to the

Immanent Critique

of

Arithmetic

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Foundation

Version:

Last Updated: 23 February 2004 C.E./B.U.E.

First Distributed: 25 November 1999 C.E./B.U.E.

Omni-Copyright_{}1999 C.E./B.U.E. by F.. Copyright to original portions of this work is hereby granted to all persons.E.D

Note:

This essay is apartiallyself-exemplifying exposition of, anda record of an ongoing self-critique of,the ideas advanced herein. It is a'meta-dynamical'and'meta-evolving'conceptual object. Edition index [self-edit iteration number] and last revision date are stated on the title page; the most recent changes are coded as plum-colored text. The time sequence of changes in the form / content of this essay is predicted to be both an illustration and an instantiation of the meta-model of ontological meta-dynamics that this essay explores, as well as of the 'homeomorphic defect' of that meta-model. We expect that successive editions of this document will document an'ideo-onto-dynamasis'rather than an'ideo-onto-stasis'; a'meta-evolving ideo-ontology'; a'multi-meta-ontic idea-cumulum'; an expanding, and ever 'thickening', increasingly inter- and intra-connected, 'inter-acted' network of'inter-implicatory','inter-determinate','inter-generative'ideas, elaborated on a mounting count of'metafinite','meta-fractal'scales, all exemplifying a"non-standard",Contra-Boolean logic; theontologicallydynamicallogic of the dialectical 'law" of cognition signified by the 'ideo-ontological', "pure-qualitative",_{}-algebraicequationinx^{2}_{}x.

This writing is an unpublished work, and one which is not sold or exchanged for remuneration or commercial gain of any kind, but is distributed «samizdat» to selected individuals and organizations, on a donation basis, free of charge. This work is a potential contribution to the collective creative property of the Terran human species: assimilate, disseminate, critique,and surpassat will. We, the authors, seek hereby to further neither our monetary riches, nor our public power, nor our personal fame. What we want, money cannot buy. We hope, with your help, to build a better us, and to help do our infinitesimal part in building a better universe ["infinitesimal" differences can matter, as nonlinear dynamics demonstrates]. More monetary wealth will not buy that betterment. More political power cannot impose it. More fame would mainly distract from it. We hope that you have chosen, or will choose, to build a better you. We hold that this choice entails the profoundest consequences for one's life, as well as for the lives of others. We also hold that such choices belong to you alone. We wish to share, with you, the forthcoming conceptual riches. We will rejoice, and we will be compensated, if you teach us in turn, help us to correct our errors, and thus advance the common-wealth of all beyond this offering. We also request our readers' forgiveness in the areas of our many shortcomings, some of which, though determined to strive ceaselessly to overcome them, we will never, in a lifetime, overcome. We, the authors, are not publicly accessible, but will endeavor to continue private transmittals to you if you indicate publicly, however cryptically, and we recommend that it be cryptically, your desire that we should do so. We want not that our existences, let alone our egos, should be an impediment to that great reverberating propagation of new cognitions, and of emergingnew formsof cognition, of which this essay is, at best, an incomplete, imperfect, transitory, and transitional manifestation. We therefore happily forego personal credit, and, by thus renouncing in advance the [remote] possibility of any notoriety resulting thereby, hope also to retain more lifetime for the continuation of this work.

Dialectical ideography as set forth herein is interpreted variously as:

This essay, in addition to that of ideogramic, pictogramic, and phonogramic symbolization, draws also upon the power of neo-mythological, allegorical, and mythopoeic — that is, of(1) a calculus of', encompassing anquanto-qualitative change'explicit,ideographicalarithmetic for the[ie]dimensional unitor metrical "sof classicalmonads"", and, thereby, 'semantifying' the "meaningless" singularities [finite-time "infinite" values] ofdimensional analysis"the "unsolvable" [in part, because of those very singularities] nonlinear integrodifferential equations and their solution-functions through their metricalespecially'using those new, explicit 'metrical qualifiers' of thisre-qualification'', concretizing and operationalizing Plato's «dimensional arithmetic'» and Diophantus'arithmoi monadikoi_{};

(2) an alternative, onto-logical;contra-Boolean algebra

(3) an ideographic, 'onto-dynamical' "symbolic logic" for the state-space/control-parameter-space'of 'meta-finite', conversion-singularitymeta-dynamics'';self-bifurcation'

(4)as well as a [a mathematics for modeling the history of mathematical ideaspsycho-]andhistorical algebrafor modeling the 'meta-evolution' of the sciences generally; an ideography for the [psycho-]history of ideas; an ideography of the 'meta-dynamical' logic of conceptual self-innovation and self-development; a 'philosophical algebra' or trans-Leibnizian «arithmeticcharacteristica universalis»; an arithmetic and algebra of innovative conception or of the creative conceptual process;

(5) a rules-system for an ideographical language ofin self-transcending [meta-]systems;qualitative self-escalation

(6) a generic algorithm for the'operation regress; for a trans-Hegelian, autopoiesic version of themeta''operation; and for a "dynamical", 'temporalized', diachronic,aufheben''meta-evolutionary'version of the Russellian/Gödelian logical types hierarchy;

(7) a model for a 'meta-fractal', non-Cantorian, of 'theory of totalitiesmeta-finite' arithmetics, and of the "foundations" of mathematics;

(8) an arithmetic, algebra, geometry, and analysis built on certain", i.e., on thenon-standard natural numbers"'Gödelian 'meta-natural' numbers', a space of"hypernumbers"'of2nd degree', 'made up out of "standard", '1st degree' natural numbers, instantiating those"non-standard models of first order Peano arithmetic"whose possibility isimplied by the first-order conjunction of Gödel's completeness and incompleteness theorems, asby the Löwenheim-Skolem theorem,constructing thereby an ', 'de-Parmenideanized' actualization of Plato'sontologically dynamical'"arithmetic of dialectics", the «».arithmoi eidetikoi

psycho-historical— symbolization to aid in the conveyance of its most urgent messages. Dialectical ideography is, we believe, a humble but potent seed. As with the several non-Euclidean geometries that arose from the failed attempts to prove the absoluteness of Euclid's geometry, thesenon-Parmenidean,contra-Boolean, andcontra-Cantorianonto-logicaland onto-dynamicalarithmeticsand theiralgebras of dialecticsmay bear fruit for humanity only if germinated through the intra- and inter-personal dialogue, and dialectic, of assimilation, critique, refutation, and supersession. We have avoided broadcast publication and indiscriminant distribution of this essay. We wish to base its circulation, and the selection of its recipients, upon our best judgment of its potential value to each candidate recipient. The taking to heart of the ideas "graphed", both ideographically and narratively, herein, can produce profound transformation in the very identity of the person so taking. Panic in response to perception of the early signs of such transformation in others may elicit, from some perceivers, a violent reaction. In particular, the intimations of the'meta-human',, implications of theΔh'cumulum'of human[oid] evolution is profoundly disturbing to some. We are therefore transmitting this document only to those whom we perceive, via their own published writings, to be already verging on similar or related conceptions as a result of their own protracted'. We have also decided not to disseminate the most "dangerous" of the results to date. We believe that you are eminently capable ofself-meta-evolution''re'-discovering these results, if you have not yet discovered them already. Should you do so, we urge that you treat them, and their dissemination to others, with utmost care. The system, more accurately, the systems, ofdialectical ideographyglossed herein continue toevolve and 'meta-evolve'rapidly in our research.They burgeon beneath our feet. We expect to exercise a similar restraint and discretion in any future progress reports which we may send your way. We therefore lodge the Omni-Copyright statement above together with this countervailing caveat: we recommend that you disseminate this document, its ideas, and/or related ideas of your own discovery, with careful judgment.Give the friends of humanity a head start vis-a-vis their adversaries.

Dialectics should inculcate humility. "Perfection" is not a final meta-statethat can be finally manifested, but an open-ended, 'uncompleteable', asymptoticprocess, moving from greater to lesser imperfection. We realize that conceptual 'homeomorphic defect' is inescapable for cognizing beings such as ourselves. Even at best, we must always be partly wrong. Even at best, one cannot be finally, completely, and wholly right. One's mental constructs cannot ever be the truth, the whole truth, and nothing but the truth. But one may be right enough for one's time, for one's moment, for one's role, and for one's part; rightenoughto help one's contemporaries to live through, and beyond, one's time, and thus, potentially, to enjoy the privilege, the pain notwithstanding, of a vital ['life-ful'] and willing participation in the succeeding epoch of imperfection.

Prescripts: Motive

Part I. Prolegomena

Kernel

Model

The Productive Force of Language

Why 'Ideography'?—Beyond Bones and Stones

Why the 'Ideographic Title' {x^{2}_{}?x}

Why 'Immanent Critique'?

Why Arithmetic?

Why 'Dialectical'?

Briefings on_{}_{ }Part II. The Meta-Evolution Of Arithmetics

[The Cognitive Psycho-Historical Dialectic of Arithmetic to date]

The Nonlinearity Barrier—The Current Psycho-Historical Impasse of the Mathematical Sciences

The Psycho-Archaeology of Number: The Pre-History of Arithmetic—From Iconic 'Tokenology' to Cuneiform Literacy

The Psycho-Historical Pattern of Arithmetical Progress and 'The Gödelian Ideo-Meta-Dynamic'

A Symbolic History of Arithmetic—Q-based Models of the Historical Meta-Evolution of the Number Concept

- Models of History; 'Meta-Arithmetical' Models of the History of Arithmetic; 'Mathematical' Models of the History of Mathematics

The Mesopotamian 'Tokenology', Diophantus'_{}, and The Elision Of The Qualifiers [Confinement of Arithmetic to Plato's «»Arithmoi Monadikoi]

- Psycho-Historical Analysis of the Reification and Fetishism of the Pseudo-Subjectivity Abstract "Quantity" in the Formation of The Nonlinearity Barrier

- The Historical Trend of the Gradual [Re-]Emergence of "Hypernumber" [
Convolute] Qualifiers in Higher Arithmetic, Higher Algebra, and Analysis

AQ-based Psycho-Historical Model of the Emergence of_{}?

A Dialectical-Ideographic, Psycho-Historical Model of the Nonlinearity Barrier and of its Overcoming? 'Philosophical Algebra'—A Meta-Dynamical Logic of Conceptual Self-Bifurcation: Ideographic Models of the Meta-Evolutions of Ideas?

- Critique of Hegel's Critique of the Conceptual Potential of Mathematics (An «Aufheben» 'Ideo-Meta-Fractal'; An 'Ideo-Onto-Dynamical', 'Multi-Meta-Ideo-Ontic Cumulum' of 'Mathic' Ideas)

- Dialectical Ideography as Candidate Trans-Leibnizian
Characteristica Universalis?What Is 'The Science of Mathematics'—'Ideometry via Ideography'? The 'Linguistic' Approach to "The Foundations of Mathematics"

The 'Gödelian Ideo-Meta-Dynamic', the_{}Conjecture, the Ever-Burgeoning 'Knowledge-Deficit', and the Ontological Status of the 'News'Part III. The Arithmetics Of Meta-Evolution

['Meta-Systematic-Dialectical', Categorial-Progression Exposition of some Systems of Dialectical Arithmetic]

Healing the Parmenidean / Zenoan Psycho-Historical Trauma [Set-Theoretic Model of 'The Gödelian Meta-Dynamic'; Inescapability of 'Meta-Dynamism']

- Derivation of Dialectical Ideography by
reductio ad absurdumSelf-Refutation of the Implicit 'Parmenidean Postulate' of "Self-Evident" Set Theory?- The 'Standard Paradoxes' — The "Reflexive" Paradoxes of Formal Logic / Set Theory and Nonlinear ['Self-Reflexive' / 'Self-Refluxive'] Dynamical Systems
- 'Non-Standard Paradoxes' — The Set of All Sets: From Paradox to Paradigm [The 'Meta-Fractal' Onto-Dynamics of its "Logical Types" Progression]

'The Method of [Re-]Flexions'—'Ideo-Gram-matical' Principles of a Comprehensively Operatorial Ideography

Nonlinear Numbers—The 'Aufheben' Evolute Product, [Self-]Multiplication of [Ontological] Qualities: Contra-Boolean Arithmetic & Contra-Boolean Algebra

- The [Re-]Emergence of the [Evolute] Qualifiers
- The Relations of Qualitative [Ontological] Inequality and of Qualitative [Ontological] Succession ['Qualitative Total Order'] & Plato's «
Arithmoi Eidetikoi»- Stages of Number-Concept Meta-Evolution Internal to
_{}and Beyond: From Ontic Pure-qualification to Ontic Quanto-qualification to Onto-metrical Quanto-qualification

_{}as "Non-Standard Natural Arithmetic" [Models for "Gödelian" 'Meta-Natural' Numbers]? The Fundamental Theorem of Dialectical Ideography?

'Convolute' Re-Qualification of the Einstein Momentum Equation: Specimen of 'Quanto-Qualitative Computation'?; Intimations of an Inter-Stellar Drive?

(Quantitative) Continuum [Hypothesis] & [Quanto-Qualitative] Cumulum [Hypothesis]—QModels of "Non-Standard", Non-Cantorian 'Metafinite' Arithmetics?

'Cumulativity' & 'lrrepeatability' in the Time Cumulum: The Time-parameter as Ultimate Dependent Variable

- Psycho-Historical Analysis of the Reification and Fetishism of the Pseudo-Subjectivity Abstract "Time" in the Formation of the Nonlinearity Barrier
- Chrono-Poiesis: 'The Method of Refluxions' and the Constitution/[Re-]Production of Time by the [Self-]Action / [Self-]Change / Self-Duality of all Eventities
- On the ['Meta-Fractal'] 'Chronodynamics' of Meta-Evolving Bodies

The Varieties of Quanto-Qualitative Ideography: Unit Interval vs. Full-Multiplicity 'Qualified' Arithmetics—_{}vs._{}&_{}and Beyond

Part IV. Applications

Taxonomic Level One Application of_{}and_{}—A Dialectical Model of Nature/'Physis' [«Aufheben» Structure/'Meta-Fractality' of the Physical Universe]

Deeper Level Applications of_{}and_{}

- Case Example: Level Two Onto-Dynamics — Meta-Evolution of the Human-Social Relations of [Expanding Human-Society Self-Re-]Production

- The 'Law' of the Tendency of Successful Advanced Industrial Capitalist Democracies to Plutocratic Totalitarian Degeneration — The Political-Economic "Law of Motion" of Modern Society.
- 'Equitism': Publics' Management of 'Externality Equities', New Paradigm for Preserving & Advancing Democratic
-Political Governance.Econo

Applications of_{}—Breaching the Nonlinearity Barrier

- Time for Closed-Form Solutions
- The Paradox of Singularity, the Fetishism of Infinity, and the Principle of Metafinity
- The Method of 'Re-Qualification' and Singularity 'Semantification': Conversion-Singularity Meta-Dynamics and the Restoration of the Qualifiers'

- Psycho-Historical Analysis of the Reification and Fetishism of the Pseudo-Subjectivity "Natural Laws" in the Formation of the Nonlinearity Barrier

- Psycho-Historical Analysis of the Reification and Fetishism of the Pseudo-Subjectivity "Chance" in the Formation of the Nonlinearity Barrier
- "Chaos" is a Misnomer: The Pythagorean Psycho-Historical Trauma Revisited and 'Meta-Re-Stimulated'
- The Healing of the Pythagorean Trauma and the Order of "Chaos" — Never-Repeating Decimals and Never-Repeating Time-Series / Trajectories
- The Order of the Primes and the Order of "Randomness" — Toward a New Paradigm of Order for Nonlinear, Self-Reflexive Dynamics and Meta-Dynamics

Self-Applications [Autocritiques]:_{}vs._{},_{}vs._{}, &_{}vs._{}[Some Aspects of TheirHomeomorphic Defect]

Part V. The'via Singularity' Self-Bifurcating MetasystemsParadigm ofQuanto-Qualitative,'Meta-Fractal' MetadynamicsPostscripts: Epilogues

Objective.

Hypothesis A.The "Law" of the Tendency of Ecological Primary Productivity To Decline?Hypothesis B.TheGreaterversus theLessor ThreatHypothesis C.The Principle of The Planetary Selection TestHypothesis D.The Fusion BarrierandThe Nonlinearity BarrierDesiderata.

a. Working Conviction:The Dialectics / Nonlinearity / Self-Reflexivity Link.

b. Working Conviction:The Nonlinearity / Set-Theoretical Paradoxes Link: Dialectics as Logos/Logy/Logic of 'Self-Reflexive / Self-Refluxive Functions'.

c. Working Conviction:Self-Action Yields 'Quanto-Qualitative' 'Meta-Finite' Growth.

d. Working Conviction:Self-Operation of 'Operators'as Ideographic Representation of'Onto-Genic'and 'Onto-Generative''Self-Reflexion' /'Self-Refluxion'.

e. Working Conviction:The Parmenidean Roots ofThe Nonlinearity Barrier.

f. Working Conviction:Goal: A Non-Reductionist Ideography of the 'Quanto-Qualitative' 'Self-Dynamics' ofNatural(Including'Human-Nature-al') 'Meta-Systems'.

g. Working Conviction:The'Evolute','Cumulative', and'Irrepeatable'Character of Cosmological Evolution / 'The Energy-Time/Space-MatterCumulum'.

h. Working Conviction:Retention and 'Semantification' of "Hybrid", Cross-Product Terms.

i. Working Conviction:Goal: An Ideography of the «Aufheben» Operation.

j. Working Conviction:The Pseudo-Agents "Pure Quantity", "Time", "Natural Law", & "Chance".The Concept of Meta-Finite, Conversion-Singularity '

k. Working Conviction:Self-Bifurcation'.The Concept of '

l. Working Conviction:Self-Duality' or 'Di-act-icity'.

Timing.