for appropriate universes of discourse, implicitly contains all of the wherewithal for 'The Gödelian Dialectic' [see below].

But why is this '''set of all sets''' a 'self-changing' '''idea-object'''; an '''idea-object''' that itself induces change in itself; an '''idea-object''' that itself causes itself to expand, qualitatively, 'ideo-ontologically'; an idea-object that is also an 'idea-subject', or agent of change, with respect to itself; an 'idea-entity' that "won't stay still" in your mind, once your mind constructs it; that forces itself to grow, and that is, thus, an 'idea-eventity', an 'ideoauto-kinesis»'?
 
This [finitary] '''Set of All Sets''' is '''forced''', in order to fulfill its own definition, the definition of its very self — viz., that it contains "All" sets — indeed, forces itself into continual expansion of its contents, of its '''elements''', of its '''membership''' — forces itself into continual qualitative self-expansion, not by the 'internalization' of anything '''external''' to it, because it already contains all of the '''ur-objects'''/"logical individuals" that found the entire universe of discourse in question, but, rather, on the contrary, via the continual 'self[-and-other subsets]-internalization', the 'internalization' of what is already '''internal''' to it, of what it already '''contains'''; the 'internalization' of itself as a whole — of its own "improper subset" — as well as of all of the "proper subsets" of itself. 
 
This '''set of all sets''' is '''forced''' to do so by its own nature/essence/'essence-iality'/essentiality/necessity; by its own '''self'''; by its own name/description/definition, i.e., by the 'intra-duality', or 'self-duality' and 'indivi-duality', of its every '''state''' of existence — because it always, in every "moment", "still" excludes those very sets which constitute its own "power set", its own subsets, among which is that set which is its own "improper" subset, namely, none other than itself.  But it is not, per its name/definition, supposed to exclude any [finite, '''constructible'''] sets at all
 
But, each time it 'internalizes' all of its subsets, including itself, it thereby transforms itself into a new, qualitatively different, qualitatively expanded set, with a different set of subsets — a qualitatively different "power-set" — all of which are not yet included in itself, among its "elements". 
 
Therefore, it must, each time'internalize' its own subsets again. But, in so doing, each time, it changes itself again, thus bringing a new, different set of [its] subsets into [potential] existence.  And so, it must actualize that potential existence, by self-/power-set-'internalizing'  again... .
 
[Indeed, one gets the same, 'ideoauto-kinesis»' result, if one simply considers the "universal set" itself as "the set of all objects" [of the universe in question], provided that one grants that 'idea-objects' — such as each "predicate" that is denoted "extensionally", in set theory, by the set of all objects that share the quality denoted by that predicate — are included among the "objects" referenced by the sub-phrase "all objects".  One, again, obtains a self-expanding 'ideo-onto-dynamasis' in the form of a 'predicates-dynamasis', or 'predico-dynamasis'].
 
This '''set of all sets''' is, thus, a logical/conceptual/mental 'self-force' that [en]forces the continual, mounting, selfaufheben» 'self-internalization' of itself and of all of its [other] subsets, thus driving its qualitative self-expansion.   
 
This '''set of all sets''' is, therefore — 
 
(1) The very object which expresses and stands for the "essence"/"quality" that all sets have in common, per set theory's way of expressing such qualities, such that, e.g., the number two is represented by the set of all sets which have exactly two members, and the color "green" is represented by the set of all objects that look green. However, contrary to the onto-statical proclivities of most "Standard" set-theorists, that quality turns out to be none other than an that of an uninterrupted movement of self-inclusion, of self-subsumption, of self-involution, of selfaufheben» 'self-internalization' ;
 
(2) The vehicle of an immanent critique of [Parmenidean] set theory itself, via a «reductio ad absurdum» refutation of Standard Set Theory's implicit 'Parmenidean Postulate' — the belief that sets, and their elements, and, indeed, that all mathematical, idea-objects, must be characterized by eternal «stasis» or changelessness;
 
(3) a set-theoretical model of the 'dialectic' itself; of a generic 'Meta-Monadology'; of what we will come to call, below, an 'auto-kinesic', 'ideo-onto-dynamical', 'Qualo-Peanic', 'ideo-meta-fractal'-constructing, 'meta-finite' 'self-progression'; an 'archeonic consecuum-cumulum', driven by a succession of selfaufheben» 'self-internalizations'  which are also 'meta-«monad»-izations'.

Sets of logical type 3 contain at most sets of sets of base objects, e.g.:

{ a, b, {a}, {b}, {a, b}, {{a}}, {{b}}, { {a, b} }, { {a}, {b} }, ... , { {a}, {a, b} }, { {b}, {a, b} } }.

Those elements of the latter set denoted by

{ {a}, {a, b} } and { {b}, {a, b} }

are called "ordered pairs", also written

<a, b> and <b, a>,

respectively, because for them, unlike for sets in general, order of listing matters:

{a, b} = {b, a},

but

{ {a}, {a, b} }  ≡ <a, b> ≠ <b, a> ≡ { {b}, {a, b}},

in fact, in general

<a, b> <b, a>.

Herein '≡' denotes 'equals by definition'.

Thus, if we take "natural" numbers to be our 'base [idea-]objects', then sets or "classes" "of" or "containing" such numbers would be of logical type 1, classes "of" or "containing" classes [of such numbers] would be of logical type 2, and classes of classes of classes [of such numbers] would be of logical type 3, and so on.

Example 0: The Gödelian Dialectic

Kurt Gödel, the contributor of, arguably, the greatest leaps forward in the science of logic since classical antiquity, described an 'axiomatic dialectic' of mathematics, albeit in "['early-']Platonic", 'a-psychological' and 'a-historical' terms, hence also in 'a-psycho-historical' terms, as follows:

"It can be shown that any formal system whatsoever — whether it is based on the theory of types or not, if only it is free from contradiction — must necessarily be deficient in its methods of proof. Or to be more exact: For any formal system you can construct a proposition — in fact a proposition of the arithmetic of integers — which is certainly true if the system is free from contradiction but cannot be proved in the given system [the foregoing summarizes Gödel's "First Incompleteness Theorem" — F.E.D.]. Now if the system under consideration (call it S) is based on the theory of types, it turns out that exactly the next higher type not contained in S is necessary to prove this arithmetic proposition, i.e., this proposition becomes a provable theorem if you add to the system the next higher type and the axioms concerning it." [Kurt Gödel; "The Present Situation of the Foundations of Mathematics (*1933o)", in S. Feferman, et. al., editors.; Kurt Gödel: Collected Works (Volume III: Unpublished Essays and Lectures); Oxford University Press (NY: 1995); page 46; bold, italics, underline, and color emphasis added by F.E.D.].

Again:

"If we imagine that the system Z  [a formal, logical, propositional-/predicate-calculus system inclusive of "Natural" Numbers'  Arithmetic, not the full system of the positive and negative Integers, and zero [which is both, [or neither] positive [n]or negative], standardly also denoted by ZF.E.D.] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding comprehension axioms, we obtain a sequence (continuable into the transfinite) of formal systems that satisfy the assumptions mentioned above, and it turns out that the consistency (ω-consistency) of any of these systems is provable in all subsequent systems. Also, the undecidable propositions constructed for the proof of Theorem 1 [Gödel's "First Incompleteness Theorem" — F.E.D.] become decidable by the adjunction of higher types and the corresponding axioms; however, in the higher systems we can construct other undecidable propositions by the same procedure. ...To be sure, all the propositions thus constructed are expressible in Z (hence are number-theoretic propositions); they are, however, not decidable in Z, but only in higher systems..." [Kurt Gödel; On Completeness and Consistency (1931a), J. van Heijenoort, editor; Frege and Gödel: Two Fundamental Texts in Mathematical Logic; Harvard University Press (Cambridge: 1970); page 108; bold, italic, underline, and color emphasis and [square-brackets-enclosed commentary] added by F.E.D.].

The cumulative «aufheben»-progression of '''conservative extensions''', i.e., the advancing 'ideo-cumulum' of axiomatic systems which Gödel describes above was viewed ahistorically by him. Gödel, as a 'Parmenidean', and a professed "mathematical Platonist" [in the sense of the earlier rather than of the later Plato; see below], didn't intend this 'meta-system' — this cumulative diachronic progression of [axioms-]systems — to serve as a temporal or psycho-historical model of the stages of human mathematical understanding, as reflective of the stages of the self-development of humanity's collective cognitive powers as a whole; of the knowledges to which each such epoch of those powers renders access, and of the "historically-specific" ideologies [or pseudo-knowledges] to which human thinking is susceptible within each such epoch.

But we do wish to explore its efficacy as such. Note how, as Gödel narrates above, each successor system «aufheben»-contains its immediate predecessor system, and, indeed, all of its predecessor systems; how each higher logical type «aufheben»-contains all predecessor logical types. Can Gödel's theory of this cumulative, 'evolute', «aufheben» progression of axioms-systems, which we term 'The Gödelian Dialectic', or 'The Godelian [Idea-Systems' Ideo-]Metadynamic', provide at least an idealized [i.e., a distorted] image of actual history, of the actual psycho-historical struggle, process, and progress of mathematical aspects of the self-development of a humanity's collective cognitive capabilities, hence of its knowledges and ideologies? We shall see.

[First: A Note on Notation. We delimit major hypotheses — typically textual, and whose text will be denoted generically, here, by ellipsis dots, '...' — as follows: ...  [though the majority of the material, so enclosed or not, remains conjectural], vs. [proven] theorems, derived deductively from explicit premises, via .... Single quote-marks, '...', enclose 'self-quotes' of our own coinages. Double quote-marks, "...", enclose exact quotes of others. Triple quote-marks, '''...''', enclose approximate, paraphrased quotes of others. Double 'angle marks', «...» , enclose non-English words, whether transliterated or rendered in their native alphabets. We often use 'word-embedded parentheticals' to 'appropriate the ambiguities' in current English usages, so as to amplify the meaning of a given phrase or sentence, by creating two [or more] distinct, but semantically parallel, or semantically convergent, readings of that "single" line.  For example, the phrase 'the dialectic of human-social formation(s)' is intended to invoke two distinct but convergent readings:  (1) ' the dialectic of human-social formation', and (2) 'the dialectic of human-social formations'. Likewise, the phrase 'The Dialectic of [the] Human[ized Portion of] Nature' is meant to evoke two 'mutually-supplementary' readings:  (i) 'The Dialectic of Human Nature', and (ii) 'The Dialectic of the Humanized Portion of Nature', i.e., of the portions of nature containing the '''self-objectifications''' of humanity, etched and inscribed into the [pre-human-]natural material via human labor. As a final example, consider the phrase 'the social-relations-of-[human-society/social-relations [self-]re-]production'. It is designed to evoke four convergent readings, namely (a) the social relations of production; (b) 'the social relations of social reproduction'; (c) 'the social-relations of social-relations-self-reproduction', and; (d) 'the social relations of (a) human-society's self-production'.  Moreover, the concept of '''the social relations of production''', in the Marxian tradition, is paired with that of '''the social forces of production''', so that this multiple reading is also linked to that of the phrase 'the social  [self-]force(s) of [human-society/social-relations [self-]re-]production [/[[self-]re-]productivity'[/'[self-]transformativity']. Throughout the quoted passages included below, all bold, italic, underline, and color emphasis — unless otherwise noted locally — has been added by F.E.D..  Wherever color emphasis occurs in this text, whether within quotations or otherwise, the color-coding standard is as follows: (-1) red text color signifies entities which conduce to the 'meta-catabolic', annihilatory species of opposition; (0) black text color signifies neutrality or the complementary species of opposition [e.g., north versus south poles of a magnet]; (+1) the blue text color signifies that which conduces to the supplementary, 'successory', or 'supercessory'  species of opposition [e.g., 'contra-thesis' succeeds, supplements, and opposes thesis; 'uni-thesis' succeeds, supplements, and opposes both thesis and 'contra-thesis'].

We use an eight-segment spectrum of increasing intensity of emphasis: (1) plain text; (2) underlined plain text; (3) italicized text; (4)
underlined italic text; (5) bold-faced text; (6) underlined bold-faced  text;  (7) italicized bold-faced text, and; (8) underlined italicized bold-faced text].

Each of Gödel's "undecidable" propositions of arithmetic that plague each 'epoch' of this formal axiomatic expansion are propositions each asserting the unsolvability of a different, specific "diophantine" [referencing Diophantus' «Arithmetica»; see more on this below] unsolvable equation. I.e., each "Gödel formula" or "Gödel sentence", which asserts the self-incompleteness-or-self-inconsistency of its axioms-system, "deformalizes" to one asserting the unsolvability of a specific "diophantine equation":

"... The Gödel sentence φ... asserts its own undeducibility from the postulates....Deformalizing φ... we see that under the standard interpretation it expresses a fact of the form [for every n-ary list of number-components of x such that each number-component is a member of the set of 'diophantine' or "Natural" Numbers in use F.E.D.] ...ƒx ≠­ gx... , where ƒ and g are n-ary polynomials....An equation ƒx = gx, where ƒ and g are two such polynomials, is called diophantine [see below for further information regarding Diophantus of Alexandria  F.E.D.] .... By a solution of the equation we mean an n-tuple α of natural numbers such that ƒα = gα... . So φ... asserts the unsolvability of the...equation ƒx = gx, and the proof of [Gödel's "First Incompleteness Theorem" — F.E.D.] produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates..." [Moshé Machover; Set Theory, Logic, and their Limitations; Cambridge University Press (Cambridge: 1996); pages 268-269].

Per the standard modern definition, a "diophantine equation" is an equation whose parameters [e.g., coefficients] and whose solutions are restricted to the "natural" numbers. Each "Gödel sentence"-encoded equation truly is unsolvable within the given axioms-system. However, the proposition that it is so, cannot be deductively proven within that axioms-system — but it can be so proven within the next axioms-system, its immediate successor the latter being created through the «aufheben» 'self-internalization'  of the '''vanguard''', 'meristemal', highest [in logical type] set idea-objects of the universe of discourse of the predecessor axioms-system. It can also be so proven within all subsequent successor-systems, created by yet-further such «aufheben» 'self-internalizations'.

If the "logical individuals" or 'arithmetical idea-objects' "existing" per the "comprehension axioms" of a given axioms-system are limited to "natural" numbers, classes of "natural" numbers,..., all the way up to classes of classes of... of "natural" numbers, e.g., to 'class-objects' up to a given "logical type", then the next system will cumulatively expand those '''existential''' limits by one step, to include also classes of classes of classes of classes of... of "natural" numbers, i.e., 'class-objects' of still higher "logical type". Each successive higher class-inclusion of previous 'class-objects' can model [including via adjunction of its corresponding "comprehension axioms", defining the 'computative behavior' of these new entities] a new kind of arithmetical 'idea-object'; indeed, a new, higher kind of number. Thereby, this qualitative expansion of each predecessor axioms-system, in the formation of its successor axioms-system, this adjunction of the additional, "comprehension axioms" to the previous, predecessor axioms, corresponds to a qualitative expansion of the 'idea-ontology', of the 'arithmetical ontology', i.e., of the 'number-ontology' of that axioms-system.

symbol 12 Specifically, the diophantine equation that was unsolvable as such within the predecessor axioms-system may itself become solvable, albeit in a non-diophantine sense, within the next [as well as in all subsequent] successor axioms-systems in this cumulative sequence of axioms-systems, precisely by means of these next new kinds of numbers, which will not be 'diophantine numbers', i.e., not "natural" numbers. symbol 12


We can see a kindred  'unsolvability-to-solvability dialectic'  at work in the following examples.

The equation [2 + x = 2 or x = 2 2] states a paradox:  how can the addition of a number, x, produce a result, a sum, that is not bigger than that 'known' number, here 2, to which that "unknown" number, x, is added? Given the N «genos» of number, addition always means increase, never no increase. This equation is not solvable within the system of arithmetic of the cardinal, or sometimes, "Natural", numbers, N ≡ {1, 2, 3, ...}, but is solvable, by the 'non-diophantine number'' 0, within the 'ideo-ontologically' expanded system of the "Whole numbers", W {0, 1, 2, 3, ...} [Adjunction of the zero concept may seem trivial to us, yet it entailed a great and protracted conceptual travail for our ancient Mediterranean ancestors, and, with respect to issues surrounding division by zero, and the related issues of singularity, remains fraught with unresolved problems, "even" among we moderns today!].

The equation [2 + x = 1] states a paradox: how can the addition of a number, x, produce a result, a sum, that is less than that 'known' number, here 2, to which that "unknown" number, x, is added? Within the W «genos» of number, addition always means a change that increases, or, at minimum, that results in no change at all, but it never means a decrease. The latter equation thus finds no number among the "Wholes" to solve/satisfy it, but it does so among the "integers" or '''integral''' numbers, the expanded numbers-set Z ≡ {...,  3,  2,  1, 0, +1, +2, +3, ...}, which is a qualitatively, that is, 'ideo-ontologically expanded, new-kinds-of-numbers-expanded, meaning or 'meme-ing'-of-"number"-expanded, semantically-expanded universe-of-discourse of "Number", vis-à-vis  the preceding «genos», the W universe-of-discourse. The equation is solved/"satisfied" by the 'non-diophantine number' 1.

Next, the equation [2x = 1] also states a '''paradox''': how can the multiplication of any number, namely that of the "multiplicand", denoted here by the algebraic "variable" or "unknown", x, by another, known, number, the "multiplier", produce a product which is less than that "multiplier", here 2? Multiplication, within the Z «genos» of number, always produces a 'product' which is either increased in absolute value relative to the "multiplicand" "factor", leaves the multiplicand unchanged, or turns it into zero. But Z multiplication can never turn a 2 into a 1. Such an equation is not solvable within the system of arithmetic of the "integers", Z. This equation is solvable, however, via 'ideo-ontological expansion' to encompass the qualitatively different system of arithmetic of the "Quotient numbers", "ratio-numbers", "ratio-nal" numbers, or "fractions", denoted by Q, i.e., by an expansion that encompasses yet a new kind of 'non-diophantine number', the 'split a-tom' ['cut uncuttable'], 'monad-fragment', or "fractional value" +1/2:

Q ≡ {...2/1, ...3/2...1/1, ...1/2... 0/1, ...+1/2...+1/1, ...+3/2...+2/1, ...}.

The [algebraically] nonlinear equation [x2 = 2] states a '''paradox''' too: it requires x to be a kind of number which is 'both odd and even at the same time' [see the classic «reductio ad absurdum» proof of the "ir-ratio-nality" of √2]. It is not solvable '''rationally'''. It is solvable via 'ideo-ontic' expansion to the "Real" numbers, this time by two distinct 'non-diophantine numbers', given the nonlinear, "2nd degree" character of this "unsolvable" equation, rather than by just one 'non-diophantine number', as were the preceding, [algebraically] linear, degree 1 "unsolvable" equations/'''paradoxes'''.  The two solutions are the "irrational" values √2 and +√2:

R {...π...3...e...2,...√2...1...0...+1...+√2...+2...+e...+3...+π...}.

Finally, for the purposes of this letter, the nonlinear equation [x2 + 1 = 0] states a '''paradox''' as well: it implies that x = +1/x, requiring a kind of number whose additive inverse, x, equals its multiplicative inverse, 1/x or x1, whereas, among "Real" numbers, 2  ≠­ 1/23  ≠­ 1/3, π  ≠­ 1/π, etc. It is not solvable or "satisfiable" within any of the foregoing «gene» of number, or of arithmetics, up through that of the "Real" numbers.

It is 'non-diophantinely' solvable, via expansion to the '''Complex''' numbers, denoted C {R + R·1}, by, again, and for the same reason, two 'non-diophantine' numbers, known as the "pure imaginary" numbers, x = +1 = 0 + 11 +i, and x = 1 = 0  11 i. And more. ...

Note how each successor «genos», or universe, of number «aufheben», contains all of its predecessor universes of number, or is a '''conservative extention''' of all of its predecessor '''universes'''.

Such a ['Qualo-Peanic' or 'Meta-Peanic'] «aufheben» 'consecuum' of «gene» evinces part of the essence of what we mean by a 'dialectic'; by a 'dialectical process', or by a 'meta-dynamical, meta-system-ic, meta-evolutionary self-progression of systems', 'self-launching' from an originating, «arché» system. But how might this potentially infinite progression of «gene» and 'species' of number, required for equational solvability, map to "sets of sets of ... of sets"?

One way that sets of higher "logical type" can model [] higher, later kinds of [non-diophantine] numbers is as ordered pairs of earlier kinds of numbers / earlier kinds of sets.

We already noted that ordered pairs can be modeled via certain kinds of sets. "Integers", for example, can be modeled as ordered pairs of "whole numbers", i.e., as sets of logical type 2 if we take the "whole numbers" as 'base objects' with the integers being defined, via their "comprehension axioms", as differences, i.e., as subtractions, viz., as:

{{1}, {1, 0}}  ≡  <1, 0>  ↔  1 0  =  +1  ≠  {{0}, {1, 0}}  ≡  <0, 1>  ↔  0 1  =  1.

Rational numbers can then be modeled as ordered pairs of integers, defined, via their "comprehension axioms", as divisions, e.g.,

<+1, +2>  ↔  (+1)÷(+2)  =  +½  ≠  <+2, +1>  ↔  (+2)÷(+1)  =  +2/1.

Thus, they also translate to ordered pairs of ordered pairs of "whole numbers", or to 'sets-of-sets of sets-of-sets' of "whole numbers", that is, to 'sets-of-sets of "whole numbers" 'squared', meaning that these sets-of-sets operate upon these very sets-of-sets themselves, per a 'multiplicand-ingestion' set-product rule, so:

+½  ↔  <+1, +2>  ↔  {{+1}, {+1, +2}↔   ‹‹1, 0>, <2, 0&raquo;  ↔   {{<1, 0>}, {<1, 0>, <2, 0>}},

which translates to

{{ { {1}, {1, 0} } },{ { {1}, {1, 0} }, { {2}, {2, 0} } }},

which is a class-object of logical type 4, i.e., of logical type 22, or "two squared", w.r.t. the "whole numbers" taken to be the 'base-objects'. Further on, ordered pairs of "Real numbers" may model Complex numbers, C, viz.:

<1.000..., 2.000...>    1 + 2i  <2.000..., 1.000...>   2 + 1i,

such that C can be modeled as the two-dimensional space of a special kind of direction-denoting, as well as magnitude-denoting, "'directed line segment"', or "'vector"'.

... and so on, to the Quaternions (H), the Octonions (O), the Clifford numbers (K), and the Grassmann numbers (G), etc. ....

Thus, the "rational" numbers may be grasped as "analogues", and as 'meta-fractal similants', of the integers, and the integers as 'meta-fractal similants' of the whole numbers.

Even though these successive numbers-systems are of different kind, differing in quality, their base 'idea-objects', or numbers — their universes of discourse — may be constructed in and as different 'epochs' in a progressive-cumulative, 'self-iteration' of one and the same «aufheben» operation of 'self-internalization',  of 'self-incorporation', of 'self-subsumption', of 'self-combination', or of 'self-combinatorics', of sets, i.e., of ordered pairs.

This number-systems progression, formed by the self-iteration of the 'ordered pairs of' or 'sets of' operations, constructs a "logical" or '''idea-object''' version of what we term an «aufheben», 'meta-fractal [ideo-]cumulum' — '[meta-]fractal' because it constructs a structure which is self-similar at successive "scales"; 'meta[-fractal]' because these "scales" are not purely quantitative, as they are for "mere" fractals, but are 'quanto-qualitative', or 'quanto-ontological'.

That is, such a 'cumulum' [self-]constructs as a diachronic '[ideo-]metaarithmos»', and persists as a synchronic '[ideo-]superarithmos»', made up out of a heterogeneous multiplicity of '[ideo-]«arithmoi»'-as-'metamonads», and also such that each constituent '[ideo-]«arithmos»', in turn, is 'populated' via a different kind of '[ideo-]«monad»'; in this case, by different kinds of number [based upon different kinds of unit[y]].

This may be seen in that the later 'similants' involve adjunctions of new [idea-]ontology, new, higher logical types of sets; new kinds of sets; new kinds of ordered pairs; new kinds of numbers, qualitatively different from all of their earlier 'similants', not just quantitatively different therefrom, because «aufheben»-'containing' all of their earlier 'similants', and thus also 'meta-fractally' 'scale-escalated' with respect to all of their earlier 'similants'.

We also find, in the history of nature to date, physical, 'external-objective' 'meta-fractal' structures; a 'physio-cumulum', or 'physio-metaarithmos»', made of multiple 'physioarithmoi»', of different kinds of 'physiomonads»': molecules as 'meta-atoms', each made up out of a heterogeneous multiplicity of atoms; atoms as 'meta-'subatomic-"particles", each made up out of a heterogeneous multiplicity of sub-atomic "particles", etcetera.

Thus, we hold that

(1) the 'internal', 'inter-subjective', 'idea-object-ive', mathematical-progress-driving, conceptual process of 'The Gödelian Dialectic', i.e., of 'ideo-onto-dynamasis' [as modeled, using the '''algebra''' of dialectical ideography, via generalized self-multiplication, 'quadraticity', or 'ideo-onto-dynamis' to appropriate Diophantus' term for '''squaring''', i.e., the second, or "quadratic", "degree", or "power", of a variable],

and;

(2) the 'external', "objective", "natural" process driving the self-development of 'physical' Nature', or «physis» of '''pre-human Nature''', and of '''extra-human Nature''',  as well as of '''human Nature''' i.e., 'physio-onto-dynamasis', share a similar,  «aufheben»/'meta-fractal' logic i.e., a generally single, singular dialectical logic — or pattern of 'followership'.

'Dialectical Models' of such 'ideo-onto-dynamasis' are addressed in Supplement A to this letter [see: "Primer, Overall Prefatories & Abstract An Introduction to Dialectical Arithmetic" (PDF)].

'Dialectical Models' of such 'physio-onto-dynamis' are addressed in Supplement B to this letter [see: Supplement B (Part II) — Including a Dialectical-Mathematical Model of The Dialectic of Nature (PDF)].

A fuller exploration — and a '''dialectical model''' — of the 'Gödelian Ideo-Meta-Evolution', observed in the [psycho-]history of arithmetics, is forthcoming in Part II. of Dialectical Ideography: The Meta-Evolution of Arithmetics].

The Nonlinearity Barrier

Of course, all of the above algebraic equations may, today, appear "trivial", having long since been solved by our remote ancestors.

But are there still "unsolvable equations in our own day?

Are there still new kinds of numbers, beyond G, yet to be discovered?

If Gödel is right, that this 'dialectic' of incompleteness/undecidability/unsolvability is "inexhaustible"; [potentially] "continuable into the transfinite", then there must be.

If so, how far has this 'Gödelian dialectic' progressed, to date, in Terran human history? As mapped into the history of the collective human psyche per its 'collective, anthropological/psyche-ological, psycho-historical conceptual readiness-gradient', how far along into it are we as of today?

Does our present stage of this 'Gödelian dialectic' have any scientific relevance?

And, if there are, today, still, some equational «insolubilia», does their solution — garnered by moving into the next higher stage of this 'Gödelian dialectic' — have any practical value, e.g., engineering value; any urgent technological application; any contribution to make to the growth of the society-productive forces of humanity, i.e., to the 'qualo-quantitative' self-productivity of the human species?

symbol 12 Yes to all. symbol 12

Indeed, the very equations which formulate this humanity's most advanced collectively-recognized formulations of its "laws" of nature are generally of a type called nonlinear [partial] differential equations.

They also remain, for the most part — especially when they are nonlinear — unsolved, typically even a century or more after their first formulation/discovery.

They are also often and without proof — simply declared to be, not just 'so far unsolved', but [forever] "unsolvable" in "exact" or "analytical" or "closed" "form".

This conclusive-sounding phrase is actually anything but — it merely means that their solutions apparently cannot be expressed in terms of the "elementary" or 'fundamental' "algebraic" and 'trans-algebraic', or "transcendental" functions or operations currently recognized as such, as "elementary", even if their solutions can be expressed in '''open form''', involving [potentially] "infinite sums", i.e., [potentially] "infinite series" or [potentially] "infinite degree polynomials" — ever improvable approximators — made up out of finite and "closed-form" terms.

The "unsolvability" or so-called "non-integrability" of these nonlinear differential equations may also mean that the "integration", or solution, of these equations encounters zero-division "singularities", which apparently lead to "function-values of infinite magnitude", so that their solution "diverges" or attains "infinite" or "undefined" values corresponding to finite values of the time parameter; that the "limit" of their "infinite series" sums, forming their integrals, appears to be without [finite] quantitative limit; appears to be quantitatively "limitless" or 'un-limit-ed'.

This '''Nonlinearity Barrier''' of modern mathematical science massively blocks this humanity's capability for further scientific and technological/engineering advance around its entire perimeter with the un-known; with its present 'un-knowledge':

"That is the way I explained non-linearity to my son. But, why was this so important that it had to be explained at all? The complete answer to this question cannot be given at present, but some people feel that the answer, if known, would shake the very foundations of mathematics and science... practically all of classical mathematical physics has evolved from the hypothesis of linearity. If it should be necessary to reject this hypothesis because of the refinements of modern experience, then our linear equations are at best a first and inadequate approximation. It was Einstein himself who suggested that the basic equations of physics must be non-linear, and that mathematical physics will have to be done over again. Should this be the case, the outcome may well be a mathematics totally different from any now known. The mathematical techniques that might be used to formulate a unified and general non-linear theory have not been recognized... we are now at the threshold of the nonlinear barrier." [Ladis Kovach; "Life Can Be So Nonlinear" in American Scientist (48:2; June 1960); pages 220-222].

No less than the founding problem of modern, '''mathematico-science''' — a problem that was also a central focus and motivation of ancient science — today takes the form of a system of nonlinear integro-differential equations which have, to this day, in both their Newtonian and Einsteinian, General Relativistic versions, remained essentially unsolved [the 1991, slow convergence, "open-form", singularity-"infinitely"-delaying/evading i.e., planetary-collisions-infinitely-delaying/evading Qiu-dong Wang series solution notwithstanding], because of their nonlinearity.

This founding problem is the fundamental problem of astronomy, the problem of the mutual-determination, and other-objects-mediated-self-determination, of the motions of celestial objects, when any more than two such objects are admitted into the mathematical model of the celestial cosmos:

"The n-body problem is the name usually given to the problem of the motion of a system of many particles attracting each other according to Newton's law of gravitation. This is the classical problem of mathematical natural science, the significance of which goes far beyond the limits of its astronomical applications. The n-body problem has been the main topic of celestial mechanics from the time of its inception as a science. The fundamental dynamical problem for a system of n gravitating bodies is the investigation and pre-determination of the changes in position and velocity that the [bodies] undergo as the time varies. However, this is a complex non-linear problem whose solution has not been possible under the present-day status of mathematical analysis." [G. F. Khilmi; Qualitative Methods in the Many-Body Problem; Gordon & Breach (1961); page v].

Indeed, the models of nature that modern mathematical science has favored are profoundly flawed and misleading in crucial aspects of their 'descriptics' of nature, due to this specific inadequacy of the mathematics that Terran humanity has evolved so far:

"It is an often-stated truism that nature is inherently non-linear. Biological systems particularly are full of ... non-linearities ... The reason that we go to the trouble of building linear models when we are really interested in non-linear systems is that we then acquire the power to evaluate the dynamic performance of the system analytically.... In fact, we can analytically solve for the response of a linear system to any conceivable input function, however complicated." [Bernard C. Patten; System Analysis and Simulation in Ecology (volume I); Academic Press (NY: 1971); page 288].

However, in the non-linear domain:

"In general, the analytical study of non-linear differential equations has been developed only to a very limited extent, owing to the inherent mathematical difficulties of the subject. There does not exist, in this field, a suitable technique for attacking general non-linear problems as they arise in practice." [John Formby; An Introduction to the Mathematical Formulation of Self-Organizing Systems; Van Nostrand (NY: 1965); page 115].

General non-linear integrodifferential equations cannot presently be solved in "closed form", because the ['''elementary'''] functions that would solve them have so far "resisted" discovery and formulation within the extant tradition of Terran human mathematics:

"... the assumption of linearity in operational processes underlies most applications of analysis to the problems of the natural world. ... Nature, with scant regard for the desires of the mathematician, often seems to delight in formulating her mysteries in terms of nonlinear systems of equations ... the theory of functions ... has been developed largely around classes of functions in which the linearity property is an essential factor ... most non-linear equations define new functions whose properties have not been explored nor for which tables exist...". [Harold T. Davis; Introduction to Nonlinear Differential and Integral Equations; Dover (NY: 1962); pages 1, 7, and 467].

In the light shed by the foregoing statements, the oft-decried '''mechanistic''' bias of mathematics, and of modern science in general, is seen in altered perspective. This new perspective is strengthened by the observation that the more 'organitic' and "organismic" qualities of Nature, which classical "mechanism" excludes — phenomenologies such as those of non-equilibrium and [meta-]evolutionary [meta-]dynamics; of holistic, synergetic "whole-more-than-sum-of-parts" organization; of the qualities of self-determination and self-development, and of sudden and qualitative self-change — find a native and potent expression in the non-linear domain.

It thus emerges that science has been '''mechanistic''' only to the extent that it has failed to be scientific enough — failed to be empirical enough, or true-enough-to-observation/-experience. Mathematics has been "mechanistic" and 'linearistic' only to the extent that it has failed to be mathematical enough. Modern science and applied mathematics have fallen short of a more adequate description of experiential/empirical truth through neglect of the immanent truth already enshrined within themselves.

Not even mechanics itself is truly '"mechanistic"':

...Mechanics as a whole is non-linear; the special parts of mechanics which are linear may seem nearer to common sense, but all this indicates is that good sense in mechanics is uncommon. We should not be resentful if materials show character instead of docile obedience. ... Although mechanics is essentially non-linear, it is little exaggeration to say that for 150 years only linear mechanics and its mathematics were studied. It became standard practice, after deriving the equations for a phenomenon, to replace them at once by a linear so-called "approximation". It would be wrong to regard this mangling as being in the original tradition of mechanics...". [C. Truesdell; "Recent Advances in Rational Mechanics" in Science (127:3301; 04-April-1958); page 735].

Closed-form-function solutions for our nonlinear-equation-expressed "laws" of nature would provide ready-calculation of global solutions, for the total domain of initial conditions. A "computer simulation solution" or "numerical solution" — the only kind of "solution", if any, presently available for most of these non-linear "laws" of nature — merely "simulates" some of the implications of the unsolved equation, and is limited to a single solution-trajectory or solution-history, from a single initial condition, a single "point" or "starting state", leaving all other starting points unsolved-for.

Such simulation-"solutions" also suffer severe limitations of computer calculation time [computation-speed] and storage capacity [memory space], as well as all of the limitations of the computational and "qualitative" [in-]accuracy of "numerical" algorithms, particularly with regard to the detection of "essential" singularities.

Nonlinearity, «Auto-Kinesis», and Dialectic

If "nonlinearity" is the root of this mathematical difficulty and "intractability", if "nonlinearity" is the cause of this present "closed-form unsolvability", then what does "nonlinearity" signify?

In its deepest meaning, nonlinearity signifies what Plato called «autokinesis» [see below]. Differential-equation "nonlinearity" is the mathematical name for the mathematical, 'equational', pure-quantitative modeling of 'self-motion', of 'self-change', of 'self[-induced] movement'; of 'self-induced change-of-state'; of "self-reflexiveness" and of 'self-reflexive', "self-referential" action; of 'self-developing process', or 'self-developing eventity'!

The equations that these presently unformulated function-formulae solve are called "nonlinear" because, in them, the unknowns [which, for integro-differential equations, are function-unknowns, not single number-values as are the solutions of algebraic equations], the so far unfathomed solving-functions, appear, with their function-values, or with the values of their derivative-functions, and/or with the values of their integrals, operating upon themselves, and/or operating upon one another.

symbol 12 This signifies the dynamical or time-like [indeed, 'chronogenic'] interaction and self-interaction of the underlying actualities that these functions mime. symbol 12

Linear integro-differential equations, the kind that have been easily solved by Terran mathematicians for so long, are characterized, in contrast, by function-unknowns which occur in "isolation", singly, independently, without interaction, operating neither upon self nor upon any other unknown / to-be-solved-for function(s).

A typical non-linear "ordinary" or "total" differential equation is an ideographical 'state-ment', that asserts, or "states", in effect, that the instantaneous velocity of evolution of the generic, pure-quantitative value of the state of the system modeled by that equation for any, generic time-value, t — the state represented by x(t), thus denoting the generic state-function-value of the dynamical function-unknown to be solved for — is proportional to a higher power of that unknown, generic state-value itself, denoted x(t)n, n > 1, i.e., to a multiplicative self-application / self-operation / 'self-flexion' or 'self-re-flexion'; to a self-multiplication, of that state-value. Such a pure-quantitative self-multiplication signifies either a 'self-magnification' or 'self-diminution' of the value(s) of the state-variable(s) for every ['non-Boolean'] value of [the state-variable-value components of the state-"vector" function,] x(t).

Such an equation describes a system whose "evolution" is at least partially 'autokinesic', self-driven; self-propelling in its state-space or 'space of states' — an imagined space or conceptually-constructed space in which every point denotes a different possible state of that system.

For example, non-linear differential equation models of predator-prey population/bio-mass dynamics within an ecological system often contain a population-size self-limiting Verhulst "self-interaction term". This term involves adding in, e.g., a self-multiplication of Ni(t), where Ni(t) denotes the population-count-as-state of the ith species as a function of time, t, such that a minus sign is applied to that 'self-product[ion]', thus providing a [negative] contribution to the "instantaneous" rate of growth of that species' population-count — where that rate of growth is here defined as the metric for the rate of evolution or velocity of evolution of that species' statewith respect to time, as the time 'count' advances:

"The nonlinear correction term is referred to as a "self-interaction...term" [which term we also term 'self[-re]-flexive' ['''bent back upon itself'''] or 'self[-re]-fluxive' ['''flowing [from self] back to self'''] — F.E.D.], of the form Ni(t)2 ... where the terms of the form Ni(t)Nj(t), i ≠ j, also quadratically nonlinear, are referred to as "mutual interaction terms" [which terms we also term 'hetero-flexive' or 'allo-flexive', meaning '''other-bent''', or '''bent by other-than-self''' — F.E.D.]." [R. Dutt, P. K. Ghosh; "Nonlinear Correction to the Lotka-Volterra Oscillation in a Predator-Prey System" in Mathematical Biosciences (27; 1975); pages 9-16].

The equations of Einstein's mathematical model of "universal gravitation", the equations of his General Theory of Relativity, are non-linear  precisely because they must model the non-a-tom-istic, 'self-reflexive', 'auto-kinesic', self-changing 'self-interactivity' of the cosmos-encompassing gravitic '''field''':

"...an interaction is non-linear if the total force exerted by several bodies is not the sum of the forces each would exert if acting alone. Why is the gravitational-inertial interaction non-linear? The reason is a fundamental one. We saw at the end of the preceding chapter that all forms of energy have mass and so act as a source of gravitation and inertia. This is true, not only of matter and of light, but also of gravitational potential energy. We know that this form of energy has a real physical significance; it has to be included in a total energy balance. ... This means that when two bodies act together as a source, in addition to their individual masses we must take their mutual gravitational potential energy as a source. The total force is then not the sum of the individual forces. ... It follows that the exact interaction between [better, amongF.E.D.] an arbitrary number of bodies is going to have a complicated form. Indeed, as we shall see, it has not been possible to formulate this interaction in an explicit way. In consequence, our previous calculation of the total inertial force due to all the matter in the universe is neither strictly correct nor easily correctable. We can only hope that our linear approximation gives an answer that has the correct order of magnitude. ... In view of all these difficulties, how was Einstein able to write down a law general enough to specify all the properties of the nonlinear gravitational-inertial interaction? The answer is that he wrote down the local properties of the interaction, using the field point of view. From this the global properties of the interaction between distant bodies can be calculated in principle, although in practice no one has been able to do this exactly even for just two bodies, except in the limit when one of them has a mass negligible compared with the other.... It is instructive to look at this self-interaction of the gravitational field from a slightly different point of view...inertial forces act on gravitational waves and, if the Principle of Equivalence is correct, so must gravitational forces. ... This shows how essential is the self-interaction of gravitation. ... It is one manifestation of the fact that gravitation acts on "everything".... We then have a self-interacting gravitational field satisfying a non-linear field law." [D. W. Sciama; The Physical Foundations of General Relativity; Doubleday (New York: 1969); pages 55-62].

Thus, for the past 300+ years human knowledge and industry have been partially paralyzed and vitiated by a perennial failure to "solve" general nonlinear integro-differential equations, that is, to attain the means by which the vast potential knowledge that especially the "laws of nature" equations among them encode can be explicitly extracted and practically applied.

Key instances of this incapacity include the Newton gravity-equations for more than two mutually-gravitating bodies, the Einstein universal gravity field equations of General Relativity just addressed in the quotation above, the Navier-Stokes equations of electro-neutral liquid/gaseous hydrodynamics, and the '''electro-magneto-hydrodynamics''' of the Maxwell-Boltzmann-Vlasov equation for electro-dynamically non-neutral, '''magneto-hydro-dynamical''' "plasmas", e.g., for superheated, ionized gasses — the very media in which nuclear fusion reactions, self-sustaining over 'mega-macroscopic' spatial and temporal scales, are observed to occur in extra-human nature, e.g., in the central core-regions of stars.

The Nonlinearity Breakthrough and the Fusion Breakthrough

The prospect of fusion power epitomizes the vast scientific, industrial-technological, and social benefits — potentially human-social progressre-igniting, and human-species-continuation-enabling — of a Nonlinearity Breakthrough.

A 'closed-form', global solution of the "highly-nonlinear" Maxwell-Boltzmann-Vlasov plasma equation should enable direct computation of control-parameter-values corresponding to self-sustaining nuclear fusion reaction regimes, while also providing other physical/engineering insights, all leading rapidly to the design and construction of commercially super-competitive fusion power reactors, utilizing low-radioactivity or 'no-radioactivity', virtually pollution-less fuel cycles [e.g., the Hydrogen-Boron fuel-regime].

Consider the intensely 'auto-kinesic', nonlinear character of a "plasma":

"A plasma is a gas of charged particles, in which the potential  energy of a typical particle due to its nearest neighbor is much smaller than its kinetic energy. The plasma state [also termed plasma phaseF.E.D.] is the fourth state of matter: heating a solid makes a liquid, heating a liquid makes a gas, heating a gas makes a plasma. (Compare the ancient Greeks' earth, water, air, and fire). The word plasma comes from the Greek «plásma», meaning "something formed or molded." It was introduced to describe ionized gases by Tonks and Langmuir [in a 1929 paper — F.E.D.]. More than 99% of the known universe is in the plasma state." [Dwight R. Nicholson; Introduction to Plasma Theory; Wiley (NY: 1983); page 1; angle-brackets added by F.E.D.].

The motion of the non-neutral, electrically-charged «plásma» "particles" just described continually, dynamically generates a changing magnetic field — a magneto-dynamical field of forces. The locations and concentrations of the electric charges of these "particles", changing moment by moment, due to that same motion, also continually change the «plásma»'s electric field of forces, sustaining an electro-dynamical field.

A «plásma» thus generates, by the continual motions of its electrically-charged constituents, an overall or collective electro-magneto-dynamic field of forces. That field of forces, in addition to acting on anything external to the plasma, also acts  self-reflexively and self-refluxively  — on the «plásma» itself, also because its constituent "particles" are electrically-charged, rather than being mostly electrically neutral[ized], as in typical gas-phase matter.

The «plásma»'s field continually changes the motions of its constituent "particles", and thus changes the electric/magnetic fields they are generating. Their changing positions/motions thus change their collective field, which changes their collective motions, which again changes their field, which again changes their motion. ... "Nonlinear self-consistent motions" are thereby possible, whereby the plasma-internal, self-generated field also guides the «plásma»'s "particles" to reproduce their very pattern of flow by which they generate that «plásma» field which, in turn, generates that «plásma» flow..., thus fomenting a sustained self-re-iteration; a self-consistent "state" of motion; a 'consistent-with-self' ['consistent-with-continuation-of-self'] / 'consisting-of-self', and 'self-reproducing' motion of the «plásma».

Indeed, the "asymptotically stable" solutions of the «plásma» equations, corresponding to actual sustainable flows [to 'spatio-temporal attractors', as contrasted with "measure zero", virtually unobservable "transients"] must typically have this dynamically "self-consistent", 'self-reproducing' character.

Along with any additional influences, acting upon it from its outside, from its 'externity', the «plásma» 's ' internity' thus interacts with itself, 'self-reflexively' and 'self-refluxively' driving its own internal motion, hence, its 'auto-morpho-genesis' and 'auto-meta-morphosis' as a 'subject-verb-object-identical eventity'. The «plásma» phase of atomic/pre-atomic matter-energy, thus, at least in part, exemplifies a 'self-forming content' — a 'self-shaping', 'self-molding' phase of matter-energy, characterized by 'self-[induced-]plasticity' as well as by 'other-[induced-]plasticity'.

The text Dialectical Ideography, in its Section 1. b., entitled Why Dialectical?, glosses a hypothetical fusion reactor design, which F.E.D. has dubbed the 'Cyclonetron' design [link:  Dialectical Ideography, Part 1. b. — Why Dialectical? pp. 25-30].

This design hypothesis is based upon the propensity or proneness of '''flowing media''', or '''rheids''' whether they be the ~electro-neutral liquid or gaseous "rheids" described by the [nonlinear] Navier-Stokes system of equations, or the electrically-charged, 'electro-magneto-active' and 'electro-magneto-self-active' plasma "rheids" described by the [nonlinear] Maxwell-Boltzmann-Vlasov [vector-]equation(s), and by the Klimontovich equation to "spontaneously" generate, i.e., to 'self-generate', '''cyclonic''', or '''self-re-entering''', vortices'''toroidal vortices'''.
 
This '''genericity''', by "'rheids''', with respect to '''toroidal-vortical flow-forms''', may be related to the ubiquitous, '''particle-like''', '''solitary wave''', or '''soliton''', solutions that have been derived, in closed form, for a very large and diverse variety of one-dimensional, nonlinear, partial-differential wave-equations. 
 
'Neo-closed-form' '''toroidal vortex''' solutions, or '''cyclonic vortex''' solutions, may be the true three-dimensional generalization of those one-dimensional, '''solitonic''', "nonlinear wave" solutions.  We conjecture that non-pulsed, continuous fusion reactions may be sustainable at the heart of the "eye" of a plasma toroidal vortex, or plasma 'micro-hurricane', induced into '''self-precipitation''' in a plasma control-chamber whose parameter-vector of control-parameter-value set-points is appropriately tuned.

toroidal_vortex

Some Russellian, Gödelian, and Boolean Clues to the Nonlinearity Breakthrough

This 'self-reflexive' nature of 'external', "physical" processes is also instantiated and mirrored in the 'human-subjectivity-internal', mental process realms of formal logic and mathematical logic. '''[Self-]Reflexiveness''' is, according to Bertrand Russell, the very heart of the '"insoluble"' set-theoretical and semantical "paradoxes" that immanently plague and that self-violatively afflict formal, mathematical logic and set-theory:

"In all  the above contradictions (which are merely selections from an indefinite number) there is a common characteristic which we may describe as self-reference or reflexiveness." [Bertrand Russell, Alfred North Whitehead, Principia Mathematica to *56; Cambridge University Press (NY: 1970); page 61].

What we call the N dialectical ideography, our 'modern resumption' of Plato's ancient «arithmos eidetikos»; the initial, «arché» dialectical 'meta-arithmetic' which — together with its dialectical 'meta-algebra', and its own 'Qualo-Peanic' «sequelae» — we are exploring, belongs to a domain of '''Non-Standard Models''' of the Peano "Natural Numbers".   That is, the N  'meta-numbers', or 'dialectors', which are arithmetical/algorithmic «aufheben» operators, are 'Peanic'.  They comply with the first four, first-order Peano Postulates, the standard axioms for the "standard" "Natural" Numbers.  Even so, these 'dialectical meta-numbers' constitute an 'ideo-«arithmos»' which differs qualitatively — 'ideo-ontologically' — from the 'ideo-«arithmos»' of the "standard" "Natural" Numbers.

The possibility of such non-standard models of the axioms of '''natural''' arithmetic was '"predicted"' both by the Löwenheim-Skolem theorem, and by the joint applicability of the Gödel Completeness and Incompleteness theorems to the "first order" axiomatic system of the Peano "Standard" "natural numbers" arithmetic:

"Most discussions of Gödel's proof [of his '''First Incompleteness Theorem''' — F.E.D.] ... focus on its quasi-paradoxical nature. It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves. It is particularly enlightening to consider together both the completeness and incompleteness theorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility. The confusion arises from the two different senses in which the term "complete" is used within logic. In the semantic sense, "complete" means "capable of proving whatever is valid", whereas in the syntactic sense, it means "capable of proving or refuting [i.e., of "deciding" — F.E.D.] each sentence of the theory". Gödel's completeness theorem states that every (countable) [and ω-consistent — F.E.D.] first-order  theory, whatever its non-logical axioms may be, is complete in the former sense: Its theorems coincide with the statements true in all models of its axioms. The incompleteness theorems, on the other hand, show that if formal number theory is consistent, it fails to be complete in the second sense. The incompleteness theorems hold also for higher-order formalizations of number theory [wherein the Godel completeness theorem no longer holds at all, neither semantically nor syntactically — F.E.D.]. If only first-order formalizations are considered, then the completeness theorem applies as well, and together they yield not a contradiction, but an interesting conclusion. Any sentence of arithmetic that is undecidable must be true in some models of Peano's axioms (lest it be formally refutable [as it would be were it true in no models of the Peano axioms — F.E.D.]) and false in others (lest it be formally provable [as it would be were it true in all models of the Peano axioms — F.E.D.]). In particular, there must be models of first-order Peano arithmetic whose elements do not "behave" the same as the natural numbers. Such non-standard models were unforeseen and unintended but they cannot be ignored, for their existence implies that no first-order axiomatization of number theory can be adequate to the task of deriving as theorems exactly those statements that are true of the ["standard" — F.E.D.] natural numbers." [John W. Dawson, Jr.; Logical Dilemmas: The Life and Work of Kurt Gödel; A. K. Peters (Wellesley, MA: 1997); pages 67-68].

The Löwenheim-Skolem theorem has similar implications:

"The research begun in 1915 by Leopold Löwenheim (1878-c. 1940), and simplified and completed by Thoralf Skolem (1887-1963) in a series of papers from 1920 to 1933, disclosed new flaws in the structure of mathematics. The substance of what is now known as the Löwenheim-Skolem theory is this. Suppose one sets up axioms, logical and mathematical, for a branch of mathematics or for set theory as a foundation for all of mathematics. The most pertinent example is the set of axioms for the whole numbers. One intends that these axioms should completely describe the positive whole numbers [i.e., the "Natural numbers, NF.E.D.] and only  the whole numbers. But, surprisingly, one discovers that one can find interpretationsmodels — that are drastically different and yet satisfy the axioms. Thus, whereas the set of whole numbers is countable, or, in Cantor's notation, there are only 0 [pronounced as either aleph-null  or aleph-noughtA.-i.-D.] of them [i.e., there are only the minimal  infinite number of them — only an «arché» infinity of them — according to Cantor's theory of an endlessly-escalating progression of infinities, starting with the infinity that he denoted by the ideogram 0, thence progressing to 1, then to 2, etc. — F.E.D.], there are interpretations that contain as many elements as the real numbers [ = 1 elements, per the Cantor Continuum Hypothesis — F.E.D.], and even sets larger in the transfinite sense. The converse phenomenon also occurs. That is, suppose one adopts a system of axioms for a theory of sets and one intends that these axioms should permit and indeed characterize non-denumerable collections of sets. One can, nevertheless, find a countable (denumerable) collection of sets that satisfies the system of axioms and other transfinite interpretations quite apart from the one intended. In fact, every consistent set of ['''first-order''' — F.E.D.] axioms has a countable model [from a '''finitist/constructivist''' point-of-view, a model potentially involving 0 "logical individuals" in its '''universe''' [of discourse], but no more than that — F.E.D.] .... In other words, axiom systems that are designed to characterize a unique class of mathematical objects do not do so. Whereas Gödel's incompleteness theorem tells us that a set of axioms is not adequate to prove all the theorems belonging to the branch of mathematics that the axioms are intended to cover, the Löwenheim-Skolem theorem tells us that a set of axioms permits many more essentially different ['qualitatively different', 'ideo-ontologically different', unequal in a non-quantitative sense — F.E.D.] interpretations than the one intended. The axioms do not limit the interpretations or models. Hence mathematical reality cannot be unambiguously incorporated in axiomatic systems.* *Older texts did "prove" that the basic systems were categorical; that is, all the interpretations of any basic axiom system are isomorphic — they are essentially the same but differ in terminology. But the "proofs" were loose in that logical principles were used that are not allowed in Hilbert's metamathematics and the axiomatic bases were not as carefully formulated then as now. No set of axioms is categorical, despite "proofs" by Hilbert and others.... One reason that unintended interpretations are possible is that each axiomatic system contains undefined terms. Formerly, it was thought that the axioms "defined" these terms implicitly. But the axioms do not suffice. Hence the concept of undefined terms must be altered in some as yet unforeseeable way. The Löwenheim-Skolem theorem is as startling as Gödel's incompleteness theorem. It is another blow to the axiomatic method which from 1900 even to recent times seemed to be the only sound approach, and is still the one employed by logicists, formalists, and set-theorists." [Morris Kline; Mathematics: The Loss of Certainty; Oxford University Press (NY: 1980); pages 271-272].

Per our view, then, the real import of Gödel's completeness theorem, in the context of his incompleteness theorems, is that as evidenced in the ubiquity of "non-standard models" interpretational and axiomatic alternativity 'abounds'.

The form of 'alternativity' most directly implied by Gödel's incompleteness theorems for higher-than-first-order axioms-systems, as set forth by Gödel in the quotations above is a '''diachronic''' alternativity; is that of what we call 'Mathematico-Auto-Dynamasis', or axioms-system 'Ideo-Onto-«Auto-Kinesis»'.

It is that of the ineluctable 'intrinsity' of number concept 'ideo-dynamasis'.

It is that of the continual, «aufheben» emergence of new 'number-ontology', driven by the diophantine-equational unsolvabilities within each predecessor axioms-system stage/epoch; within each epoch of that diachronic, [psycho-]historical 'axioms meta-system', or «aufheben» systems-progression, of systems of mathematics as a whole.

It is that of the 'ideo-«auto-kinesis»' forming that «aufheben» 'consecuum', or 'consecuum' of "conservative extensions" of an expanded successor axioms-system for each predecessor axioms-system stage/epoch.

All of this constitutes that "inexhaustible" [Gödel] self-driven «aufheben» progression of the human-conceptual [axiomatic] systems of mathematics of which Gödel wrote.

This «aufheben» progression itself, is, indeed, a [meta-]Peanic or 'Qualo-Peanic' succession or 'consecuum'.

It is modelable by a "non-standard" interpretation of Peano Arithmetic, i.e., of the "Natural" Numbers arithmetic, '''non-standardly''' re-interpreted as based upon a realm of 'meta-numbers' which comply with the first four, "first order" Peano Postulates for the "Natural" Numbers, but which are unquantifiable [ontological] qualifiers, as distinct from the "unqualified quantifier" numbers which found the "Standard" interpretation of Peano "Natural" Arithmetic.

We denote the rules-system of 'dialectical arithmetic' that arises from this "non-standard model" of the first-order Peano Axioms via the ideogram N.

That form of 'alternativity' is, in short, that which results from the logical immanence of the conceptual and [psycho-]historical 'Dialectic of Mathematics', within each predecessor mathematical axioms-system, which 'self-forces' each such system to form an alternative, '''successor-system''' the next system, 'consecutive-successive' to it itself.

The ['self-exploratory',] system-'''evolutionary dynamics''' within each such predecessor axioms-system's stage/epoch formalized as [attempted] 'theorem-proving' and [attempted] 'equation-solving' — mount, at length, to the 'meta-evolutionary', 'meta-dynamics' level mount until they eventually spill-over, beyond the boundaries of that predecessor axioms-system's system-identity, into the constitution of an unprecedentedly-new, qualitatively different, 'ideo-ontologically-expanded', successor axioms-system's system-identity, encompassing new «gene» and new «species» of number.

This intrinsic 'Ideo-Metadynamic' of each human idea-system's self-induced «aufheben», or 'dialectical', multi-system, diachronic systems-progression is also the very meaning of our central symbol, .

The algebra of the '''non-standard''' N initial 'dialectical arithmetic' is also describable as a 'contra-Boolean algebra'. The calculative phenomena of this N 'contra-Boolean algebra' are quite contrary to those which, in Boolean algebra, mirror formal logic.

Boolean algebra models 'ideo-onto-stasis'.

The N 'contra-Boolean arithmetic', and its 'contra-Boolean algebra', model [e.g., the Gödelian] 'ideo-onto-dynamasis'.

This 'contra-Boolean' arithmetic and algebra issue, as such, from the 'strong contrary' of the Boolean-algebraic axiom which Boole termed '"the fundamental law of ['Dianoic' — F.E.D.] thought"', and which Boole also termed the '''law of [exo-]duality'''.

The 'contra-Boolean' axiom of the N 'rules-system', which we call the 'fundamental rule of dialectical thought', the 'rule of intra-duality', or 'principle of self-duality', describes the consequences of '''squaring''', or «dynamis», i.e., of generalized self-multiplication / self-operation / self-application / 'self-re-flexion' / 'self-re-fluxion', among the N species of 'dialectical meta-numbers', 'ontological qualifiers', «aufheben» operators, or 'dialectors'.

Summarizing, then, the foregoing, we have found that the N 'dialectical ideography' is an arithmetic of 'Gödelian-Skolemian meta-Natural Numbers', i.e., of 'meta-numbers' which are 'Peanic', in that they satisfy the first four, "first-order" Peano Postulates for the "Natural" Numbers, despite also exhibiting the 'strongly contra-Boolean' characteristic in their self-operation and mutual operation — in their '[self-][inter-]action', '[self-]application', or '[self-][re-]flexion', i.e., in their '[self-][re-]flexive', '[self-]operative', or 'generalized-multiplication operation' self-multiplicative behavior.  Moreover, that 'generalized-multiplicative' behavior constitutes an ideographical, arithmetical, algorithmic «mimesis» of the '''self-«aufheben»''' operation which resides at the heart of all dialectical process.

This initial, initiating dialectical arithmetic, models, in diametrical contrast to the "Standard Natural" arithmetic of 'unqualified quantifiers', the operations of an 'idea-realm', or 'ideo-«arithmos», of 'unquantified and, indeed, unquantifiable qualifiers', interpretable to describe trans-Russellian and, indeed, trans-Gödelian '[onto-]logical types', or [ontological] qualities, ontological categories, and 'meta-qualities' [modelable via 'meta-predicates', or 'predicates of predicates'], which beget new, but 'meta', '[onto-]types'; new such [meta-]qualities, even/ever higher in 'meta' degree.  These 'numeralic ontic qualifiers' accomplish this via their algorithmic «mimesis» of the Marxian-Hegelian-Platonic «aufheben» operation.

Dialectical 'Meta-Monadology'

This self-proliferation of 'onto-predicates', 'ontological qualifiers', or 'ontological categories' may be "interpreted" or "modeled" as resulting from the quantitatively growing population counts/population densities of the «monads» of the «arithmoi» that the N 'meta-numerals', by means of the ontological categories to which they are assigned, are used to represent.

This self-proliferation may be conceived as arising from the self-expanding self-reproduction, or self-accelerating 'auto-catalysis', of those monadic populationsarithmoi» of «monads»/'''units'''/'''logical''' [and physical] individuals.

This self-proliferation is one, then, which increasingly 'consequents' in those «arithmoi»s' thus self-intensifying their own 'self-environment' or 'self-surroundment', their 'self-[inter-]action self-densification' or 'intra-action self-densification', as well as their 'inter-mutually other-inter-active self-densification'.

The single 'self-surroundment' kind of 'self-densification' arises via the thus increasingly 'self-frequentizing' mutual confrontation of their individual eventities/«monads», or, in other words, via the 'self-deepening «auto-dynamis»', [or 'self-squaring'] 'self-confrontation' within the physical-spatially distributed, multiple loci of concentration of their own class/category [e.g., due to their concrete, physical-spatially distributed '«arithmos» of «arithmoi» meta-dynamics', or 'superarithmos» meta-dynamics', or 'cumulum meta-dynamics' ].

Thus, for this view, the definition of that «genos» known as '''Arithmetic''' expands.

'''Arithmetic''' becomes the general study of «arithmoi», whether they be 'ideoarithmoi»' or 'physioarithmoi»'/physis»-arithmoi»'/'physicalarithmoi»'.

'''Arithmetic''' 're-becomes', then, the generic study of the '«auto-dynamis»' of '''populations''' of «monads» of every «genos» and «species», and of their [trans-Leibnizian] 'Meta-Monadology' of their self-«aufheben» processes of 'auto-meta-monadization', or of the 'self-internalizations' of sub-populations of the predecessor  «monads» of a given «arithmos» «genos».

Such selfaufheben» 'self-internalizations' give birth to a qualitatively, ontologically new, higher, unprecedented successor '[meta-]«arithmos»'/population/system, constituted out of new '[meta-]«monads»', constituting a higher/more-inclusive 'ontological type', and, thereby, to a new «genos», of new, unprecedented, ontology, in which each typical «monad» of the successor-system «arithmos» is a 'meta-unit', made up out of a heterogeneous multiplicity of the units/«monads» of the predecessor system/«arithmos».

Thus, in terms of the 'physio-ontology', or '«physis»-ontology', of the 'Meta-Monadological', 'meta-fractal', 'self-«aufheben»' '''Dialectic of Nature''''

Each typical '''sub-atomic particle''' [meta1-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''pre-nuclear particle''' [meta0-]«monads» [e.g., of different kinds of '''quarks'''].

Each typical '''atom''' [
meta2-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''sub-atomic particle''' [meta1-]«monads».

Each typical '''molecule''' [
meta3-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''atom''' [meta2-]«monads».

Each
typical '''prokaryotic cell''' [
meta4-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''molecule''' [meta3-]«monads».

Each
typical '''eukaryotic cell''' [
meta5-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''prokaryotic cell''' [meta4-]«monads».

Each
typical [
meta6-]«monad» '''multi-cellular organism''' [each 'meta-bion' ] each '''multi-eukaryotic''' animal ['''meta-zoan'''] or plant ['''meta-phytan'''] '''biological individual''' is a 'self-«aufheben»' 'self-internalization'  of a heterogeneous multiplicity of '''eukaryotic cell''' [meta5-]«monads».

And, each typical '''animal society''' [
meta7-]«monad» is a 'self-«aufheben»' 'self-internalization' of a heterogeneous multiplicity of '''meta-bion''', or '''meta-biotic''', '''meta-zoan''' [meta6-]«monads».

The humanly/presently-known cosmos is the 'cumulum' of these multiple «arithmoi» of '[meta-]«monads»' is the 'superarithmos»' formed by this '''multitude'''-ofarithmoi», or «arithmos»-of-«arithmoi» — when we grasp each «arithmoi» of that '''multitude'''-of-«arithmoi» as an [ontological-categorial] [super-]unit, or '[super-]«monad»', in its own right.

The present cosmos is the cumulative 'multi-population physical-spatial meta-distribution', or distribution of population distributions, of these qualitatively, ontologically different «monads», plus of their 'hybridizations' or mutual-/self-organizations, as both 'ontological [hetero-]conversion' and 'ontological [self-]conversion' formations.

Thus, also, in terms of the 'ideo-ontology' of number and of the 'ideo-onto-dynamasis' of number as already recounted above.

Each typical instance of the integer «genos» of number is modelable as an «aufheben» self-internalization of two whole numbers in the set-theoretical, ordered-pairs sense, i.e., as a 'meta-whole-number,' made up out of a heterogeneous multiplicity [an ordered pair] of whole numbers.

Each typical instance of the rational-number «genos» of number is modelable as an «aufheben» self-internalization of two integer-numbers in that same sense, i.e.,, as a 'meta-integer', made up out of a heterogeneous multiplicity [an ordered pair] of integer-numbers. ...

And, each typical instance of the complex-number «genos» of number is modelable as an «aufheben» self-internalization of two "Real"-numbers, as a meta-"Real" number, also in that sense, i.e., as made up out of a heterogeneous multiplicity [an ordered pair] of "Real"-numbers.

Each typical 'meta-Natural-number', or non-standard "
Natural-number" each 'dialector' «monad», or '''qualitative unit'''/'ontological qualifier', of the N dialectical arithmetic is modelable as an «aufheben» self-internalization, 'self-subsumption', or 'self-subscriptization' of a single N-number, or ["standard"] "Natural" number.

Note that each such post-«arché» "Natural" Number, beyond the '''singleton''' «monad»/unit denoted by '1' itself, is itself a "Natural" «arithmos» of multiple '1' units/«monad»s a Platonic «arithmos monadikos» in its own right, e.g., 2 = 1 + 1, 3 = 1 + 1 + 1 = 2 + 1, 4 = 1 + 1 + 1 + 1 = 3 + 1, etc., and thus also, an «aufheben» conservation/elevation/cancelation of its immediate predecessor «arithmos».

aufheben_natural_numbers

Among the most startling revelations of our research — startling, at least initially — was the discovery of, and the formation of new mental "eyes" that could discern, the evidence of '''non-standard''' 'Peanicity', first-order Peano Postulates conformance by 'qualifiers' instead of by '''quantifiers'''   and of the ubiquity of this 'Peanicity', not only in pre-human and extra-human nature at large, but also in the structure of core artefacts of the 'Meme-nome' or '''Phenome''' of human nature, of the human part of nature, including of those 'artefactual', symbolic systems which constitute the core of human literacy.

Our exposition of these discoveries, below, employs several new words, or 'neologia', which are defined, briefly, at its outset, with the proviso that the exposition itself will enrich, deepen, and extend this initial account of their meanings.

By an 'archeonic consecuum', we mean a descriptively discontinuous, temporally-deployed or diachronic succession of '''consecutive''' discrete existences [with no description of any other such existences "in-between" each consecutive pair of such existences]; a [self-driven] progression of existences which are '''internal''', mental constructs, or '''external''', physical [self-]constructs, and which are such that this [self-]progression has a beginning or originating such [self-]construct, an «arché» «arithmos», but does not necessarily have any pre-definite ending or final/ultimate such [self-]construct/«ultime» «arithmos».

By the term 'meta-monadic', we refer to a sequence of mental or physical [self-]constructions in which each most-recently-emerged «arithmos» is such that its '''units of population''', '''individuals''', or «monads» are themselves each made up out of, as «aufheben» [self-]internalizations of, the '''units of population''', or «monads», of their immediate / reverse-consecutive predecessor [self-]construction, in an 'archeonic consecuum' of [self-]successive / [self-]progressive such [self-]constructions.

By the terms 'qualo[-quanto]-fractal' and 'meta-fractal', we refer to the «aufheben» diachronic structure of a 'meta-monadic archeonic consecuum', as well as to the «aufheben» synchronic structure of any [post-«arché»] individual [self-]constructions within that 'consecuum', such that this '[self-]constructure' thus manifests, as its '''moments''', either a durationally-scaled / time-size-scaled / time-"length"-scaled, diachronic finite similarity-regress, or a space-scaled / spatial-size-scaled synchronic finite similarity-regress — or,  as a whole, 'diachronico-synchronically' a [quanto-]qualitative gradient of '''scales'''/'''levels''' of qualitatively, ontologically distinct mutual similarity, in both its diachronic and its synchronic directions/'''dimensions'''.

To better grasp the 'synchronic meta-fractal' structure that is generated, as a by-product, by the 'diachronic meta-fractal' self-movement/self-process of the 'ontologically-dynamical' '''dialectic of nature''', consider your own body-mind, as it exists at this very moment, as you read these words.

An adult human body is, biologically speaking, the result of an «arché» zygotic embryo's 'auto-dynamasis'.

Such a body is a single 'meta-biotic', "meta-zoan",  'meta-eukaryotic' unit, or [meta[n]-]«monad».

That human body is a 'meta-monadic super7-organism', or 'super7-system'; a complex[ified] unity; a unified sub-whole, 'sub-hol', '''sub-totality''', 'sub-cosmos', or '''micro-cosm''' [cf. Leibniz; cf. Bohm] of the '''macro-cosmos''' itself, in the following exact sense.

Your body is a 'meta7monad»' of the «arithmos»/population of the Terran human species, made up out of an ordered, «arché» 'embryo-dynamically' self-organized, heterogeneous multiplicity/«arithmos» of 'super6-system' organs as its immediate, '[meta6-]«monads»'.

Each typical organ of your body is, in turn, made up out of an ordered, «arché» 'embryo-dynamically' self-organized, heterogeneous multiplicity/«arithmos» of 'super5-systemeukaryotic cells as its immediate, '[meta5-]«monads»'.

Each typical eukaryotic cell of your body is, in turn, made up out of an ordered, self-organized, heterogeneous multiplicity/«arithmos» of 'super4-systemprokaryotic cell-ancestored intra-cellular organelles as its immediate, [meta4-]'«monads»'.

Each typical intra-cellular organelle of your body is, in turn, made up out of an ordered, self-organized, heterogeneous multiplicity/«arithmos» of 'super3-systemmolecules as its immediate, '[meta3-]«monads»'.

Each typical molecule of your body is, in turn, made up out of an ordered, self-organized, heterogeneous multiplicity/«arithmos» of 'super2-system' atoms as its immediate,  '[meta2-]«monads»'.

Each typical atom of your body is, in turn, made up out of an ordered, self-organized, heterogeneous multiplicity/«arithmos» of 'super1-system' sub-atomic "particles" [e.g., "protons" and "neutrons"] as its immediate, '[meta1-]«monads»'.

Finally [to this humanity's present consensus knowledge], each typical sub-atomic "particle" of your body is, in turn, made up out of an ordered, self-organized, heterogeneous multiplicity/«arithmos» of '[super0]-systempre-nuclear "particles" [e.g., "electrons" and "quarks"] as its immediate — and ultimate, «arché» — '[meta0-]«monads»'.

Furthermore, each human body-mind is a part of the 'super8-system' of a geo-local, [self-]«aufheben»-conserved 'animal-societies', of humans-as-animals, which, having become 'self-internalized', as 'animal-society' «monads» — via the 'co-formation', or 'mutually-induced formation', of a new level of 'internity' — together, in '''inter-mutual domestication''',  with a heterogeneous multiplicity of  other '''meta-zoan''' 'animal-societies' as co-«monads», and with 'meta-phytan' plant communities, formed the  'super9-system' of human[-led] 'meta-societies' which, today, populate the Terran 'noo-bio-[atmo-hydro-litho-]sphere' as its so-far-ultimate cosmological '''meristem'''/'meta-fractal' scale/level/layer.

'Meta-fractal, meta-monadic, archeonic consecua' include discrete, «arché»-ic-state-determined [self-]«aufheben» systems-progressions, punctuated by the '''immanent revolutions''' or '''self-revolutions''' — the 'meta-finite' ontological self-conversion singularities — that [self-]convert each successive '''predecessor''' system in such a progression into its next, '''consecutive''' successor system.

They include also the 'synchronico-diachronically' «aufheben»-structured, systematic-/'dynamical-taxonomic' «genos»/«species» progressions of Hegelian and trans-Hegelian, time-dependent truth-valued, dynamical/'meta-dynamical' [dialectical] '''disjunctive syllogisms'''.

They include «arithmoi»-progressions — or '''populations-progressions''' — of [self-]«aufheben»-created, 'self-internalization'-created, self-ordering, self-organized, 'self-systematized' populations of 'metanmonads»'.

They include [self-]«aufheben», diachronic 'self-successions'/'self-progressions' of human-social systems — of human-social formations of '''social relations of [human-society/social relations self-[re-]]production'''; of predecessor human-social systems — each one 'historical singularity-generating' and '[immanently-]self-revolutionizing', and thereby birthing their next '''consecutive''' successor human-social systems/formations, in an historical process which instantiates the «autokinesis»/Platonic, and the Hegelian/Marxian, '''historical dialectic'''.

They include the unity of all of the above.

Core-artefact, cultural, linguistic, cognitive, '''Phenomic''' innovations of human history, as mentioned above, are '''first-order Peanic'''; are constituted as '[self-]«aufheben», meta-fractal, meta-monadic archeonic consecua', each one therefore modelable via suitably defined, "non-standard", Peano-eligible '''successor functions'''.  What was initially so startling about this discovery?

What was so startling, at first, is that these ''successions'', or, indeed, '''progressions''' — dialectical/'auto-kinesic', or not — are describable by the first four, "first order" Peano Postulates, whose original intension was to axiomatize the [so-called] "Natural" numbers arithmetic, and nothing more / nothing other.

Peanicity

Consider a '''standard''' version of these four "Peano Postulates", intended to describe the '''Natur[e-]al  [Numbers] Succession''', as rendered by the following four sentences, beautiful in their simplicity:

[P1.]  1 is a "Natural" Number.

[P2.]  The successor of any "Natural" Number is also a "Natural" Number.

[P3.]  Distinct "Natural" Numbers have distinct successors.

[P4.]  1 is not the successor of any "Natural" Number.

These four Peano axioms may be restated in a deliberately "genericizing" fashion — one that intimates, more explicitly than does their above, '''standard''' version, how these axioms can span, to encompass, both extremes of the range of differences between those sentences' "standard", 'pure, unqualified quantifier' interpretation, for N, and their 'doubly-opposite', '''non-standard''', 'pure, unquantifiable ontological qualifier' interpretation, the latter for the «arché» of the dialectical arithmetics, herein denoted by N.

Consider such a 'genericized' version of the "first order" Peano axioms, re-rendered below [wherein we use '&' to abbreviate the word '''and'''; 'to abbreviate the phrase '''is an lement of''', or '''is a member of'''; '' to abbreviate the word '''implies''', or the phrase '"if ... is true, then ... is also true'''; 'to abbreviate the phrase '''equals by definition'''; '' to abbreviate the phrase '''is not equal to'''; '~' to abbreviate the word '''not'''; '' to abbreviate the phrase '''there exists''', and; '|' to abbreviate the phrase '''such that'''].

Also, in the re-rendition below, we use a 'singly-underscored Q', namely NQ, to denote the '''space''' or "set" of 'dialectical meta-numbers', NQ { q1, q2, q3, ... }, whereas N denotes the entire arithmetic, the full rules-system, including rules of operation for '''generalized addition''' and for '''generalized multiplication''', etc., of which the list of the  'dialectical meta-numbers' themselves forms only a component. Note too, that NQ     Q, where Q denotes the '''space''' or set of the rational numbers: Q { -2/1, ..., -3/2. ..., -1/2, ..., 0/1, ..., +1/2, ..., +3/2, ..., +2/1,... }.
 

 Any '''Natur[e-]al''' diachronic Succession/[self-]progression of «arithmoi», including any '''diachronic, historical dialectic''', and also any '''synchronic, systematic dialectic''', is such that —

[P1'.] The '«arché»-ic «monad»', originating unit, or «arché» — modernly graspable as also a '''singleton''' '«arché»-ic «arithmos»', or 'ur-«arithmos»', in its own right — is a member of the '''Natur[e-]al''' diachronic Succession/progression/'consecuum' of «arithmoi», or of assemblages of units/«monads», denoted by X.

Example 1:
    for
X = N ; 1 N

Example 2:
    for X = NQ ; q1 
NQ

[P2'.]  The successor «arithmos», '''population''', or multitude of individuals/units/«monads», denoted by
X, of any «arithmos», denoted by x, that is itself a member of this '''Natur[e-]al''' Succession/progression/'consecuum' of «arithmoi», denoted X, is also a member of X.

Example 1:
    for
X = N , and for x = n,
    if  n
N, and if the successor of n is denoted by s(n),
    and is such that s(n)
= n + 1,
    then
s(n) is also an element of N :
    [n
N  [s(n) N].


Example 2:
    for X = NQ ;
    [
qn
NQ   [ s[qn]    qs(n)    qn+1    NQ ]


[P3'.]  Any two, distinct «arithmoi», belonging to
X, have distinct successors.


Example 1:
    for X = N ;
    [n, m
N] & [n m]  [s(n) s(m)]

Example 2:

    for X = NQ , given n, m N;
   
[ [ qn, qm NQ ] & [ qn  qm ] ] [ s[ qn ] s[ qm ] ],
    indeed,
for X = NQ ;
    [ qn, qm NQ ] & [ n m ] qn qm ] & [ s[ qn ]  s[ qm ] ],
    where the '
neogram', ' ', denotes the relationship of 'non-quantitative inequality', or 'qualitative inequality'.


[P4'.]  The '«arché»-ic «monad»'/'ur-«arithmos»' of
X has no predecessor in X.

Example 1:
    for
X = N ;
    ~[ x N]   |   [s(x) = 1 ]

Example 2:
    for
X = NQ ;
    ~[
qx NQ |  [ s[ qx ]  =  
qs(x)  =  q1 ]

These axioms describe the succession/'consecuum' of the "Natural Numbers".  They also describe the succession/'consecuum' of '''Natural''' 'meta-numbers' which form the NQ progression.

The NQ 'consecuum' is not the 'counting consecuum' of the N progression — the latter being a progression, at most, of an ['onto-statical'] 'quanto-dynamasis' within an ontological, qualitative 'onto-stasis'.

The NQ 'consecuum' is the 'consecuum'  of/for 'ontological dynamicity', of/for the principles of change/expansion in the extant kinds of being, i.e., a 'consecuum' for describing/modeling ['ideo-' and 'physio-'] 'onto-dynamasis'.


A level up beyond that description, these axioms also describe the originating, or «arché», 'dialectical arithmetic' — noneother than the N arithmetic itself — of the succession/progression of the 'dialectical arithmetics', or of the 'dialectical ideographies', which itself constitutes yet another instance of a '
Peanic', «aufheben», 'meta-fractal', 'meta-monadic', '«arché»-onic consecuum', as addressed in the book entitled Dialectical Ideography.

That is, these axioms also describe the 'meta-number consecuum' of N, the '''simplest/'''least complex''', and '''most abstract'''/'''least thought-concrete''' system of 'dialectical arithmetic' itself, plus the dialectical succession/progression of further, more '''complex''', more '''thought-concrete''' systems of 'dialectical arithmetic' that issues from it, and by it, as the «arché» of the 'dialectical arithmetics'.

In short, these axioms also describe the [likewise] 'Qualo-Peanic', 'meta-fractal', 'meta-monadic', self-«aufheben», and 'ideo-auto-kinesic' 'archeonic consecuum' 'self-progression' of the systems of 'dialectical ideography' — in short, a '''systematic''' and 'meta-systematic' dialectic of 'dialectical arithmetics'.

Thus, this succession/progression of 'dialectical arithmetics' does not form a 'self-exception'; does not constitute an exception to the 'Peanic' pattern of succession encoded beginning in the «arché» 'dialectical arithmetic', N of that very succession/progression.  The progression of the 'dialectical arithmetics' is '''self-consistent''' also in this sense, that it 'consists of ['similants' of] [it]self', and describes itself, in an instantiation of the principle of 'self-including modeling'.

The abstract pattern of progression encoded in NQ is that of '''dialectic''' in general.

The succession, or 'consecuum', of the rules-systems of the 'dialectical arithmetics' also instantiates that pattern, is a particular case thereof. The pattern of progression of the 'dialectical ideographies' is itself a dialectical pattern — that of the 'meta-system-atic dialectic' of the systems of the 'dialectical ideographies'.

That particular pattern, too, is a particular case of the general, generic, «genos» dialectical — 'Qualo-Peanic', 'meta-fractal', 'meta-monadic', 'archeonic consecuum' pattern — or self-«aufheben» pattern, of 'self-internalization'-driven 'ideo-«autokinesis».

In Summary:  '''Dialectic''' is 'Meta-Peanic' or 'Qualo-Peanic' in the sense of a '''non-standard''', qualitative/ontological, '''onto-genesis'''/'onto-dynamasis'/'onto-dynamical' interpretation of the "first order" Peano axioms. Thus, 'dialectic' exhibits an '''arithmetical structure''', or «arithmos» structure, otherwise widely familiar today only in the form of the 'core-dianoic', '''purely-quantitative''', reductionistic, 'linearistic' realm of counting and of the '''counting numbers''' of N. The latter has, heretofore, by many accounts, seemed to constitute the extreme of opposition to 'dialectic', especially/usually in the sense of a rigid radical dualism of mutual, external opposition, rather than of a dialectical antithesis of mutual [internal-/intra-]opposition, within a [sub-]totality that encompasses both, and that calls forth synthesis.

Some "Non-Standard" Examples of '[Qualo-]Peanic' Progression

As we noted above, initially, this was a shocking discovery for us.

Subsequently, as our research has deepened, this finding has come to seem, to us, to be an essential one — 'essence-ial', necessary, inescapable, and inevitable as yet a further testament to a cosmological reality which is '''everywhere dense with dialectic'''.

Viewing this matter '''psycho-historically''', it would seem that a predominantly 'dianoiac' Terran humanity discovered the
dialectic, at first, in its 'least dialectical' form — in its least-discernible, exchange-value-experience-inculcated, reductionistic, and exclusively 'Quanto-Peanic' form. That is, this '''prehistoric''' humanity discovered dialectic mainly as the '''purely-quantitative''' «aufheben» operation of — in effect — the Peano successor function as model for the function of counting [using '
' to abbreviate the phrase '''for All''', or '''for every''', and '
' to abbreviate the phrase '''goes to''', or to abbreviate the word '''becomes''']:

[n N ]:  [ n    s(n)  =  n + Δ n + 1     n ].

Note, in the above, how the Peano succession-operation, s(n) = n + 1, is an «aufheben» operation — is a '''conservation''' of the operand, n, but also a '''negation''' of n, in that s(n) ≠ n, or  ~[ n = s(n) ], and also an '''elevation''' of n, i.e., to a "higher level" of quantity [only]. The process denoted s(n) '''elevates''', '''escalates''', '''surpasses''', '''supersedes''', and '''transcends''' the «arithmos» of n units/«monads», but only quantitatively, by adding one more qualitatively-homogeneous, qualitatively-identical, qualitatively-indifferent unit one more '''purely quantitative''' «monad» to the «arithmos» denoted n, so that 'n + 1' names the very next, consecutive, bigger «arithmos», or assemblage of «monads»/assemblage of units/assemblage of 1s, just one step beyond the n «arithmos».

We began, in the course of these discoveries, to notice a human, linguistic, cultural, artefactual, '''Phenomic''' «mimesis» of pre-human/extra-human 'Natur[e-]al Dialectic'. We began to notice a human «mimesis» of the self-«aufheben», 'auto-kinesic', 'meta-«monad»-ological', 'meta-fractal-izing' '«arché»-onic consecuum' pattern of ontological innovation, otherwise manifest so robustly and so comprehensively in ancestral/pre-human, and in contemporaneous/extra-human, Nature.

This «mimesis» turned out to be ubiquitous among and within the artefacts of human inventiveness, of 'human-natur[e-]ality', and, specifically, within the very core of human 'linguistic-technology' and 'thought-technology', already from ancient times, though often in 'reductionistic', 're-dianoic' forms. We will address, below, the two most prominent examples of this «mimesis», which reside at the heart of literate human culture globally.

Example 1: The 'Meta-Monadology' of the Hindu-Arabic Numerical Notation

Preponderantly 'dianoic' humanity, early on, discovered a 'dianoic shadow' of the dialectical, self-«aufheben» principle of 'Meta-Monadology' — of the principle of 'self-internalization'; of diachronic [and therefore also of synchronic] 'self-meta-fractal-ization', and of «monads»' 'self-metamonad»-ization', or of «monads»' self-'monado-dynamasis' — in a key non-dialectical, non-auto-kinesic, exchange-value-modeling form.

That is, humanity discovered a 'reductionistic shadow' of these principles, via the place-value/zero-as-place-holder principle, and ideographical-linguistic notation and thought-technology, of the Hindu-Arabic or 'Indo-Arabic' numerals-system.

Thus, for example, 10, or 101, is the Hindu-Arabic way of shorthand-denoting the next, consecutive in "order of magnitude", meta1monad», or 'meta1-unit', to 1, or 100, as «monad»/unit, or as 'meta0monad»/meta0-unit, i.e., as '«arché»-«monad»' or '«arché»-unit'. This first 'meta-unit', denoted by 10, is made up out of a homogeneous multiplicity of [exactly ten] units, or meta0-units, i.e., of the kind of 'ur-units' denoted by 1.

The next consecutive "order of magnitude" successor 'meta-unit' to the 10 'meta-unit' is represented, in the Hindu-Arabic notation-system, by 100, or 102.  This 'meta-unit', e.g., for '''counting in units of one hundred''', denotes, with respect to the «arché»-unit', 1, a 'meta-metamonad»', or meta2monad», such that this 'meta2-unit', 100, is made up out of a homogeneous multiplicity of [again, exactly ten] 'meta1-units'; ten 10s.

The next consecutive "order of magnitude" successor 'meta-unit' to the 100 'meta-unit' is represented, in the Hindu-Arabic notation-system, by 1000, or 103.  This 'meta-unit', e.g., for '''counting in units of one thousand''', denotes, with respect to the «arché»-unit', 1, a 'meta-meta-metamonad»', or meta3monad», 'meta1-' to the 'meta2-' unit, denoted by 100, such that this 'meta3-unit', 1000, or "1M", is made up out of a homogeneous multiplicity of [again, exactly ten] 'meta2-units'; ten 100s.

aufheben_indo_arabic

And so on . . . That is, in this 'consecuum' — of «arithmoi»-becoming-new-«monads» in their own right — the '''succession function''', call it S, is such that, for any such [meta-]unit, denoted U, its successor-unit is S(U) = 10U.

Note the 'onto-statical', reductionistic persuasion of this "purely-quantitative", pure, unqualified quantifiers numerals-system 'Meta-Monadology'. Thus, the '«arithmos»-becomemonad»', 10, conceptually, reduces absolutely to an assemblage/aggregate/sum of units 1+1+1+1+1+1+1+1+1+1, which 10 internalizes, stands for '''nothing but''' ten 1s, for "nothing but" an «arithmos» or simple aggregate of ten 1 units, without any '''qualitative residual''', and without any 'incremental number-ontology' being created by the 'metamonad»'-ization' of these ten 1s: 10 is "nothing but" an aggregation of [ten] 1s. There is no sense of any difference in '''kind of being''', of any qualitative difference, in 10 vs. 1. There is a difference of "scale", of "order of magnitude", together with a mutual similarity — that of both 1 and 10 being 'units', or «monads».  Still, any difference between the two is conceived as being strictly a difference of magnitude, a difference of quantity only; a "purely-quantitative" difference:  10 > 1.

This first example of a 'Meta-Monadology' — as with the second example, to follow, below — does not constitute a 'dialectic' in our sense. It does not constitute a 'dialectic' in that sense, because it lacks the character of «autokinesis». It lacks an '''agent''' of action, an entity that belongs in the '''subject'''-slot of a sentence, acting upon an entity named in the '''object'''-slot of that sentence, describing that agent's most essential activity, that "object" naming the agent itself, again, an agent/subject acting thus 'self-reflexively' and 'self-refluxively', back upon itself, the source of the action, in the mode of an activity, named by a verb in that a sentence, which is noneother than another name for that same agent/subject/object, naming an essential mode of activity of that self-same agent/subject/object/verb.

That is, this example lacks the character of such a 'subject-verb-object-identical eventity'.

Its structure is that of an «aufheben» and 'meta-monadic' progression/'archeonic consecuum' — { 1, 10, 100, 1000, . . . }, instead of the { 1, 2, 3, . . . } of Peano's {sτ(1)}  — but not that of a self-«aufheben» self-progression. Its Hindu-Arabic {Sτ(1)} progression is animated and propelled by a force of human cognition, by a human mind as subject, which is posited as external to those numerals, that it acts upon, mentally, or in '''the space of the mind'''', as its objects.  Those numerals are presented as forming an inert, inanimate system of 'statical' idea-objects — not as one of 'idea-subjects', i.e., of 'idea-agents', 'idea-subject/objects', or of 'ideo-eventities' — and as a static realm of idea-objects which constitutes only a partial objectification of human cognition as subject, as a whole. This system of numerals is not posited as an 'autokinesic ideo-eventity', not as a cognitive action of human cognition as a whole, as subject or agent, acting upon itself as a whole, with itself thus as its own object — not as 'object-ification' of itself as a whole.

The Hindu-Arabic numerals-system progression is not given as 'self-animated'; not given as '''self-propelling''', by way of an '''internal force''', '''immanent force''', or '''self-force''' of its own 'intra-duality'; i.e, by way of its own '''dialectical/ontological/existential self-contradiction''' [not the same as '''propositional self-contradiction'''], or 'internal contra-kinesis'.

We encountered, above, such '''self-forces''' at work, in the physical sense, in the fecundity of the «arché» populations of pre-nuclear "particle" «monads» that launched '''The Dialectic of Nature''', i.e., the «arché»-«arithmos» 'proto-subject'/agent of cosmological 'meta-evolution' which, together with its «sequelae», self-constructed the entire cosmos as we know it today.

We saw such at work, also, above, in a logical sense, in Example 0 (above), in the definitional self-force, or logical self-force, of the [finitary] '''Set of All Sets''' 'ideo-eventity'. We found it to be, or being, '''forced''', in order to fulfill its definition that it is to contain "All" sets into continual 'ideo-ontological', 'ideo-onto-dynamical', 'ideo-auto-kinetical', qualitative self-expansion, via continual 'self[-and-other subsets]-internalization'. We found it to be, or being, '''forced''' by its own nature/essence/'essence-iality'/essentiality/necessity; by its own '''self'''; by its own name/description/definition, i.e., by the 'intra-duality', or 'self-duality' and 'in-divi-duality', or 'internal division duality', of its every state of existence — because it always, in every "moment", "still" excludes those very sets which constitute its own "power set", its own subsets, among which is that set which is its own "improper" subset, namely, none other than itself.

Each time it internalizes all of its subsets, including itself, it thereby transforms itself into a new, qualitatively different, qualitatively expanded set, with a different set of subsets, a different "power-set". Therefore, it must, each time, internalize its own subsets again —

Sτ+1  =   Sτ2   =   Sτ + 2Sτ,

i.e.,

Sτ+1  =   Sτ2   =   Sτ 2Sτ,

or

Sτ+1  =   Sτ2   =   Sτ    P[Sτ],

or

Sτ+1  =   Sτ2   =   Sτ united with the Power-Set of Sτ,

a set-equation whose solution is

Sτ   =  S02^τ   =   U2^τ, wherein 2^τ    2τ

in a self-driven process of the 'Meta-Monadology' of the Set that contains, given a suitable definition of the Universe [of discourse] i.e., of the Universal Set, U S0 all of the wherewithal of 'The Gödelian Dialectic'.

aufheben_set_of_all_sets

The Hindu-Arabic, numeralic 'Meta-Monadology' is constructed via '''exterior forcing''' by an '''other''' conceived as external to that 'ideo-«arithmos»' by the supervising mental agent of our own mental embodiment of the rules of that arithmetical ideography.

It is not a 'consecuum' that is 'self-constructed', via an 'internal, self-directed force', not via a self-sourced and self-impinging, self-refluxive force, arising from the 'intra-duality', 'self-duality', internal 'self-antithesis', or 'self-other-ness' / 'otherness-to-self' of its own «arché»-«arithmos». The Hindu-Arabic 'consecuum of «arithmos»'  does not generate its own '«sequelae»', its own continuation, 'extention', or '''succession''', by '''interior forcing''' of itself as a whole, by itself as a whole, [agent or] subject-to-itself-as-object, object-to-itself-as-subject  [or agent] subject/verb/object identical.

Nonetheless, this progression of «arithmoi» 1, 10, 100, 1000, ... still does exhibit the 'meta-monadic' structure, the '''[quanto-fractal]''', or scaled-self-similarity structure, and the 'Peanic' structure of a ['''non-standard''', but still 'purely-quantitative'] 'archeonic consecuum'.

All '''dialectical self-progressions''' are 'Peanic', 'meta-fractal', 'meta-monadic', «aufheben», 'archeonic consecuum' of «arithmoi»-of-«monads» progressions. However, not all 'Peanic', 'meta-fractal', 'meta-monadic', «aufheben», 'archeonic consecuum'  of «arithmoi»-of-«monads» progressions are '''dialectical'''.

Example 2: The 'Meta-Monadology' of Phonetic Writing Systems

Even prior to the innovation of the Hindu-Arabic notation for numeration, literate portions of Terran humanity had encountered another 'artefactual' 'archeonic consecuum'. They had — thereby, i.e., by virtue of their literacy — already, however [un]knowingly or [un]intentionally, inculcated themselves with experiences thereof. They had done so by means of another major, and marvelous, innovationby means of another key instantiation of the generic «aufheben» structure of 'metamonad»-ization' and of 'meta-fractal-ization'.  They had done so, that is, by means of a structure 'characterize-able' as a scaled, [quanto-]qualitative [and finite] self-similarity regress.

This, earlier '''Phenomic''' emergence of an artefactual 'meta-«monad»-ology' was one of a 'crypto-[quanto-]qualitative', ontologically-pluralist, and 'quasi-non-reductionistic', character. It was not, as with the later emergence of the Hindu-Arabic numerals' '''ideogramic''' notational 'Meta-Monadology', one of a 'purely-quantitative', ontologically monolithic, reductionistic character.

This earlier "cultural" or "technological" innovation is, again while being, indeed, 'Qualo-Peanic', 'onto-generative', and '''ontologically non-reductionist''' in its structure also not one which constitutes a 'dialectic' per our definition.

True, it exhibits a 'Peanic', 'meta-fractal', 'meta-monadic', «aufheben» that is, an 'internalization' of predecessor «monads» 'archeonic consecuum'  progression-structure. It exhibits this as an expanding, scaled self-similarity structure, constituted and continually re-constituted, diachronically, as a progression, and, synchronically, persisting as an [expanding] 'meta-anatomy'.

However, its '''energizing principle''' is not a '''self-energizing principle'''.  It lacks the character of an «auto-kinesis», as did, also, the "orders of magnitude" 'meta-units' structure of the Hindu-Arabic numerals-progression, 10 100 1000 . . ..

It exhibits, instead, the character of an enduring 'allo-kinesis', caused by an "other" than itself; by a to-it-external, human-collective subject, creating/driving/energizing its creation and reproduction from without it, from outside it.  It — this system of 'ideo-artefacts', or 'psycho-artefacts' — is not the subject [agent] of its own development. Its «aufheben» progression is not a selfaufheben» self-progression.

Nevertheless, it does constitute a potential preparation — even if largely an unconscious and unintentional one — a potential '''psycho-historical''' self-pre-conditioning of its users for the capability to perceive such; to perceive 'dialectic'. This capability blossomed forth into explicit consciousness, especially, in Ancient times, in the most advanced reaches of the late-Platonic tradition, and, since then, in '''modern times''', in especially, those of the Hegelian and Marxian traditions.

That earlier innovation to which we alluded, just above, is noneother than that of the phonetic lexicological principle of written prose language — of humanity's phonetic, '''phonogramic''', alphabetic writing systems. Let us, then, take phonetic characters, phonograms, the '''letters''' themselves of a given alphabet — e.g., the {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} of the present English text — as the «arché»-«monads» of the «arché»-«arithmos» of phonetic writing.

A typical, phonetically-rendered written '"word" is an ordered, organized '''internalization''' of a multiplicity/sub-population/sub-«arithmos» of the full «arithmos» of the "letters" — of the '''phono-grams''', or sound-of-speech-valued characters/symbols — of the alphabet in question.  Per this principle, each typical "word" is a 'self-«aufheben»' of "letters", is a 'meta-symbol', 'meta-character', or 'meta-letter' — a 'meta1monad»', or 'meta1-unit', with respect to phonetic symbols, or "letters", as 'meta0monads»/-units'. Each typical "word" is a 'meta-[phono-]gram', made up out of a heterogeneous multiplicity of '''[phono-]grams''', by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of '''phono-grams'''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "letters" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "words" as its units/«monads».

Next, each typical "sentence" of phonetic writing is an ordered, organized '''internalization''' / 'uni[t-]ification'  of a multitude/sub-population/sub-«arithmos» of "words", forming the 'new interior'; the corresponding, combined new 'internity ' and new 'externity', of a "sentence" as a 'new unit' of writing, and a new "'scale"'/'''level''' of text, beyond/above the "word"-unit level/scale [and also, even more so, beyond/above the "letter"-unit level/scale].  Each typical "sentence" is a 'self-«aufheben»' of "words", is a 'meta1monad»', or 'meta1-unit', with respect to "words" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "letters" taken as 'meta0monads»/-units'. That is, each typical "sentence" is a 'meta-word', made up out of a heterogeneous multiplicity of "words", by means of the '''internalization''' — or '''interiorization''' forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''words''.  These '''internalizations''' of the former,'''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "words" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "sentences" as its units/«monads».

Next, each typical "paragraph", or "passage", of phonetic writing is an ordered, organized '''internalization''' / 'uni[t-]ification' of a multitude/sub-population/sub-«arithmos» of "sentences", forming the 'new interior'; the corresponding, combined new 'internity' and new 'externity', of a "paragraph" or "passage" as a 'new unit' of writing, and a new "'scale"'/'''level''' of text, beyond/above the "sentence"-unit level/scale [and also, even more so, beyond/above the "word"-unit level/scale, and the "letter"-unit level/scale].  Each typical "paragraph" is a 'self-«aufheben»' of "sentences", is a 'meta1monad»', or 'meta1-unit', with respect to "sentences" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "words" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "letters" taken as 'meta0monads»/-units'. That is, each typical "paragraph", or discretized "passage", of text is a 'meta-sentence', made up out of a heterogeneous multiplicity of "sentences", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''sentences''.  These '''internalizations''' of the former,'''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "sentences" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "paragraphs" as its units/«monads».

The next successor — in this 'Qualo-Peanic', 'meta-fractal', 'meta-monadic', «aufheben», 'archeonic consecuum' of «arithmoi» — each typical "chapter" of phonetic writing, is an ordered, organized '''internalization''' / 'uni[t-]ification' of a multitude/sub-population/sub-«arithmos» of "paragraphs", forming the 'new interior'; the corresponding, combined new 'internity' and new 'externity', of a "chapter" as a 'new unit' of writing, and as a new "'scale"'/'''level''' of text, beyond/above the "paragraph"-unit level [and also, even more so, beyond/above the "sentence"-unit level, the "word"-unit level, and the "letter"-unit level]. Each typical "chapter" is a 'self-«aufheben»' of "paragraphs", is a 'meta1monad»', or 'meta1-unit', with respect to "paragraphs" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "sentences" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "words" taken as 'meta0monads»/-units',and is a 'meta4monad»' with respect to "letters" taken as 'meta0monads»/-units'.  That is, each typical "chapter" is a 'meta-paragraph', or 'meta-passage', made up out of a heterogeneous multiplicity of "paragraphs", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''paragraphs''.  These '''internalizations''' of the former,'''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "paragraphs" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "chapters" as its units/«monads».

Consecutively next — in this 'consecuum' of «arithmoi» — each typical "book" of phonetic writing, is an ordered, organized '''internalization''' / 'uni[t-]ification' of a multitude/sub-population/sub-«arithmos» of the «arithmos» of "chapters", forming the 'new interior'; the corresponding, combined new 'internity' and new 'externity', of a "book" as a 'new unit' of writing, and as a new "'scale"'/'''level''' of text, beyond/above the "chapter"-unit level [and also, even more so, beyond/above the "paragraph"-unit level, the individual "sentence"-unit level, the individual "word"-unit level, and the individual "letter"-unit level]. Each typical "book" is a 'self-«aufheben»' of "chapters", is a 'meta1monad»', or 'meta1-unit', with respect to "chapters" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "paragraphs" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "sentences" taken as 'meta0monads»/-units', and is a 'meta4monad»' with respect to "words" taken as 'meta0monads»/-units', and is a 'meta5monad»' with respect to "letters" taken as 'meta0monads»/-units'.  That is, each typical "book" is a 'meta-chapter', made up out of a heterogeneous multiplicity of "chapters", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''chapters''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "chapters" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "books" as its units/«monads».

Progressively next, and finally — in this «aufheben» progression  of «arithmoi» — each typical "library" of phonetic writings, is an ordered, organized '''internalization''' / 'uni[t-]ification' of a multitude/sub-population/sub-«arithmos» of the «arithmos» of "books", forming the 'new interior'; the corresponding, combined new 'internity' and new 'externity', of a "library" as a 'new unit' of writing, and as a new "'scale"'/'''level''' of text, beyond/above the "book"-unit level [and also, even more so, beyond/above the individual "chapter"-unit level, the individual "paragraph"-unit level, the individual "sentence"-unit level, the individual "word"-unit level, and the individual "letter"- unit level]. Each typical "library" is a 'self-«aufheben»' of "books", is a 'meta1monad»', or 'meta1-unit', with respect to "books" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "chapters" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "paragraphs" taken as 'meta0monads»/-units', and is a 'meta4monad»' with respect to "sentences" taken as 'meta0monads»/-units', and is a 'meta5monad»' with respect to "words" taken as 'meta0monads»/-units', and is a 'meta6monad»' with respect to "letters" taken as 'meta0monads»/-units'. That is, each typical "library" is a 'meta-book', made up out of a heterogeneous multiplicity of "books", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''books''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "books" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "libraries" as its units/«monads».

The word 'dialectic' names a universal, and, in general, 'meta-fractal', and 'meta-monadic' principle. The word 'dialectic' names a «monads»-as-subjects [or agents] selfaufheben» principle, of «autokinesis», one known, e.g., at least in the late-Platonic tradition, since ancient times, B.C.E.

aufheben_phonetic_systems

Example 3: The 'Meta-Monadology' of 'Computerware'

The standard '''architecture''' of today's 'computerware' presents an «aufheben» 'Meta-Monadology' which is, in some ways, a hybrid of elements of Example 1 and of elements of Example 2.

The «arché»-«monad» of the «arché»-«arithmos» of today's 'computerware' is the Boolean/binary "bit".

A typical, phonetically-rendered written "computer byte" is an ordered, organized '''internalization''' of the two-valued, binary/Boolean multiplicity/sub-population/sub-«arithmos» of "bits", valued either "0" or "1", "OFF", or "ON", of the computer hardware/software unit in question. Per this principle, each typical "byte" is a 'meta1monad»', or 'meta1-unit', with respect to "bits", as 'meta0monads»/-units'.  Each typical "byte" is a 'self-«aufheben»' of "bits", a  'meta-bit', made up out of a value-and-position-heterogeneous multiplicity of "bits", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of "bits".  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "bits" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "bytes" as its units/«monads».

Next, a typical "computer word" is an ordered, organized '''internalization''' of a multitude/sub-population/sub-«arithmos» of "computer bytes", forming the '''new interior'''; the corresponding, combined new 'internity' and new 'externity', of a "computer word" as a 'new unit' of 'computerware', and a new "'scale"'/'''level''' of EDP entity, beyond/above the "byte"-unit level/scale [and also, even more so, beyond/above the "bit"-unit level/scale].  Each typical "computer word" is a 'self-«aufheben»' of "bytes", a 'meta1monad»', or 'meta1-unit', with respect to "computer bytes" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "computer bits" taken as 'meta0monads»/-units'. That is, each typical "computer word" is a 'meta-byte', made up out of a heterogeneous multiplicity of "bytes", by means of the 'internalization' — or 'interiorization' forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''bytes''.  These 'internalizations' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "computer bytes" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "computer words" as its units/«monads».

The next successor — in this 'Qualo-Peanic', 'meta-fractal', 'meta-monadic', «aufheben», 'archeonic consecuum' of «arithmoi» — a typical "computer command" of 'computerware', is an ordered, organized '''internalization''' of a multitude/sub-population/sub-«arithmos» of "computer words" — with at least one "computer word" as '''operator''', and another "computer word" as '''operand''' forming the '''new interior'''; the corresponding, combined new 'internity' and new 'externity', of a "computer command" as a 'new unit' of 'computerware', and as a new "'scale"'/'''level''' of EDP entity, beyond/above the "computer word"-unit level [and also, even more so, beyond/above the "computer byte"-unit level, and the "computer bit"-unit level].  Each typical "computer command" is a 'self-«aufheben»' of "computer words", is a 'meta1monad»', or 'meta1-unit', with respect to "computer words" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "computer bytes" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "computer bits" taken as 'meta0monads»/-units'.  That is, each typical "computer command" is a 'meta-computer word', made up out of a heterogeneous multiplicity of [at least two] "computer words", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''computer words''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "computer words" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "computer commands" as its units/«monads».

Consecutively next — in this 'consecuum' of «arithmoi» — a typical "computer program" of "computer software", is an ordered, organized '''internalization''' of a multitude/sub-population/sub-«arithmos» of the «arithmos» of "computer commands", forming the '''new interior'''; the corresponding, combined new 'internity' and new 'externity', of a "computer program" as a 'new unit' of 'computerware', and as a new "'scale"'/'''level''' of EDP entity, beyond/above the "computer command"-unit level [and also, even more so, beyond/above the individual "computer word"-unit level, the individual "computer byte"-unit level, and the individual "computer bit"-unit level].  Each typical "computer program" is a 'self-«aufheben»' of "computer commands", is a 'meta1monad»', or 'meta1-unit', with respect to "computer commands" taken as 'meta1monads»/-units', and is a 'meta2monad»' with respect to "computer words" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "computer bytes" taken as 'meta0monads»/-units', and is a 'meta4monad»' with respect to "computer bits" taken as 'meta0monads»/-units'.  That is, each typical "computer program" is a 'meta-command', made up out of a heterogeneous multiplicity of "commands", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''computer commands''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "computer commands" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "computer programs" as its units/«monads».

Progressively next — in this «aufheben» progression  of «arithmoi» — a typical "software system" of "computer software", is an ordered, organized '''internalization''' of a multitude/sub-population/sub-«arithmos» of the «arithmos» of "computer programs", forming the '''new interior'''; the corresponding, combined new 'internity' and new 'externity', of a "software system" as a 'new unit' of 'computerware', and as a new "'scale"'/'''level''' of EDP entity, beyond/above the individual "computer program"-unit level [and also, even more so, beyond/above the individual "computer command"-unit level, the individual "computer word"-unit level, the individual "computer byte"-unit level, and the individual "computer bit"- unit level].  Each typical "software system" is a 'self-«aufheben»' of "computer programs", is a 'meta1monad»', or 'meta1-unit', with respect to "computer programs" taken as 'meta0monads»/-units', and is a 'meta2monad»' with respect to "computer commands" taken as 'meta0monads»/-units', and is a 'meta3monad»' with respect to "computer words" taken as 'meta0monads»/-units', and is a 'meta4monad»' with respect to "computer bytes" taken as 'meta0monads»/-units', and is a 'meta5monad»' with respect to "computer bits" taken as 'meta0monads»/-units'. That is, each typical "software system" is a 'meta-program', made up out of a heterogeneous multiplicity of "programs", by means of the '''internalization''' — or '''interiorization''', forming a '''new [unit's] exterior''' — of a 'sub-«arithmos»' of the «arithmos» of ''computer programs''.  These '''internalizations''' of the former, '''old''', predecessor «monads» of the former, '''old''', predecessor «arithmos» — of "computer programs" — give rise to a new «arithmos», a new assemblage, of new «monads»/units:  the successor «arithmos» that has "software systems" as its units/«monads».

aufheben_computerware

Example 4: The 'Meta-Monadology' of the Platonic Dialectic

Plato, in his "Socratic" dialogues, often formulated '''dialectic''' in terms of, e.g., the «genos» as 'meta1-taxon' of a philosophical taxonomy of all ideas — in our terms, i.e., of a systematic/philosophical-'''taxonomic''' 'ideo-ontological' category/concept — of each «genos» as «eide»-unit, «eide»-«monad», or as «ιδεα»-«μονας»['idea-monad']/«ιδεα»-unit of his dialecticalarithmoi eide-tikoi»'.  Plato posited his «genos» 'idea-units' as 'meta1monads»', or 'meta1-units', to his «species» 'taxons'/'taxa'; to his «species» 'idea-«monads»'/'idea-units', taken as 'meta0-taxons' or 'meta0-taxa' .  He posited the «arithmoi» of «genos» 'idea-units' as 'metaarithmoi»' to the «arithmoi» of «species» 'idea-units', with each «species» 'idea-unit' conceived as a 'meta0-unit' or 'meta0monad»' in relation to the «genos» to which that «species», by its nature ['nature-ally'] belonged, i.e., due to its qualitative/conceptual content [please now 're-see' the extended quote from Jacob Klein's book at outset of this letter].

Per this categorial construct, each typical/nonsui generis» 'idea-«genos»' was viewed as being a 'meta-«species»', [synchronically and eternally] «aufheben» made up out of — but also as 'ideo-ontologically' irreducible to — a heterogeneous multiplicity, or «arithmos» ['«arithmos eide-tikos»'], of qualitatively differing, ontologically-distinct 'idea-«species»', in an 'idea-universe' structured as an [at least] two-level 'meta-fractal' of '«arithmoi eide-tikoi»' of 'idea-«species»' undergirding '«arithmoi eide-tikoi»' of 'idea-«gene»'.

Each typical 'idea-«genos»' in this total structure of knowledge/philosophy — this universal structure/taxonomy of ideas — stands, per Plato, relative to its constituent 'idea-«species»', as an «aufheben» '''conservation''' / '''containment''' / 'internalization', and, also, as an «aufheben» '''negation''' / '''elevation''', of the multiple 'idea-«species»' belonging to that 'idea-«genos»'; of the multitude of 'idea-«species»' implicitly '''inside''' that 'idea-«genos»' — i.e., its «arithmos» of '«eide»-«species»' [Plato, of course, did not employ the later, German, word, "«aufheben»", in his descriptions of dialectics, but the concept so-named today is present in his written descriptions, at least in a synchronic, 'eternalized' form.].

aufheben_idea_genos

For the modern mind, and for its modern arithmetic [which, unlike the ancient arithmetic, has expanded 'ideo-ontologically' to include, e.g., 0 and 1 as numbers], an analogous, further such «aufheben» relationship — indeed, a 'meta-fractal' relationship, in our terms — exists between each related, higher 'idea-multitude' — or '«arithmos eide-tikos»' — of 'idea-«gene»', as 'idea-«monads»', and the 'supergenos»' higher-unit, constituted out of them as their 'meta1genos»' relative to the '«gene»' level taken as the 'meta0monads»/-units', 'base/«arché» level', or as their 'meta2-taxon'/'meta2-unit', relative to the «species»-level taken as the 'meta0monads»/-units base/«arché» level'.  ... and so on, in finite regress, up into as many higher levels/'meta-fractal scales' of 'ideo-taxonomy' as are requisite to the classification of the extant universe of philosophical ideas.

aufheben_super_genos

Likewise, an analogous, further such «aufheben» relationship — a 'meta-fractal' relationship — exists between each related, lower 'idea-multitude' — or '«arithmos eide-tikos»' — of 'idea-subspecies»', as 'idea-monads', and their 'super-subspecies»' higher-unit — or, simply, their «species» higher-unit — constituted out of them as their 'meta1-subspecies»' relative to the 'subspecies»' level taken as 'meta0monads»/-units', 'base/«arché» level', or as their 'meta¯1-taxa'/'meta¯1-units', relative to the «species»-level taken as the 'meta0monads»/-units', 'base/«arché» level'. ... and so on, in finite regress, down into as many lower, deeper,  negative-exponent levels/'meta-fractal  scales' of 'ideo-taxonomy' as are requisite to the classification of the extant universe of philosophical ideas.

aufheben_species

This 'meta-fractal', 'synchronic-«aufheben»', '''systematic''', '''encyclopedic''' structure of philosophical ideas, although, and, in part — and in a way that perhaps appears paradoxical — precisely because its '''idea-objects''' or 'idea-units', are 'a-tomistic' — '''un-cuttable''' — is '''hol-istic''', rather than '''reductionistic'''.

It is so because any given, typical, «genos»'s level, per Plato, is not reducible to its constituent «species'» level, to the level of its «species'» «arithmos eidetikos», even though that «genos» [«aufheben»-]'''contains'''/'internalizes', and is constituted by, that «arithmos». «Genos» and «species» were seen, by Plato, as being ever '''co-present''', '''co-eval''', and 'co-presenting' to the '''eye''' of the human mind; as ever concurrently existing, ever-contemporaneous, but also as qualitatively distinct levels of 'idea-ontology'; of 'idea-being'.

Example 5: The Historical-Dialectical 'Meta-Monadology' of Human-Social Formation(s)

The next example of an «aufheben» progression which exhibits the 'Qualo-Peanic', 'meta-fractal', 'metamonad»-ic' 'archeonic consecuum' process/structure which we have been discussing, does qualify as a 'dialectic'  per our definition. It constitutes a 'dialectical model'  which is also [an aspect of ] the story of Terran humanity itself, to-date.

 This additional example is one of a 'physical dialectic', or '«physis» dialectic'  [although the Ancient mind might have called it '''the [his]story of «anti-physis»'''] — indeed, an aspect of 'The Dialectic of Nature'  within its 'human-social' epoch. It is the 'dialectic of human-social formation(s)', in terms of the «monads» and the «arithmoi» of human settlement/governance structures. We view these successive '''social formations''' [Marx], as 'archaeological/meta-geomorphological sedimentary layerings', and as 'meta-geological formations' of the Earth's surface — 'human-natur[e-]al', 'megalithic meta-encrustations of the Earth's crust'.

The 'meta-dynamics' of the 'meta-evolution' of these [this] '[meta-]dynamical [meta-]system[s]' of such human-social formation(s) constitutes an «autokinesis», and an «auto-onto-dynamasis» at the level of 'human-social ontology',  Their systems-progression, or 'diachronic meta-system', is a 'selfaufheben» self-progression' of 'Qualo-Peanic', 'meta-fractal', 'metamonad»-ic', 'archeonic consecuum' process/structure, when we grasp each of its successive «arithmoi» of human social formation «monads» as a 'collective human subject[-ivity], or 'agent[-ivity]'.

This systems-self-progression is therefore one that qualifies as an 'historical-dialectical process' per our definition. The reader is referred to this Introductory Letter's Supplement B (Part III, page B-23) for an N-ideographical rendition of the 'dialectical model' rendered narratively below [linked at: [Supplement B Part III - including a Psycho-Historical Model of The Dialectic of Human Nature (.pdf)].

A. "Bands".  Let us define the «arché»-«arithmos» of human settlement formations to be that of the non-settlement-pattern '''population''' of small, mobile, '''nomadic''' "bands" of [proto-]human predators/foragers/scavengers/hunter-gatherers.

Note:  By '''population''', in this context, we do not mean the count of [proto-]human "biological individuals", whether of the typical or average "band", or of the totality of all "bands" extant as of a particular value of some time parameter. The [minimally 'memetically-emerged', phenomically 'proto-ic', proto-]human individual is not the unit, or «monad», of counting for this 'dialectical model' narrative. The "band" itself, whatever its size in terms of [proto-]human individuals, is that unit, or «monad». This model thus eschews the usual '''human individual-ism / atomism / reductionism'''.

Suppose that the '''population''' of the "bands" «arithmos» — the '''population''' of which each individual "band" is a «monad»/unit — reproduces itself with expansion, grows in certain localities of the planetary biosphere.  Then, as the 'monadic population' of the "bands"-as-«monads» 'densifies' itself in those localities, a condition of '''critically''' high "bands" density may arise, which we term the 'self-surroundment' of the "band" «monad», the 'self-environment' of the "bands", or 'surroundment-/environment-by-likes', created, for the "bands", by the "bands". This condition would arise, first and especially, within the 'centerward' sub-population of "band" «monads» of each of the key/core such localities, or 'meristemal'/'''vanguard''' social-relations-innovation '''nucleation zones'''.  This means that there has arisen a condition of "bands" densely surrounded by [other] "bands" at the heart of each such locality. This condition would have thereby supplanted, in intensity/'intensivity', within those key/core loci, the 'precedingly-dominant' condition of the 'surroundment' of the "bands" «monads» only by the accumulated 'monadic populations' of various scales/levels/layers of pre-human-natur[e]-al ontology, especially of the immediate ontological predecessor of the 'taxonomy level one' "human societies" «arithmos», in the form of the «arithmos» of 'multi-meta-zoan' "animal societies", and of 'multi-meta-phytan' plant communities.

A new innovation in the human-social settlement/governance patterns' taxonomy of 'socio-ontology' is thereby seeded.

The former condition was dominated by and characterized by 'merely-hybridizing' reactions/inter-actions, 'ontological conversion' 'hetero-/inter-actions', of "band" «monads» with the accumulated 'monadic populations' of various scales/levels/layers of 'pre-human-natur[e]-al' ontology. The new condition — in the 'ontological innovation nucleation zones' — is dominated by and characterized by 'self-hybridizing' interactions, 'self-interactions', or 'intra-actions', of "band" «monads» with [other] "band" «monads», which become more and more frequent/increasingly 'self-frequentized', as the '''population density''' of "band" «monads» grows therein.

The formerly-dominant modes of monadic interaction — of 'ontological other-conversion', or 'hetero-conversion' — had partially converted pre-human-natur[e]-al biomass into [proto-]human 'socio-mass', in the form of this 'monadic population' of the "bands" «arithmos».  This process of '''bio-mass''' to 'socio-mass' ontological conversion was 'self-catalyzed' by, or 'auto-catalyzed' by, and 'self-celerated', in proportion to the presence of, and to the density of/'physical-spatial concentration' of, the therefore '''[self-]expanding''' "bands" «arithmos».

But as the — therefore growing — 'physical-spatial concentration' of the «monads» of the "bands" «arithmos», in the key/core '''nucleation zones''' of human-social formation, crosses a "critical mass"/'''critical density''' threshold, the process of 'ontological hetero-conversion', of past monadic sub-populations, into the growing "bands" monadic population, shifts. It shifts into a new and previously unprecedented process, of the nascent 'ontological self-conversion' of [part of] the burgeoning "bands" «arithmos» 'socio-ontology', by that burgeoning "bands" «arithmos» 'socio-ontology'; its 'self-conversion' into the 'socio-ontology' of a new, 'self-involutively higher', previously unexampled '''onto-logical type''', a new increment of 'socio-ontological' innovation in the history of human-social formation(s).

That is, the 'self-frequentization' of this new mode of action — of '''self-inter-action''', or '''intra-action''' — of "bands" with "bands", then, as it exceeds its critical frequency/density threshold, precipitates the irruption of yet a new, previously non-extant, previously non-existent 'meta-fractal' scale/level/layer of human settlement/governance patterns and practices, namely, that of the multi-"band" — episodically semi-sedentary — "camp" human-social formation(s).

A "camp", grasped as a human-social unit/«monad», is a '
meta1monad»', 'meta1-unit', or 'super1-unit', relative to a "band", grasped also as such a human-social unit/«monad».

Each typical "camp" is a meta-"band", made up out of a [local-][sub-]«arithmos» of "bands", i.e., made up out of a heterogeneous multiplicity of "bands", by means of a 'selfaufheben» self-internalization' of that local, predecessor «arithmos» of "bands" as predecessor «monads».

This 'selfaufheben»' self-operation — of an «arithmos» of "band" «monads», as collective human-social '''subject'''/agent of [self-]action, acting/operating upon /within itself, via the "band" «monads» operating among themselves — gives rise to an ontologically, qualitatively, behaviorally new and different, previously unprecedented «arithmos», one that has "camps" as its «monads»:  the «arithmos» of the — initially multi-"band" — "camps".

'Ideographized' / 'ideogramized' "shorthand" summary [in the following formula, b denotes the ontological category of the "bands" «arithmos» and c denotes the ontological category of the "camps" «arithmos»]:

b  →  b[ b ]   b '''of'''  b  =  b2   b + Δb   b + c .

[Link to supplementary information:  If you would like more information about the rules of 'qualitative calculation' that are used in the "shorthand" expression above i.e., about the rules of 'ontological multiplication', 'multiplication of qualities' ['multiplication of ontological qualifiers'], 'categorial multiplication', or '«aufheben» multiplication' then click on the following link:  http://www.dialectics.org/archives/pdf/Fract1-1.pdf, and scroll down to page 4]

aufheben_arche_camp

B. "Camps".  Suppose, as the next, consecutive emergence in this 'Qualo-Peanic' 'selfaufheben» succession' / 'consecuum-cumulum' of human-social emergences, that the '''population''' of the "camps" «arithmos» — the '''population''' of which each individual "camp" is a «monad»/unit — reproduces itself with expansion, grows in certain localities of the planetary biosphere.  Then, as the 'monadic population' of the "camps"-as-«monads» 'densifies' itself in those localities, a condition of '''critically''' high "camp" density may arise, which we term the 'self-surroundment' of the "camp" «monads», the 'self-environment' of the "camps", or the 'surroundment-/'environment-by-likes', created, for the "camps", by  the "camps". This condition would arise, first and especially, within the 'centerward' sub-population of "camp" «monads», within each of the key/core such localities, or 'meristemal'/'''vanguard''' social-relations-innovation '''nucleation zones'''.  This means that there has arisen a condition of "camps" densely surrounded by [other] "camps" at the heart of each such locality.

This condition would have thereby supplanted, in intensity/'intensivity', within these key/core loci, the 'precedingly-dominant' condition of the 'surroundment' of the "camp" «monads» by their immediate-predecessor, 'inverse-consecutive' «monads», of the «arithmos» of "bands".

A new innovation in the human-social settlement/governance patterns' taxonomy of 'socio-ontology' is thereby seeded.

The former condition was dominated by, and characterized by, 'merely-hybridizing' reactions/inter-actions, by 'ontological conversion' 'hetero-/inter-actions', of "camp" «monads» with immediate predecessor, "band" «monads».

The new condition — in the 'ontological innovation nucleation zones' — is dominated by, and characterized by, 'self-hybridizing' interactions, 'self-interactions', or 'intra-actions', of "camp" «monads» with[in]/upon [other] "camp" «monads», such 'self-actions' or 'self-operations' becoming more and more frequent/increasingly 'self-frequentized', as the '''population density''' of "camp" «monads» grows therein.

The formerly-dominant modes of monadic interaction — of 'ontological other-conversion', or 'hetero-conversion' — had partially converted the still-extant "band" 'socio-ontology'/'socio-mass' into "camp" 'socio-ontology'/'socio-mass'. This process of 'ontological hetero-conversion' of [part of] the remaining «monads» of the precedingly-self-manifested «arithmoi» — of the "band" «arithmos» — is 'auto-catalyzed' by, and 'celerates' itself, in proportion to the presence of, and to the density of/'physical-spatial concentration' of, the therefore '''[self-]expanding''' "camps" «arithmos».

But as the — therefore growing — 'physical-spatial concentration' of the «monads» of the "camps" «arithmos», in the key/core '''nucleation zones''', crosses a "critical mass"/'''critical density''' threshold, the process of 'ontological hetero-conversion', of past monadic sub-populations into the growing "camps" monadic population, shifts.

It shifts into a new and previously unprecedented process. This new process is that of the nascent 'ontological self-conversion' of [part of] the burgeoning "camps" «arithmos» 'socio-ontology', by that burgeoning "camps" «arithmos» 'socio-ontology', into something else:  its 'self-conversion' into the 'socio-ontology' of a new, 'self-involutively higher', previously unexampled '''onto-logical type''', a new increment of 'socio-ontological' innovation in the history of human-social formation(s).

That is, the 'self-frequentization' of this new mode of action — of '''self-inter-action''', or '''intra-action''' — of "camps" with "camps", then precipitates, as it exceeds its critical frequency/density threshold, the irruption of yet a new, previously non-extant, previously non-existent 'meta-fractal' scale/level/layer of human settlement/governance patterns and practices, namely, that of the — multi-"camp" — "village" human-social formation(s).

A "village", grasped as a human-social unit/«monad», is a 'meta1monad»', 'meta1-unit', or 'super1-unit' relative to a "camp", grasped also as such a human-social unit/«monad», and a 'meta2monad»' relative to a "band", grasped also as such a human-social unit.

Each typical "village" is a meta-"camp", made up out of a [local-][sub-]«arithmos» of "camps", i.e., made up out of a heterogeneous multiplicity of "camps", by means of a 'selfaufheben» self-internalization' of that local, predecessor «arithmos» of "camps" as predecessor «monads».

This 'selfaufheben»' self-operation — of a sub-«arithmos» of "camp" «monads», as collective human-social '''subject'''/agent of [self-]action, acting/operating upon/within itself; via its "camp" «monads» operating among themselves — gives rise to an ontologically, qualitatively, behaviorally new and different, previously unprecedented «arithmos», one that has "villages" as its «monads»:  the «arithmos» of multi-"camps", i.e., of the — initially 'many-camp' — "villages".

'Ideographized' / 'ideogramized' "shorthand" summary [in the following formula, v denotes the ontological category of the "villages" «arithmos»]:

b + c  →  [ b + c ][ [ b + c ] ]   [ b + c ]2   [ b + c ] + Δ[ b + c ]   b + c + qcb + v.

[Note:  '''Hybrid''', 'ontological conversion formation', '''real subsumption''' «arithmoi», such as that denoted by qcb, above, are not depicted in the '«aufheben» diagrams' included in this Introductory Letter. Such hybrid or cross-product terms denote partial or total dialectical synthesis formations which convert the «monads» of earlier-emerged into the «monads» of the most recently-emerged ontological categories.]

aufheben_arche_village

C. "Villages".  Suppose, as the next, consecutive emergence in this 'Qualo-Peanic' 'selfaufheben» succession'/'consecuum-cumulum' of human-social emergences, that the '''population''' of the "villages" «arithmos» — the '''population''' of which each individual "village" is a «monad»/unit — reproduces itself with expansion, grows in certain localities of the planetary biosphere/emergent "noosphere".

Then, as the 'monadic population' of the "villages"-as-«monads» grows and 'densifies' itself in those localities, a condition of '''critically''' high "villages" density may arise, which we term the 'self-surroundment' of the "village" «monad», the 'self-environment' of the "villages", or the 'surroundment-/environment-by-likes', created, for the "villages", by the "villages".

This condition would arise, first and especially, within the 'centerward' sub-population of "village" «monads» of each of the key/core such localities, or 'meristemal'/'''vanguard''' social-relations-innovation '''nucleation zones'''.  This means that there has arisen a condition of "villages" densely surrounded by [other] "villages" at the heart of each such locality.

This condition would have thereby supplanted, in intensity/'intensivity', within these key/core loci, the 'precedingly-dominant' condition of the 'surroundment' of the "village" «monads» by their immediate-predecessor, 'inverse-consecutive' «monads», namely, by the «monads» of the «arithmos» of "camps".

A new innovation in the human-social settlement/governance patterns' taxonomy of  'socio-ontology' is thereby seeded.

The former condition was dominated by and characterized by 'merely-hybridizing' reactions/inter-actions, by 'ontological conversion' via hetero-/inter-actions', of "village" «monads» with immediate predecessor, "camp" «monads», and with [any] still-persisting earlier-predecessor «monads» — predecessor «monads» as yet unassimilated to any higher '''degree''' of ontological 'self-involution' / 'self-internalization' / 'self-complexification' — i.e., with "band" «monads».

The new condition — in the 'ontological innovation nucleation zones' — is dominated by and characterized by 'self-hybridizing' interactions, 'self-interactions', or 'intra-actions', of "village" «monads» with [other] "village" «monads», which become more and more frequent/increasingly 'self-frequentized', as the '''population density''' of "village" «monads» grows therein.

The formerly-dominant modes of monadic interaction — of 'ontological other-conversion', or 'hetero-conversion' — had partially converted the still-extant "camp" 'socio-ontology'/'socio-mass' into "village" 'socio-ontology'/'socio-mass', as well as converting part of any still-extant "band" 'socio-ontology'/'socio-mass' into "village" 'socio-ontology'/'socio-mass'.

This process of 'ontological hetero-conversion' of [part of] the remaining «monads» of the precedingly-self-manifested «arithmoi» — of the "camps" «arithmos», and of the "bands" «arithmos» — is 'auto-catalyzed' by, and 'celerates' itself, in proportion to the presence of, and to the density of/'physical-spatial concentration' of, the therefore ['''self-]expanding''' "village" «arithmos».

However, as the — therefore and thereby growing — 'physical-spatial concentration' of the «monads» of the "villages" «arithmos», in the key/core '''nucleation zones''', crosses a "critical mass"/'''critical density''' threshold, the process of the 'ontological hetero-conversion/assimilation, of earlier-manifested monadic sub-populations into the growing "villages" monadic population, shifts.

It shifts into a new and previously unprecedented process, of the nascent 'ontological self-conversion' of [part of] the burgeoning "villages" «arithmos» 'socio-ontology', by that very burgeoning "villages" «arithmos» 'socio-ontology', into yet a different 'socio-ontology': its 'self-conversion' into the 'socio-ontology' of a new, 'self-involutively higher', previously unexampled '''onto-logical type''', a new increment of 'socio-ontological' innovation in the history of human-social formation(s).

That is, the 'self-frequentization' of this new mode of action — of '''self-inter-action''', or '''intra-action''' — of "villages" with "villages", then, as it exceeds its critical frequency/density threshold, precipitates the irruption of yet a new, previously non-extant, previously non-existent 'meta-fractal' scale/level/layer of human settlement/governance patterns and practices, namely, that of the — multi-"village" — "chiefdom", or "tribal", human-social formation(s).

A "chiefdom", grasped as a human-social unit/«monad», is a 'meta1monad»', 'meta1-unit', or 'super1-unit' relative to a "village", grasped also as such a human-social unit/«monad»; a 'meta2monad»' relative to a "camp", grasped also as such a human-social unit; a 'meta3monad»' relative to a "band", grasped also as such a human-social unit.

Each typical "chiefdom" is a meta-"village", made up out of a [local-][sub-]«arithmos» of "villages", i.e., made up out of a heterogeneous multiplicity of "village" «monads», by means of their coalescence into a new 'internity/externity', i.e., by means of a 'selfaufheben» 'self-internalization' of that local, predecessor «arithmos» of "villages" as predecessor «monads».

This 'selfaufheben»' self-operation — of an «arithmos» of "village" «monads», as collective/holistic human-social '''subject'''/agent of [self-]action, acting/operating upon/within itself, via its "village" «monads» operating among themselves — gives rise to an ontologically, qualitatively, behaviorally new and different, previously unprecedented «arithmos», one that has "chiefdoms" as its «monads»:  the «arithmos» of — multi-"village", 'meta-village' — "chiefdoms".

'Ideographized' / 'ideogramized' "shorthand" summary [in the following formula,  f denotes the ontological category of the "chiefdoms" «arithmos»]:

b + c + qcb + v 
[
b + c + qcb + v ][ [ b + c + qcb + v ] ]   =
[
b + c + qcb + v ]2   =
[
b + c + qcb + v ] + Δ[ b + c + qcb + v=
b + c + qcb + v + qvb + qvc + qvcb +  f .

aufheben_arche_chiefdom

D. "Chiefdoms".  Suppose, consecutively next in this 'selfaufheben» self-progression'; in this '[Qualo-]Peanic' succession of human 'socio-onto-dynamasis', that the '''population''' of the "chiefdom" «arithmos», or "tribes" «arithmos» — the '''population''' of which each individual "chiefdom" is a «monad»/unit — reproduces itself with expansion, grows in certain localities of the planetary biosphere/emergent "noosphere".

Then, as the 'monadic population' of the "chiefdoms"-as-«monads» 'densifies' itself in those localities, a condition of '''critically''' high "chiefdom" density may arise, which we term the 'self-surroundment' of the "chiefdom" «monad», the 'self-environment' of the "chiefdoms", or the 'surroundment-/environment-by-likes', created, for the "chiefdoms", by the "chiefdoms".

This condition would arise, first and especially, within the 'centerward' sub-population of "chiefdom" «monads» of each of the key/core such localities, or 'meristemal'/'''vanguard''' social-relations-innovation '''nucleation zones'''.  This means that there has arisen a condition of "chiefdoms" densely surrounded by [other] "chiefdoms" at the heart of each such locality.

This condition would have thereby supplanted, in intensity/'intensivity', within these key/core loci, the 'precedingly-dominant' condition of the 'surroundment' of the "chiefdom" «monads» by their immediate-predecessor, 'inverse-consecutive' «monads», the "village" «monads» of the «arithmos» of "villages".

A new innovation in the human-social settlement/governance patterns' taxonomy of 'socio-ontology' is thereby seeded.

The former condition was dominated by and characterized by 'merely-hybridizing' reactions/inter-actions, 'ontological conversion' 'hetero-/inter-actions', of "chiefdom" «monads» with their immediate predecessor «monads», and with [any] still-persisting earlier-predecessor «monads» — predecessor «monads» as yet unassimilated to any higher '''degree''' of ontological 'self-involution' / 'self-internalization' / 'self-complexification' — i.e., '''inter-actions''' with "village" «monads», with "camp" «monads», and with "band" «monads».

The new condition — in the 'ontological innovation nucleation zones' — is dominated by, and characterized by, 'self-hybridizing' interactions, 'self-interactions', or 'intra-actions', of "chiefdom" «monads» with [other] "chiefdom" «monads», which become more and more frequent/increasingly 'self-frequentized', as the '''population density''' of "chiefdom" «monads» grows therein.

The formerly-dominant modes of monadic interaction — of 'ontological other-conversion', or 'hetero-conversion' — had partially converted still-extant "villages" 'socio-ontology'/'socio-mass' into "chiefdom" 'socio-ontology'/'socio-mass'; [part of] still-extant "camps" 'socio-ontology'/'socio-mass' into "chiefdom" 'socio-ontology'/'socio-mass', and [part of] [any] still-extant "bands" 'socio-ontology'/'socio-mass' into "chiefdom" 'socio-ontology'/'socio-mass'.

This process of 'ontological hetero-conversion', or subordination/assimilation, of [part of] the remaining «monads» of the precedingly-self-manifested «arithmoi» — of the "villages" «arithmos», of the "camps" «arithmos», and of the "bands" «arithmos» — is 'auto-catalyzed' by, and 'celerates' itself, in proportion to the presence of, and to the density of/'physical-spatial concentration' of, the therefore '''[self-]expanding''' "chiefdoms" «arithmos».

But as the — therefore and thereby growing — 'physical-spatial concentration' of the «monads» of the "chiefdoms" «arithmos», in the key/core '''nucleation zones''', crosses its own "critical mass"/'''critical density''' threshold, the process of 'ontological hetero-conversion', of past monadic sub-populations into the growing "chiefdoms" monadic population, shifts.

It shifts into a new and previously unprecedented process, of the nascent 'ontological self-conversion' of [part of] that burgeoning "chiefdoms" «arithmos» 'socio-ontology', by that burgeoning "chiefdoms" «arithmos» 'socio-ontology', into another 'socio-ontology': its 'self-conversion' into the 'socio-mass' and 'socio-ontology' of a new, 'self-involutively higher', previously unexampled '''onto-logical type''', a new increment of 'socio-ontological' innovation in the history of human-social formation(s).

That is, the 'self-frequentization' of this new mode of action — of '''self-inter-action''', or '''intra-action''' — of "chiefdoms" with "chiefdoms", then precipitates, as it exceeds its critical frequency/density threshold, the irruption of yet a new, previously non-extant, previously non-existent 'meta-fractal' scale/level/layer of human settlement/governance patterns and practices, namely, that of the — initially multi-"chiefdom" — "city-state" human-social formation(s).

A "city-state", grasped as a human-social unit/«monad», is a 'meta1monad»', 'meta1-unit', or 'super1-unit' relative to a "chiefdom", grasped also as such a human-social unit/«monad»; is a 'meta2monad»' relative to a "village", grasped also as such a human-social unit; is a 'meta3monad»' relative to a "camp", grasped also as such a human-social unit, and is a 'meta4monad»' relative to a "band", grasped also as such a human-social unit.

Each typical "city-state" is a meta-"chiefdom", or 'meta'-"tribe" — often founded as the result of a local alliance of "tribes"; an alliance of formerly disparate and formerly mutually-warring local "tribes" against more distant "tribes" — i.e., is made up out of a [local-][sub-]«arithmos» of "chiefdoms" or "tribes", i.e., made up out of a heterogeneous multiplicity of "chiefdoms", by means of a 'selfaufheben» 'self-internalization' of that local, predecessor «arithmos» of "chiefdoms" as predecessor «monads».

This 'selfaufheben»' self-operation — of an «arithmos» of "chiefdom" «monads», as collective/holistic human-social '''subject'''/agent of [self-]action, acting/operating upon/within itself, via its "chiefdom" «monads» operating among themselves — gives rise to an ontologically, qualitatively, behaviorally new and different, previously unprecedented «arithmos», one that has "city-states" as its «monads»:  the «arithmos» of — [initially] multi-"tribe", multi-"chiefdom" — "city-states".

'Ideographized' / 'ideogramized' "shorthand" summary [in the following formula,  s denotes the ontological category of the "city-states" «arithmos»]:

b + c + qcb + v + qvb + qvc + qvcb + f 
[
b + c + qcb + v + qvb + qvc + qvcb + f ]2  =
b
+ c + qcb + v + qvb + qvc + qvcb +  f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s.

aufheben_arche_city_state

E. "City-States".  Suppose, as the next, consecutive emergence in this 'Qualo-Peanic' 'selfaufheben»' succession/'consecuum-cumulum' of human-social emergences, that the '''population''' of the "city-states" «arithmos» — the '''population''' of which each individual "city-state" is a «monad»/unit — reproduces itself with expansion, grows in certain localities of the planetary biosphere/emergent "noosphere".

Then, as the 'monadic population' of the "city-states"-as-«monads» 'densifies' itself in those localities, a condition of '''critically''' high "city-state" density may arise, which we term the 'self-surroundment' of the "city-state" «monad», the 'self-environment' of the "city-states", or the 'surroundment-/environment-by-likes', created, for the "city-states", by the "city-states".

This condition would arise, first and especially, within the 'centerward' sub-population of "city-states" «monads» of each of the key/core such localities, or 'meristemal'/'''vanguard''' social-relations-innovation '''nucleation zones'''.  This means that there has arisen a condition of "city-states" densely surrounded by [other] "city-states" at the heart of each such locality. This condition would have thereby supplanted, in intensity/'intensivity', within these key/core loci, the 'precedingly-dominant' condition of the 'surroundment' of the "city-state" «monads» by their immediate-predecessor, 'inverse-consecutive' «monads», namely, the «monads» of the «arithmos» of "chiefdoms".

A new innovation in the human-social settlement/governance patterns' taxonomy of 'socio-ontology' — of human 'socio-systematics', or 'socio-taxonomics' — is thereby seeded.

The former condition was dominated by, and characterized by, 'merely-hybridizing' reactions/inter-actions, 'ontological conversion' 'hetero-/inter-actions', of "city-state" «monads» with their immediate predecessor «monads», and with [any] still-persisting earlier-predecessor «monads» — predecessor «monads» as yet unassimilated to any higher '''degree''' of ontological 'self-involution'/'self-internalization'/'self-complexification' — i.e., interactions with "chiefdom" «monads», with "village" «monads», with "camp" «monads», and with "band" «monads».

The new condition — in the '''ontological innovation nucleation zones''' — is dominated by and characterized by 'self-hybridizing' interactions, 'self-interactions', or 'intra-actions', of "city-state" «monads» with [other] "city-state" «monads», which become more and more frequent/increasingly 'self-frequentized', as the '''population density''' of "city-state" «monads» grows therein.

The formerly-dominant modes of monadic interaction — of 'ontological other-conversion', or 'hetero-conversion' — had partially converted, or subordinated, still-extant "chiefdoms" 'socio-ontology' / 'socio-mass' [in]to "city-states" 'socio-ontology' / 'socio-mass'; still-extant "villages" 'socio-ontology' / 'socio-mass' [in]to "city-states" 'socio-ontology' / 'socio-mass'; [any] still-extant "camps" 'socio-ontology' / 'socio-mass' [in]to "city-states" 'socio-ontology' / 'socio-mass', and [any] still-extant "bands" 'socio-ontology' / 'socio-mass' [in]to "city-states" 'socio-ontology' / 'socio-mass'.

This process of 'ontological hetero-conversion' of [part of] the remaining «monads» of the precedingly-self-manifested «arithmoi» — of the "chiefdom" «arithmos», of the "villages" «arithmos», of the "camps" «arithmos», and of the "bands" «arithmos» — is 'auto-catalyzed' by, and 'celerates' itself, in proportion to the presence of, and to the density of/'physical-spatial concentration' of, the therefore '''[self-]expanding''' "city-states" «arithmos».

However, as the — therefore, thereby growing — 'physical-spatial concentration' of the «monads» of the "city-states" «arithmos», in the key/core '''nucleation zones''', crosses its "critical mass"/'''critical density''' threshold, the process of 'ontological hetero-conversion', of past monadic sub-populations into the growing "city-states" monadic population, shifts.

It shifts into a new and previously unprecedented process, of the nascent 'ontological self-conversion' of [part of] the burgeoning "city-states" «arithmos» 'socio-ontology', by that burgeoning "city-states" «arithmos» 'socio-ontology', into something new: its 'self-conversion' into the 'socio-mass' and the 'socio-ontology' of a new, 'self-involutively higher', previously unexampled '''onto-logical type''', a new increment of 'socio-ontological' innovation in the history of human-social formation(s).

That is, the 'self-frequentization' of this new mode of action — of '''self-inter-action''', or '''intra-action''' — of "city-states" with "city-states", then precipitates, as it exceeds its critical frequency/density threshold, the irruption of yet a new, previously non-extant, previously non-existent 'meta-fractal' scale/level/layer of human settlement/governance patterns and practices, namely, that of the multi-"city-states" "empire" human-social formation(s), e.g., the Incan, Mayan, Aztec, Babylonian, Egyptian, Persian, Athenian, Carthaginian, Macedonian, and Roman "empires".

Note:  The word "empire" is used, herein, only in its earlier meaning, to describe a pre-nation-state, multi-city-state governance/settlement formation, primarily of the Ancient-historical world.

This social formation involved city-state-colonization, and tributary, etc., conquest, [partial-]enslavement, and/or other subjugation of, especially, other/rival "city-states", by a dominant/'''central''' "city-state", as seen, for example, in those "empires" that emerged, in the Mediterranean planetary 'sub-hemisphere' of planet Terra, during the period of '''classical antiquity''', such as the rapacious/parasitical "empires" centered upon the transiently-dominant/'''central''' "city-states" of Akkad, Persepolis, Athens, Carthage, Macedon, Alexandria, and Rome.

The word "empire" is not used herein to describe that qualitatively, ontologically, behaviorally, and '''categorially''' different — systematically/taxonomically different — and 'meta-fractally' higher scale/level/layer of the later-to-emerge imperialist formations of the "nation-state" epoch.

The "nation-state" epoch of '''[national] empires''' the "nation-state" epoch/scale/level/layer of multi-nation-al, or multi-"nation-state", 'nation-al/colonial imperialisms is higher in 'ontic dimensionality' — but nonetheless 'meta-fractally analogous to' — those earlier-to-emerge formations of the multi-city-state "empires" epoch.

The former, national, '''empires''', are typically centered in a single, dominant transitional/mercantile-capitalist, or industrial-capitalist, "nation-state", such as the rapacious/parasitical inter-[proto-]national imperialisms centered upon the transiently-dominant '''central''' [proto-]"nation-states" of medieval Portugal, medieval Spain, and the Dutch United Provinces, or, later, those of modern France, England, Russia, Germany, Italy, Japan, the so-called "Soviet" Union, the so-called "Peoples' " "Republic" of China, and last, but far from least, in rapacity — the North American United States.

An "empire", grasped as a human-social unit/«monad», is a 'meta1monad»', 'meta1-unit', or 'super1-unit' relative to a "city-state", grasped also as such a human-social unit/«monad»; is a 'meta2monad»' relative to a "chiefdom", grasped also as such a human-social unit; is a 'meta3monad»' relative to a "village", grasped also as such a human-social unit; is a 'meta4monad»' relative to a "camp", grasped also as such a human-social unit, and is a 'meta5monad»' relative to a "band", grasped also as such a human-social unit.

Each typical "empire" is a meta-"city-state" — often founded via the military and commercial conquest, by a single, '''central''', dominant "city-state", such as ancient Macedon, or ancient Rome, of a multitude of other "city-states" — a 'meta-"city-state" ', made up out of a [local-][sub-]«arithmos» of "city-states", i.e., made up out of a heterogeneous multiplicity of "city-state" «monads», by means of a 'selfaufheben» 'self-internalization' of that local, predecessor «arithmos» of "city-states" as predecessor «monads».

This 'selfaufheben»' self-operation — of an «arithmos» of "city-state" «monads», as collective human-social '''subject'''/agent of [self-]action, acting/operating upon/within itself, via its "city-states" «monads» operating among themselves — gives rise to an ontologically, qualitatively, behaviorally new and different, previously unprecedented «arithmos», one that has "empires" as its «monads»:  the «arithmos» of multi-"city-state" "empires".

'Ideographized' / 'ideogramized' "shorthand" summary [in the following formula,  e denotes the ontological category of the "empires" «arithmos»]:

b + c + qcb + v + qvb + qvc + qvcb + f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s 
[ b + c + qcb + v + qvb + qvc + qvcb + f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s ]2   =
b + c + qcb + v + qvb + qvc + qvcb + f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s +
qsb + qsc + qscb + qsv + qsvb + qsvc + qsvcb + qsf + qsfb + qsfc + qsfcb + qsfv + qsfvb + qsfvc + qsfvcb + e .

aufheben_arche_empire

F.  "Empires".  We have arrived at a relative '''ultimate''', or «terminus», of this particular, finite [self-]regress, or finite 'self-progression', of human-social formation(s), in terms of both full 'self-internalizations', and of full 'self-metamonad»-izations'. The multi-city-state '''clash of empires''' was, in Earth's history, not the only driving force — or even the principal driving force — of the next irruption of a new ontological category and «arithmos» of human-social formation(s).

The planetary '''population''' and 'densification'/physical-spatial '''concentration''' of such multi-city-state "empires"-as-«monads», remained too "few and far between", particularly in relation to the attained level of transportation technology of their epoch.

The global interaction of "empires" with [other] "empires" remained rather tenuous, infrequent, and rarefied, across physical-spatial/duration-of-travel temporal distances that were large in relation to the transport capabilities of those times.

In the larger Mediterranean world of the ancient Occident, after the Roman "empire" subjugated, and incorporated into itself, most of the Carthaginian and Macedonian "empires", as well as most of the pre-"city-state" human-social terrains of the rest of western and eastern Europe, and of the islands later to be known as the British Isles, it was not so much inter-"empire" as 'intra-[Roman-]"empire" ' processes that precipitated the path which, at length, led to the appearance of the new, previously-unprecedented «arithmos» of "nation-states".

It was more the internal «lysis», into Eastern and Western Roman "empires", and other aspects of the internal, immanent, 'meta-catabolic', interior 'self-dis-organization' — the human-social-reproductive 'self-entropy' accumulation — of the ancient Roman [in]human-social-formations, in parallel, in counter-point, and in '''co-evolution''' with their mutually-destructive 'hybrid' interactions with the surrounding "barbarian" 'mobile-chiefdoms', etc., that led to the collapse of "empires" in Europe/'Mediterranea'. The result was an Occidental "Dark Ages" that protractedly-delayed the emergence of the next, "nation-state" epoch/'socio-ontology' and stage of human-social formation, which eventually arose from out of the ruins of "empires".

In that sense, then, the [[mercantile+]-capitalist] "nation-states" «arithmos» might even be categorized as a new «arché»-«arithmos», of a new, separate 'dialectic' of human-social formation(s).

All we can say, in favor of the post-"empires" continuity of the old 'dialectic' of human-social formation, is that each typical "nation-state" is a partial 'meta-"empire" ', made up out of a [local-][sub-]«arithmos» of 'ruin-ed' '''empire-fragments''', i.e., made up out of a heterogeneous multiplicity of the, often-overlapping, remains of the social, memetic terrains of multi-city-state "empires", by means of an '«aufheben» 'internalization' of the local debris of fallen "empires" as predecessor «monads».

G. "Nation-States".  Even so, were such a — "nation-states" «arithmos» as «arché» — model expected to see its own 'consecuum', for even one next, successor «arithmos», then a major new '''singularity''' in Terran, human-social history, must also be expected.

All of the epochs of human-social formation as yet considered, and 'dialectically-derived', or 'selfaufheben»-derived', in the above-rendered model-narrative so far, were confined to the theatre of a single planet's "geo"-logical ['''planet-ological''', planetary-]formation platform.

Is it still plausible to assume that the entirety of this phase of the 'self-meta-evolution' of the cosmos — of this human[oid] part of 'The Dialectic of Nature' — namely, 'The Dialectic of [the] Human[-ized Portion of] Nature', should be forever confined to a single planet for each and every '''human[oid] species''' that arises in a given stellar/planetary system of a given galaxy?

If the consecutively-next «arithmos», after the "nation-states" «arithmos», is the «arithmos» of '''world-poli''', and not that of a singleton '''world-polis''', e.g., on planet Terra alone — a single «monad», instead of a new «arithmos» of such '''global-poli''', or '''planetary-poli''' «monads» — then the "ecosphere" of Terran humanity, of the Terran human[oid] «species», must first expand beyond the confines of planet Terra.

If, locally, there is to be even a mere pair of '''planetary poli''' «monads», i.e., in the 'Terra-proximate' part of the cosmological «arithmos» of '''planetary poli''', then the vast human '«species»-project' of the colonization and '''Terra-re-forming''' of, e.g., Mars, must emerge in Terran humanity's future.

Indeed, competent human-«species» extinction-risk 'human-social risk management' would require such a '''diversification''' of the [initially-]Terran human «species»' 'planets portfolio': an outspreading of our «species»' population to the nearby planets of this solar system, and, eventually, beyond.

We might thus be led to frame the hypothesis that human[oid] societies that continue the 'consecuum' of 'self-progressive' cosmological «auto-kinesis» — of cosmological '''self-evolution''' and 'self-meta-evolution' — also continue into a 'multi-planetary' phase/stage of human-social formation(s), with respect to this 'taxonomy level two' view of cosmological 'onto-dynamasis'.

We might also wonder what the emergence of such '''planetary engineering''' levels of growth of the human '''social-productive forces''' and capabilities might have to do with a possible 'self-continuation' of the «arithmoi»-'consecuum' of 'The Dialectic of Nature' with respect to its 'taxonomy level one' view, already broached above, viz.

pre-nuclears sub-atomics atoms molecules prokaryotic cells eukaryotic cells meta-biota animal societies human societies . . . .

We mean a 'self-continuation' beyond the epoch/ontological category/«arithmos» that has "human societies" as its «monads», to a possible, consecutively next epoch/ontological category/«arithmos» of the '''meta-human'''.

The [Self-]Growth of the [Human-]Social [Self-]Force(s) of  Human-Society/Human-Social-Relations] [Self-[Re-]]Production / [Self-[Re-]]Productivity / ['Self-Transformativity']

Note:  If the above-narrated 'dialectical model' of the history of human-social formation(s) is one which appropriates the Marxian theory of '''social [meta-]evolution''', then the driving force of the 'socio-ontological' epochal transitions that it portrays must be '''the growth of the social forces of production'''.

Indeed, it is so, implicitly.

The growth of the numerosity and the density of the 'meristemal' «monads», assumed at every step of this model, is, precisely, the product of the growing effectiveness, efficiency, and productivity of the 'quanto-qualitatively', 'quanto-ontologically' changing repertoire of human, social-reproductive practices, within every stage.

The '''social force(s) of production''' concept does not refer only to the [labor-]time/durational-productivity, and other-inputs-productivity, of human activity with respect to the products, the "goods" output thereby, conceived atomistically, and in isolation.

If these "goods" are "good", are truly "goods", and not '''bads''' — if they truly represent use-value, not merely from the atomistic, immediate-subjective, "individual consumer" point of view, but also from the 'mediate-subjective' point of view of maintaining and advancing human-social organization, human-social order, '''human-social negative entropy''' — then their human, social consumption, including by the human producers of such "goods", must both sustain, and add to, the productivity of further goods production, by those human producers, which, in toto, adds to the self-support capability of the human population, and 'quanto-qualitatively' advances the '''human-nature''', the '''meme-pool''', the '''Phenome''' of that, therefore 'quanto-qualitatively' growing population of such producers, etc.

The Marxian '''social forces of production''' must, thus, refer to the level/rate/scale of the 'quanto-qualitative' 'self-productivity' of humanity; 'human socio-mass self-productivity' — including both the biological embodiment of human 'subject-ivity', or 'agent-ivity', and its "objectified", artefactual concomitants and accoutrements.

The growth of the human-social self-force of production — as the growth of the 'self-[re-]productivity' of the 'human forces of human-society self-expanding, auto-catalytic self-[re-]production', is expressed in the expansion, and physical-spatial 'self-densification', of the populations of social-formation «monads» at every level/'meta-fractal' scale addressed by the preceding narrative.

For an explicit N-ideographical, 'dialectical model' of the 'self-meta-evolution' of the human-social forces of production, see this Introductory Letter's Supplement B. [forthcoming].

The model of the 'self-meta-evolution' human social formation(s) rendered narratively and in dialectical-ideographical "shorthand" above, is supplementary to the one which Marx formulated.

Marx's model of human-social 'meta-evolution' described [what we would term] 'the historical-dialectical meta-evolution' of the "social relations of production"  as driven by the growth of "the social forces of production".  The model above owes more to archaeological and anthropological findings that have accrued since Marx's lifetime.

But suppose we go outside Marx's known writings to define the first two epochs of these "social relations of production".

Suppose that we take the quasi-ecological human-social relations of human-social re-production of the foraging/scavenging/hunting-and-gathering "band", or '''predation''', to be the predominant mode of human-social reproduction characterized by the immediate Appropriation of "raw" nature, with only a near-vanishing contribution of human labor to the refinement of such directly-appropriated natural products for human consumption and as the relation characterizing the first, «arché» epoch of human-social relations of production 'meta-evolution'.

Suppose, further, that we take the human-social relations of the production and [re-]distribution of Goods/obligatory Gifts as characterizing the second epoch of human-social relations of production 'meta-evolution', involving human-labor-improved-for-human-use, human-labor-modified proto-human appropriations of/as nature.

We then have, per our hypothesis, the following historical-dialectical progression of the 'meta-evolution' of human-social relations of production, driven by the development of the human-social forces of production

"raw" Appropriation-relations  Goods/obligatory Gifts-relations  → Commodity/Barter-relation  → Money/Circulation-relation  → «Kapital»-relation  → generalized «Equity»-relation  → . . .  .

You may access an explicit N-ideographical, 'dialectical model' of the 'meta-evolution' of the human-social relations of production in these terms, via clicking on the following link to this Introductory Letter's Supplement B":  Supplement B Part III - including a Psycho-Historical Model of the Dialectic of Human Nature (.pdf), and scroll down to pages 24  through 33.

Note:  The above-rendered model is not one of  '''a purely object-ive dialectic without subjects/agents'''.

The above-rendered model is also not one which naively, after the fashion, and the fantasies, of "rugged individualist" 'human-biological-individual atomism', takes the individual human as the invariable, trans-historical subject/agent of all of human history, acting in accord with a fixed and invariable, rigidly genomically-determined "human nature".

This model's subjects are '''social-relations-of-production''' holisms, collectives of human social-individuals, at progressively-advancing levels of '''Phenomic''' self-development, whose concerted and relationally-varying actions produce the qualitatively, ontologically different consequences seen at each level/layer/scale of this human-made 'model of human history'.

The behaviors and consequences of which "nation-states" are capable differ dramatically, qualitatively, ontologically, and 'meta-finitely' from the behaviors and consequences of which multi-city-state "empires" are capable, which, in turn, differ dramatically, qualitatively, ontologically, and 'meta-finitely' from the behaviors and consequences of which multi-village "chiefdoms" are capable, which, in turn, differ dramatically, qualitatively, ontologically, and 'meta-finitely' from the behaviors and consequences of which multi-camp "villages" are capable, which, in turn, differ dramatically, qualitatively, ontologically, and 'meta-finitely' from the behaviors and consequences of which multi-band "camps" are capable, which, in turn, differ dramatically, qualitatively, ontologically, and 'meta-finitely' from the behaviors and consequences of which proto-human "bands" are capable.

We don't find, for example, "bands", or "camps" — or even "villages" or "chiefdoms", for the most part — producing monumental, architectural works of stone, though some 'megalithic'  construction may inhere in the 'religio-politico-economic' dynamics of the "chiefdoms" «arithmos».

True, a relatively stable human genome, and including the psychological predispositions to which that genome inclines each human biological individual, is «aufheben»-conserved in each and every one of the 'socio-meta-fractal'/'socio-ontological' archaeological layerings and horizons narrated above. But this genome is also «aufheben»-conserved within a dramatically different '''phenome''', a drastically different system of motivational, rewards and punishments, positive/negative reinforcement structures and processes, in each and every 'socio-ontologically' distinct, '''social-relations-of-[human-society- / social-relations self-re-]production'''  human-social formation narrated above. The [sentential] subject [and the sentential object, and the 'essence-ial' sentential verb] content of the cosmological, 'dialectical «auto-kinesis»', 'self-evolve(s)', and 'self-meta-evolve(s)', with the 'self-progression' of that very 'dialectic', as does that 'dialectic' itself.

Example 6:  The 'Synchronico-Diachronic' Dialectic of Democratic Self-Governance Structure-Process

The Plan of governance of Foundation Encyclopedia Dialectica instantiates the principle of democratic self-governance of human organizations which we term 'Base-ocracy'.

Human-social self-organization structures/processes of this kind are base-constructed, base checked-and-balanced, and base-controlled.

The F.E.D. Plan, for example, requires that each member of the «Genos», Generic, or '''General''' Council be majority+-elected by the whole base of the «Species», Specific, or '''Special''' Council to which that member also belongs, and in which that member works, and must remain "in the trenches"
a participating member, in good standing with their peers in, with, and of that Special constituent Council — in order to continue as a member of the General Council.  This rule applies even to our founding members including Karl Seldon without exception.

Each member of the General Council must be at least majority-elected by, and, in writing, explicitly policy-mandated by, their Special Council.

Each such elected, mandated delegate to the General Council is recallable, by majority-vote of the assembly of the members of their Special Council, at any time, e.g., for violation of their mandate.

All of this aims to ensure that the General Council reflects the true policy-«Genos» of the «Arithmos» constituted by the Special Councils as «Monads»
thus reflecting the true policy-totality of the Foundation as a whole.


The psycho-historical model/idealization of the '''dialogic''' dialectic of deliberation within such Special Councils, as within the General Council, can be represented as follows, distilling manifold psycho-historical field observations of such human-social processes.

When a session of policy dialogue, or policy deliberation, opens, preparatory to a decision, or '''act''', of the Council, as collective subject, the first person to speak may set forth a '''thesis''', or '''hypothesis''', as to the '''sense of the whole''' Council, as to what policy/action it will support, and should therefore adopt.  That thesis, in those [rare] instances where the initial speaker captures, in this opening statement, an expression of the full view, '''truth''', and intent of all of the members of the Council, acceptable to all members of the Council, would also be the 'uni-thesis' or '''dialectical synthesis''' of the views of that Council, on the issue under deliberation, at that moment in the [psycho-]history of that Council.  The test of such an expression is that it be followed by silence in terms of further 'contra-thesis' expression, and thence by unanimous consent and adoption by the Council.

However, typically, the first voice is unable to encompass the views of the Council in its totality.

The work of dialogue and deliberation the work of dialectic is needed, required, and necessary, for the Council, by the Council, to discover/forge its explicit self-knowledge, and expression, of its own truth for the psycho-historical '''moment''' and issue at hand, taking into account the new information about its own state, and about the state of the world in which it inheres, that is continually emergent, due to the continual 'onto-dynamasis' of that world taking into account the relevant / '''moment-ary''' '''state of the totality'''.

Typically, the formulation expressed by the first speaker excludes part of that truth, in the minds of other members of the Council.  That statement
because of its incompleteness[es], provokes a statement in response, by, e.g., the second to speak.  This second policy-proposal assertion is 'contra-thesis' to the initial '''thesis'''.  Further speakers may clarify/elaborate that 'contra-thesis'.  Or, the third speaker may attempt to unite the mutually-supplementary content of the first '''thesis''' and of the 'first contra-thesis', in a 'first uni-thesis'.  Or, a speaker subsequent to the third voice will typically attempt this.  That 'first uni-thesis' will typically still strike many members of the Council as insufficient, provoking their expression of  a 'second contra-thesis'.

This 'dialogue-ic' dialectic will continue to self-iterate and spiral, building / «bildung»  the explicit self-knowledge, and situation-knowledge, of the assembly, until a speaker is able to achieve a final synthesis, final 'uni-thesis' formulation, for that psycho-historical moment an 'nth uni-thesis' that '''provokes''' only silence, assent, and adoption, instead of an '(n+1)st contra-thesis', on the part of the rest of the Council.

The person who is able to state the '''moment-ary absolute''', 'silencing uni-thesis', during deliberations on the updated mandate for a given Special Council's delegate to the General Council, may be a natural choice for election as that delegate.

The person, the Council-member, able to conceive, know, express, and thus achieve the '''synthesis''' of a given Special Council mandate tends to vary from moment to moment, and from crisis to crisis.

aufheben self-governance

To the extent that private-capitalist Board's of Directors actually embody the "one share of capital equity stock, one vote" capital equity principle of "shareholder democracy", of "stockholder democracy", of '''capital-owner/-contributor democracy/voting power in proportion to the capital-value contributed''', or of 'internality equity' rather than honoring that principle only "in the breach" then the structure/process of Board Committee and Sub-Committee praxis will approximate such a dialectic/dialogic process/structure, as distorted by the inequalities of influence and voting-power reflecting their differential capital-equity capital-value contributions and ownership among the various stockholder directors, plus by the influence of the "insider-directors" from "appointed"  senior management.

The structure/process of the disposition-management of 'externality equities' as a new class of collective, public property; of their economic-democratic governance, as proposed by F.E.D. as the «arche» of the post-capitalist social relation of production of 'generalized equity', in an 'Equitist' society of actualized '''political-economy''' of comprehensive, or political-economic, democracy also incarnates these dialogic dialectic principles.

The Planetary, or Global Association of Public Directors [GAPD] is a 'meta1monad»'/'meta1-unit' of the «arithmos» of the Continental Associations of Public Directors «monads»/units.

Each Continental Association of Public Directors is, in turn, a 'metamonad»'/'meta-unit' of the «arithmos» of the National Associations of Public Directors [NAPD] «monads»/units for the nation-states located within that Continent.

Each  National Association of Public Directors [NAPD] is, in turn, a 'metamonad»'/'meta-unit' of the «arithmos» of the Regional Associations of Public Directors [RAPD] «monads»/units, for all of the Regions within the nation-state of that NAPD unit.

Each Regional Association of Public Directors [RAPD] is, in turn, a 'metamonad»'/'meta-unit' of the «arithmos» of the Provincial/State Associations of Public Directors «monads»/units, for the "Provinces", or "[nation-state sub-]States", constituting that Region.

Each Provincial/State Association of Public Directors is, in turn, a 'metamonad»'/'meta-unit' of the «arithmos» of the County Associations of Public Directors [CAPD] «monads»/units, for all of the Counties geographically "contained in" that "Province"/"State".

Each County Association of Public Directors [CAPD] is, in turn, a 'metamonad»'/'meta-unit' of the «arithmos» of the Local or Municipal Associations of Public Directors [MAPD] «monads»/units, for all of the localities / cities, geographically constituting that County.

Each Local or Municipal Association of Public Directors [MAPD] is, in turn, a 'metamonad»'/'meta-unit' for the «arithmos» of the local/municipal public-elected public directors, serving on the 'externality-equities'-managing, second, public, "grass-roots" Boards, of Public Directors '«monads»'/'units' within each local/municipal operating unit of each externalities-producing firm impacting those local publics those publics which are the public stakeholders of that local/municipal geographical area.

Those public stakeholders thus publicly, collectively own, and hence have right of disposition over, the externality equities generated by the externalities the "external costs" and "external benefits" imposed upon those citizens by those firms' operating units' local ['''production'''] activities.

Those public stakeholders, or collective 'externality-equity owners, via their elected Boards of Public Directors, collectively set the 'externalities budget', of the externalities that they will have to experience, for the annual operating plan of each such operating unit, in continual, often adversarial negotiation  / litigation / arbitration with the conventional, 'internality-equities' Board, or with the '''capital-equities Board'''-appointed "Executive Committee", or "Management Committee, of each such local operating unit.

However, it is crucial, to the essence of political-economic democracy, to note that, in this model, Associations of Public Directors do not elect, nor mandate nor even, necessarily, supply, from out of their own ranks the majority-elected, explicitly mandated, and recallable delegates who constitute the "higher", more '''generic''', Associations of Public Directors.

Only the entire public the entire, geographical electoral-base of citizens for each level of association of public directors, elects, and mandates, and supplies, the delegates to the next-"higher" Association of Public Directors.

The process-goal of this governance-structure is to institute a 'meta-monadological', 'meta-fractal', nested, '''direct-democratic''', "grass-roots"-democratic, constitutionally, juridically-empowered, society-wide network of interlocking dialogues about the '''human-social morphology''' and 'socio-morpho-dynamics' / 'socio-morpho-genesis' / 'socio-meta-morph-osis' of a rapidly capital-relation-tyranny-transcending human society, and about the total magnitude, and about the distribution / allocation, of the accumulation of capitalist-firm-generated externalities that human-society will permit to be inflicted upon itself, by itself.

This economic-democratic governance process/structure is designed to help overcome the utter prostitution by Big Corporate and Rocke-Nazi Plutocracy "Big Money" that otherwise vitiates the "popularly-elected" representative-democratic legislatures, executives, and judiciaries of conventional capitalist political-only "democracy", especially during the most "advanced", '[partial] state-capitalism/totalitarianizing' stages in the historical accumulation / expropriation / concentration / consolidation / centralization process of capital-value ownership/control/power monopolization by an ever-shrinking minority of the human population.

For more on the concepts of 'Comprehensive, or Political-Economic, Democracy', of 'Externality-Equity', of 'Generalized Equity', of '''Equitism''', and of 'Equitarian Society', simply click on the following link:  Supplement B Part IV - addressing 'Equitism' as the Successor System to Capitalism (.pdf).

For a related view, regarding the traditional concepts of '''true soviets''', '''free associations of producers''', or '''producers' councils" of '''democratic communist''', or of "council communist" structures of direct-democratic human-social self-governance, simply click this link:

http://www.lust-for-life.org/Lust-For-Life/WorkersCouncilsAndEconomics/WorkersCouncilsAndEconomics.htm.

Note:  For reasons of space, the Local, Provincial, and Continental levels / layers / scales of this 'meta-monadological', 'meta-fractal' self-governance structure/process or human-social self-governance 'eventity' are not separately depicted in the «aufheben» diagram below, as would be necessary for a more complete view of the proposed governance structure/process.

 
aufheben externality equities

Example 7: The Historical Dialectic of Cosmological Natural History — The Dialectic of Nature

BEGIN
END

Example 0 Revisited: The 'Qualo-Peanic

Meta-Monadologies' of 'The Gödelian Dialectic'

Suppose that we apply the '''«arithmos» of «monads»''' concept to 'The Godelian Dialectic', at the level of a given, single axioms-system within its progression of axioms-systems 'meta-evolving, diachronic axioms-meta-system'].  Then each typical individual axiom of that axioms-system may be taken as a «monad», or '''unit''', of that axioms-system grasped as the «arithmos», or '''assemblage''', of its axioms.
Each such 'axiom-«monad»', or 'axiom-unit', is thus heterogeneous [
] with respect to each of the other 'axiom-«monads»', or axiom-units', of that axioms-system «arithmos».  That is, we are not dealing, in this case, with some kind of Platonic «arithmos monadikos», in which each  'axiom-«monad»', or 'axiom-unit', is an "identical",  "homogeneous" '''replica[nt]''' / "replication" of every other  'axiom-«monad»' or 'axiom-unit'.  Were we dealing with such a "homogeneous" «arithmos monadikos» of axioms, then each axiom would be indistinguishable from any/every other axiom in that axioms-system/axiomsarithmos». Such 'identicality' and '''redundancy''' of axioms would be useless in the case of an axioms-system.

We are dealing, on the contrary, with a special kind of «arithmos eidetikos», in which each axiom — as with Plato's original, universal, dialectical «Arithmos Eidetikos», with respect to his '''universal «eide»''' themselves — is unique.

So, viewing any given post-«arché» axioms-system in the axioms-systems-progression, or 'diachronic axioms-meta-system', as an «eide-arithmos», with its component axioms as its «eide-monads», we can now readily see aufheben character of this component of 'The Godelian Dialectic'.  We can see that, as per Gödel, this axioms-system «aufheben»-contains all of the axioms of its predecessor axioms-systems in that Gödelian progression, while also «aufheben»-negating and «aufheben»-elevating its predecessor — contributing one or more new "comprehension axiom(s)", that was(were) unprecedented / non-existent / not found, prior to it, in this Gödelian sequence / progression.

Likewise, all of this axioms-system's axioms will be "contained" / «aufheben»-conserved in all subsequent  / successor axioms-systems in such a Gödelian systems-progression.

This «aufheben» relationship between consecutively-successive axioms-systems in the Gödelian progression of axioms-systems is illustrated in the Graphic below.

aufheben_godel_axioms

Suppose, on another, deeper, more concrete level, we apply the '''«arithmos» of «monads»''' concept to 'The Godelian Dialectic', namely, at the level of the individual numbers that, as '''logical individuals''', and as '''idea-objects''', make up the number "space", or '''number-set''', associated each a given axioms-system in a Gödelian diachronic 'axioms-meta-system', i.e., a Gödelian-dialectical progression of axioms-systems.

Then, each typical individual number, of that space may be viewed as a «monad», or '''unit''', of that number-space, with that number-space therefore viewed as the «arithmos» of such numbers, i.e., as the '''assemblage''' of such numbers, and where each individual number is itself modeled as a set, that is, as an ordered pair, in the manner that we have already explored, above.

Each typical such 'numbermonad»' is thus — in terms of its '''set-model''', or '''ordered-pair model''' — heterogeneous [ ] with respect to each of the other '''numbers-asmonads»'''.That is, from this viewpoint, we are not dealing, in this case, with some kind of Platonic «arithmos monadikos», in which each  'number-«monad»', or 'number-as-unit', is an "identical",  "homogeneous" '''replica[nt]''' / "replication" of every other  'number-«monad»' or 'number-as-unit' in the "space" of numbers, or "set" of numbers, associated with the given axioms-system.  Were we dealing with such a "homogeneous" «arithmos monadikos» of numbers-as-units, then each  'number-«monad»' or 'number-as-unit' would be indistinguishable from any/every other 'number-«monad»' or 'number-as-unit'  in that '''numbers-set''' / '''numbers-space''' / '''numbers-system''' / 'numbersarithmos»'. Such 'identicality' and '''redundancy''' of 'number-«monads»' would be useless in the case of an axioms-system built to model/describe an arithmetic.

We are dealing, on the contrary, with a special kind of «arithmos eidetikos», in which each 'number-«eide-monad»' — as with Plato's original, universal, dialectical «Arithmos Eidetikos», with respect to his '''universal «eide»''' themselves — is unique.

Moreover, at the level of the sets, or ordered pairs, which model each successive new kind of number, numbers of their predecessor kind are inside each of the number of this successor kind, as we have seen above.

This «aufheben» relationship between consecutively-successive numbers-systems in the Gödelian progression of numbers-systems is illustrated in the Graphic below.

That is, suppose we view '''set-elements''' [including '''set-elements''' that are themselves, already, sets] as the «monads», or '''units''', of the set which contains them, that containing-set being viewed also, therefore, as the «arithmos» of these, its set-elements.

aufheben_godel_numbers

Then, considering any post-«arché» number-space, in the progression of number-spaces that is part and parcel of 'The Godelian Dialectic', as «arithmos», and its component number-"points" as the «monads» constituent of that «arithmos», we can see the «aufheben» character of the number-space component of 'The Godelian Dialectic'.  We can see that, as per Gödel, this number-space, or numbers-system, «aufheben»-contains all of the numbers of its predecessor number-space / numbers-systems in that Gödelian progression, while also «aufheben»-negating and «aufheben»-elevating its predecessor number-space / numbers-system — contributing a new kind of number, of a kind that was unprecedented / non-existent / not found, prior to it, in this Gödelian sequence / progression of axioms-systems, and associated number-spaces / numbers-systems.

Each typical number of the number-space of the given axioms-system — i.e., of the given stage / '''epoch''' of' The Godelian Dialectic' / Arithmetical axioms-system progression / Arithmetical axioms-systems diachronic 'meta-system' — "contains", or «aufheben»-conserves, a heterogeneous multiplicity — an "ordered pair" — of numbers of its 'inverse-consecutive', immediate-predecessor number-space, of the immediate-predecessor axioms-system of Arithmetic, at the level of that typical number's "sets of ordered pairs" model / representation.

In other words —

If we view such a typical number, of a given successor number-space / axioms-system, as modeled by a set, then each such number of higher '''degree''' / "logical type", is revealed to be made up out of a heterogeneous multiplicity of [two] sets of lower '''degree'''/"logical type", from the predecessor number-space «arithmos» / axioms-system.

If we view such a typical number, of a given successor number-space «arithmos» / axioms-system, as modeled by an ordered pair, then each such number is revealed to be a 'meta-ordered-pair', made up out of a heterogeneous multiplicity of [two] ordered-pairs from the predecessor number-space «arithmos» / axioms-system.

If we view such a typical number, of a given successor 'numbers-space' «arithmos» / axioms-system, as an "element" of a given "successor" numbers-space, or numbers-set', as «arithmos», then each such number is revealed to be a 'meta-sub-element', made up out of a heterogeneous multiplicity of the 'sub-elements' of the successor 'numbers-space', which 'sub-elements' are "elements" proper for the predecessor numbers-space as «arithmos» of the predecessor axioms-system.

If we view such a typical number, of a given successor 'numbers-space' «arithmos» / axioms-system, as a "member", or "point", of that given "successor" numbers-space, or numbers-set', viewing this numbers-space, or numbers-set,' that number's «arithmos», then each such number is revealed to be a 'meta-member', made up out of a heterogeneous multiplicity of the 'members' of the predecessor numbers-space, or numbers-set, as the «arithmos» for those '''members'''.

We may also view this progression of 'numbers[-"points"]-sets', or 'numbers-spaces', '''geometrically''', as per the following graphic.

number_spaces_progressions

Summarizing the mental-objects 'intra-observations' set forth above, then, we find that each typical number-«monad» of the successor Number-space, Nτ+1, of successor axioms-system of Arithmetic, Aτ+1, is a 'meta-number' with respect to some two typical number-«monads» of the predecessor Number-space, Nτ, of the predecessor axioms-system of Arithmetic, Aτ.

Drawing together the two major, 'intercoordinated' «aufheben»-processes which make up 'The Godelian Dialectic' — namely, the «aufheben»-process of the «arithmoi» of the successive axioms-sets, and the «aufheben»-process of the successive numbers-sets — we can diagram 'The Godelian Dialectic', as a whole, as follows —

aufheben_godel_dialectic_as_whole

The Examples Considered as a Whole

Consider the examples of 'Peanic', 'meta-fractal', 'meta-monadic', «aufheben», 'archeonic consecua', or of successions/progressions of «arithmoi», set forth above, as a whole, namely —

. . . sub-«species» «species» «gene» super-«gene» . . .

for the 'meta-fractal', 'meta-monadic', «aufheben» structure of Plato's philosophical '''dialectical arithmetic of ideas''', or «arithmoi eidetikoi»,

letters words sentences paragraphs chapters books libraries

for the '«aufheben», meta-monadic', 'meta-fractal' structure of some key 'psycho-artefacts' of phonetic literacy,

computer bits  computer bytes  computer words  computer commands  computer programs  software systems

for the 'meta-fractal', 'meta-monadic', «aufheben» structure of some key 'psycho-artefacts' of modern 'computerware',

Local/Municipal Electorates  County Electorates  Provincial Electorates →
Regional Electorates
 
 National/Continental Electorates  Global Electorate . . .

for the '«aufheben», meta-monadic', 'meta-fractal'  structure of the publicly-elected, base-elected, mandated, recallable delegates to the various scales of the Associations of Public Directors which would institutionalize the new, constitutionally-established collective property rights per the Externality Equity principle, as part of the new social relation of production of 'Generalized Equity' at the heart of Equitism, the politically-and-economically democratic successor system to Capitalism, as hypothesized by F.E.D.,

1 10 100 1000 . . .

for the Indo-Arabic numerals-system's progression of '''purely-quantitative''', "orders of magnitude" [meta-]units,

1 2 3 4 . . . ,

for the 'pure unqualified  quantifiers' generic 'Quanto-Peanic' progression of the "Natural" «arché» of the "Standard" Arithmetics,

q1 q2 q3 q4 . . . ,

for the 'pure unquantifiable ontological qualifiers' generic 'Qualo-Peanic' progression of the «arché» 'arithmetic of dialectics', and

S0U S1 S2 S3 . . . ,

for the [finitary] '''Set Of All Sets''' 'idea-process-object', self-iterating self-«aufheben» 'ideo-«autokinesis», or 'mental eventity' — 'the dialectic of the '''Set'''-concept',  then, for the 'Meta-Axiomatics' of 'The Gödelian Dialectic', the «aufheben» progressions of the Axioms-systems of the successive systems of Arithmetic,

A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 . . . ,

and, the progression of their corresponding numbers-Systems, or numbers-Spaces,

S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 . . . ,

and, finally, the pedagogical, 'meta-systematic-dialectical' progression within the Axioms-systems of "purely-quantitative" arithmetic [which, by the way, immanently emerge their own special kind of "'hypernumber''' qualifiers, beginning, explicitly, with the '''qualitative unit''', i, of the Complex-numbers system of arithmetic, herein denoted by C ], that we utilized to illustrate that 'Gödelian Dialectic', namely:

N W Z Q R C H O K G . . . .

Note how all of them tend to strike even the 'less Parmenidean', modern mind as mainly static-"synchronic" structures, already constructed, already completed, and "ready-made"; as present structures — as single-slice-of-time, '''history cross-sectional''', 'meta-anatomical' structures — rather than as "diachronic", historical processes.

It is, of course, true that such "all-present-at-once", self-similar level-structures, or 'synchronic meta-fractals', can be presented in a diachronic discourse, e.g., via a ''categorial progression" method of exposition, or '''method of presentation''' — which '''mode of presentation''' is often termed a '''systematic dialectic'''.

However, only the 'dialectic of human-social formation(s)' example, as a 'taxonomy-level twopart of 'The Dialectic of Nature', 'inside' the '''human societies''' «arithmos» of the 'taxonomy-level one' '''Dialectic of Nature''', as well as that 'taxonomy level one' '''Dialectic of Nature''' itself,  viz. —

human societies:

"bands" "camps""villages" "chiefdoms""city-states" "empires" "nation-states" . . . ,

and

pre-nuclears sub-atomics atoms molecules prokaryotic cells
eukaryotic cells
meta-biota animal societies human societies . . .

— tend to be immediately perceived as 'diachronic-historical', or, in our terms, as instances of '''historical dialectic'''.

The F.E.D. paradigm for '''historical dialectics''' is a paradigm of diachronic, 'auto-kinesic', 'systems self-progression' — of 'selfaufheben»' diachronic 'meta-systems' each of whose system-constituents is quantitatively evolving, and also is qualitatively/ontologically 'meta-evolving'/'''self-revolutionizing'''; which are dynamical and 'meta-dynamical'; which are diachronically 'meta-fractal', and which, at every stage, in every epoch of their '''self-development''', exhibit also the 'synchronic meta-fractal' structure of an (n+1)-level 'super-system', or 'supern-system'.

This paradigm sees, synchronically, «arché» systems ['super0-systems'], perhaps «aufheben»-contained in 'super1-systems', which may, in turn, be «aufheben»-contained in 'super-super-systems', or 'super2-systems', which may, in turn, be «aufheben»-contained in 'super-super-super-systems', or 'super3-systems', . . ., which may, in turn, be «aufheben»-contained in 'supern-systems', such that N n > 3.

Alternatively, and still synchronically, this paradigm of '''historical dialectics''' sees each single system ['sub0-system'] as, potentially, «aufheben»-containing an «arithmos» of 'sub1-systems', which may, in turn, be each «aufheben»-containing an «arithmos» of 'sub-sub-systems', or 'sub2-systems', which may, in turn, each be «aufheben»-containing an «arithmos» of 'sub-sub-sub-systems', or 'sub3-systems', ..., which may, in turn, each be «aufheben»-containing an «arithmos» of 'subn-systems', such that N n > 3.

However, all of these synchronic '''moments''', and their synchronic «aufheben» structures, are constituted diachronically system-self-created historically.

They are so created via the «aufheben» 'self-internalization' processes of '«physis»-ics', or '''physics'''; of physically-spatially-extended/[self-]deployed local populations of «monads», of therefore [self-]progressively, 'meta-«monad»-ically' different kinds, different qualities, different beingdifferent ontology.

This paradigm therefore sees diachronic 'meta-[superx-]system' systems-'self-«aufheben» self-progressions':

«arché» system  'super1-system'  'super2-system'  . . . 'supern-system'  'supern+1-system'    . . .


The above sets forth the generic 'meta-[supern-]system' paradigm, which is explicitly and concretely addressed by the later, '''higher''' 'dialectical ideographies' of 'the dialectic of dialectical ideographies'.

This paradigm can also be rendered as follows:

«arché» level /layer /scale

'synchronic meta-fractal with 2 mutually-similar scales /levels /layers 

'synchronic meta-fractal with 3 mutually-similar scales /levels /layers 
  . . .


The later, '''higher''' 'meta-numerals' of the later, '''higher''' dialectical arithmetics '''spontaneously''' achieve ideographical syntax augmentations which conduce to higher complexities/'thought-concretenesses'/'potential descriptive richnesses' for the modeling of such 'meta-[
superx-]systems'.

They do so by means of ideographical, 'numeralic', [quanto-]qualitative «monads»/[quanto-]qualitative units in the mathematical sense of such 'unit-y'.

These higher 'meta-numeralic' units are syntactical structures which themselves arise in a 'meta-system-atic dialectical', 'Peanic', 'meta-fractal', 'meta-monadic', 'archeonic consecuum' 'self-«aufheben» self-progression' of «arithmoi».

That is, they arise in a '''meta-systematic-dialectical''', '''categorial progression''' exposition of the systems of 'dialectical arithmetic' i.e., of the systems of  'dialectical ideography' at its pre-algebraic level.


This progression of 'numeralic' '
ideoarithmoi»' begins with the N «arithmos», of 'unqualified quantifiers', as «arché»-thesis. This '''thesis''' then calls forth, out of the Gödelian/Skolemian 'self-duality' of the "standard" N
arithmetic, its "non-standard" NQ 'contra-«arithmos»', of 'unquantified ontological qualifiers', as this progression's 'first contra-thesis'  to that '«arché» thesis'. Therefore, together, '«arché» thesis' and 'first contra-thesis' evoke this progression's 'first uni-thesis', the NU 'uni-«arithmos»', of 'quantifiable ontological qualifiers'. Thence, moving on, out of the 'self-duality' of the NQ «arithmos», we move to a new, 'second contra-thesis' «arithmos», of 'unquantifiable metrical qualifiers', denoted NM, '
contrarizing'  to all that has preceded it, and so on ..., viz.    

N NQ NU NqQN NM NqΜN NqΜQ NqΜU . . .

The following graphic provides a pictorial summary of the above-encoded 'meta-systematic dialectic of the systems of dialectical arithmetic', modeled in the language of the initial dialectical arithmetic, denoted herein by N, and includes explicit representation of all 'ideo-ontologically hybrid', 'full and partial '''synthesis''' terms' of this dialectical categorial-progression / systems-progression, through its fourth '''epoch'''.

aufheben_dialectical_ideography

The fuller exposition of this 'dialectic of dialectical ideographies' is the office of Part III. of the book Dialectical IdeographyThe Arithmetics of 'Meta-Evolution', which is not yet extant. For a more detailed preview of its content, via texts already available on this web site, click on the link(s) below:

1.  Encyclopedia Dialectica  'Fractment' #1:  Generic Dialectical Interpretation of the N  Ideography.
2
Encyclopedia Dialectica  Brief #1:  The N
, or Pure Qualitative, Dialectical Algebra as «Characteristica Universalis»: How to Use it as an «Organon» for Discovery; Section 3, Example 3; Meta-System-atic Dialectic The Dialectic of The Rules-Systems of Dialectical Ideography itself.

The portions of this largely algorithmically-generated [via the '«aufheben» algorithm'] 'dialectical' succession of 'dialectical ideoarithmoi»' that have been explored by, and 'semantified' by, the F.E.D. research, so far, turn out to be "spontaneously" constructed, in the formation of their 'meta-numerals', or 'numeralic meta-«monads»', so as to syntactically capture, or '''mirror''', the above-described synchronic 'meta-[supern-]system' structures.

They also "spontaneously" capture the diachronic processes of self-constitution / self-construction of such 'meta-[supern-]systems'.

They discover a '''natural''', easy 'meta-continuity', bridging the gulfs of the ontologically self-revolutionary nonlinear differential equation "singularities" — of the apparently absolute, "infinite" discontinuities of 'self-bifurcation', via 'self-conversion'-completion-driven zero-division which separate '''predecessor systems'''  from their next, consecutive '''successor systems'''  within the typical 'diachronic meta-system'  systems-progression.

They do so by means of multi-level, nested 'self-subscriptization', and also by means of '[quanto-]qualitative fraction', [finite] "continued fraction"-like finite-regress 'syntactics'.

These later 'dialectical arithmetics'  model these 'meta-[supern-]system' processes/structures explicitly, with ever greater explicitude as this 'Qualo-Peanic' succession/progression of systems of ' dialectical ideography'  proceeds/succeeds itself/'self-supersedes' and, repeatedly, 'self-revolutionizes'.

These successive 'higher meta-numerals' achieve this increasing capability for modeling 'historical-dialectical' processes of nature, or 'processes of existence', by means of their open-ended 'self-progression' into increasingly determinations-enriched, descriptive-capacity augmented, multi-leveled and non-reductionist ideographical languages 'holistic notations'.  These notations are capable of tracking 'purely-quantitative', 'onto-statical'"state-space" "dynamics" and '''co-evolutions'''.  They are also, simultaneously, capable of tracking 'quanto-qualitative', onto-dynamical, '''state-space''' / 'control-parameter-space meta-space' 'meta-dynamics' and 'co-meta-evolutions', of all 'n+1 levels/layers/scales' of each, successive 'supern-system' with each level/layer/scale tracked explicitly and separately in terms of the inter-mutual interactions and 'inter-determinations' of each such '''level''' with every other such '''level''', as well as with[in] itself.

Capitalist and state-capitalist ideologues, as well as other reductionist 'anti-humanists' and 'anti-human-liberationists', tend to see in such 'Meta-Peanic', 'meta-evolutionary', 'meta-dynamical', 'meta-finite', diachronically self-constructed, 'meta-monadological', 'meta-fractal', ontologically multi-scaled, multi-leveled, multi-layered, nested «aufheben» structures only synchronic, frozen, '''eternally-static''' so they hope!!! "hierarchies".

The Dialectics Barrier

This re-discovery of Plato's 'dialectical arithmetic' emerged also, for us, in the context of the most advanced development of Plato's thinking, as embodied in his final dialogues, beginning with The Parmenides.

In those dialogues, Plato advances beyond his earlier, 'Parmenideanic' eternal «stasis» of the "Forms", to embrace «autokinesis», and the primacy of this "self-motion" over "derived motion", or other-induced, externally-induced change:

"The dialogues of the Socratic period provide that view of the world usually associated with Plato. The period of transition and criticism, and the final synthesis, are little noted; nor does the transition occur by an abrupt break, but rather by a pointing up of difficulties, and an introduction of new emphases. ... The Parmenides can be taken as signaling the change. In this dialogue Socrates is unable to defend his Doctrine of Ideas. The problem of the utter difference between time and eternity sets the problem. As creatures of time it seems that we should have no capacity to know the universal forms, nor can we have, then, any connection with the universal God, or He with us. ... Where the Republic and Phaedo stressed the unchanging nature of the soul, the emphasis in the Phaedrus is exactly reversed. In this dialogue, the soul is the principle of self-motion, and we are told that the soul is always in motion, and what is always in motion is immortal. The difference now between spirit and matter is not changelessness in contrast with change, but self-motion [i.e., that which we have termed 'self-re-flexive' or 'auto-flexive' 'nonlinearity', á la Hegel's "being for itself", i.e., self-'beholding' beingF.E.D.], the essence of the soul, in contrast with derived motion [our 'merely flexive' or 'allo-flexive' "nonlinearity" — or, in some models, linearity á la Hegel's "being-in-itself" or "being-for-another", i.e., a kind of being which may be 'beheld' by other being, but which itself 'beholds' neither other being nor itself/its own being — F.E.D.]. The emphasis on self-motion is continued even in the Laws, Plato's final dialogue." [William L. Riese; Dictionary of Religion and Philosophy: Eastern and Western Thought; Humanities Press, Inc.(New Jersey: 1980); pages 442-443].

The «insolubilia» of our epoch include, as we have already encountered in the quote from Bertrand Russell, above, not only the "natural laws"/"laws of motion"/'laws of change'-formulating nonlinear, total and partial integro-differential equations of "mathematics proper", but also the '''self-referential''', semantically 'self-reflexive', 'self-refluxive', "impredicative", '''nonlinear [quadratic] propositional function''' [cf. Bertrand Russell] paradoxes of set-theory and formal logic, with their 'mentally auto-kinesic' set and proposition idea-objects or 'idea-eventities'. These include the alternately, 'instantaneously' self-ingesting and self-disgorging "Russell Set" [of all sets which are not members of themselves], the anciently-known, truth-value self-oscillatory, thus "limit-cycle" like, "pseudomenon" of Epimenides [modernly reduced to the sentence "This sentence is false."] — "limit-cycles" being a principal, "self-oscillatory" state-space solution-trajectory unique to nonlinear differential equations, impossible for linear differential equations — and, more substantively, the cumulative, ontologically self-expanding, definitionally self-propelling, 'logically auto-kinesic', ever self-/power-set internalizing '''Set Of All Sets''', i.e., the set-theoretical, "extensional" definition of the "intension" of the set-concept itself [as arising from a perhaps '''potentially infinite''', but actually finite, universal set of actually-constructed "logical individuals"].

Their common characteristic? Their common characteristic is «autokinesis».

Thus, 'The Nonlinearity Barrier', as our contemporary, applied-mathematical 'Insolubilia Barrier', is, in terms of the Russellian formulation [of, we claim, its roots or reflections in mathematical formal logic and set theory], none other than the ideological obstacle of  'The Reflexiveness Barrier''The Self-Reference Barrier' or 'The Self-Reflexivity Barrier'.

It is also, in terms, especially, of the ultimate formulation of the ancient Orient, 'The Refluxiveness Barrier' 'The Self-Refluxivity Barrier', that is, 'The «Karma» Barrier' — the barrier of explicit recognition of what that tradition calls '''The Law of [the Reflux of] Action [back upon its source]''' or '''The Law of «Karma»'".

In terms, especially, of the ultimate [Platonic] formulation of the ancient Occident, it is 'The «Autokinesis» Barrier'.

It is, in short, none other than 'The Dialectics Barrier'.

As such, it is a barrier immanent within both ancient and modern mathematico-scientific ideology an ideological barrier never yet, to-date, wholly breached.

Consider, then, five hypotheses:

(1)  That contemporary Terran humanity, as per our conjecture above, resides within that 'Gödelian epoch' of the 'psycho-historical' 'meta-evolution' of mathematics — and of the core human 'Phenome' of human cognitive powers generally in which the very equations of «autokinesis», which are also the extant mathematical formulations of dialectical process, though standardly unrecognized as such, namely, the general, especially the current "laws"-of-nature-formulating nonlinear integro-differential equations, constitute the «insolubilia», the "unsolvable equations", in some sense "diophantine" [w.r.t. this, see the work on Hilbert's Tenth Problem].

symbol 12
(2)  That these equations belong to the "deformalization" of an incompleteness-asserting 'Gödel formula'  for the present epoch of the «de facto» meristemal axioms-system of contemporary mathematics, i.e., for the current stage of the Gödelian incompleteness/undecidability/unsolvability dialectic of mathematics.

(3)  That this "deformalization"  is, in effect, a proposition asserting that most nonlinear integro-differential equations are unsolvable, not only in the diophantine sense, but also in terms of the so far extant non-diophantine, higher kinds of number, indeed, in terms of the entire current stage of expansion of the number concept, of the conceptual, 'ideo-ontological' expansion of the implicit axiomatic system of the kinds of number, and of their operatorial, 'arithmetical logic', so far officially admitted into arithmetic and its higher mathematical octaves.

symbol 12 (4)  That this unsolvability-asserting proposition is true but undecidable, unprovable within the current «de facto»
meristemal axiomatic system of mathematics; 'undeducible' from its «de facto» axioms, but that this proposition becomes decidable/provable/formally demonstrable within the Gödelian next axioms-system of mathematics, via the adjunction of the next increment of '''comprehension axioms''', for the next 'self-internalization', or logical type level, of sets, and, thereby, for the next new kind of [non-diophantine] number which certain sets from among that next higher logical type of set can model.

Lastly:

symbol 12 (5)  That this new kind, or '[ideo-]ontological category' ['ideo-onto'], of number [in some way] also supplies the new kinds of '''elementary transcendental functions''' [cf. G. N. Watson] required to solve generalized total and partial nonlinear integro-differential equations in an expanded, new definition of '''closed-form''', '''analytical''' solution. symbol 12

Consider also, the following questions connected to these hypotheses:

If this is so, then how do the 'dialectors' or 'dialectical meta-vectors'; the new kinds of 'meta-numbers', including '[Qualo-]Peanic' but 'contra-Boolean', 'dialectical numbers', or '«aufheben»-operation-modeling numbers' — the new species of 'Gödelian/Skolemian' "Non-Standard", 'meta-Natural' Numbers of our «arché» 'dialectical arithmetic' — fit into this picture?

Also, in what way, if any, can the 'meta-numbers' of the N arithmetic, and its «sequelae», be modeled using sets given that, for example, sets of ordered pairs have been used to model the other kinds of numbers so far surfaced?  That is, what is the set theoretical translation of the dialectical ideographies?

Do those 'dialectical meta-numbers' provide even a prelude to that new system of arithmetic essential to these new kinds of "elementary transcendental" 'nonlinear' solution-functions?

Where does the axioms-system of the N arithmetic, and the axioms-systems of the other '''epochs''' of the dialectical ideographies, fit/fall in the '''psycho-historical''',  «aufheben» progression of arithmetics that we outlined above, namely:

''Natural'' Numbers "Whole" Numbers Integers (Z) → "Rational" Numbers (Q) "Real" Numbers "Complex" Numbers Quaternions (H) →  Octonions Clifford Numbers (K) → Grassmann Numbers . . .

Would these be functions capable of formulating «autokinesis»; of describing 'diachronic fractals', or never exactly repeating, but ever-self-similar, 'self-powering', '''self-squaring''' [cf. Mandelbrot; «[auto-]dynamis», cf. Diophantus], self-iterative, and self-accelerating [or temporally, 'chronogenically' fractal-scaling, diachronically-scaling] temporal progressions [i.e., with 'temporal acceleration' as their principle of progressively-diminishing-duration fractal duration-scaling]; of describing 'self-reflexive action'; 'auto-kinetic', 'self-developing process', or 'self-developing eventity', and of formulating a full-regalia theory of 'self-reflexive functions'; of the operations of 'generalized «aufheben»-operator [self-]multiplication' or '[self-re-]flexion'?

We shall see.

The Immanence of a "Non-Standard", Alternative Mathematics within the Official Foundations of "Standard" Mathematics

After his Completeness and Incompleteness theorems, the final crowning achievement of Gödel's life's work, supplemented by that of Paul Cohen circa mid-century last, addressed the "undecidability" of two propositions, "The Axiom of Choice" [an axiom of Zermelo-Fraenkel set theory, the system of nine axioms widely considered to constitute the foundation for all of modern mathematics], and "The Generalized Continuum Hypothesis" [a candidate additional, tenth axiom for that system, and a generalization of Cantor's Continuum Hypothesis regarding the first two 'scales' of "actual infinity" in Cantor's [purportedly] ''actually infinite'' progression of '''actually transfinite''' cardinal quantities].

Gödel's and Cohen's work together demonstrated that either of these propositions, or their contraries, in any combination, are compatible, as axioms, with the remaining eight Zermelo-Fraenkel axioms, i.e., are "independent" of, or Gödel-undecidable from, those eight Axioms [just as Euclid's Fifth or Parallels Postulate is "independent" of, or Gödel-undecidable from, the other postulates of Euclidean Geometry].

These demonstrations dealt a devastating blow to the absolutist ambitions of logicists, formalists, and 'setists' alike:

"Actually, Cantor went much further. He hypothesized that the order of infinity of the irrational numbers immediately follows that of the rationals. That is, he believed that there is no order of infinity that is both higher than that of the rational numbers and lower than that of the irrational numbers. This statement became known as the Continuum Hypothesis, and the work of Kurt Gödel and Paul Cohen in the twentieth century established that it is impossible to prove this hypothesis within the rest of mathematics. The Continuum Hypothesis stands alone (with some equivalent restatements) opposite the rest of mathematics, their respective truths independent of each other. This remains one of the most bizarre truths in the foundations of mathematics." [Amir D. Aczel; Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem; published by Four Walls Eight Windows (NY: 1996); page 139n].

***

"None of the proposed solutions of the basic problems of the foundations [of mathematics — F.E.D.] — the axiomatization of set theory, logicism, intuitionism, or formalism — achieved the objective of providing a universally acceptable approach to mathematics. In ... 1951 ..., Gödel proved that ... the continuum hypothesis ... is consistent with the Zermelo-Fraenkel system of axioms (without the axiom of choice). In 1963, Paul J. Cohen ... proved that ... even if one retained the axiom of choice in the Zermelo-Fraenkel system, the continuum hypothesis could not be proved [or dis-proved — F.E.D.]. These results imply that we are free to construct new systems of mathematics in which either or both of the two controversial axioms are denied. All of the developments since 1930 leave open two major problems: to prove the consistency of unrestricted classical analysis and set theory, and to build mathematics on a strictly intuitionistic basis or to determine the limits of this approach. The source of the difficulties in both of these problems is infinity as used in infinite sets and infinite processes. This concept, which created problems even for the Greeks in connection with irrational numbers and which they evaded in the method of exhaustion, has been a subject of contention ever since and prompted Weyl to remark that mathematics is the science of infinity." [Morris Kline; Mathematical Thought from Ancient to Modern Times (volume 3); Oxford University Press (NY: 1972); pages 1208-1209].

***

"The two independence results [of Gödel and CohenF.E.D.] mean that in the Zermelo-Fraenkel system the axiom of choice and the continuum hypothesis are undecidable [in the Gödelian sense  — F.E.D.].... There are then many mathematics [i.e., 'alternativity abounds'! F.E.D.]. There are numerous directions in which set theory (apart from other foundations of mathematics) can go. ... As for the continuum hypothesis, here one ventures into the unknown and, whether one affirms or denies it, significant consequences are not known as yet. ... Just as the work on the parallel axiom led to the parting of the ways for geometry, so Cohen's work on these two axioms about sets leads to a manifold parting of the ways for all of mathematics based especially on set theory, though it also affects other foundational approaches. It opens up several directions that mathematics can take but provides no obvious reason for preferring one over another." [Morris Kline; Mathematics: The Loss of Certainty; Oxford University Press (NY: 1980); pages 268-270].

***

"Although these results [of Gödel and Cohen — F.E.D.] have a formal analogy with geometry, the situation is quite different, since it is possible to set up the different kinds of geometry from a unified standpoint, namely that of general set theory. However, there is no unified principle for founding the different, mutually-exclusive systems of set theory [unproven, dogmatic assertionF.E.D.]. According to the present state of affairs, such principles of a mathematical nature do not even seem to exist, because a higher mathematical abstraction than that of set theory is absolutely inconceivable [unproven, symbol 12 clinically hysterical symbol 12 assertionF.E.D.]. Gödel himself expressed the view that the development of set theory will lead to new axioms, which will allow the continuum hypothesis to be disproved." [W. Gellert, H. Küstner, M. Hellwich, H. Kästner, editors; The VNR Concise Encyclopedia of Mathematics; Van Nostrand Reinhold (NY: 1977); page 722].

Joining the eight (8) remaining Zermelo-Fraenkel Axioms with each consistent alternative to the candidate-tenth, generalized  'Cantor Axiom'  and/or to The Axiom of Choice — both of which are 'paradox-breeding', anti-holistic, '''point-reductionist''', or 'point-atomistic', 'dimensionality-denying' propositions — may yield a new, 'ideo-ontologically' different, 'dimensionality-affirming', "Non-Standard", or 'Contra-Cantorian', 'Anti-Point-Atomistic', '''Non-Reductionist''' Mathematics, with an 'Holistic Notation', holistic syntactically, as well as symantically.

Each such 'Neo-Mathematics' is, logically, equally as self-consistent as, yet qualitatively different from, existing, "Standard" Mathematics, just as the several Non-Euclidean Geometries differ qualitatively but self-consistently from the Euclidean.

Might one of these 3+ possible Non-Standard axiomatic "set" theories, arithmos» theories', or '''totality''' theories, perhaps one that incorporates, as replacement axioms, negations of both The Axiom of Choice and The Generalized Continuum Hypothesis, provide, if elaborated, '''short-cuts''' to 'The Nonlinearity Breakthrough'?

We shall see.

Dialectical «Characteristica Universalis»: An Ideography for Example 5

The narration, or '''words-model''', provided above, in the section entitled 'Example 5The Historical-Dialectical Meta-Monadology of Human-Social Formation(s)', of the 'selfaufheben» self-progression' of 'human[oid]-social geo-demo-graphic forms', required a lengthy exposition, with much repetition. That repetition reflected the similarity-aspect among the various levels/scales in the 'meta-fractal' structure — formulated using "phonetic", '''phonogramic''' symbology — to bring out that combination of '''the unprecedented''' and '''the recurrent''' which are both inherent, as a '''qualitative helicity''', in the 'Peanic', 'meta-monadic', 'meta-fractal cumulum' which that '''historical-dialectical self-progression''' builds up.

Using the shorthand of the '''algebra''' — i.e., of the '''character-language''', or of the '''characteristic''', or «characteristica», or of the '''idea-graph-y''' — of the N '''«Characteristica Universalis»''' [cf. Leibniz], we can abbreviate, or 'syntactically concentrate', that 'phonogramic' exposition, via the higher potential, '''intensional''' 'semantic density' of its ideographical symbols. This was illustrated within the text of the cited section, by the ideographical-dialectical formulae accompanying and preceding each graphic included in that section.  The purpose of this section is to provide a fuller exposition of the workings of those formulae, including of the "hybrid" terms that were not depicted in those graphics.

Under abbreviation via '''translation''' of the phonetic narrative into dialectical-ideographical formulae, this story of human-social formation, this '''herstory''' and '''hisstory''' of the historical, diachronic self-«bildung» of humanity, of the self-building 'meta-monadology' of the «arithmoi» — of the 'meta-fractal consecuum-cumulum' — of human-social settlement-governance complexes, looks like this:

b 

b + [ c  qbb ]  →

b + c + qcb + [ v  qcc ] 

b + c + qcb + v + qvb + qvc + qvcb + [ f  qvv ] 

b + c + qcb + v + qvb + qvc + qvcb + f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + [ s  qff ]  →

b + c + qcb + v + qvb + qvc + qvcb + f + qfb + qfc + qfcb + qfv + qfvb + qfvc + qfvcb + s + qsb + qsc + qscb + qsv + qsvb + qsvc + qsvcb + qsf + qsfb + qsfc + qsfcb + qsfv + qsfvb + qsfvc + qsfvcb + [ e  qss ] . . . .

The above 'dialectical-ideogramic' rendering of the story takes all of seven lines, versus more than 200 lines for the 'phonogramic' rendering!

However, normally, absent further explication, it would not be immediately, intuitively obvious as to how to read / de-code this compressed, ideographical rendering.  The remainder of this section is allocated to explicating the workings, and the meanings, of the ideography above.

The point of departure for this explication  is a decoding of the "intensions" of the '''colorized''' and 'self-hybrid' terms of the inhomogeneous sums of «arithmoi» denoted by the lines of ideography above. This decoding requires an explicit definition of the 'socio-ontological categories' of this story of human-social formation — and of the assignments of their names, and of their '''algebraic''' or '''characteristic''', mnemonic/abbreviative symbols, like b, c, etc. — phonograms '''Shanghai-ed''' into abbreviative service as ideograms to the minimally-interpreted 'meta-numerals', or 'qualitative numerals' [the 'generic ontological qualifiers' / «aufheben» operators] of the generic, core numbers-space, NQ  q1q2q3, ... }, of the N Rules-System of Dialectical Arithmetic.  That is, let's begin with a decoding of just the 'historical contra-thesis' category-assignments / «arithmoi»-assignments:

qbb or qc or c denotes the «arithmos» that has "camps" as its «monads»             q2;
qcc or qv or v denotes the «arithmos» that has "villages" as its «monads»            q4;
qvv or qf or f denotes the «arithmos» that has "chiefdoms" as its «monads»        q8;
qff or qs or s denotes the «arithmos» that has "city-states" as its «monads»        q16;
qss or qe or e denotes the «arithmos» that has "empires" as its «monads»           q32; ...,
wherein, 
qb or b denotes the «arithmos» that has "bands" as its «monads»         q1.

You may have noticed the gaps between the successive numerical subscripts of the qn generic, '''purely-qualitative''' 'meta-numbers' assigned to the successive «arithmoi» as listed above.  Only those NQ generic 'meta-numbers' that have the form q2^w, for w W  { 0, 1, 2, 3, 4, 5, ... }, for consecutive members of W from w = 0 through w = 5 [ 20 = 1; 25 = 32 ], are assigned, above.

Question:

Why are these assignments "non-consecutive"?

Partial Answer:

Filling in, between each pair of '''successive''' 'self-hybrid' «arithmos» symbols, from qbb c  q2 onward, are one or more of — are an increasing number of — interpreted ideograms denoting '''hybrid''' «arithmoi»/ontological categories, representing various '''[mostly '''partial'''-]synthesis''', '''complex unity''', or 'ontological conversion' structures/processes, of the symbolic form qzy..., each denoting a possible process-structure which '''synthesizes''', or produces, «monads» of ontological category/«arithmos» z,or qz, from the '''raw materials''' of the «monads» of y..., or qy....

All of the structures denoted in the '''inhomogeneous sums''', or 'cumula', of «arithmoi», symbolized by the lines of formulae above — '''hybrid''' and 'self-hybrid' alike — are posited as being merely possible existences, possible manifestations, that may or may not be instantiated, or actualized, in any particular instance of the generic process of human[oid]-species social history that is, per hypothesis, modeled by this sequence/succession/[self-]progression of series of interpreted 'meta-number unit[ie]s.  We interpret the abstract simplicity of the NQ meta-numbers space as being sufficient to describe, not a deterministic space of certainties, nor even a space of pre-determined probabilities, but, instead, as an ontological 'Possibility-Space'.

We will address only partially the general nature of such '''hybrid''', '''mixed''', or '''intermediate''' «arithmoi» in this Introductory Letter.  And, instead of exploring, in this introductory venue, the specific natures and meanings of the specific '''inter-mediating''', 'ontological conversion-formation', 'ontological synthesis-formation', '''real subsumption''', or '''hybrid''' «arithmoi» of «monads», predicted — or, rather, '''post-dicted''', '''reconstructed''' — by the 'dialectical-ideographical model' of Example 5 elaborated in this Section, i.e., by such terms as  qcb q3qvb q5qvc q6, and  qvcb q7, etc. — we use groups of '''ellipsis dots''' — '...' — to register our '''elision''' of the symbols for these «arithmoi» of «monads», below.

Given these provisos, we obtain the following symbolization of a temporal, diachronic, historical 'consecuum' — sequence, succession, or progression — of, generally, multiarithmos», multi-«genos», 'multi-ontic' '''accumulations''', or 'cumula', of «monads». That is, the 'ontological meta-states' of the universe of discourse of human-social formation(s), predicted — or '''retro-dicted''' — by this dialectical model, are 'multi-ontological' — consisting of «monads» of more than one [meta-]monadic, or 'onto-logical', type, in all '''epochs''' except for the initial, «arché», '''epoch''', the '''epoch''' denoted by τ = 0, whose '''post-dicted''' 'ontological meta-state' is given by the singular value b, or qb.  The resulting story-symbolization is therefore:

b 
b
+ c  →

b
+ c + ... + v 

b
+ c + ... + v + ... + f 

b
+ c + ... + v + ... + f + ... + s  →

b
+ c + ...+ v + ... + f + ... + s + ... + e .....

The «arithmoi»-symbols — in the succession of series, or '''sums''', of «arithmoi»-symbols, above — have qualitative denotation only. By the foregoing statement, we mean that these «arithmoi»-symbols are entirely devoid of any explicit, specific quantitative determinations. They do carry the implicit, generic quantitative determination that each denotes an «arithmoi», i.e., an assemblage of qualitative «monads», or units, consisting, by definition, of a multiplicity of «monads» — consisting of more than one «monad».

However, this 'qualitativity' notwithstanding, the succession above is a 'dialectical-mathematical' model. It is the result of an '''algorithmic calculation'''. At least, it is so, once the assignments have been decided. The assignments in question are those of the specific, hypothesized Possibility-Space's 'self-hybrid' 'socio-ontological categories', or '''onto-logical types''', of human-social formation(s), to the generic 'ontological qualifier meta-numbers' of the NQ generic 'Possibility-Space'. Note that the 'socio-ontological types' of human social formation(s) in question are the hypothesized 'socioarithmoi»'.

Clarity as to the conduct of this calculation requires that a few simple rules regarding 'the elements of generic ontological qualifier computations' be specified in greater detail before we proceed to describe that specific 'qualitative calculation' — of 'qualitative algorithmics' — as a whole.

The first such rule prescribes the operation of '''addition''', generalized to encompass the generic 'ontological qualifier meta-numbers' of the N dialectical arithmetic.  The rule, and its sub-rules, for this '''generalized addition''' operation, are as follows  —

Rule 1
'Generalized Addition' of 'meta-numerals' interpreted as generic 'ontological qualifier meta-numbers', or 'kind-of-being qualifier meta-numbers':

The NQ 'pure qualifiers' are '''non-linear''', '''non-additive''', or '''non-addable''' [cf. Plato's «asumblêtoi» numbers:  "From the Pythagoreans, ton arithmon nomizontes arkhen einai — who consider number to be the first principle (Ar. Met. 986a15) — number played a great part in metaphysics, especially in Plato's unwritten doctrines, involving obscure distinctions of e.g. sumblêtoi and asumblêtoi — addible and non-addible numbers." Urmson, J. O.; The Greek Philosophical Vocabulary; Gerald Duckworth & Co., Ltd.; London: 1990, pp. 31-32.]

PROBLEM BEGIN

Rule 1a

Two or more '''alike-in-kind''' 'ontological qualifiers', denoting two or more '''alike-in-kind''' 'ontological categories' — i.e., denoting, repetitiously, a single «arithmoi» of «monads» — occurring with (a) '+' sign(s) placed between them, being thus mutually '''redundant''' — and thus expressing an ideographical pleonasm — may be replaced by a single occurrence of their symbol.  That is, two or more '''alike-in-kind''' 'ontological qualifiers', occurring with a '+' sign placed between each successive pair, do not add in the sense that their '''addition''' is '''idempotent'''. The '''sum''' resulting, in idempotent addition, is just a singular occurrence of the '''redundant''' symbol:  x + x  ≠ 2x; instead x + x  = x.  This will be familiar for those already conversant with the '''Boolean arithmetic'' undergirding Boolean algebra0B + 0B  =   0B, and, perhaps more surprisingly, 1B + 1B  =   1B.

Rule 1b

Two or more '''unalike-in-kind''' 'ontological qualifiers', denoting two or more '''unlike-in-kind''' 'ontological categories', i.e., two or more qualitatively distinct «arithmoi» of «monads», occurring with (a) '+' sign(s) placed between them, being thus mutually ontologically heterogeneous, or different in terms of the kind-of-being that they represent — different in the '''unit''' [«monad»] of '''measure''' / '''unit''' [«monad»] of '''account''' that they represent in this sense — cannot combine, or "reduce", to any other, singular 'ontological qualifier' existing at the '''taxonomic level''', or '''meta-fractal scale''', of the discourse which conjoins them.  They therefore do not add in the sense that their '''addition''' is '''non-amalgamative'''.  Sums of ontologically heterogeneous 'ontological qualifiers' are irreducible:  if x  y, then x + y  =  x + y.

Some Illustrations of Rule 1b

(1) The proverbial '''apples + oranges''' is an example of a '''non-amalgamative''' or '''inhomogeneous''' sum [note, however, that, by breaking out of the implicit universe of discourse, and level of discourse, of the foregoing "sum"; by moving higher into abstraction, to the level, or 'meta-fractal scale', of the «genos», e.g., of '''fruit''', where the distinction between the "apples" «species» and the "oranges" «species» of that «genos» is lost to view, made invisible, indiscernible, or unmentionable, by regressing to the more abstract/less determinate ontological category of '''fruit''', which subsumes the ontological categories of "apples" and "oranges" as implicit sub-categories of itself, this inhomogeneous sum can be translated into the — more abstract / less concrete / less complex, homogeneous sum:  '''(1)fruit + (1)fruit  =  (2)fruit'''. ].

PROBLEM END

(2)  Consider the so-called "real" unit, which we denote here by r [ +1], in its conjoinment with the so-called '''imaginary unit''', i [ +-1], within the numbers-space of the so-called '''Complex Numbers''',   C { R + R·+1 } { R + R·i }, wherein R denotes the space, or set, of all '''Real numbers'''.  There does not exist, in all of C space, anywhere in the C space, any such number as z1 xr + 0i, or as z2 0r + yi, such that, e.g.,  1 + i = z1 or 1 + i = z2.  The number 1, or r, and the '''hypernumber''' i, do not mix; do not amalgamate; do not add '''amalgamatively'''.

Likewise, there is no generic 'meta-number', z, in NQ, and given x  NQ    y, and x   y, such that x + y  =  z.

Note
In NQ, if x  ≠ y, then x  y.  Quantitative difference is not expressible within the [ideographical] language of NQ. That is, there are no 'meta-numbers' in the space NQx  NQ   y, such that x  <  y, or such that x   >  y.

The second rule prescribes the operation of '''multiplication''', generalized so as to encompass the generic 'ontological qualifier meta-numbers', or '''kind-of-being''' 'qualifier meta-numbers', of the N dialectical arithmetic.  This second rule, as set forth below, defines what we name the '«aufheben» evolute product' of 'ontological qualifier meta-numbers' of the NQ «species» in general.  This rule stipulates the operation of the '''multiplication of ontological qualities''' as follows —

Rule 2
'Generalized Multiplication' for NQ  'meta-numbers' interpreted as 'ontological qualifiers': why and how x  ×  y means /models 'x «aufheben» y'.

If x qx and y qy are both in NQ  —  x  NQ    y, or, equivalently, qx   NQ    qy  —  and qx  qy, or x  y, then y  ×  x  x + Δ[ y; x ]    q×  qx   qx + Δ[ qy; qx ]      qx + qyx  ≡  x + qyx, and y  qyx  x, or, equivalently, qy   qyx   qx.


Likewise, x  ×  y   y + Δ[ xy ]     qx  ×  qy     qy  + Δ[ qx; qy ]      qy + qxy  ≡  y + qxy, and x   qyx   y, or, equivalently, qx   qxy  qy.

In terms of the 'minimally-interpreted', Natural Number subscripted, generic NQ  'meta-numbers', this product-rule means the following:

If j and k are Natural Numbers, members of the set/space N, then

qj  ×  qk   ≡  qk + qj+k, and
qk ×  qj   ≡  qj + qk+j.

Note
In NQ, under this product-rule, qxy = qyx.  The symbol qxy, and the symbol qyx, connote an ontological category which combines the connotations/intension of the ontological category denoted by y, or qy, with the connotations/intension of the ontological category denoted by x, or qx.

This '«aufheben» evolute product rule' is termed '''evolute''' — as opposed to '''convolute''' — because the '''multiplicand''', '''argument''', or '''operand'' — namely, the x in y × x, or y x ], or simply yx  ≡  x + qyx — is added back, in the expression for the outcome of this operation — in the '''product''' / '''function-value''' / '''result-of-operation''' — so that x is not '''erased''', or ''covered over / covered up''', as a result of the multiplication operation, but is '''present''' in that result as a separately-visible term. The source of the product, its '''past''' — the '''multiplier''', and, even more directly so, the '''multiplicand''', that created it — is visible in the '''present''' of their outcome, their '''product'''.

This '«aufheben» evolute product rule' is termed '''«aufheben»''' because, per it, the '''multiplicand''', '''argument''', or ''operand''' — e.g., the x in y × x, or y x ], or simply yx  ≡  x + qyx — is:

(1) as we have seen, in terms of its '''evolute''' quality, '''conserved''' or '''preserved''' as a presence in the product;

(2) '''canceled''', '''annulled''', or '''[determinately, or concretely, as opposed to abstractly] negated''', i.e., determinately changed, changed in its '''determinations''', or qualities / nature / character[istics], in this example, by y, or qy. The operator, y, acts so as to add the quality qyx to that pre-existing quality, x or qx, upon which qy has operated, or with which qy has interacted, or '''mixed''', thus creating or producing a qualitative change, in that qyx  qx, so that also x + qyx  qx, and;

(3)  '''elevated''' to the '''advanced''', or '''progressed''' level of adding a '''complex unity''', denoted by qyx, a term whose meaning / intension is co-determined by both the multiplier, y, or qy, and the multiplicand, x or qx — by both '''function''' and '''argument''', '''operator''' and '''operand''', as indicated by the encoding/symbolism. That symbolism, i.e., contains references to both y and x at the subscript level, in the '''elevation term''', '''qualitative increment term''', or '''ontological increment term''', qyx   Δ[ y; x ] in this example.

Summarizing
Per this '''product rule''', or 'qualifier multiplication rule', in the result of the expression y × x, or qy  ×  qx, namely, x + qyx, or qx + qyx

(a) y '''conserves''' x, in that x re-appears explicitly in the result, both as an x term added back, non-amalgamatively, to the other component of that result, qyx, and also, in that x, re-appears in that other component, in a '''subsumed''' position — '''subsumed''' by y;

(b) y '''negates''' x, in that the result of y acting upon x, namely x + qyx, is qualitatively different from x, i.e., x + qyx    x, and, indeed, x + qyx  x, thus affirming that  ¬x  =  x + qyx ], i.e., affirming that it is not the case that x  =  x + qyx ];

(c)  y '''elevates''' x, in that the result of y acting upon x, namely x + qyx, is a richer construct / intension / meaning than is either qy or qx alone; is more determinate richer in determinations less abstract, more concrete, more complex, that is qy by itself, or qx by itself.

Thus, in toto, the NQ  'meta-number' expression y × x, or yx ] the latter read off as '''y of x''' signifies 'y «aufheben» x'.

If x qx and y qy are both in NQ  —  x  NQ y, or, equivalently, qx   NQ qy  —  and '''synonymously''', i.e., if qx = qy, or equivalently, if  x = y, then y  ×  x  =  x  × x  =   qx  ×  qx   '''x of x''' x2   qx2   x + Δ[ x; x ]   x + Δ[ x ]    x + Δx     qx  + Δ[ qx; qx ]     qx  + Δ[ qx ]     qx  + Δqx     qx + qxx  ≡  x qxx, and x    qxx, or, equivalently, qx ≠ qxx; indeed, x  qxx, or, equivalently, qx  qxx.  Boolean algebra's '''Boolean arithmetic''' features not only idempotent addition, but idempotent multiplication as well.  If x is a Boolean variable, not only does x + x  =  x, as in 0B + 0B =  0B, and 1B + 1B =  1B, but also x × x  =  x, as in 0B × 0B =  0B, and 1B × 1B =  1B.  Indeed, Boole called his rule, that x2 = x, "The Fundamental Law of Thought". The arithmetic of the NQ 'meta-numbers', on the contrary, is 'contra-Boolean' in the strong sense that, if x is a variable ranging within NQ, then not only is x2  ≠  x, as in x2 >  x, or x2 <  x, but, in a deeper sense than that of quantitative inequality — one of ontological inequality — x2 x.

In terms of the 'minimally-interpreted', Natural Number subscripted, generic NQ  'meta-numbers', this '''special-case''' product-rule means the following:

If n is a Natural Number, a member of the set/space N, then
qn  ×  qn   ≡  qn + qn+n 
=  qn + q2n.

F.E.D. characterizes a product of the form  x × x, or x2, or xx ], as 'the selfaufheben» of x', because, in such a product:

(i)    x '''conserves''' x, in that x re-appears explicitly in the product, and does so in a double way, both by way of the x term being added back, non-amalgamatively, to the other component of that result, namely qxx, and also in that other component, by way of x as the '''doubled''' constituent of its subscript, and hence in a '''subsumed''' position — '''subsumed''' by x itself; thus constituting a 'self-subsumption' of x.

This is often interpretable as connoting a ['meta-monadizing'] 'self-internalization' of the «monads» of the x «arithmos», so as to form an ontologically new, unprecedented «arithmos», one whose «monads» are 'meta-«monads»' of the «monads» of the x «arithmos», as illustrated in the many examples of 'meta-monadologies' above.

(ii)    x '''negates''' its '''argument''', its '''operand''', its object — which, in this case, is x again, itself! — in that the result of x acting upon x, namely x + qxx, is qualitatively different from x, i.e., x + qxx    x, and, indeed, x + qxx  x, thus affirming that  ¬x  =  x + qxx ], i.e., affirming 'not x  =  x + qxx ]';

(iii)    x '''elevates''' x, in that the result of x acting upon x, namely Δx, or x + qxx, is a richer construct / intension / meaning than is qx alone; is typically interpretable as a ['meta-fractally'] '''higher''' «arithmos» than that denoted by x, in that the «monads» of Δx, or x + qxx, are 'metamonads»' of the «monads» of x.

Thus, in toto, the NQ  'meta-number' expression x × x, or xx ] — the latter read off as '''x of x''' — signifies 'x «aufheben» x', or 'the selfaufheben» of x'.

Dialectical-Mathematical, '''Psycho-Historical''' Shorthand for a Core Content of this Letter

The key terms of a major argument embedded in this Introductory Letter, in terms of the assignment [] of those terms to the first four 'meta-numbers' of the N dialectical arithmetic, used as unquantified, and 'unquantifiable', '[ideo-]ontological qualifiers', for the construction of an N  '''shorthand''' model of that argument, are the following

'«arché» thesis'  :       q1     qA ≡  A

denotes the «mentalité» of Ancient, Hellenistic humanity, as an 'ideo-ontological category';

'first contra-thesis' :    q2     qAA ≡ qM  ≡ M

denotes the «mentalité» of Modern humanity, as the protracted result of the self-confrontation, and of the immanent  [or self-]critique, of the Ancient «mentalité», forming the '''Phenome''' of '''Modernity''';

'first uni-thesis' :    q3     qMA

denotes that reconciliation, "'complex unity'", or '''dialectical synthesis''' of the Modern with the Ancient «mentalité» / '''meme pool''' / '''Phenome''' which is proposed in this letter; the re-assimilation of the Ancient «mentalité» and «problematique» to the Modern, '''hybridizing'''  and '''bridging'''  the two;

'second contra-thesis'  :    q4     qMM

denotes the self-confrontation, and immanent [or self-]critique, of the «mentalité» of '''Modernity''', as psycho-historical,  'ideo-ontological category'; fruition of the 'self-«aufheben»' self-negation of M.

After the '«arché» thesis  is posited, the '''systematic dialectic''' method of presentation of the model specified above proceeds as a cumulative one as a self-progressing 'cumulum-sum' of the terms stated above, in keeping with Hegel's principle of dialectical exposition:  "at each stage of its further determination it raises the entire mass of its preceding content, and by its dialectical advance it not only does not lose anything or leave anything behind, but carries along with it all it has gained and inwardly enriches and consolidates itself" [from Hegel's Science of Logic, page 840, as quoted on page 7 of Tony Smith's The Logic of Marx's Capital, published by SUNY Press (New York: 1990)].


Given the definitions/'''interpretations''' of the above-stated assignments, the 'Argument Generator', '[Method-of-]Exposition Function', or '[Method-of-]Presentation Operation'  for the heuristic, '''intensional-intuitional'''  unfoldment of this key content can be compactly expressed via the following formulae, per the rules of N 'qualifier addition', and 'qualifier multiplication', including its rules for the 'self-multiplication of qualifiers' 

qA ]two_to_the_taufor τ taking on the following values in succession:  0 1 2,

i.e.:

qA ]two_to_the_zero qA ]two_to_the_one [  qA ]two_to_the_two     [ q]     [ qA + qM ]     [ qA + qMqMA + qMM ].

argument_stages


A key element of the '''psycho-historical''' opposition between the Ancient
«mentalité» versus the Modern, emphasized in this letter, and explored at depth in Jacob Klein's Greek Mathematical Thought and the Origin of Algebra, cited at the outset, is manifested in the contrasting ways in which the Ancient versus the Modern «mentalités» perform the abstraction of ["Natural"] Number.

Consider, for example, a list of instances of 'quanto-qualitative'  statement-fragments such as the following

". . . two fish . . .";
". . . two loaves of bread . . .";
". . . two figs . . .";
". . . two yards of cloth . . .".

In this list, the word "two" functions as the '''quantifier''', and the words or phrases "fish", "loaves of bread", "figs", and "cloth" function as 'ontological qualifiers', while the word "yard" functions as a 'metrical qualifier'  [or as '''metrical unit  name''', i.e., as metrical «monad» name].

The Modern collective «mentalité», in constructing its arithmetic, simply extracts, abstracts, or 'genericizes' only the '''quantifier'''  from such phrases, 'notating' it via the Indo-Arabic ideogramic symbol '2'.

The Ancient collective «mentalité», in constructing its arithmetic its '''«Arithmoi Monad-ikoi»'''  ideographically extracts, abstracts, or 'genericizes' BOTH the '''quantifier''' and the 'qualifiers'  from such phrases.

Thus, Diophantus, in his «Arithmetiké», using its proto-ideographical notation for the '''«Arithmoi Monad-ikoi»''', would generically represent all occurrences of the word for "two" in such phrases, by the modified [bar-capped], and thereby 'proto-ideogram-ized', Greek letter /'''phonogram''', barred_beta.

But he would also represent all of the 'qualifier' occurrences in such phrases as the above e.g., "fish", "loaves of bread", "figs", "cloth", and "yard" as also forming a part of arithmetic, by means of his "syncopated" symbol for the abstract / generic qualitative unit, or «monad» his symbol for any one of the «Monads» of the [modified, fractions-admitting] Platonic '''«Arithmoi Monad-ikoi»''',  circle_over_M.barred_beta3.

Thus, Anciently, the arithmetical content of the four phrases above would all be represented, [proto-]ideographically, not just by the single symbol, '2', as per the Modern «mentalité», but by a dual proto-ideographical symbolization / notation, circle_over_M., denoting 'two generic units/«Monads»'.

This element of the opposition between the Ancient and the Modern «mentalités» can be visualized as follows

aufheben_meta-arithmosology

Our '''psycho-historical''' hypothesis to account for this opposition is the following:  that it manifests the «mentalité»-forming effects of the far deeper and more developed immersion of Modern humanity, vis-a-vis that of Ancient Hellenistic humanity, in the experience of exchange-value exchange as predominating social relation of production, an experience which, by increasingly obscuring the nature and even the very existence of the «Monad» of monetary-, and of capital-, value, unconsciously inculcates that 'Elision of the Qualifiers', and that 'Fetishism of Pure Quantity', which Modern mathematics so pervasively epitomizes.

[Note: The second 'contra-thesis', denoted qMM, is only hinted at in the expositions within this letter, with some of the elements of the implicit intensionality of the qMM symbol being addressed separately, in the sections, and in the contexts, where they are proximate, in content, to related elements of the main exposition.  But the unity of the multitude of elements implicit in the symbol qMM, and for which qMM stands, is not presented systematically, or as such, herein.  The presentation of the qMM 'contra-thesis' as a totality belongs to the sequel — to the supplementary texts to this letter, and to Dialectical Ideography itself].

The Ancient Alexandrian 'Proto-Renaissance' and the Civilizational Collapse into the Last Dark Age

The final  Platonic synthesis, which centered upon the idea of «autokinesis», failed to become a sufficient "material force", within or beyond the 'meristemal' developments of Hellenistic civilization pioneered by the ancient Alexandrians. That final synthesis proved to be insufficient to drive a [psycho-]historical-dialectical 'contra-Parmenidean'/'trans-Parmenidean' revolution in Hellenistic philosophical and proto-scientific theory, let alone in Hellenistic human social praxis.

Had it succeeded in so becoming, then perhaps the Occidental "Dark Ages", ten centuries of pseudo-Christian, theocratic totalitarianism and endless Inquisitorial atrocity, might have been averted, with accession to the higher system/'meta-attractor' of industrial capitalist civilization attained, ten centuries earlier! Admittedly, given the devlopment of the human social forces of production, and of the human social relations of production, in Ancient 'Mediteranea' up to that time, such a possibility was of low probability, at best.

Our present 'Dim Ages' civilization is but six centuries out of the ten centuries' "Dark Ages" into which that Ancient World collapsed. It holds the potential to open upward and outward into a future Age Of Light. Yet, and precisely for that reason, already the forces of those who fear that Light are gathered to close humanity back down again; to drag us all back into the abyss of a new and this-time-final Darkness.

We hold that a 'meta-fractal analogy' — a 'quanto-qualitatively' scaled, diachronically-deployed [self-]similarity — obtains between the human-social 'meta-state' of Mediterranean late antiquity, with its 'meristem' at ancient Alexandria, and that of our contemporary 'late modernity' global civilization, with its 'meristemal emergences' in a multiplicity of locales worldwide.

Mediterranean late antiquity verged upon 'industrial renaissance', the "take-off" 'meta-state' of an "industrial revolution", before Roman-Imperialized pseudo-Christian genocide plunged that world into a thousand years' abyss of Darkness.

The Roman imperium viciously repressed the early Christian movement for its dissent from the prevailing, Pagan, imperial ideology and way of life. That imperium found that early Christian movement increasingly both threatening to its power, and difficult to destroy. It also found potential in this movement for the ideological engineering of a new, state-crafted, state-perverted false religion, one of far greater 'ideological-efficiency' than Paganism.

That imperium therefore eventually adopted and imposed that perversion, switching from the repression and slaughter of Christians to the repression and slaughter of Pagansespecially of the "learned" ones, in whom it also perceived a growing threat to its rule.

It may be hard for us, '''with our noses to the grindstone''' of proximate concerns, to foresee the other side of the 'meta-evolutionary leap' upon which our own civilization is lately verging, and given an ideological vantage point, still buried deep within prehistoric times [in Marx's sense].

Further on, we will attempt to 'solve for' some features of that next system of civilization, using the new tools of the N dialectical algebra [to jump to this solution-attempt, click on the following linkSupplement B Part IV - addressing 'Equitism' as the Successor System to Capitalism (.pdf)

But, for now, perhaps it would be helpful — toward forming, via that analogy, a concrete imagination/vision of that other side of our own potential leap forward to consider some descriptions of the human-social 'meta-state' of late antiquity from the vantage of our side of its leap, a leap that was not completely aborted, but "merely" delayed, for ~50 generations, by the Occidental Dark Ages:

"The period following the Peloponnesian War was one of political disunity among the Greek states, rendering them easy prey for the now strong kingdom of Macedonia which lay to the north. King Philip of Macedonia was gradually extending his power southward and Demosthenes thundered his unheeded warnings. The Greeks rallied too late for a successful defense and, with the Athenian defeat at Chaeronea in 338 B.C.[E. — F.E.D.], Greece became a part of the Macedonian empire. Two years after the fall of the Greek states, ambitious Alexander the Great succeeded his father Philip and set out upon his unparalleled career of conquest which added vast portions of the civilized world to the growing Macedonian domains. Behind him, wherever he led his victorious army, he created, at well-chosen places, a string of new cities. It was in this way, when Alexander entered Egypt, that the city of Alexandria was founded in 332 B.C.[E.]. ... From its inception, Alexandria showed every sign of fulfilling a remarkable future. In an incredibly short time, largely due to its very fortunate location at a natural intersection of some important trade routes, it grew in wealth, and became the most magnificent and cosmopolitan center of the world. ..." [Howard Eves; An Introduction to the History of Mathematics (3rd edition); Holt, Rinehart & Winston (NY: 1969); pages 112-113].

***
"After Alexander the Great died in 323 B.C.[E.], his empire was partitioned among some of his military leaders, resulting in the eventual emergence of three empires, under separate rule, but nevertheless united by the bonds of the Hellenistic civilization that had followed Alexander's conquests. Egypt fell to the lot of Ptolemy. ... He selected Alexandria as his capital and, to attract learned men to his city, immediately began the erection of the famed University of Alexandria. This was the first institution of its kind. ... Report has it that it was highly endowed and that its attractive and elaborate plan contained lecture rooms, laboratories, gardens, museums, library facilities, and living quarters. The core of the institution was the great library, which for a long time was the largest repository of learned works to be found anywhere in the world, boasting, within forty years of its founding, over 600,000 papyrus rolls. It was about 300 B.C.[E.] that the university opened its doors and Alexandria became, and remained for close to a thousand years, the intellectual metropolis of the Greek race [and not of the Greek "race" alone, but of the Occidental Afro/Euro/Near-Asian hemisphere of humanity entire! — F.E.D.]." [Ibid.; page 113].

***

"No other city has been the center of mathematical activity for so long a period as was Alexandria from the days of Euclid (ca. 300 B.C.[E.]) to the time of Hypatia (A.D. 415 [C.E.]). It was a very cosmopolitan center, and the mathematics that resulted from Alexandrian scholarship was not all of the same type. ..." [Carl Boyer, Uta Merzbach; A History of Mathematics (2nd edition); John Wiley & Sons, Inc. (NY: 1991); page 178].

***
"About 290 B.C.[E.] Ptolemy Soter built a center in which scholars could study and teach. This building, dedicated to the Muses, became known as the Museum, and it housed poets, philosophers, philologists, astronomers, geographers, physicians, historians, artists, and most of the famous mathematicians of the Alexandrian Greek civilization. Adjacent to the Museum, Ptolemy built a library, not only for the preservation of important documents but for the use of the general public. This famous library was said at one time to contain 750,000 volumes, including the personal library of Aristotle and his successor Theophrastus. Books, incidentally, were more readily available in Alexandria than in classical Greece because Egyptian papyrus was at hand. In fact, Alexandria became the center of the book-copying trade of the ancient world. The Ptolemies also pursued Alexander's plan of encouraging a mixture of peoples, so that Greeks, Persians, Jews, Ethiopians, Arabs, Romans, Indians, and Negroes came unhindered to Alexandria and mingled freely in the city. Aristocrat, citizen, and slave jostled each other and, in fact, the class distinctions of the older Greek civilization broke down. The civilization in Egypt was influenced further by the knowledge brought in by traders and by the special expeditions organized by the scholars to learn more about other parts of the world. Consequently, intellectual horizons were broadened. The long sea voyages of the Alexandrians called for far better knowledge of geography, methods of telling time, and navigational techniques, while commercial competition generated interest in materials, in the efficiency of production, and in improvement of skills. Arts that had been despised in the classical period were taken up with new zest and training schools were established. Pure science continued to be pursued but also applied science." [Morris Kline; Mathematical Thought from Ancient to Modern Times (volume I); Oxford University Press (NY: 1972); pages 102-103].

***
"About 290 B.C.[E.] Ptolemy Soter built a center in which scholars could study and teach. This building, dedicated to the Muses, became known as the Museum, and it housed poets, philosophers, philologists, astronomers, geographers, physicians, historians, artists, and most of the famous mathematicians of the Alexandrian Greek civilization. Adjacent to the Museum, Ptolemy built a library, not only for the preservation of important documents but for the use of the general public. This famous library was said at one time to contain 750,000 volumes, including the personal library of Aristotle and his successor Theophrastus. Books, incidentally, were more readily available in Alexandria than in classical Greece because Egyptian papyrus was at hand. In fact, Alexandria became the center of the book-copying trade of the ancient world. The Ptolemies also pursued Alexander's plan of encouraging a mixture of peoples, so that Greeks, Persians, Jews, Ethiopians, Arabs, Romans, Indians, and Negroes came unhindered to Alexandria and mingled freely in the city. Aristocrat, citizen, and slave jostled each other and, in fact, the class distinctions of the older Greek civilization broke down. The civilization in Egypt was influenced further by the knowledge brought in by traders and by the special expeditions organized by the scholars to learn more about other parts of the world. Consequently, intellectual horizons were broadened. The long sea voyages of the Alexandrians called for far better knowledge of geography, methods of telling time, and navigational techniques, while commercial competition generated interest in materials, in the efficiency of production, and in improvement of skills. Arts that had been despised in the classical period were taken up with new zest and training schools were established. Pure science continued to be pursued but also applied science." [Morris Kline; Mathematical Thought from Ancient to Modern Times (volume I); Oxford University Press (NY: 1972); pages 102-103].

***

"The mechanical devices created by the Alexandrians were astonishing even by modern standards. Pumps to bring up water from wells and cisterns, pulleys, wedges, tackles, systems of gears, and a mileage measuring device no different from what may be found in the modern automobile were used commonly. Steam power was employed to drive a vehicle along the city streets in the annual religious parade. Water or air heated by fire in secret vessels of temple altars was used to make statues move. ...Water power operated a musical organ and made figures on a fountain move automatically while compressed air was used to operate a gun. New mechanical instruments, including an improved sundial, were invented to refine astronomical measurements." [Ibid.; page 103].

***

"Proclus refers to Heron as mechanicus, which might mean a mechanical engineer today, and discusses him in connection with the inventor Ctesibius, his teacher. Heron was also a good surveyor. ... The striking fact about Heron's work is his commingling of rigorous mathematics and the approximate procedures and formulas of the Egyptians. On the one hand, he wrote a commentary on Euclid, used the exact results of Archimedes (indeed he refers to him often), and in original works proved a number of new theorems of Euclidean geometry. On the other hand, he was concerned with applied geometry and mechanics and gave all sorts of approximate results without apology. He used Egyptian formulas freely and much of his geometry was also Egyptian in character. ...His applied works include Mechanics, The Construction of Catapults, Measurements, The Design of Guns, Pneumatica (the theory and use of air pressure), and On The Art of Construction of Automata. He gives designs for water clocks, measuring instruments, automatic machines, weight lifting machines, and war engines." [Ibid.; pages 116-117].

Consider in this regard also the circa 80 B.C.E., purportedly «anthyphairesis»-employing complex-gearing-based Orrery-computer known as the "Antikythera Mechanism". [see, for example: D. H. Fowler; The Mathematics of Plato's Academy; Clarendon (NY: 1987); pages 223, 364].

"The highest point of Alexandrian Greek algebra is reached with Diophantus. ... His work towers above that of his contemporaries; unfortunately, it came too late to be highly influential in his time because a destructive tide was already engulfing the civilization. Diophantus wrote several books that are lost in their entirety. ...His great work is the ArithmeticaDiophantus says, comprises thirteen books. We have six [surviving in Greek; 4 more were recently found, in Arabic translation, possibly translations of Hypatia's commentaries on books 4 through 7, rather than of Diophantus' originals — which, F.E.D.] ... One of Diophantus' major steps is the introduction of symbolism [i.e., of proto-ideographyF.E.D.] in algebra.... The appearance of such symbolism is of course remarkable but the use of powers higher than three is even more extraordinary. The classical Greeks could not and would not consider a product of more than three factors because such a product had no [then-known — F.E.D.] geometrical significance [i.e., given the apparently 3-and-no-more/no-less-dimensional physical space of our world — F.E.D.]. On a purely arithmetical basis, however, such products do have a meaning; and this is precisely the basis Diophantus adopts." [Ibid.; pages 138-139].

***

"The death of Archimedes portended what was to happen to the entire Greek world. In 216 B.C.[E.] Syracuse allied itself with Carthage in the second Punic war between that city and Rome. The Romans attacked Syracuse in 212 B.C.[E.]. While drawing mathematical figures in the sand, Archimedes was challenged by one of the soldiers who had just taken the city. Story has it that Archimedes was so lost in thought that he did not hear the challenge of the Roman soldier. The soldier thereupon killed him, despite the order of the Roman commander, Marcellus, that Archimedes be unharmed." [Ibid.; page 106].

***

"The fate of Hypatia, an Alexandrian mathematician of note and the daughter of Theon of Alexandria [the redactor of Euclid's ElementsF.E.D.], symbolizes the end of the era. Because she refused to abandon the Greek religion, Christian fanatics seized her in the streets of Alexandria and tore her to pieces." [Ibid.; page 181].

***

"From the standpoint of the history of mathematics, the rise of Christianity had unfortunate consequences. Though the Christian leaders adopted many Greek and Oriental myths and customs with the intent of making Christianity more acceptable to converts, they opposed pagan learning and ridiculed mathematics, astronomy, and physical science; Christians were forbidden to contaminate themselves with Greek learning. Despite cruel persecution by the Romans, Christianity spread and became so powerful that the emperor Constantine (272-337 [C.E.]) was obliged to consign it a privileged position in the Roman Empire. The Christians were now able to effect even greater destruction of Greek culture. The emperor Theodosius proscribed the pagan religions and, in 392 [C.E.] ordered that the Greek temples be destroyed. Pagans were attacked and murdered throughout the empire. Greek books were burned by the thousands. In that year Theodosius banned the pagan religions, the Christians destroyed the temple of Serapis [in Alexandria — F.E.D.], which still housed the only extensive collection of Greek works. It is estimated that 300,000 manuscripts were destroyed. Many other works written on parchment were expunged by the Christians so that they could use the parchment for their own writings." [Ibid.; pages 180-181].

***

"In 529 [C.E.], the Eastern Roman emperor Justinian closed all the Greek schools of philosophy, including Plato's Academy. ...The final blow to Alexandria was the conquest of Egypt by the upsurging Moslems in ... 640 [C.E.]. The remaining books were destroyed on the ground given by Omar, the Arab conqueror: "Either the books contain what is in the Koran, in which case we do not have to read them, or they contain the opposite of what is in the Koran, in which case we must not read them." And so for six months the baths of Alexandria were heated by burning rolls of parchment. After the capture of Alexandria by the Mohammedans, the majority of the scholars migrated to Constantinople, which had become the capital of the Eastern Roman Empire. Though no activity along the lines of Greek thought could flourish in the unfriendly Christian atmosphere of Byzantium, this flux of scholars and their works to comparative safety increased the treasury of knowledge that was to reach Europe eight hundred years later. It is perhaps pointless to contemplate what might have been. But one cannot help observe that the Alexandrian Greek civilization ended its active scientific life on the threshold of the modern age. It had the unusual combination of theoretical and practical interests that proved so fertile a thousand years later. Until the last few centuries of its existence, it enjoyed freedom of thought, which is also essential to a flourishing culture. And it tackled and made major advances in several fields that were to become all-important in the Renaissance: quantitative plane and solid geometry; trigonometry; algebra; calculus; and astronomy. It has often been said that man proposes and God disposes. It is more accurate to say of the Greeks that God proposed them and man disposed of them. The Greek mathematicians were wiped out. But the fruits of their work did reach Europe..." [Ibid.; Morris Kline; Mathematical Thought from Ancient to Modern Times (volume I); page 181].

Analyzing recent trends in ideology, within the overall pattern of recent Terran human '''psycho-history''', and correlating their hidden initiators, just discernible at the shadowy edges of events, with the key 'contra-temporal terms' of the '''psycho-historical equations''' of dialectical ideography [Note: Such 'reaction[ary] terms' emerge explicitly only beginning in the Z stage of the dialectic within the   dialectical ideography. These terms lack explicit or separable  'expressibility'  within  both of Z's predecessor ideographical-dialectical languages, namely, within both the N and the W languages], we conjecture as follows [for we can claim no direct, "insider knowledge" of these matters]:

Global Strategic Hypotheses — Towards A Strategy for Humanity:
A Gloss on Some Recent Chapters of the 'Hidden History' of The Human Species

symbol 12 Now, once again, the further development of our this-time-global human civilization, of widespread prosperity, society-wide education, and political democracy; of mathematics, science, and competitive-capitalism-driven technological progress, "the growth of the productive forces", i.e., of humanity's 'self-productivity', threatens the power of today's imperium: of the iron-fisted 'invisible hand'; of the 'invisible dictatorship' of the oil/finance plutocracy; of the global 'Dictatorship of Petroleum'.

In particular, the advent of nuclear fusion technology alone would wipe out key financial foundations of their rule.

In general, fixed capital-value accumulates in increasing ratio to circulating capital-value as the accelerating historical accumulation of capital-value, and the acceleration of capital-value-productivity-increasing technological advances which 'capital-value-profit-ism' incents and incites, proceeds, thereby acceleratedly expanding the cross-section of vulnerability of especially the greatest conglomerations of accumulated fixed-capital-value, which are the monopoly of the capitalist plutocracy — threatening the asset-value of both their directly-owned fixed capital equipment, and of their banks' massive loans, which financed fixed-capital construction/purchase by others — to technological or obsolescence de-valuation or value-obliteration, potentially wiping out the economic foundation of the capitalist plutocracy's political power, which power they value above all else.

This is the economic foundation of today's secret, hidden, ruling ideology of 'capitalist anti-capitalism'.

As a result, that capitalist plutocracy has turned anti-capitalist; has turned against science and technology, against industrial progress, and even against economic growth in so far as they all benefit the majority of humanity.

The innermost, ruling faction of this petroleum/finance imperium, the faction of the plutocracy that we call, for '''psycho-historical''' reasons, the 'Rocke-Nazis', has turned ever-more-absolutely against the popular culture of scientific, technological, public-educating, and middle-class, competitive capitalism.

The Rocke-Nazis have concluded that Marx was, essentially, correct: that what the growth of the social-productive forces brings namely an increasing[ly] middle-class, politically-democratic, competitive-capitalist, increasingly educated, increasingly inventive, increasingly prosperous, and therefore increasingly independent [more difficult to prostitute] humanity is increasingly plutocratically, timocratically ungovernable; is ungovernable by them, despite all of their vast wealth and power.

Nor does this faction hope, any longer, that a negative-population-growth, negative-economic-growth, neo-feudal New Dark Age can, alone, even with a thorough 'de-education' and 're-peasantization' of the surviving masses under their crypto-aristocratic rule, make humanity again governable by them for long.

The 1990s global "technology-boom", its [former-]Third-World 'industrialization boom', and, especially, its global 'democracy boom' after the sweeping, swift, and amazingly non-violent fall of some of the most vicious and violent totalitarian dictatorships the world has ever seen, the Stalinist dictatorships of Russia/Eastern Europe, terrified them to their craven core.

Widespread prosperity, education, political democracy; widespread popular knowledge of mathematics and science, competitive-capitalism-driven technological progress, and global cultural renaissance will lead to, among many things which reflect blessings for humanity, but that are anathema to the Rocke-Nazis, the emergence of ever more new billionaires; new potential Rocke-Nazis rivals; new Steve Jobses, new Ross Perots, new Bill Gateses, new George Soroses, who may not choose to join the Rocke-Nazis 'Country[ies]-Club', or to play by the Rocke-Nazis' rules.

The Rocke-Nazis felt compelled to mobilize some of the vast quantities of liquidity that are under their ownership or control, to engineer risky "asset-pricing bubbles", through their usual 'reverse money-laundering' techniques, to bring these potential rivals down — the Japanese economy, the Asian "Tigers", the California/Silicon Valley high-tech industry, the internet "dotcoms", etc.

To understand the current Rocke-Nazi 'humanocidal culminant', it must first be understood, especially in '''psycho-historical''' terms, how and why coercive power is an absolute addiction for the Rocke-Nazis, the only thing that, bottom-line, really matters to them; how no amount of murder is, to them, too high a price to pay to retain it. They would, can — and do! — murder their own mothers if mother gets in the way of power. The hyper-deSadean, Sadism-inducing tenor of their upbringing is also key.

This newer capitalist anti-capitalism has lately converged, in their deliberations, with an older neo-/crypto-Malthusian, neo-/crypto-Social-Darwinist human anti-humanism.

They have concluded that the human genome itself, with the proneness to 'exo-genetic' learning that it engenders, the capacity for innovation — and for revolt — that it bestows, has got to go [except, of course, for them, with their technologically-extended life-spans].

This ruling faction is therefore rapidly readying its 'meta-fractally analogous' pseudo-solution; its 'humanocidal' Final Dark Age; its genomically-engineered '''Final Solution to the Humanity Problem''' [for which its [initially] servant-dictator Hitler's "Final Solution to the Jewish Problem", and its [initially] servant-dictator Stalin's gulags, were but partial and local dress rehearsals].

They are preparing a genetically-engineered, successor 'designer disease' pandemic to their mere "population control" AIDS virus, and arranging for a global pandemic that will, quite quickly, once they unleash it, wipe out the genomic human race globally. [For more on AIDS as a Rocke-Nazi, multi-genocidal, ''genetically-engineered'' 'designer disease', see pages 21-33, et passim., in Dr. Leonard G. Horowitz's Emerging Viruses (Tetrahedron, Inc.; Sandpoint: 1999) as well as Dr. Robert Strecker's "AIDS Is Germ Warfare", as well as his The Strecker Memorandum, published by The Strecker Group (Los Angeles, 1988)].

Only the Rocke-Nazis will have the antidote. And we, their present human slaves, will be replaced by the race of genomically-engineered, congenitally-servile, pandemic-immune, chimaeric slave drones, '''chromosomically incapable''' of deep learning, of technological innovation, or of revolt, that their secret laboratories are now so frantically fabricating, with exponentially expanding expenses.

Ruling classes have always had a "hate-love relationship" with their slave-classes, have always hated the resistance of their slaves to those ruler's continual rape of their slaves lives; hated their slaves' ever-present threat of revolt, and slaughtered as many of their slaves as they could afford to lose when revolts did erupt. But those rulers also abjectly depended upon those slaves for the work and the services which sustained those rulers' "lavish" lives.

The Rocke-Nazis now believe that they can put an end to all of this once and for all; that human [bio-]technology has reached, at last, a level such that they can finally afford to exterminate virtually all of their present human "slaves"; the vast majority of the human race.

The Rocke-Nazis are 'the humanocidal culminant', the ultimate canalization of the total history of human tyranny. But, perhaps, the very idea of engineering a sub-humanoid species that can work but not learn, serve but not innovate, and survive, being victimized and violated daily, and yet not revolt, is a contradiction in terms. Perhaps the 'partial humanocide' which the Rocke-Nazis intend can only actualize / eventuate, if "successful", in a 'total genomocide': 'human-species-suicide'. And, perhaps, the desperate, life-despairing 'annihilism' of the pseudo-Islamicist "suicide bombers" is but one more sign of this 'humanocidal contra-zeitgeist'.

Former German military intelligence operative Hitler, initially assigned to spy on the "National Socialist" party, was later re-assigned, with massive financing, to take over that party and the German nation, and to steer Germany into war with Russia.

The progenitors of the Rocke-Nazi faction in the City of London and in New York feared the threat to their global dominance that a German-Russian alliance, combining German science and industry with Russian science and labor-/natural-resources, would pose.

But Hitler soon perceived his potential to replace his masters with himself, made a pact with Stalin [to postpone their showdown 'til later], and turned on his [former] masters to the West. Those erstwhile masters had eventually to endure great costs to themselves, e.g., to ally with 'Franken-dictator' Stalin, and to institute the somewhat pro-labor "New Deal" reforms in the U. S., to enlist their middle/working classes in the armed overthrow of their former henchman Hitler, who now threatened the nascent Rocke-Nazis' very survival.

The lesser "servant-dictators" spawned worldwide by the Rocke-Nazi imperium so ubiquitously ever since also tend, like Hitler before them, quite regularly, indeed, "lawfully", to transform themselves into 'Franken-dictators'. Need we mention Ngo din Diem, the Shah of Iran, Marcos, Noriega, Idi Amin, Milosevich, Saddam Hussein, etc., ad nauseam?  They tend to turn on their Rocke-Nazi masters like Frankenstein's monster turned on its creator; seeking to become the masters of their [former] masters; seeking to make their masters 'former', by seizing the imperium for themselves.

The Rocke-Nazi apparatus must therefore suddenly reverse gears, and rush to forcibly remove their turncoat dictators, often at great ideological, political, and economic cost to the Rocke-Nazis. That is one of their greatest weaknesses.

After World War II, it was only the horrific parasitism of the Rocke-Nazi's CIA-installed, CIA-sponsored, and CIA-underwritten servant-dictatorships across the globe [with their CIA-trained, CIA-funded, CIA-armed secret police, and their CIA-trained rape/torture/death-squad government terrorists, who, on CIA orders, massacred several generations of pro-democracy nationalists throughout the Third World], that made 'pseudo-socialism', i.e., Stalinist proto-state-capitalism, so-called "Communism" — which at least promised a little industrialization in return for its brand of totalitarian viciousness — look marginally more attractive.

The U.S. government/CIA foreign policy of enforcing "non-Communist" police-state dictatorships throughout the Third World thus served as the primary recruiting force for this "Communism", keeping its moldering corpse seemingly alive for so long, far beyond its "natural" life-span, thus keeping vast corporate welfare payments to the "defense" industry/'''military-industrial complex''', and to the Rocke-Nazi banks which financed the deficit spending supporting that corporate welfare system, alive as well.

But recently, the Rocke-Nazis were stricken with horror by the largely non-violent overthrow of the Stalinist dictatorships at their core, in Russia and Eastern Europe. They — aided and abetted by their University of Chicago hired liars, their "shock treatment" social-torturers — engineered, and continue to engineer a horrific revenge and punishment upon the Russian people for their democratic aspirations, subjecting them to Weimar-Germany-like conditions in an effort to discredit democracy, and in the hopes of turning the Russian people into neo-Nazis.

But the Rocke-Nazis know only too well that they have to be exceedingly careful, and not move too quickly, as they move toward their ultra-horrific and ultra-criminal goal. Were a misstep on their part to awaken the people too early from their divided-and-conquered, Balkanized mentalities and foci — from their ideological slumber — then the Rocke-Nazis would face the same kind of nonviolent, overwhelmingly popular overthrow to which their Stalinist brethren in Russia and Eastern Europe recently succumbed.

The stakes would be so much higher, and the popular motivation for that overthrow so vastly the greater — the interdiction of 'humanocide' — than even in the case of that overthrow of the utter viciousness of Stalinist 'pseudo-socialism'.

Before they can get their '''Final Solution to the Humanity Problem''' in place, they will prepare the way for it economically — hence politically — by crushing the global middle class, including by collapsing majority U.S. living standards to Third World levels. They are gearing up for the massive middle class repression that this will require. In the United States, the heartland of democracy, the last vestige of the New Deal "social safety net", the Social Security system, is being targeted for destruction, along with the remnants of the private pension system. The police state rule requisite to repressing the public reaction to the crushing of middle class living standards is nearing readiness, via the Rocke-Nazis' USA "PATRIOT" Act [whose true name is "The USA TRAITORS' Act"], and via their new, KGB-clone "Department of Homeland Security" and domestic/foreign "Intelligence Czar", plus their recent FCC-foisted monopolization of the mass media, and the federally-imposed nationwide installation of unauditable, hackable electronic voting machines.

The Rocke-Nazis reason that, if they can crush what remains of democracy at the center, they can quickly mop-up its motley mementos everywhere else soon thereafter. The "PATRIOT" Act's unconstitutional "legalization" of spying by the new 'U. S. secret police' on church meetings and other public meetings of U. S. citizens; of secret police surveillance of U. S. citizen's reading choices at libraries and bookstores, with prosecution of librarians and bookstore owners who reveal that spying; of secret searches of citizens homes, without notification; of declaring as "terrorists" citizens who merely assemble peaceably to petition their government for redress of grievances, plus the claim that usurper/"President" Bush has the right to violate, at whim, the FISA law, U.S. anti-torture laws, and any other laws, by "executive order" decree, and to declare any U. S. citizen an "enemy combatant", thus to strip that citizen of all civil liberties, to incarcerate that citizen indefinitely, without due process of law, without any access to judicial review, to torture that citizen if the usurper/"President" so decides, and to execute that citizen upon conviction by a "kangaroo court" of military officers, proceeding in secret, without even the right of an attorney for the defense, has already publicly proclaimed usurper/"President" Bush to be an absolute dictator, with only feeble public opposition to-date.

The training and inculcation of U. S. citizen soldiers — volunteer/professional, and national guard — by U. S. "intelligence", in the warrant-less invasion of Iraqi homes, and the lawless, trial-less incarceration, torture, and execution of Iraqi citizens, will soon come home to roost.

The video tapes documenting the worst of the Rocke-Nazi crimes against humanity committed at Abu Graib, and throughout Iraq and Afghanistan, and in other nations as well, under the auspices and the orders of U. S. "intelligence", have not been shared with the American people by "their" government. We were shown only the humiliations. The videos of the tortures, the rapes, and the murders of Iraqi citizens have been withheld from us.

Soon it will be American citizens, stateside — any who dare to disagree with the gathering Bush/Rocke-Nazi dictatorship — whose homes will be invaded in the wee hours; who will be incarcerated without warrant, without due process of law; tortured, raped, and murdered by the Bush/Rocke-Nazi 'New Gestapo' — if the American people fail to awaken from their denial, and to put a stop to the Bush/Rocke-Nazi plutocratic-theocratic, totalitarian destruction of their nation, and of the rest of the world.

The American people face an horrific 'karmic reflux': If they continue to allow their Rocke-Nazi-prostituted, Rocke-Nazi-perverted government to visit servant-dictator rapacious tyrannies upon their fellow human beings in the Third World and, indeed, worldwide, then they will reap what they have sown; they will reap a like rapacious tyranny in their own lives, in their own homes, in the USA.

They will so reap because the predictably desperate, suicidal, terrorist violence of the reaction to those tyrannies by those fellow human beings — by those remaining Third World citizens who are left alive after the pro-democracy nationalists have been massacred by those tyrannies under CIA direction — including those survivors' counter-attacks upon the U. S., will be used, by the Rocke-Nazi-captured U.S. government, as an excuse to complete the «de facto» repeal of the U. S. Constitution and Bill Of Rights already underway, and to institute a fundamentalist, terrorist, anti-feminist, totalitarian, pseudo-Christian theocratic police-state in the U.S., as a 'karmic' mirror-image of the fundamentalist, terrorist, anti-feminist, totalitarian, pseudo-Islamic theocratic police-states that the Rocke-Nazis have already engineered and provoked in the Third World, via especially the Saudi branch of their global Dictatorship of Petroleum.

Along the way to their '''Final Solution''', the Rocke-Nazis will turn all of Terran human society, everywhere, into Beirut, into Lebanon, into Palestine; into Afghanistan, into Iraq — into One Big Concentration Camp/Extermination Camp, globally.

The Rocke-Nazis will try to turn the whole world into that into which they have already converted large parts of the world — into Pol Pot Cambodia, into Sarajevo, into Bosnia, into Chechnya, into Somalia, into Haiti, into East Timor, into Columbia, into Argentina, into Ruwanda, into Myanmar, into Darfur, into post-tsunami Bonder Ace, into post-earthquake Kashmir, into post-Katrina New Orleans, into Lebanon. All these have been but their dress rehearsals.

They are masters of mass manipulation, masters at dividing and conquering, at profiling the American people mass-psychologically for the weaknesses that they can exploit; masters at the [cf. Bamford] "Operation Northwoods"-style fabrication of "events", by which they can fool us into..., e.g., going to war over which end of the egg we open [cf. Jonathan Swift].

Merely political checks and balances against governmental abuse of political power, such as those built into the U. S. Constitution, fail once the unchecked economic power of the core capitalist plutocracy as a "lawful" consequence of "The Historical Tendency of Capitalist Accumulation" [Marx, Karl; Capital, Volume I, Chapter 32]; of the hyper-centralization, hyper-consolidation, and hyper-concentration of capital-value ownership via their "lobbyists", etc., burgeons to the point where it can prostitute the executive, legislative, and judicial branches of government, buying them out lock, stock, and barrel! Only an accession to comprehensive, political-economic democracy, holds out any hope of also restoring merely political democracy  [For an exposition of the F.E.D.-predicted successor '''social relation of production''' residing at the heart of the successor system to '''capital-ism''' — i.e., the present human-social system, founded upon the '''capital-relation''', or '''wage-labor relation''' — the relation of alienated [sold] labor/alienated [sold] life-time — a successor social relation that we term 'generalized equity', beginning with the institution of 'externality equity', plus an exposition of the economic-democratic core of the resulting 'Equitist' Society, enabling, also, the restoration and fulfillment of political democracy, see Supplement B to this Letter, via the following link: Supplement B Part IV - addressing 'Equitism' as the Successor System to Capitalism (.pdf) ].

If the people of the United States lean toward the "New Republican Party" [more aptly termed the 'Rape-the-Public-an' Party, for its policy of the rampant rape and pillage — privatization, crony-confiscation, and other "primitive accumulation" — against all public assets, and, on the other side, of the foisting "socialization" of private crony-corporate costs upon the taxpayers of the working/middle class], then the Rocke-Nazis will try to destroy the U. S. middle class, and its remnant democratic institutions, via a new round of "primitive accumulation" at the expense of that middle class — via the looting of the U. S. taxpayers' Treasury, the Social Security "Trust Fund", and of corporate private pension-plans; via all of the corporate-welfare-proliferating, hyper-rich hyper-re-enriching/middle-class impoverishing policies that we have seen ever since the Bush «coup d'état». The Rocke-Nazis will destroy public education with their "'All Children Left Behind''' inculcation of just one exam — aimed at enforcing ignorance of all else — and with their anti-scientific, pseudo-Christian "Creation Science/Intelligent Design" suppression of even what little still remains of science education in the U.S. public schools.

There is, ever-increasingly, throughout the epoch of the predominance of the capital-value-relation among, and over, all of the other historically-created, and still conserved, human-social relations of human society re-production — and, hence, presently-burgeoning more than ever before — a  fundamentally healthy human reaction against the rampant growth of social atomization, social alienation, anomie, and demoralization, engendered and continually accelerated, throughout the capital-epoch, by the capital-relation-immanent, ever-intensifying substitution of the cash nexus for all other forms of human involvement and connexion; of "the exchange-value" for all other species of human value, in this ever-increasingly 'capital-morphic' '''world of strangers''', this [pseudo-]society of self- and other-estrangement, of universal [self-]alienation, i.e., of universal [self-]selling, of universal prostitution.

The Rocke-Nazis will continue to invest the billions of dollars of labor-created value, extracted out of the hides of all of the rest of us, but confiscated/controlled by the Rocke-Nazis, pouring it into their massive "ideological engineering" projects, to pervert this humane reaction into a mass recruiting tool for the pseudo-Christian, Inquisitorial, mass-murderous, theocratic-totalitarian ideology that they are concocting.

The Rocke-Nazis do this so as to foist an experience of authoritarian pseudo-community — as an «ersatz» substitute for the [seemingly nowhere any longer available] genuine article upon the Rocke-Nazi-impoverished, former U.S. middle-class; the down-sized, out-sourced, pharmaceuticals-poisoned, desperate and despairing American workers  via the Rocke-Nazi's lavishly-financed, pseudo-Christian, theocratic-totalitarian, racist, females-denigrating, and genocidal pseudo-churches so as to yet more drastically deepen the American people's already-egregious ideological enslavement.

The Rocke-Nazis are ever trying to twist this humane reaction into yet further fodder for their New Dark Ages pseudo-Christianity; for a fundamentalist, theocratic-totalitarian dictatorship, with its 'New Inquisition', mirror-opposing the maniacal pseudo-Islamic fundamentalism that they have also covertly created, via, especially, the Saudi branch of their global 'Dictatorship of Petroleum', with its massive ideological engineering and bankrolling of the Wahhabist perversion of the humane core of Islam.

After the "Soviet" Union self-dissolved, the Rocke-Nazis desperately needed a new "national security threat", to "justify" the re-escalation of "Defense" spending corporate welfare, of Federal deficits and public debt, with the vast taxpayer-borne burden of the debt-service payments they generate [more corporate welfare to the giant Rocke-Nazi banks, which "loan" the U. S. government the money to finance the deficit], and to help "justify" their «de facto» repeal of the U. S. Constitution and Bill Of Rights.

Of course, any 'Rape-the-Public-an', oil-executive-stuffed regime worthy of the name "will look the other way" on anti-price-fixing law enforcement when its buddies in the Rocke-Nazi oil companies raise oil prices, due to "oil shortages" from "greater demand", of course, despite the end of the Iraq oil embargo, and the return of the huge Iraqi oil supply to the world market! So the Bush 'public tax' cuts for the rich are seconded by his 'private tax' hikes for the rest of America, via the Rocke-Nazi oil companies' "taxation without representation".

It's so much easier now than ever before, today, for the Rocke-Nazis to rule, and to race toward their 'Final Solution', via the ideological infrastructure they have lately fabricated, and now operate, via their 'Rape-the-Public-an', mostly unwitting, and, to the Rocke-Nazis, «untermenschen» agents; the 'drooling greedies'; the "Christian", money-worshipping, Mammon-idolizing, monetary-status-obsessed  "New Republicans", than it was before, trying to foist their 'Meta-Nazi', "People are Pollution"  neo-pseudo-religion of 'Earth-ism'  via the Democratic Party.

After all, the Rocke-Nazis' ancient, Roman-imperial progenitors succeeded in bringing off a Dark Ages once before, using an earlier version of pseudo-Christianity [one also diametrically opposed to the actual teachings of Jesus of Nazareth, in theory and, even more massively more so, in practice [see, for example:  Rocco Errico; And There Was Light; Noohra Foundation (1998)]].

The "Old", true Republicans, and traditional conservatives in general, heeding the wisdom of the U. S. Founding Fathers, and 'Founding Mothers', about how human beings cannot be trusted with unchecked power; about how '''power corrupts, and absolute power corrupts absolutely''', worried about, and fought against, excessive, unbalanced concentrations of power, in the state [which they called "Communism"], and in private, mega-corporate behemoths as well.

Not so, the "new" Rape-the-Public-ans!

These new Rape-the-Public-ans are all for 'Corporate Stalinism'; 'Private and Privatized Stalinism'; the global, public dictatorship of the private bureaucracies which rule those "inter-national", "multi-national", "trans-national" — indeed 'meta-national'"mega-corporation" formations which are, after all, global command economies internally, even if they [pretend to] "compete" against one other, in so-called "free markets", externally.

Should the people of the United States, in revulsion, reject the Bush Rape-the-Public-ans, and lean back toward the "Democratic" Party, the Rocke-Nazis will simply switch ideological gears, and endeavor to collapse U. S. living standards to Third World levels, destroying the U. S. middle class, and its remnant democratic institutions, via "Carbon Taxes" and other "zero [i.e., negative] economic growth", "zero [i.e., negative] population growth" policies.

The Rocke-Nazis will then endeavor to continue to destroy public education via a recrudescence of their anti-productive-forces, anti-science, anti-technology, neo-Luddite pseudo-religion of '''Earth-ism'''.

The Rocke-Nazis  will continue — at an accelerated rate — to pervert the healthy popular reaction against capitalism's rampant, accelerating production of pollution, etc., externalities, and its rapacious destruction of the "natural basis" [Marx] of human society, into a "back to nature" social atavism; a "People are Pollution", neo-Nazi, pro-Dark Ages neo-/crypto-feudalism, "justifying" their accelerating global genocides — engineered to eventuate in virtually-complete global 'humanocide' — on pseudo-"ecological" grounds. The Hitlerian/Goebbelsian "Big Lie" of the "Global Warming" hoax will be used to dupe what's left of "The Left" into assisting the Rocke-Nazis in crushing even what little still remains of First-World working-class living standards down to Third World levels, via "Carbon Taxes", and the like, for, e.g., the '''90% reduction in per capita energy use''' for which the groupies of Rocke-Nazi whore Al Gore are calling.

Different ideological means, same ends.

The Rocke-Nazis are not just serial homicidal maniacs: they are 'serial genocidal maniacs'.  Indeed, they are '''humanocidal maniacs'''!

The Rocke-Nazis are not merely psychopathic, nor even merely sociopathic, in the 'Neroan'/'Caligulan' sense — they are 'cosmopathic' — hyper-self-degenerated human beings who have transformed themselves into '''humanocidal''' maniacal monstrosities in nominally human form, fanatically hell-bent on their attempt to be "victorious" via violation of the fundamental order and "lawfulness" of the cosmos.

The Rocke-Nazi puppet-masters remain hidden from public view.  Adolph Hitler — a mere henchman and servant-dictator of theirs, initially, until he turned on them, attempted to seize their power for himself, and nearly succeeded at it — is their best known public instantiation.

Hitler's fate, destiny, or «karma» is exemplary of their "lawful" fate as well.

Hitler's "1000 year Reich" lay a smoking ruin in little more than 10.  Hitler himself lay a self-murdered corpse, a stinking pile of charred and smoking bones.

However, in the process of incurring his own accelerated demise, he also reduced Germany, and much of the rest of the world, to a wasteland of rubble as well, murdering millions before the 'self-refluxion' of his multi-genocidal maniacal actions re-impinged, in kind, upon its source.

Likewise, with his erstwhile masters, the Rocke-Nazis.  There is no doubt that they too will destroy themselves in short order.

The real issues are these —

How deeply will the rest of humanity allow itself to be duped by the Rocke-Nazis — into partial complicity with their '''People are Pollution''' ideologies, and actions — and for how much longer?

Therefore
— how much of the rest of humanity will the Rocke-Nazis take down with them?

AND, MOST IMPORTANTLY OF ALL:  WHAT ARE YOU
GOING TO DO ABOUT IT???

Some commentators have documented [including Julian Simon, who met with an untimely demise] how these "Small is Beautiful", anti-population, and pro-poverty movements are bankrolled by the biggest of the big  — and the richest of the rich — i.e., by the Rocke-Nazi-run foundations and corporations.

The people of the United States, in cooperation with the rest of the people of this world, must put a stop to the Rocke-Nazis' "'Final Solution"', or the Rocke-Nazis will surely "put a stop to" all of us — to our futures, to our children's futures, and to our children's children's futures.

If the Terran human species cares enough for and about itself, then it is capable of thwarting the Rocke-Nazis' plan, creating, in the same process, a this-time-global renaissance of humanity.

Those who do not care enough are, defenselessly, headed down the drain of history, into the sewer of history, along with the Rocke-Nazis.

For, should the Rocke-Nazis "succeed", then planet Earth will have failed its 'meta-Darwinian, Human-Phenomic/-Genomic Planetary Selection Test'.

Then, before much longer, this whole planet — nöosphere and biosphere alike will perish in a final 'Milankovitchian' Ice Age which is otherwise already built-in, to the variations in Earth's orbital geometry with respect to the Sun, coupled with the photosynthesis-depleted/fossil-fuel-accumulation-depleted carbon dioxide content of Earth's present atmosphere: a final, "Snowball Earth" Ice Age that only such a global renaissance, a renascent leap forward in the human-social productive forces — in human cognition, in science, in the technology of planetary 'econo-ecological' self-management, and in civilization entire — could have averted!

What the rest of Terran humanity now faces, in its face-off with the Rocke-Nazis, is a global, planet-wide test of its moral fitness to survive of the '''fitness''', not just of its unconsciously-achieved human Genome, but of the attained level of the «aufheben» '''complex unity''' of that human Genome with its unconsciously-and-consciously-achieved human Phenome; i.e., including the "fitness" of its 'extra-Genomic', exo-chromosomically-acquired and -transmitted, and partly exo-somatic/'arte-fact-ual' '''culture''' and 'objective-collective body' the very core of its potential humanity!

For each of us, as individuals — as potentially "historic" beings, as opposed to ending up as merely "pre-historic" beings, in Marx's sense, and, thus, ultimately, as opposed to ending up as merely '''sub-human beings''' for each potential human being, to become actually human, the phenotype must overcome the genotype within us.

For the Terran human species as a potentially "historic" species, not just as a "pre-historic" species, in Marx's sense, and thus, ultimately, as opposed to ending up as a failed, '''sub-humanity'''  for our species, as a potential planetary humanity, to become a true humanity, the Phenome must overcome the Genome, with the word "overcome" understood, as always, in its «aufheben» meaning.

It is dangerous to assume that the Bush/'Meta-Nazi' regime will relinquish power at the end of Bush's term.

An incident may be manufactured, á la Hitler's burning of the Reichstag, and "Operation Northwoods" [ see J. Bamford, Body of Secrets; Anchor (New York: 2001); pages 80-91], which the regime will use to "justify" suspension of elections, of habeas corpus, of the posse comitatus law, and of even today's tattered remnants of the U. S. Constitution and Bill Of Rights generally, with imposition of martial law, occupation of key U. S. cities by federal troops and private corporate mercenary forces, trial of U.S. citizens by military tribunal kangaroo courts, and concentration camp "internment" and extermination of "enemy citizens" and "enemy civilians".

It should be obvious, by now, to all who are not in hysterical denial, that this regime, and the sinister forces behind it, are operating on a strict schedule, a 'timetable to totalitarianism'. The secret executive orders are already in place. Only if Bush becomes so unpopular so rapidly as to threaten the Rocke-Nazis with exposure and total de-legitimation will they scuttle him, as they did, on an earlier, slower track to totalitarianism, with Nixon. symbol 12

Plans

We plan to distribute the Briefings from Part I. of Dialectical Ideography to this website, under separate cover [see below], as circumstances permit.

As noted at the outset of this letter, these distributions of text are part of our implementation of 'The Seldonian Imperative' — the ethical imperative to act presently so as to help avert, if possible, the collective agony of foreseeable future catastrophes of contracted social reproduction of human-social-reproductive collapse and the wars, and/or the genocides, and the "dark ages" which manifest them, or, if it is not possible, or too late, to completely avert said catastrophes, to act so as to help reduce both their severity and their duration.

These distributions also advance 'The Seldon Project', the project to develop, and to make more broadly accessible, for Terran humanity, as the "Moment Of Truth" and "Hour Of Destiny" of its 'meta-Darwinian Planetary Selection Test' fast approaches, the new «organon» of 'dialectical ideography': a modern, cumulatively-higher, helical re-emergence of Plato's «arithmoi eidetikoi», as [re-]discovered by Dr. Seldon.

If you find even a little merit in even some of the new ideas emergent in the works transmitted to this website, then we urge you to put those ideas to good use in your own work, with or without attribution to this source.  We would prefer for '''psycho-historical''' reasons that you exercise the non-attribution option.

The works of this Seldonian '''school of thought''' are increasingly, and openly, if slowly, becoming accessible on the world wide web.  However, only those who have already developed at least one interest that resides along at least one of the paths that lead to these works will easily find them. Otherwise, and for others, they are hard to find, like a needle in a haystack. 

These distributions of texts form part of our efforts to help this humanity to actualize its «eu-catastrophic» potential for a world-wide renaissance of material, emotional, intellectual, and spiritual prosperity, founded, in part, upon the humanity-wide diffusion of this new «organon» of 'nonlinear logic', and, thus, upon all of the 'psyche-ological', conceptual, mathematical, scientific, engineering/technological, and social evolutionary/social 'meta-evolutionary' accompaniments of that advanced cognitive level. These include the advent of 'subatomic power', starting with the harnessing of Hydrogen, protonic fusion power, the greatest human-social productive force ever to get '''on the agenda''' for socio-technological mastery by Terran humanity so far.

Dialogically yours,

Hermes de Nemores

for
Foundation Encyclopedia Dialectica,
F.E.D.-mobile



Supplementary Documents

Links to texts that are supplementary to this Introductory Letter, and that are presently posted to the www.dialectics.org website, are provided below —




Release History

v1.0
original printing and mailing
25-November-1999


v2.0
put onto the http://www.dialectics.org website in PDF format
16-January-2006


v3.0
revised text first put onto http://www.adventures-in-dialectics.org website in HTML format
30-May-2007


v4.N
HTML (with added graphics) versions continually revised and put onto http://www.adventures-in-dialectics.org website
from August-2007 to present





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