Encyclopedia Dialectica Brief #2 —
Meta-Monadology: The Universality of the «Aufheben» Process
by Anonymous for F.E.D
December 17, 2008


What this text addresses, in toto, is the '''psycho-historical''', cognitive revolution that F.E.D. is offering. It's deeper title, then, should be:

Announcing a Cognitive Revolution, Necessary Precursor to Humankind's Self-Liberatory Self-Transformation Beyond the Capital-Relation as Predominant Social Relation of Re-Production of Modern Society

As I state at the outset of Part II of this text:

"I believe that experience with compact, explicit dialectical models; with the 'algorithmic dialectics' and the 'dialectical algorithmics' of, e.g., the Q categorial ideography, can help to catalyze, in the 'experiencer', a cognitive revolution. It can help to catapult that experiencer [deeper] into '''the dialectical operations stage of adult human cognitive development'''. That, at least, has been my experience.

I also hold that such a revolution in cognition, diffusing in widening waves, rippling throughout the global populace, is prerequisite to a successful, liberatory transition to the higher polity — to the political-economic democracy — of 'democratic-communist' society".

I.e., to the 'externality-equities based economic democracy that F.E.D. advocates [for more on this, see http://www.equitism.org].

The rest of the content of Part II is a specimen of the positive fruition of a Marxian, immanent critique of the core of Modern Science — i.e., of an evocation of a determinate [self-]negation of the capital epoch's science of arithmetic, yielding, as its positive outcome, the discovery and presentation of an arithmetic for Marxian dialectics.

The 'cognitive revolution' created by the assimilation of the resulting "new «organon»" for science is catalyzed and facilitated by the new, partially-algorithmic, partially intensional/intuitional, and interpretive/connotative, Dialectical Heuristic of F.E.D.'s new, and Trans-Leibnizian, Dialectical «Characteristica Universalis».

This Mathematics of Dialectics, fruit of the Historical Dialectic of Mathematics, and of the dialectical, Immanent Critique of Modern Mathematics, provides, at long last, the heretofore missing precision ground-work of a non-reductionist, non-atomistic paradigm for a 'Trans-Modern Science' — for a comprehensive Scientific and Mathematical Alternative to Reductionism.

The communication strategy applied in Part II may be of interest to the reader.

It essays to communicate the unfamiliar — in this case, both Systematic Dialectics, and the compact, mnemonic Modeling thereof in the ideographic language of F.E.D.'s new dialectical algebra — via a partial reduction of unfamiliarity. It seeks to convey both of these "esoteric" topics via their illustration in the context of a topic which is exoteric to the readers of the http://www.dialectics.org and of the http://www.adventures-in-dialectics.org websites; well-known to them simply by virtue of their literacy.

The text of Part II illustrates the narrative "method of presentation" [Marx] of systematic dialectics, and the compact modeling thereof via the new Dialectical Ideography, for the familiar case of Phonetic Writing Systems.

In this process, Part II moves beyond insubstantial and external, undialectical critiques of Modern Science, to immanent critique. It locates the foundation of its critique in the very heart of the object of criticism, in this case, in the 'self-duality' of ordinary arithmetic. By then developing that immanent ground of critique, i.e., of the self-critical, determinate self-negation of standard arithmetic — of 'it-self' by 'it-self' or in accord with 'it-self'Part II achieves a positive development that both supersedes and critically conserves the content of that object of criticism — Modern Mathematics — in the «aufheben», dialectical manner.

Part II follows F.E.D.'s lead, essaying to aid the reader's discernment of this immanent ground of the critique of arithmetic also via the '''external''', 'psycho-historical' evidence of 'psycho-archaeology'. It does so, in this case, by resurrecting some terminology and concepts — in particular, that of «arithmos» and that of «monad», or, respectively, that of "number" [of qualitative units] and that of [qualitative] "unit" — from Ancient Mediterranean humanity's development of mathematics, up to the start of the last Occidental "Dark Ages". As we show, these Ancient developments partially survive, to this day, in vestigial and cognate forms within modern mathematical terminology and conception. They also illuminate the modern «mentalité» by contrast. Understanding the Ancient vs. the Modern views of "number", or of «arithmos», incidentally, offers an alternative, less mysterious, less mystical, less abstract-idealist-sounding, and more concrete, more sensuous view of Ancient Pythagoreanism. If the Pythagoreans held that '''Everything is number''', perhaps they meant that '''Everything is «arithmos»''' — i.e., that all is made up out of «arithmoi» of «monads»; out of populations of qualitative units. This intimates a scaled 'monadism' which may have been very far from a reductive atomism.

'Psycho-historical analysis' of the Ancient theory of arithmetic — as elaborated also in Jacob Klein's book, entitled Greek Mathematical Thought and the Origin of Algebra, as well as elsewhere — reveals a more concrete, sensuous, and qualitative scientific and mathematical «mentalité» among our Ancient ancestors, versus that of Modern Occidental humanity.

It reveals that Ancient «mentalité» to be, precisely, one not yet as pervasively impacted by the incessant experiences of exchange-value exchange as was the mathematico-scientific «mentalité» that had emerged by the time of the European Renaissance, closing the Occidental "Dark Ages". The later «mentalité» gave birth, at length, to Modern Science.

Humanity's pervasive, extensive, and intensive diffusion, and incessant daily practice, of commodity exchange — even of barter, but, especially, of money-mediated exchange of commodities, and of money-price-consciousness — has subliminally introduced a fundamental confusion regarding the dispensability of the qualitative, and about the reductive 'equatablity' of the heterogeneous, into humanity's «mentalité». This was evident already in Marx's [psycho-historical] analysis of the very «arché» form of exchange-value, the "elementary or accidental form of [commodity-]value", in his Capital (Volume One; Part 1; Chapter 1; Section 3; A. Elementary or Accidental Form of Value).

This Ancient mathematico-scientific «mentalité» can thus be leveraged to outwardly 'co-illuminate' the ground of the innermost of the immanent critiques of Modern Science, that of Modern Arithmetic: the «arché» of all of the ideology, as also of much of the social-reproductive efficacy, of the products of Modern Science.

The Modern «mentalité» is content with the purely-quantitative abstraction of, e.g., '2 + 2 = 4'.

The Ancients' was not.

As evinced in the Ancient Alexandrian «Arithmêtikê» — or '«Arithmos-êtikê»' — of Diophantus, circa 250 C.E., in which the earliest known emergence of [proto-]ideographic-symbolic algebra was recorded, the Ancients required expressions of the form ' Mβ'β' ι^σ Mδ' ' (M beta-prime beta-prime space iota caret sigma space M delta-prime), wherein the Greek letter "beta" is used to represent the number we denote today by '2', the Greek letter "delta" to represent what we today denote by '4', the abbreviation 'ι^σ' [a syncopation of the Roman «aequalis», or «αεθυαλισ» (alpha epsilon theta upsilon alpha lambda iota sigma)] to represent "equals" or '=', and the abbreviation 'M' to represent the Greek word «Μονας» (capital-mu omicron nu alpha word-terminating-version-of-sigma), or «Monad», which translates to the word "unit" in Modern English. Note that, in the Ancient expression, the qualifier, not the quantifier, comes first!

Yes, the Ancients abstracted, but they did not abstract only generic '''quantifiers''', such as the '2's and the '4' in the example above. They also abstracted a generic, symbolic qualifier, M, denoting the generic qualitative unit — ontological or metrical — with which they required every '''well-formed''' abstract quantifier to be paired.

The «monad» symbol, or unit symbol, M, could stand for a cabbage, for a cannibal, or for a king — for whatever the context of the given expression required — indifferently and 'chameleonically'. This "syncopated" symbol, M, stood for the generic «Monad» of Plato's «Arithmoi Monadikoi».

Psycho-historically, this M, this generic «Monad», is a conceptual reflexion of Money, of the Money-praxis, of the equation of the apparently qualitatively heterogeneous, in the Money-mediated exchange process — in the Money-mediated "circulation" of commodities — via Money-measured price, i.e., via the Monetary unit. The emergence of the «Arithmoi Monadikoi» theory of "logistical" arithmetic is a reflection — an only partly conscious, explicit psycho-historical consequence — of the burgeoning of the praxis of Monetized exchange within Ancient Mediterranean humanity; of the 'mentalitary', memetic, mentally-mimetic impact of Monetized commerce on the earlier-emerged mental images of the Mesopotamian 'tokenography', and of its predecessor, undergirding 'tokenology', uncovered by Denise Schmandt-Besserat. Coincidentally, 'non-cognately', but not 'unmnemonically', in English, 'M^o' is a syncopation for the word Money, as well as for the word Monad!

Diophantus's «Arithmoi Monadikoi» denoted, by M^o, proto-ideographically, an abstracted metrical unit [e.g., a unit of conventional measure, like a «stadium»], or and abstracted ontological unit [e.g., for an individual unit of some kind of being, such as a '''philosopher'''] QUALIFIER. This usage died out in the post-Dark Ages rebirth of Occidental mathematics and science. It was eclipsed in proto-Modern humanity's adoption of a more fully ideographically symbolic [Hindu-Arabic] arithmetical and algebraical notation of abstract quantity, until now — until F.E.D.

What the resulting new, precision «organon» of dialectics offers — what the new, Dialectical «Characteristica Universalis» bequeathed to humanity by F.E.D. offers — is, I think, not the blinding light of a formless, indeterminate and indulgent effulgence. The dawning of such a light, because blinding, is also, still, "that dark night in which all cows are black'' [Hegel]. It would provide no visible navigational specificity, either as a "method of discovery" [Marx], or as a "method of presentation" [Marx], post-discovery. Nor does Q offer a rigid dichotomization — e.g., of ultra-dense, impenetrable "atoms" versus vacuous void, providing a total void of creative freedom; no scope for novel exploration, nor for intuitive imagination. It is rather, a complex unity of some aspects of the two, and, precisely, a dialectical complex unity thereof, which constitutes, as a whole, an 'Algorithmic Heuristic'.

I have encountered numerous abstract negations, of late, regarding the Capital-relation-shaped ideology infecting Modern Science, among self-professedly "Marxian" circles.

The "Left", today, is deeply infected — perhaps terminally so — with the utterly contra-Marxian, anti-productive forces, anti-human, anti-progress products of the 'Rocke-Nazi' ruling class's mass production of ideology — of the viciously misanthropic, 'Meta-Nazi', pro-genocidal — indeed, pro-humanocidal — 'anti-human-population', 'mass-exterminationist', '''People Are Pollution''' ideologies of Neo-Malthusian 'Ecologism', 'Earthism', 'Greenism', 'Naturism', 'Negative[-Growth]-ism' or 'Zero[-Growth-]ism', 'Limitism', 'Smallism', 'Animalism', 'Global-Warmism', 'Peak-Oil-ism', and Neo-Luddite "Neo-Primitivism". As such, it is nothing more than a "Left" asset of the ruling plutocracy.

All of these "left" ideologies are no less the wholly-owned subsidiaries of the 'Rocke-Nazi' faction than are the so-called "right-wing" ideologies of, e.g. 'Neo-Con-ism', 'Neo-Liberalism', pseudo-Christian theocratic totalitarianism, pseudo-Islamic theocratic totalitarianism, pseudo-Judaic theocratic and Zionist totalitarianisms, pseudo-Christian 'Rapturism', etc., etc., «ad nauseam».

Given the tenor of those pseudo-Left [pseudo-]"critiques", I felt that the highest priority — for the moment — resides elsewhere than in a direct critique of such extremely anti-Marxian ideologies. All of these pseudo-Marxian, anti-science ideologies bear the markings of the 'human anti-humanism' of the ruling faction of the present ruling class. They demonstrate the potential lethality of the 'anti-psychohistory' of that faction's massive ideology-engineering operations. Those operations are designed to vitiate, eviscerate, and immobilize any actual opposition, informed by actual Marxian theory, to that faction's 'humanocidal' plans.

The priority is, IMHO, not a mere critical negation of those negations, but a 'pos-i-tion': positing specimens of the positive FRUITION of a dialectical, or immanent, Marxian critique/determinate [self-]negation of Modern Science.

Most of the "critiques" of Modern Science that I cited above amount to nothing more than an '''external''' critique of Modern Science. They lead to an '''abstract negation''' of Modern Science — to a "throwing of the baby out with the bathwater".

The pseudo-Marxian critics who purvey them say that there is no "baby" in Modern Science. They say that Modern Science contains nothing whatsoever that is a product of "universal labor" [Marx]; nothing of universal use-value; nothing "this-sided", no practical truth [Marx; Theses on Feuerbach]; no historically-generic social-reproductive efficacy. They say that nothing of these beneficial features exists at all in the Science which the human species has so protractedly created, in co-evolution with our creation, and our world-historical elaboration, of the Capital-relation.

Their fundamentally nihilistic critiques of Modern Science seek to externally negate Modern Science, e.g., on 'Naturist' and 'Animalist' grounds, leaving only an indeterminate "Nothingness".

'Retrojecting' the upshot of such 'externalist' critique into the past finds the following simile. It would be as if Marx had succumbed to such sub-dialectical, reductionist tendencies, and had simply written, in place of the «Grundrisse», and of the four volumes of «Das Kapital», etc., etc., the single sentence: "Bourgeois Political Economy is not a science; it is nothing but ideology." And, left it at that.... I feel sure that there were also whole legions of 'sub-dialecticals' in Marx's time, as before, who did, essentially, just that. We do not know their names. Deservedly so. For, in regard to any actual critique of Modern Science, they contributed, essentially, "Nothing" to humanity: "indeterminate Nothingness".

The chief virtue of dialectical, or immanent, critique is its '''radicalism'''.

Dialectical critique, immanent critique, locates the deepest root of the indictments lodged by both the external and the internal dissent from a given, ideology-afflicted science — or from Modern Science as a totality — in the innermost core, the very heart of that object of critique itself.

It locates that root in the most fundamental, «arché» category, or concept, residing at the very foundation of that object of critique, from which that ideological science flows/has developed; the "cell-form" [Marx] of its totality. It locates that root in an 'intra-duality' — in a 'self-duality' — of that very root category, or concept, itself.

That 'internal duality' partially vitiates the theoretical and practical grasp of the totality — here of Nature itself, as maximal totality — that this ideological science aims and claims to have achieved.

Then, this immanent critique unfolds outward from that innermost heart of its object, reconstructing that object as it goes, until the totality of that object has been transformed '''from bottom to top'''; '''from inside to outside'''; from core to surface; from root to branch to leaf.

In that process, what remains 'use-valuable' in what was already explicit within that object, prior to this critique, is preserved, and elevated into a new and '''higher''' — i.e., into a more-inclusive — context, where vast treasures of new 'ideo-ontology', of new conceptual advances, that had been locked and blocked by the old object, pre-critique, are evoked, and burgeon into explicitude.

The result is, not an "abstract negation", wasting all, and leaving nothing, but a '''determinate negation''', rich in its manifold of integrated, new and old determinatenesses.

A specimen of the fruition of such an immanent critique of Modern Science is what I have sought to provide in the text posted below, entitled "Meta-Monadology: The Universality of the «Aufheben» Process".

To synopsize the substance of this text, some of its key, core content is called-out as follows:

1.  Application of the "Natural Dialectors" Arithmetic to Generate Key, Core Content of Marx's «Das Kapital»

The specimen includes a demonstration of how the «arché» category of Marx's «Das Kapital», that of the Commodity, or, more specifically, that of the Commodity's Elementary Value-Form, can generate the '''Table of Content''' for a Marxian Critique of Political Economy, as a 'Systematic Dialectical' presentation. As such, that presentation is an «aufheben»-cumulative categorial progression, via the 'self-iteration', or 'iterated categorial self-subsumption / self-internalization / self-meta-monadization', of that «arché» category or "cell-form". That is, that presentation advances by means of the 'self-reflexive' self-confrontation of that «arché» category, or "cell-form", as, itself, an «aufheben» operator, under the rules of the F.E.D.'s Q dialectical arithmetic, as contra "Natural" Arithmetic.

2.  The Self-Duality of the so-called "Natural" Arithmetic of the so-called "Natural" Numbers, N = { 1, 2, 3, ... }

The "Natural Numbers", the "ultimate cell-form" of all of the mathematical systems of Modern Science, is a one-sided construct. If, in the phrase "3 pounds [of] potatoes", we describe '3' as '''quantifier''', 'pounds' as 'metrical qualifier', and 'potatoes' as 'ontological qualifier', it is clear that '3' alone, as in N = { 1, 2, 3, ... }, is an 'unqualified quantifier'. The suppressed 'intra-dual' to that one-sided interpretation of the root category of "Natural" Arithmetic is an Arithmetic of ' unquantifiable ontological qualifiers'. Such a system constitutes a 'contra-thesis' to the '«arché»-thesis' of the 'unqualified quantifiers' system of arithmetic. This 'contra-thesis' constitutes a "Non-Standard" arithmetic. It is a system of 'meta-"Natural" meta-numbers', that "obey" the first four, "first order" Peano Postulates that also describe the ordinary, "Natural", numbers, but with an entirely contrary sense. Upon examination, after 'de-suppression', it is found that this F.E.D.-discovered 'counter-arithmetic' fits, and quite spontaneously so, the description of a 'de-Parmenideanized', dynamical and 'meta-dynamical' — revolutionized — version of Plato's ancient — and, until now, mysterious«arithmoi eidetikoi», the «arithmoi» of his «dialektikê». Not surprisingly, it also fits the description of a 'contra-Boolean algebra', of a dialectical, 'contental' logic, as 'contra-thesis' to the '«arché»-thesis' of Boole's arithmetic and algebra for form-al logic. This opposite, 'contra-thesis' — of a 'dialectical symbolic logic' — flows from a deep negation of the axiom that Boole called "the fundamental law of [linear] thought".

Again, it is also found that these new "qualitative numbers", or qualitative «arithmoi», conform to the same four, "first order" axioms, or "Peano Postulates", as do the "Natural" Numbers, although with a completely contrary sense — as F.E.D. has already argued in its extant expositions. Indeed, it is also found that these first four axioms are so universal that both the "categorial progressions" of "systematic dialectic", and the 'systems progressions' of historical dialectic are also described by them; are also 'Peanic'.

This result was foreseen — in so far as the 'Dianoetic' tools of «Verstand» reasoning, and of the 20th Century burgeoning of "Mathematical Logic", can encompass them — via some of the deepest theorems of Modern Mathematics. They are (1) the Gödel Incompleteness Theorem [in conjunction with the Gödel Completeness Theorem, co-applied at the "first order" level], and (2) the Löwenheim Skolem Theorem. Each of these theorems by itself predicted, "non-constructively", the 'constructability' of "Non-Standard Models" of the "Natural" Numbers.

The N plus the contra-N systems of arithmetic go on to find a '''complex unity''', or 'uni-thesis'; that is, a 'qualo-quantitative' third category and system of arithmetic as part of their own Q-modelable — i.e., '''dialectical''' — categorial-progression/systems-progression.

That category is the category of a system of 'quantifiable ontological qualifiers arithmetic', called U.

F.E.D.'s dialectic of the dialectical arithmetics, modelable by themselves as such, then burgeons forth from there.

This immanent negation of the exchange-value-inculcated, ideological one-sidedness that infects contemporary '''capitalist arithmetic''' also brings something positive to light. It brings to positive fruition an 'ideo-ontologically' expanded, «aufheben»-expanded, qualitatively revolutionized conception of arithmetic, and an expanded, 'qualo-quantitative' transformation of mathematics in general, in the form of a general theory of «arithmoi».

By this phrase, I take it that F.E.D. means a universal theory of those assemblages, «ensembles», "sets", or "populations" of [qualitative, heterogeneous] «monads», i.e., of the [qualitative] '''units''', '''parts''', '''particulars''', '''elements''', '''members''', '''constituents''', or '''individuals''', the 'little unities', the 'little wholes' that we encounter, so ubiquitously, at every level, and scale, of our experience and cognition of this «kosmos». This development, IMHO, represents the true fruition of Ancient Pythagoreanism, at long last!

3.  The '''Set Of All Sets''' and the Immanent Critique of '''Natural''' Set Theory

Set Theory is still oft considered to provide the foundational theory for all of Modern Mathematics. The '''Set Of All Sets''' is the set-theoretical, or "extensional", definition of the "Set" idea itself. The [finitary] '''Set Of All Sets''' should therefore be the fundamental [idea-]object of a Set Theory that should also, itself, be '''realistic''', or finitary and '''constructivist'''. Instead, the '''Set Of All Sets''' has been denied by and exiled from "Standard" Set Theory, because of the "contradictions" — "formal", "propositional", but also 'contental', '''existential''', 'ideo-ontological' — which the admission of this [idea-]object entails for undialectical, 'dianoetic' Set Theory.

The '"inner unrest"', the 'non-dead-ness' — the immanent aliveness — of this set, is embodied in each human mind that constructs it inwardly, and views it via "the mind's eye". The immanent critique, grounded in the 'self-duality' of this '''Set Of All Sets''' «speci»-men of the Set «genos», and in that set's consequent '[ideo]-«auto-kinesis»', and in its ever-self-regenerated self-critique, reveals that this set turns out to be: (i) not a static, noun-like, '''dead''' 'idea-object', but a 'self-constructing idea-process', an idea[l[ized]] '''process-object''', a 'mental eventity' whose very essence — and thus the essence of the Set concept itself — is self-transformation; 'self-revolutionization'; (ii) a Modern, trans-Platonic '[ideo-]«auto-kinesis»', and also (iii) a set-theoretical model of the generic dialectic itself.

The [re-]admission of '[ideo-]«auto-kinesis»' into Set Theory — via the '''Set Of All Sets''' itself, as 'self-motile' 'Non-Standard Paradox', but also via the noticing of the related 'self-motility' of the 'Standard Paradoxes' of "Natural" Set Theory, e.g., the paradox of the '''self-oscillating''' Russell Set — as all sharing the characteristic of being 'self-motile' idea-objects, and as being, therefore, also, contra-Parmenidean counter-examples — reveals a hitherto unnoticed, immanent, deductive, formal-logical, «reductio ad absurdum» self-refutation of 'dianoetic' '''Natural''' Set Theory. It does so by way of requiring, per the rules of formal logic itself, the formal negation of '''Natural''' Set Theory's implicit, formally unstated, but pervasively presumed 'Parmenidean Postulate', the de facto axiom which asserts the Parmenidean presumption of universal 'ideo-ontostasis»': the presumption that no set-theoretical idea-object can be changing — let alone self-changing; that there is no movement — let alone self-movement — in the idea-world of mathematics.

With any luck, this essay will contribute to the conveyance of these F.E.D. discoveries to a wider public.


Table of Contents

  1. Prefatories
    Overall Title Page
    Omni-Copyright Notice
    About the Author
    Contents Map

  2. Part I:  The «Aufheben» 'Meta-Monadology' of Platonic «Arithmoi Eidetikoi» Dialectics

    Part I — Section A
    Title Page for Part I

  3. Part I — Section B
    «Aufheben» 'Self-Subsumption' / 'Self-Internalization', 'Self-Meta-Monad-ization', & 'Meta-Fractal' Structure

  4. Part II:  A Purely-Qualitative Dialectical-Mathematical Model for Systematic Dialectics

    Part II — Section A
    Title Page for Part II
    'Exemplificatory Case': The [Systematic-]Dialectic of Phonetic Writing Systems
    Key Claims for this Part
    'Qualitative Calculation' with Generic Q 'Meta-Numbers'

  5. Part II — Section B
    'Qualitative Calculation' with Interpreted Q Meta-Numbers
    'Connotogram' Self-Multiplication
    'Ontology Spaces' of Self-Growing Dimensionality

  6. Part II — Section C
    The Dimensionally-Expanding Space of Q as a 'Possibility-Space'
    The 'Cruciality' of the Initial Category Choice
    Step 0: Focusing on the Initial Category

  7. Part II — Section D
    Steps 2 through 6 of the Modeled Systematic Dialectical Method of Exposition

  8. Part II — Section E
    What Is Still Left Out
    Marxian Systematic Dialectics and Some Dialectical Models of the Immanent Critique of Political Economy
    Dialectical Modeling via a Different Variant of the Seldon Function
    The Dialectic of the Dialectic Itself, Expressed via the Q Dialectical Shorthand
    Transition to Part III

  9. Part III:  Three 'Meta-Monadological' Models in Historical Dialectics

    Part III — Section A The Dialectic of [the] Human [Part of] Nature, One View — The Historical Dialectic of the 'Meta-Monadology' of Human-Social Formation(s)