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