{ x2  x }:

Dialectical
 Ideography

A Contribution
to the
Immanent Critique
of
Arithmetic


Prescripts










Distributed «Samizdat» by
Foundation Encyclopedia Dialectica


Version: 7.02.23.2878

Formatted for HTML: 29 January 2007
Last Updated: 23 February 2004 C.E./B.U.E.
First Distributed: 25 November 1999 C.E./B.U.E.





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Omni-Copyright Notice
Omni-Copyright  1999 C.E./B.U.E. by F.E.D. Copyright to original portions of this work is hereby granted to all persons.
Note:
This essay is a partially self-exemplifying exposition of, and a 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; the ontologically dynamical logic of the dialectical 'law" of cognition signified by the 'ideo-ontological', "pure-qualitative", -algebraic inequation x2  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This writing is an unpublished work, and one which is not sold or exchanged for remuneration or commercial gain of any kind, but is distributed «samizdat» to selected individuals and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 surpass at 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 emerging new forms of 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:
(1) a calculus of 'quanto-qualitative change', encompassing an explicit, ideographical arithmetic for the dimensional unit[ie]s or metrical "monads" of classical "dimensional analysis", and, thereby, 'semantifying' the "meaningless" singularities [finite-time "infinite" values] of especially the "unsolvable" [in part, because of those very singularities] nonlinear integrodifferential equations and their solution-functions through their metrical 're-qualification' using those new, explicit 'metrical qualifiers' of this 'dimensional arithmetic', concretizing and operationalizing Plato's «arithmoi monadikoi» and Diophantus' ;

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

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

(4) a mathematics for modeling the history of mathematical ideas as well as a [psycho-]historical algebra and arithmetic for 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 «characteristica universalis»; an arithmetic and algebra of innovative conception or of the creative conceptual process;

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

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

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

(8) an arithmetic, algebra, geometry, and analysis built on certain "non-standard natural numbers", i.e., on the 'Gödelian 'meta-natural' numbers', a space of "hypernumbers" 'of 2nd degree', 'made up out of "standard", '1st degree' natural numbers, instantiating those "non-standard models of first order Peano arithmetic" whose possibility is implied by the first-order conjunction of Gödel's completeness and incompleteness theorems, as by the Löwenheim-Skolem theorem, constructing thereby an 'ontologically dynamical', 'de-Parmenideanized' actualization of Plato's "arithmetic of dialectics", the «arithmoi eidetikoi».
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 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, these non-Parmenidean, contra-Boolean, and contra-Cantorian onto-logical and onto-dynamical arithmetics and their algebras of dialectics may 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', Δh, implications of the '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 'self-meta-evolution'. We have also decided not to disseminate the most "dangerous" of the results to date. We believe that you are eminently capable of '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, of dialectical ideography glossed herein continue to evolve 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-state that can be finally manifested, but an open-ended, 'uncompleteable', asymptotic process, 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; right enough to 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.
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Prescripts:   Motive

Part I.      Prolegomena
 Part II.     The Meta-Evolution Of Arithmetics
[The Cognitive Psycho-Historical Dialectic of Arithmetic to date]
Part III.    The Arithmetics Of Meta-Evolution
['Meta-Systematic-Dialectical', Categorial-Progression Exposition of some Systems of Dialectical Arithmetic]
The Varieties of Quanto-Qualitative Ideography: Unit Interval vs. Full-Multiplicity 'Qualified' Arithmetics — vs.  & and Beyond
Part IV.    Applications
 Part V.    The 'via Singularity' Self-Bifurcating Metasystems Paradigm of Quanto-Qualitative, 'Meta-Fractal' Metadynamics

Postscripts:   Epilogues
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Objective.
Desiderata.
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 of The Nonlinearity Barrier.

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

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

 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".

k. Working Conviction: The Concept of Meta-Finite, Conversion-Singularity 'Self-Bifurcation'.

l. Working Conviction: The Concept of 'Self-Duality' or 'Di-act-icity'.
Timing.
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