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    Default Unifying Engels's Three "Laws" of Dialectics...

    RevLeft Sciences Forum Participants,

    PLEASE NOTE: Much new content, as of 24 February 2013, was added to this Vignette at the website, a part of which I have also added below.

    Below, I have excerpted a new [FONT=Arial,Helvetica,sans-serif]F.E.D. [/FONT]vignette -- Vignette #10 -- just posted to the website, entitled --

    Unification of Engels's Three "Laws" of Dialectics in the Seldonian 'Meta-Model' of 'The Dialectic of Nature'

    -- the full text of which can be reached via the following URLs --

    Happy reading!



    [FONT=Arial]F[/FONT][FONT=Arial].E.D.[/FONT] [FONT=Arial]Vignette [/FONT][FONT=Arial,Helvetica,sans-serif][FONT=Arial]#10[/FONT][/FONT] --

    Unification of Engelss ThreeLawsof Dialectics in the SeldonianMeta-Modelof The Dialectic of Nature.

    Authors Preface. The purpose of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial,Helvetica,sans-serif]#10 [/FONT]is to present the singular expression of Engels’s three “laws” of dialectics that arises in the Seldonian account of The Dialectic of Nature -- the dialectic of the maximal totality; of our cosmos as a whole.

    A Note about the On-Line Availability of Definitions of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Technical Terms. Definitions of [FONT=Arial]Encyclopedia Dialectica[/FONT] technical terms and ‘neologia’ are available on-line via the following URLs --

    -- by clicking on the links associated with each such term, listed, alphabetically, on the web-pages linked above.

    The [FONT=Arial]Encyclopedia Dialectica[/FONT] special terms most fundamental to this vignette are indicated below, together with links to their [FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial]. [/FONT]definitions --

    «arithmos» and «arithmoi»




    ‘‘‘Seldon Functions’’’

    -- definitions resources which will be expanded as the [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial]. [/FONT][FONT=Arial]Encyclopedia[/FONT][FONT=Arial] Project[/FONT] unfolds.

    I. Engelss Three Lawsof Dialectics in His Own Words.

    The storied scientist, J. B. S. Haldane, in his preface to the publication of Engels’s incomplete manuscripts, entitled Dialectics of Nature by their editors, notes that Engels, from at least [FONT=Arial,Helvetica,sans-serif]1871[/FONT], “intended to write a great book to show [quoting Engels: ] “that in nature the same dialectical laws of movement are carried out in the confusion of its countless changes, as also govern the apparent contingency of events in history.” If this book had been written, it would have been of immense importance for the development of science.” [Frederick Engels, Dialectics of Nature, International Publishers [NY: [FONT=Arial,Helvetica,sans-serif]1963[/FONT]], p.[FONT=Arial,Helvetica,sans-serif]viii[/FONT]].

    These Dialectics of Nature manuscripts were never quite finished for publication by Engels, but Engels did, in the short second chapter, entitled “Dialectics”, provide the following formulation of his threelawsof dialectics:

    ...It is, therefore, from the history of nature and human society that the laws of dialectics are abstracted. For they are nothing but the most general laws of these two aspects of historical development, as well as of thought itself. And indeed they can be reduced in the main to three:

    The law of the transformation of quantity into quality and vice versa;

    The law of the interpenetration of opposites;

    The law of the negation of the negation.

    All three are developed by Hegel in his idealist fashion as mere laws of thought: the first, in the first part of his Logic, in the Doctrine of Being; the second fills the whole of the second and by far the most important part of his Logic, the Doctrine of Essence; finally the third figures as the fundamental law of the construction of the [M.D.: Hegel’s] whole system.

    The mistake lies in the fact that these laws are foisted on nature and history as laws of thought, and not deduced from them.

    This is the source of the whole forced and often outrageous treatment; the universe, willy-nilly, is made out to be arranged in accordance with a system of thought which itself is only the product of a definite stage of evolution of human thought.

    If we turn the thing round, then everything becomes simple, and the dialectical laws that look so extremely mysterious in idealist philosophy at once become simple and clear as noonday.

    Moreover, anyone who is even only slightly acquainted with his Hegel will be aware that in hundreds of passages Hegel is capable of giving the most striking individual illustrations from nature and history of the dialectical laws.

    We are not concerned here with writing a handbook of dialectics, but only with showing that the dialectical laws are really laws of development of nature, and therefore valid also for theoretical natural science.

    Hence we cannot go into the inner interconnection of these laws with one another. [ibid., pp. [FONT=Arial,Helvetica,sans-serif]26-27[/FONT], emphasis added].

    We of[FONT=Arial] Foundation [/FONT][FONT=Arial,Helvetica,sans-serif][FONT=Arial]Encyclopedia Dialectica[/FONT][/FONT] [[FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT]] are, however, concerned, in particular, with writing a handbook of dialectics, one which is tentatively slated to appear in [FONT=Arial,Helvetica,sans-serif]2014[/FONT], under the primary authorship of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Public Liaison Officer Aoristos Dyosphainthos.

    We are also concerned, in general, with resuming and continuing the unfinished works of both Engels and Marx.

    We must also, therefore, “go into the inner interconnection of these laws with one another.”

    This [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial]. [/FONT]Vignette [FONT="Arial Bold"][FONT=Arial][FONT=Arial,Helvetica,sans-serif]#[/FONT][FONT=Arial,Helvetica,sans-serif]10[/FONT] [/FONT][/FONT]presents a brief summary of their “inner interconnection” in their more concrete context of universal natural history as the history of the maximal dialectical totality.

    Both Marx and Engels frequently criticized the “abstract” character of the natural sciences of their time, and, in particular, their abstractness in relation to the history of nature.

    Engels’s account of the three laws of dialectics, nonetheless, had to remain, precisely, abstracted from natural history, “suprahistorical” [Marx] in their formulation and presentation, because the natural science of Engels’s time had not yet penetrated with sufficient depth into the macrocosm, ‘mesocosm’, and microcosm of the present cosmological epoch, or into the reconstruction of those of past epochs of natural history, to provide a knowledge-base sufficient to support the unified formulation of The Historical Dialectic of Nature.

    This fact may have contributed to the reasons owing to which Engels decided to give priority to other projects, and never completed either the Dialectics of Nature ms., or his intended greater work on dialectics.

    But, beginning circa the [FONT=Arial,Helvetica,sans-serif]2[/FONT]nd half of the [FONT=Arial,Helvetica,sans-serif]20[/FONT]th century, that condition has begun to be remedied.

    We take advantage of that growth in knowledge, subsequent to Engels’s lifetime, herein.

    II. Unitary Operation of the Three Laws of Dialectics in Cosmos-History, Stated Narratively [FONT=Arial,Helvetica,sans-serif][FONT=Arial]&[/FONT][/FONT]In a Nutshell.

    [FONT=Arial]1[/FONT]. «Genos»: Generic Narrative of the Unified Dialectical Meta-Dynamic of Natures Self-Revolutions. The in-context operation of Engels’s first “law” of dialectics, the lawof the transformation of quantitative change into qualitative change, as concretely occurring in the main sequence of the natural history of the cosmos, is driven by the ‘auto-catalytic’ character of the populations of individuals of each successive major kind of individuals, each represented by a distinct ‘‘‘ontological category’’’, that has arisen in the self-caused course of Nature’s ‘self-meta-evolution’.

    Individuals of each newest kind make more individuals of that kind, of their own kind; they self-reproduce, expandedly, and often at an accelerated rate, at least early-on in their ‘self-evolution’.

    They do so by converting ‘onto-mass’ of other, previously ‘self-meta-evolved’ kinds, into ‘onto-mass’ of their own kind.

    The ‘qualitative growth’ of the universe, and of the ontology of the universe, that such “quantitative growth”, such accelerated expanded self-reproduction, passes into, is not precipitated just by the growing quantity, the growing population-count, the growing number, of individuals of that newest ‘onto-type’.

    It is also a matter of the growing physical-spatial concentration of that mounting quantity of such individuals.

    That rising ‘onto-density’, in the cores of that ‘onto-type’s’ nucleation-zones, where that concentration maximizes, bring individuals of that same newest ‘onto-type’ into intense interaction mainly with one another, no longer, at least not at the cores of those nucleation-zones, mainly with individuals of earlier ‘meta-evolved’ ‘onto-types’.

    Latent potentials of such interaction of similars are thus actualized to unprecedented degree.

    From such unprecedented degrees of ‘‘‘self-interaction’’’ -- or of ‘‘‘intra-action’’’ -- within the populations of the newest ‘onto-type’, a next newest ‘onto-type’ irrupts, one constituted by an unprecedented new kind of individuals, qualitatively different from those of the old newest ‘onto-type’, and qualitatively different from those of all earlier ‘meta-evolved’ ‘onto-types’, representing, thus, a new quality of cosmological being, a new ‘onto-type’ -- unprecedented, new ontology.

    Thus, quantitative change, change in the quantity and concentration/density of individuals of the earlier kind of ontology, mere quantitative growth, has abruptly given rise to a population of a new kind of individual, to qualitative change, to qualitative growth -- to ontological revolution; to yet another increment in the growth of the ontology of the cosmos.

    Quantitativeevolution has passed into qualitative meta-evolution.

    The dynamics of evolution -- quantitative expanded self-reproduction of individuals of a given, then-vanguard ‘onto-type’ -- has, via metafinite singularity, passed into the meta-dynamics of the «aufheben» birth of a new vanguard ‘onto-type’, from out of the very womb of its immediate predecessor ‘self-hybrid’ vanguard ‘onto-type’.

    And it has done so by way of the interpenetration of opposites -- not, in the case of this main sequence, historical dialectic of natural history, of ‘‘‘complementary opposites’’’, nor of annihilatory opposites, but of supplementary opposites

    [see [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial,Helvetica,sans-serif][FONT=Arial]#6[/FONT][/FONT], The Dialectic of Opposition, for background on this ideo-taxonomyof the three «species», or kinds, of oppositeness:].

    The quality of oppositeness of the old vanguard ‘onto-type’, versus its successor, may not always be experientially and affectively accessible to its human observers, which is why [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] calls [FONT=Arial,Helvetica,sans-serif][FONT=Arial]2[/FONT][/FONT]nd terms in its ideo-dialectical meta-models’ ‘contra-thesisterms, whereas [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] calls [FONT=Arial,Helvetica,sans-serif][FONT=Arial]2[/FONT][/FONT]nd terms in its physi[c]o-dialectical meta-models’ meta-physisterms.

    The formerly-latent, unmanifest potentials of the ‘‘‘self-interaction’’’ of the up-until-then newest ‘onto-type’ constitute an immanent other-ness’, an intra-duality’, or self-duality’, an internal-/self-opposition’, of that up-until-then newest ‘onto-type’, one which becomes ‘outered’, externally manifested, once the intensity of population ‘‘‘self-interaction’’’, or ‘intra-action’, breaches the threshold whereafter the next newest ‘onto-type’ becomes ‘irruptively actualized’, actualized as the 'supplementary opposite of the dominant, external face/manifestation of its predecessor vanguard ‘onto-type’.

    And this process, of new ‘onto-type’ actualization, typically by self-«aufheben» self-meta-individual-ization, i.e., by self-meta-monad-ization, is also the concrete, in-context instantiation of the dialecticalnegation of the negation.

    It is the ‘present self-meta-monad-ization of the previous self-meta-monad-ization; the ‘present self-«aufheben» of the previous self-«aufheben»’.

    [FONT=Arial]2[/FONT]. «[FONT=&quot]Species[/FONT]»: Specific Narrative of the Unitary Nature-Dialectic of a Paradigmatic Particular Example.

    The Particular, Paradigmatic Example for this Vignette: Stellar Meta-Evolution’ / “Stellar Nucleosynthesis. In the Dyadic Seldon Functionformulation of the Dialectical Theory of Everything Equation, that is, of the SeldonianCosmos-History Meta-Model, the various kinds of, and “generations” of, stars, figure as dialectical synthesis formations, i.e., as ‘partial and total uni-physes, which synthesize ions, atomic nuclei, eventual non-ionic atoms, from sub-atomic particles, mainly, at first, by a thermonuclear fusion process named stellar nucleosynthesis, which populates the later universe with higher atomic species.

    Thus, stars figure in that Meta-Model as ontological conversion-formations, i.e., as ‘ontologically hybridized formations, containing both “lower atomic speciesnuclei, generated, earlier, by cosmological nucleosynthesis, and sub-atomicparticles [e.g., “free, naked protons -- ionized Hydrogen “atoms”], and even sub-nuclear particles.

    However the purpose of this example, we will focus attention only upon the thermonuclear-fusion-conducting core of a hypothetical, Sol-like, “first generation” star, so that we can treat that stellar core as the locus of a self-«aufheben» progression of self-conversions, or auto-conversions [punctuated also by allo-conversions], of meta-physes’, of self-hybridself-meta-individualizations’ [also referred to as ‘self-meta-unitizations’] of “sub-atomic particles” as units -- of core protons [and neutrons] as units -- into Helium nuclei, ions of Helium atoms [e.g., [FONT=Arial]He++[/FONT]], as the meta-unitsof those units.

    We will describe these processes “locally”, i.e., relative to this one, individual stellar core as locus.

    Beyond, e.g., the “proton-proton [p-p] reaction, the fusion of protons/ionized Hydrogen atoms/Hydrogen isotope ions [e.g., Deuterium] into Helium nuclei, stellar nucleosynthesis will later entail, for stars of large-enough initial mass, a progression of further ‘‘‘self-interactions’’’, or intra-actions’, of their newly-arising stellar core «arithmoi» of nuclei [meta-]units of rising atomic number; of interactions” among their [meta-]units, i.e., of both auto-meta-unitization’ and allo-meta-unitization’ nuclear reactions, sometimes followed by atomic-number-reducing radioactive decay of the reaction-product nuclei.

    These nuclear reactions include Hydrogen nuclei into Helium nuclei [atomic number [FONT=Arial]2[/FONT]: auto[FONT=Symbol][FONT=Symbol] ---->[/FONT][/FONT] allo]; Helium nuclei into Carbon nuclei [atomic number [FONT=Arial]6[/FONT]: auto]; Carbon nuclei into Magnesium nuclei [atomic number [FONT=Arial]12[/FONT]: auto], Sodium nuclei [atomic number [FONT=Arial]11[/FONT]: auto], Neon nuclei [atomic number [FONT=Arial]10[/FONT]: auto], Nitrogen nuclei [atomic number [FONT=Arial]7[/FONT]: via radioactive decay], and Oxygen nuclei [atomic number [FONT=Arial]8[/FONT]: auto]; Oxygen nuclei into Sulfur nuclei [atomic number [FONT=Arial]16[/FONT]: auto], Phosphorus nuclei [atomic number[FONT=&quot] [FONT=Arial]15[/FONT][/FONT]: auto], and Silicon nuclei [atomic number [FONT=Arial]14[/FONT]: auto]; ... ending with Iron nuclei [atomic number [FONT=Arial]26[/FONT]:allo-meta-unitization’/radioactive decay].

    The main progression of such stellar core fusion “burning” is Hydrogen[proton] “burning”, Heliumburning”, Carbonburning”, Neonburning”, Oxygen burning”, and Siliconburning”.

    The nuclei units of most of the remaining chemical elements are generated by hybrid and/or ‘‘‘non-nucleosynthetic’’’ processes.

    The driver of this progression is the dialecticalintra-duality or self-duality or indivisible-duality that stars are.

    A star is the dialectical ‘‘‘complex unity’’’, the continual dueling, of a continuing self-gravitational self-implosion and of a colossal thermonuclear self-explosion, both at the macrocosmic level, opposing one another at every point within the star [sometimes called “hydrostatic equilibrium”], both arising from the very “self”, the body, of the star itself, and both tied to the opposition between the core protons’ mutually attracting nuclear force, and mutually repelling electrostatic force, at the microcosmic level, with both of these microcosmic forces sourced in the self-same core protons themselves.

    Thus, as initial core Hydrogen is completely converted into Helium in a Sun-like star, the ‘thermonuclear self-explosion’ dual momentaneously subsides, leaving the ‘self-gravitational self-implosion’ dual unopposed.

    Therefore, there begins, at this moment, an accelerating self-contraction, self-compressing the whole star, and its core, and driving-up the ‘physical-spatial concentration’ -- the "density" -- of the Helium “ash” in the core, until a critical threshold of Helium density is crossed, at which “the Helium flash”, a “runaway process” of Helium fusion -- of the “ash” resurrecting itself to “burn” again, in a new way of “burning” -- irrupts into existence, along with the first new kinds of nuclei that result from Helium fusion.

    The latter are the new, unprecedented ‘physio-ontology’ of, e.g., Carbon nuclei [new and unprecedented for this locus, for this core’s locale, at least; new for the total cosmos in the case of the very first, "first generation star", since Carbon nuclei are effectively not produced at all during the preceding epoch of "cosmological nucleosynthesis"].

    The quantitative growth of the stellar core Helium population, via Hydrogen fusion, and its quantitative densification’ due to the exhaustion of Hydrogen “fuel”, leads to that qualitative, ontological change, which is the irruption of the new ‘physio-ontology’ of Carbon nuclei, etc., as the “ash” of “Helium burning”, after the metafinite singularity of “The Helium flash”, which converts the relative ‘‘‘ash/waste/entropy”’ of core Helium nuclei into the new, relative ‘‘‘fuel/resource/negentropy’’’ of a new epoch of revived, continued stellar evolution for the particular star in question.

    When the core Helium “fuel” is itself, in turn, completely converted to Carbon, etc., “ash”, this process self-continues again, in temporal meta-fractal self-similarity to the Hydrogen/Helium pattern just described, until energy-liberating, self-explosion-dual-sustaining nucleosynthesis finds its limit with the irruption of Iron nuclei as the predominant content of the stellar core.

    . . .

    Links to definitions of additional Encyclopedia Dialectica special terms deployed in the discourse above --



    auto-negation or self-negation

    Booles Algebra



    dialectical categorial progression

    ‘‘‘dialectical contradiction’’’ versus ‘‘‘propositional contradiction’’’, etc.

    dynamics versus ‘‘‘meta-dynamics’’’,19DEC2012.jpg


    evolution versus ‘‘‘meta-evolution’’’,19DEC2012.jpg

    self-meta-monad-ization or self-meta-individual-ization

    ontological category


    Last edited by Miguel Detonnaciones; 11th March 2013 at 15:38. Reason: to paste-in additional content
  2. #542
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    Default New Content Added to F.E.D. Vignette #10: Unification of Engels's "Laws"

    RevLeft Sciences Forum Participants,

    FYI: Major expansions of the content of [FONT=Arial]F.E.D.[/FONT] Vignette [FONT=Arial]#10[/FONT], on the Unification of Engels's three "laws" of his Dialectics of Nature, have recently been added.

    For the new, full content of that Vignette, see --


    Last edited by Miguel Detonnaciones; 1st March 2013 at 08:27.
  3. #543
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    Default Dialectical-Algebraic Expression of Engels Three "Laws" of Dialectics

    RevLeft Sciences Forum Participants,

    FYI: Aoristos Dysophainthos, Public Liaison Officer of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT], has recently posted a new entry to his blog, presenting a dialectical-algebraic expression of Engels's three "laws" of Dialectics.

    For the full blog-entry, see --

    I have excerpted from this blog-entry below.



    [FONT=&quot]"[/FONT]One additional way for we of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] to celebrate, and to celebrate by succinctly summarizing, . . . [the] . . . recent Vignette -- [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial]#10[/FONT] -- on the unification of Engels’s three “laws” of dialectics, is to express those three “laws” in the dialectical ideography of the Seldonian First Dialectical Algebra, that of the [FONT=Arial]Q[/FONT] arithmetic.

    The purpose of this blog entry is to share that celebration, as that summary, with our world wide web public.

    Its first task is to communicate the meaning(s) of the dialectical-ideographical symbols used for this formulation --

    Attachment 9024

    The next task is to express Engels’s three laws of The Dialectic of Nature in terms of the symbols defined above, and in accordance with the Seldonian interpretation of those three laws, as rendered . . . in [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial]#10[/FONT].

    [Reception of the meanings of these dialectical-algebraic expressions of Engels's "laws" -- and especially of "laws" [FONT=Arial]2[/FONT] and [FONT=Arial]3[/FONT] -- will be facilitated, for the reader, if it is kept in mind that the generic dialectical-negation operator, represented by '[FONT=Arial Black]~[/FONT]' in the typography available herein, but with a more "angular" rendering in the JPEG images pasted-in to this post, can be characterized in a simple way.

    It can be characterized, syntactically, and informally, as an operation which acts like "a horizontal, leftward ditto mark", that is, it replaces itself with an exact copy of the symbol immediately to its right, a copy that is thus placed immediately to the left of that symbol that was copied from immediately to the right of the dialectical negation sign, and that dialectical negation sign is thereby removed, eliminated, supplanted by that copy.]

    After that task is accomplished, the final task for this blog-entry is to translate those ideographic renderings of Engel’s laws back into their exact prose counterparts, and to narrate some of the nuances of those ideographical formulations.

    We leave to a later presentation the exposition of the detailed inner interconnection of these laws with one another [cf. Engels, Dialectics of Nature], in part, by means of their derivation, as theorems, from the first order axioms of the [FONT=Arial]Q[/FONT] dialectical arithmetics, from the definitions of the symbols used, as above, and from several additional Principles of Nature-Dialectic.

    For now, let us simply say that the unity and interconnectedness of Engels’s threelaws is implicitly contained in the Dyadic Seldon Function itself, in its assertion that the vast qualitative, ontological diversity of our cosmos is seeded in the self-iterated self-movement of a single, ‘‘‘singular’’’, ontic category/«arithmos», as «arché» ontological category, with the term «arché» meaning 'the ultimate ancestor in a meta-genealogy', e.g., in an ontological categorial progression --

    [FONT=Arial Black][FONT=Verdana]namely, the [/FONT][/FONT][FONT=Arial Black][FONT=Verdana][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0[/FONT][FONT=Arial])[/FONT] '''seed''' ontological category in: [/FONT]

    [/FONT][FONT=Arial](t[/FONT][FONT=Arial]) = [/FONT][FONT=Arial Black][ >-|-<[/FONT][FONT=Arial](0[/FONT][FONT=Arial]) [/FONT][FONT=Arial Black]][/FONT][FONT=Arial]^(2^t)[/FONT].

    Engels’s three laws of The Dialectic of Nature, in terms of the symbols defined above, and in accordance with the Seldonian interpretation of those three laws, as rendered by . . . in [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial]#10[/FONT], look like this --

    Attachment 9025

    -- whose direct “prose” translations are the following --

    [FONT=Arial]1[/FONT]. "The [FONT=Arial]t[/FONT]-cumulum ontology increment is, explicitly, not contained in the [FONT=Arial]t[/FONT] ontology-cumulum, which, over time, turns itself into the full self-action of that [FONT=Arial]t[/FONT] ontology-cumulum, which equals that [FONT=Arial]t[/FONT] ontology-cumulum non-amalgamatively added to that [FONT=Arial]t[/FONT]-cumulum ontology increment, which sum is equal to the [FONT=Arial]t+1[/FONT] ontology-cumulum, which does explicitly contain the [FONT=Arial]t[/FONT]-cumulum ontology increment, and that [FONT=Arial]t[/FONT]-cumulum ontology increment is qualitatively, ontologically unequal to the [FONT=Arial]t[/FONT] ontology-cumulum itself."

    [FONT=Arial]2[/FONT]. "The [FONT=Arial]t[/FONT]-cumulum ontology increment is implicitly contained in the [FONT=Arial]t[/FONT] ontology-cumulum, as a not-yet-actualized potentiality, and the [FONT=Arial]t[/FONT] ontology-cumulum, over time, turns itself into its own dialectical, determinate, self-«aufheben» self-negation of itself, which equals itself non-amalgamatively and antagonistically added to the [FONT=Arial]t-[/FONT]cumulum ontology increment, which together equal the [FONT=Arial]t+1[/FONT] ontology-cumulum, which does explicitly contain the [FONT=Arial]t[/FONT]-cumulum ontology increment, which is the supplementary other to the [FONT=Arial]t[/FONT] ontology-cumulum itself."

    [FONT=Arial]3[/FONT]. "As epoch [FONT=Arial]t [/FONT]turns itself into epoch [FONT=Arial]t+1[/FONT], the [FONT=Arial]t[/FONT] ontology-cumulum turns itself into the full self-action of the [FONT=Arial]t[/FONT] ontology-cumulum, which is equal to the dialectical, determinate, self-«aufheben» self-negation of the [FONT=Arial]t[/FONT] ontology-cumulum; then, next, as epoch [FONT=Arial]t+[/FONT][FONT=Arial]1 [/FONT]turns itself into epoch [FONT=Arial]t+2[/FONT], the [FONT=Arial]t+1[/FONT] ontology-cumulum turns itself into the full self-action of the [FONT=Arial]t+1[/FONT] ontology-cumulum, which is equal to the dialectical, determinate, self-«aufheben» self-negation of the [FONT=Arial]t+1[/FONT] ontology-cumulum, which is equal to the dialectical, determinate, self-«aufheben» self-negation of the dialectical, determinate, self-«aufheben» self-negation of the [FONT=Arial]t[/FONT] ontology-cumulum."

    Commentary on Engelss First Law of Dialectics. The quantitative nature of the kind of change that passes into the qualitative, ontological kind of change -- the latter represented by [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT] -- in the dialectical-algebraic expression of law[FONT=Arial] 1[/FONT] cannot be expressed, as such, in the purely-qualitative, purely-ontological language of the [FONT=Arial]Q[/FONT] dialectical algebras, and at this cumula-of-ontological-categories scale of description.

    It can be adumbrated only, by a kind of qualitative shadow of the quantitative, in the form of the absence of explicit containment of the qualitative increment of new ontology, [FONT=Arial Black]delta[/FONT]-[FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT], in [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT], versus its full presence/containment in [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t+1)[/FONT].

    There is, at the ontological-categorial scale of description, a dialectical-algebraic expression of law[FONT=Arial] 1[/FONT] that can be formulated in the seventh, quanto-qualitative dialectical-algebra in the Seldonian dialectical progression of dialectical algebras, the '[FONT=Arial]Mu[/FONT]' algebra, that we will present later, in another venue.[FONT=&quot]


    Commentary on Engelss Second Law of Dialectics. As . . . stated in [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial]#10[/FONT] --

    The quality of oppositeness of the old vanguard ‘onto-type’, versus its successor, may not always be experientially and affectively accessible to its human observers, which is why [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] calls [FONT=Arial]2[/FONT]nd terms in its ideo-dialectical meta-models contra-thesisterms, whereas it calls [FONT=Arial]2[/FONT]nd terms in its physi[c]o-dialectical meta-models meta-physisterms.

    The formerly-latent, unmanifest potentials of the ‘‘‘self-interaction’’’ of the up-until-then newest ‘onto-type’ constitute an immanent other-ness, an intra-duality’, or self-duality’, an internal-/self-opposition, of that up-until-then newest ‘onto-type’, one which becomes outered, externally manifested, only once the intensity of population ‘‘‘self-interaction’’’, or ‘intra-action’, breaches the threshold whereafter the next newest ‘onto-type’ becomes ‘irruptively actualized’, actualized as the 'supplementary other of the dominant, external-face/-manifestation of its predecessor vanguard ‘onto-type’.

    Commentary on Engelss Third Law of Dialectics. Relative to the scale of negation of negation presented in the algebraical rendering above, at a more intensive scale of ‘[FONT=Arial]t[/FONT]ime’, ‘[FONT=Arial]t[/FONT]emporality’, or ‘[FONT=Arial]t[/FONT]-epochality’, every value of [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT], for [FONT=Arial]t > 0[/FONT], can be grasped as the product of a dialectical negation of negation.

    That is, if each value of [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT] is understood to be, already, in itself, in general, an ‘‘‘eventity’’’, and, in particular, a particular, specific negation operation -- a particular, specific dialectical, determinate, «aufheben»-negation operator -- then every specific value of [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT] falls under the general symbol [FONT=Arial Black]~[/FONT]: [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial])[/FONT] [FONT=Arial Narrow]IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT].

    Therefore, for all [FONT=Arial]t+1[/FONT], we have that [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial]+1)[/FONT][FONT=Arial Narrow] IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT], each value of [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t[/FONT][FONT=Arial]+1)[/FONT] thus instancing negation negation, viz. --

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](1)[/FONT][FONT=Symbol] [FONT=Arial]=[/FONT] [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0)[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0)[/FONT][FONT=Symbol][FONT=Symbol][FONT=Arial] =[/FONT][/FONT] [/FONT][FONT=Arial Black]~[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0) [/FONT][FONT=Arial Narrow]IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT];

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](2)[/FONT][FONT=Symbol] [FONT=Arial]=[/FONT] [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](1)[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](1)[/FONT][FONT=Symbol][FONT=Symbol][FONT=Arial] =[/FONT][/FONT] [/FONT][FONT=Arial Black]~[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](1)[/FONT] [FONT=Arial Narrow]IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT];

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](3)[/FONT][FONT=Symbol] [FONT=Arial]=[/FONT] [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](2)[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](2) [/FONT][FONT=Symbol][FONT=Symbol][FONT=Arial]=[/FONT][/FONT] [/FONT][FONT=Arial Black]~[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](2)[/FONT] [FONT=Arial Narrow]IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT];

    , and, in general --

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t+1)[/FONT][FONT=Symbol] [FONT=Arial]=[/FONT] [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT][FONT=Symbol][FONT=Symbol][FONT=Arial] = [/FONT][/FONT] [/FONT][FONT=Arial Black]~[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT] [FONT=Arial Narrow]IS CONTAINED IN[/FONT] [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT].

    It is only [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0)[/FONT], the ‘‘‘«arché»’’’, the ultimate ancestor, the opening, non-cumulum singleton/‘‘‘singularity’’’ of an historical progression of cumula, or consecuum of cumula, of ontological categories, e.g. of system categories, etc., which is not contained in the category of negation negations for its native 'meta-model'.

    I.e., it is only the founding term, [FONT=Arial Black]

    [/FONT][FONT=Arial](0)[/FONT][FONT=Symbol][FONT=Arial Black] [---> [/FONT][/FONT][FONT=Arial]q1 [/FONT][FONT=Arial Narrow]IN [/FONT] [FONT=Arial][FONT=Arial Black]W[/FONT]Q = { q0[/FONT][FONT=Arial],[/FONT][FONT=Arial] q1[/FONT][FONT=Arial],[/FONT][FONT=Arial] q2[/FONT][FONT=Arial],[/FONT][FONT=Arial] q3[/FONT][FONT=Arial],[/FONT][FONT=Arial] ... }[/FONT],

    that cannot be expressed as a product of dialectical negation-negation, and that cannot be expressed as being, itself, an ontic cumulum, in that same particular language and meta-model of [FONT=Arial]{ [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT][FONT=Arial] }[/FONT] --

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0)[/FONT][FONT=Arial Narrow] IS [FONT=Arial Black]NOT[/FONT] CONTAINED IN[/FONT] the meaning of [FONT=Arial Black]~[/FONT][FONT=Arial Black]~[/FONT]; [FONT=Symbol][FONT=Symbol]

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0) ~= [/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT] for any value of [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT], i.e., [FONT=Symbol][FONT=Symbol]

    [FONT=Arial Black]>-|-<[/FONT][FONT=Arial](0)[/FONT] has no predecessor in [FONT=Arial]{[/FONT][FONT=Arial][FONT=Arial Black]>-|-<[/FONT][FONT=Arial](t)[/FONT][/FONT][FONT=Arial Black] |[/FONT][FONT=Symbol] [FONT=Arial]t[/FONT] [/FONT][FONT=Symbol][FONT=Symbol][FONT=Arial Narrow]is IN[/FONT] [/FONT][/FONT][FONT=Arial Black]W[/FONT][FONT=Arial]}[/FONT].

    But in general, for any specific value of [FONT=Arial]t[/FONT], [FONT=Arial]t = [/FONT][FONT=Arial]0 [/FONT]included, we have[FONT=&quot] --[/FONT][FONT=Symbol]

    [FONT=Arial Black] [FONT=&quot]...

    [FONT=Arial Black][FONT=Arial](t[/FONT][FONT=Arial])[/FONT]~[/FONT][FONT=Arial](t[/FONT][FONT=Arial]) [/FONT][FONT=Arial][FONT=Arial] =[/FONT] [/FONT][FONT=Arial][FONT=Arial Black]~[/FONT][FONT=Arial](t+1[/FONT][FONT=Arial])[/FONT] [/FONT][FONT=Symbol][FONT=Arial Black]--->
    [/FONT][/FONT] [FONT=Arial Black]
    [FONT=Arial Black][FONT=Arial](t[/FONT][/FONT][FONT=Arial Black][FONT=Arial][FONT=Arial][FONT=Arial]+1[/FONT][/FONT][/FONT][FONT=Arial])[/FONT]~[/FONT][FONT=Arial](t[/FONT][FONT=Arial][FONT=Arial]+1[/FONT][/FONT][FONT=Arial]) [/FONT][FONT=Arial][FONT=Arial] =[/FONT] [/FONT][FONT=Arial][FONT=Arial][FONT=Arial Black]~[/FONT][FONT=Arial](t+2[/FONT][FONT=Arial])[/FONT] [/FONT][FONT=Symbol][FONT=Arial Black]--->
    [/FONT][/FONT] [FONT=Arial Black]
    [FONT=Arial Black][FONT=Arial](t[/FONT][/FONT][FONT=Arial Black][FONT=Arial][FONT=Arial][FONT=Arial]+[/FONT][/FONT][/FONT][/FONT]
    [FONT=Arial][FONT=Arial Black][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=Arial])[/FONT][FONT=Arial][FONT=Arial Black]~[/FONT][/FONT][FONT=Arial](t[/FONT][FONT=Arial][FONT=Arial]+[/FONT][/FONT][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=Arial]) [/FONT][FONT=Arial][FONT=Arial] =[/FONT][/FONT] [FONT=Arial][FONT=Arial][FONT=Arial Black]~[/FONT][FONT=Arial](t+3[/FONT][FONT=Arial])[/FONT] [/FONT][FONT=Symbol][FONT=Arial Black]--->[/FONT][/FONT]

    [FONT=Arial Black]~[/FONT][FONT=Arial Black][FONT=Arial](t[/FONT][/FONT][FONT=Arial Black][FONT=Arial][FONT=Arial][FONT=Arial]+[/FONT][/FONT][/FONT][/FONT]
    [FONT=Arial][FONT=Arial Black][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=Arial])[/FONT][FONT=Arial][FONT=Arial Black]~[/FONT][/FONT][FONT=Arial](t[/FONT][FONT=Arial][FONT=Arial]+[/FONT][/FONT][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial][FONT=Arial]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=Arial]) [/FONT][FONT=Arial][FONT=Arial] =[/FONT][/FONT] [FONT=Arial][FONT=Arial][FONT=Arial Black]~[/FONT][FONT=Arial](t+4[/FONT][FONT=Arial])[/FONT] [/FONT][FONT=Symbol][FONT=Arial Black]--->[/FONT][/FONT][/FONT] [FONT=&quot]...[/FONT] .[FONT=&quot]"[/FONT]


    Links to definitions of additional [FONT=Arial]Encyclopedia Dialectica[/FONT] special terms deployed in the discourse above --


    auto-negation or self-negation



    dialectical categorial progression


    ontological category

    Last edited by Miguel Detonnaciones; 10th March 2013 at 18:07. Reason: content additions
  4. #544
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    Default '''Number''' Theory, Ancient vs. Modern vs. 'Trans-Modern' [Seldonian]

    [FONT=Times,&quot;Times New Roman&quot;,serif]RevLeft Sciences Forum Participants,[/FONT]

    [FONT=Times,&quot;Times New Roman&quot;,serif]

    [FONT=Times,&quot;Times New Roman&quot;,serif]I have excerpted, below, parts of the forthcoming [/FONT][FONT=Arial,Helvetica,sans-serif]F.E.D.[/FONT] [FONT=Arial,Helvetica,sans-serif]Vignette[/FONT] [FONT=Arial,Helvetica,sans-serif]#11[/FONT][FONT=Times,&quot;Times New Roman&quot;,serif], by [/FONT][FONT=Arial,Helvetica,sans-serif]F.E.D.[/FONT] [FONT=Times,&quot;Times New Roman&quot;,serif]Public Liaison Officer Aoristos Dyosphanthos, entitled '''Number''' Theory, Ancient vs. Modern vs. 'Trans-Modern', below, for your reading and learning pleasure.[/FONT]

    [FONT=Times,&quot;Times New Roman&quot;,serif]

    [FONT=Times,&quot;Times New Roman&quot;,serif]I will add further excerpts as they become extant.[/FONT]

    [FONT=Times,&quot;Times New Roman&quot;,serif]The original text can be accessed via the following URLs --,28MAR2013.pdf

    [FONT=Times,&quot;Times New Roman&quot;,serif]Regards,[/FONT]

    [FONT=Times,&quot;Times New Roman&quot;,serif]Miguel [/FONT]

    [FONT=Times,&quot;Times New Roman&quot;,serif]P.S. I have sometimes had to render, below, Greek letters, and other Ancient Greek symbols, in ways that could be accommodated within the available typography, or even as blank character-strings -- _______ -- when I found no more adequate way to render them.[/FONT]

    [FONT=Arial]F[/FONT][FONT=Arial].E.D.[/FONT] [FONT=Arial]Vignette[/FONT][FONT=Arial,Helvetica,sans-serif][FONT=Arial] #11[/FONT][/FONT] --






    .[FONT=Times,&quot;Times New Roman&quot;,serif]

    [FONT=Times,&quot;Times New Roman&quot;,serif][FONT=Times New Roman]'[/FONT][/FONT][FONT=Arial Black]Trans[/FONT][FONT=Arial Black]-[/FONT][FONT=Arial]Modern[/FONT][FONT=Times,&quot;Times New Roman&quot;,serif][FONT=Times New Roman]'[/FONT][/FONT].

    by Aoristos Dyosphainthos

    Authors Preface. The purpose of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] Vignette [FONT=Arial]#11 [/FONT]is to present [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT]’s «Arithmos» Theory -- a psychohistorical-dialectical synthesis of Ancient «Arithmos» Theory [FONT=Arial]&[/FONT]/with Modern ‘‘‘Number Theory’’’ -- without recourse to numbers.

    A Note about the On-Line Availability of Definitions of [FONT=Arial,Helvetica,sans-serif][FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT][/FONT] Key Technical Terms. Definitions of [FONT=Arial,Helvetica,sans-serif][FONT=Arial]Encyclopedia Dialectica[/FONT][/FONT] technical terms and ‘neologia’ are available on-line via the following URLs --

    --by clicking on the links associated with each such term, listed, alphabetically, on the web-pages linked above.

    The [FONT=Arial,Helvetica,sans-serif][FONT=Arial]Encyclopedia Dialectica[/FONT][/FONT] special terms most fundamental to this vignette are indicated below, together with links to their [FONT=Arial,Helvetica,sans-serif][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT][/FONT] definitions --

    «arithmos» and «arithmoi»

    «aufheben»,%20A%20Dialectical%20%27%27Theory%20of%20Ev erything%27%27,%20Volume%200.,%20FOUNDATIONS,%20Ed ition%201.00,%20first%20published%2010DEC2011,%20Definition,%2 0AUFHEBEN,%2018AUG2011,%20JPEG.jpg



    [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT]dialectical arithmetic/algebra

    self-meta-monad-ization or self-meta-individual-ization

    -- we plan to expand these definitions resources as the [FONT=Arial,Helvetica,sans-serif][FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial]. [/FONT][FONT=Arial]Encyclopedia[/FONT][FONT=Arial] Project[/FONT][/FONT] unfolds.

    [FONT=Arial Black]q1[/FONT] [FONT=Arial]<---][/FONT] [FONT=Verdana]I[/FONT]. [FONT=Times,&quot;Times New Roman&quot;,serif]Ancient[/FONT][FONT=Times,&quot;Times New Roman&quot;,serif] «Arithmos» Theories[/FONT].

    Exemplary Ancient, Quanto-Qualitative Definitions of ‘“Number’’’ as «Arithmos»: E.g., by Euclid and by Aristotle.

    Not generally realized by we Moderns is the '''psychohistorical''' fact that the Ancients’ concept of ‘‘‘Number’’’ -- named, in ancient Greek, by the word «Arithmos», the ancient Greek word from which the modern English word Arithmetic” descends -- as proven by the ‘psychoartefacts’ that the Ancients left behind, and that have come down to us, still extant, was qualitatively, ideo-ontologically’ different from our Modern concept of “Number”.

    Summarily we can say that Ancients defined an «Arithmos» as an ‘‘‘assemblage of qualitative -- of multiple-qualities-exhibiting -- units/things of a given, single kind’’’.

    «Arithmoi» are thus bothquanto-qualitative’, and sensuous and ideational, phenomena, not “purely quantitative”, quantifier-only’ ideo-phenomena, such as, e.g., [FONT=Arial]2[FONT=Times New Roman], [/FONT]3[FONT=Times New Roman], [/FONT]4[FONT=Times New Roman], [/FONT]5[FONT=Times New Roman], [/FONT]6[FONT=Times New Roman], [FONT=Arial]and[/FONT][/FONT] 7[/FONT][FONT=Times New Roman], ... .[/FONT]

    Herein we will take, as representative of the Ancients’ concept of ‘‘‘Number’’’, in its philosophical and/or mathematically technical form, the still-extant recorded thoughts of Euclid and Aristotle.

    For Euclid [third Century B.C.E.]: “Euclid defines in the Elements, VII, 2, a number as “the multitude made up of units” having previously (Elements, VII, 1) said that a unit is “that by virtue of which each of existing things is called one.” As a unit is not composed of units, neither EUCLID nor ARISTOTLE regard a unit as a number, but rather as “the basis of counting, or as the origin [i.e., as the «arché» -- A.D.] of number.” There is an echo of this Euclidean definition in CANTOR’s definition of the cardinal number as a set composed of nothing but units ... .” [H. Hermes, et al., Numbers, Springer Verlag, [NY: [FONT=Arial]1991[/FONT]], p. [FONT=Arial]12[/FONT]].

    Note that this -- “self-evident?” -- claim that “a unit is not composed of units” posits a radical duality between multiplicity / number on the one hand, and unity on the other, i.e., between «arithmos»[FONT=&quot] [FONT=Arial]&[/FONT][/FONT] «monad».

    The Ancients excepted a single «monad» from their category of ‘‘‘number’’’ simply because a single «monad» is not an assemblage of «monads» -- is not plural.

    This claim ignores the fact that reality is rife with ‘assemblages of meta^[FONT=Arial]n+1[/FONT]-units, each of which is composes of a sub-assemblage of meta^[FONT=Arial]n[/FONT]-units -- e.g., as a population with molecules as its units is one of each of whose units is composed of atoms as its sub-units, as a population with atomic nuclei as its units is one each of whose units is composed of “sub-atomic particles” [e.g., protons] as its sub-units, and as a population of “sub-atomic particles” is one each of whose units is composed of “pre-sub-atomic particles” [e.g., quarks and gluons] as its sub-units.

    For Aristotle [circa [FONT=Arial]335[/FONT] B.C.E]: “Apart from this definition of number, which is oriented towards the idea of counting, one can find in ARISTOTLE the following statement: that which is divisible into discrete parts is called [A.D.: «plethos»] [FONT=Symbol]_____[/FONT] (multitude), and the bounded (finite) multiplicity is called the number (ARISTOTLE [1], 1020a, 7.14). The [A.D.: Ancient] Greeks thus regarded as numbers, only the natural numbers, excluding unity; fractions were treated as ratios of [A.D.: “natural”] numbers, and irrational numbers as relationships between incommensurable magnitudes in geometry ... . [Ibidem].

    Actually, the statement above is an anachronism, a ‘moderno-morphism’, and a ‘retro-projection’ of the Modern meme of “number” back upon the Ancient one: the Ancients did not hold the modern conception of the “natural” number, as pure, unqualified quantifier”.

    On the contrary, as we shall show herein, via Diophantus’s circa [FONT=Arial]250[/FONT] C.E. treatise The Arithmetica, the ancient meme of ‘‘‘number’’’ was a [FONT=Arial Black]hybrid[/FONT], quanto-qualitative’ one.

    Attending closely to the qualitative, ‘ideo-ontological’ distinction of the Ancient concept of ‘‘‘number’’’ from the Modern can enable one to solve -- with both speed and clarity -- mysteries that still baffle many scholars of philosophy, e.g. --

    arithmos: number; arithmêtikê: the science of number. Zero was unknown as a number and one also was not counted as a number, the first number being duas [A.D.: or ‘dyos’] -- two. From the Pythagoreans, ton arithmon nomizontes arkhên einai -- who consider number to be the first principle (Ar. Met. 986a15) -- number played a great part in metaphysics, especially in Plato’s unwritten doctrines, involving obscure distinctions of e.g. sumblêtoi and asumblêtoi -- addible and non-addible numbers.” [J. O. Urmson, The Greek Philosophical Vocabularly, Duckworth [London: [FONT=Arial]2001[/FONT]], pp. [FONT=Arial]31-32[/FONT]].

    The «Arithmoi Eide-tikoi» of Plato’s static, eternal dialectic, or ideo-taxonomy, were, in his conception, «arithmoi» of «Eide-Monads» -- Assemblages of «[FONT=Symbol]Idea[/FONT]»-Units -- for Plato’s reified, deified «[FONT=Symbol]Idea[FONT=Symbol][FONT=Symbol]s[/FONT][/FONT][/FONT]», which he supposed to be the immutable, perfect, Parmenidean Causes behind the imperfect copies of them which somehow constituted and conducted the dynamic flux of our sensuous world.

    Per Plato, for each such Causal «[FONT=Symbol]I[/FONT][FONT=Symbol]dea[/FONT]», call it I[FONT=Arial]1[/FONT], any perfect copy of It was redundant in terms of philosophical logic, and could not exist: I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]1[/FONT] [FONT=Arial]~=[/FONT] ‘[FONT=Arial]2[/FONT]I[FONT=Arial]1[/FONT]; instead, I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]1[/FONT] [FONT=Arial] = [/FONT] I[FONT=Arial]1[/FONT] [an algebraic property which Modern mathematics names “additive idempotency”].

    Moreover, for any two -- heterogeneous, qualitatively/ontologically distinct -- such «[FONT=Symbol]Idea[FONT=Symbol][FONT=Symbol]s[/FONT][/FONT][/FONT]», call them I[FONT=Arial]1[/FONT] and I[FONT=Arial]2[/FONT], their very apples versus oranges heterogeneity makes them “non-amalgamative” [cf. Dr. Charles Musès] if added together:

    I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]2[/FONT] [FONT=Arial]~=[/FONT] ‘[FONT=Arial]2[/FONT]I[FONT=Arial]1[/FONT], and I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]2[/FONT] [FONT=Arial]~=[/FONT][FONT=Arial]2[/FONT]I[FONT=Arial]2[/FONT]; instead,

    I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]2[/FONT] [FONT=Arial]=[/FONT] I[FONT=Arial]1[/FONT] [FONT=Arial]+[/FONT] I[FONT=Arial]2[/FONT],

    without further possibility of reduction within this language;

    “apples plus oranges” equals “apples plus oranges”, irreducibly so, at this level.

    Thus, in both ‘self-addition’ and ‘other-addition’, Plato’s «Eide»-Units are unaddible”, unsum-able’ -- «asumblêtoi».

    Also, given that the original Pythagoreans held that «arithmoi» -- ‘‘‘assemblages of qualitative, multiple-qualities-exhibiting units/things of various single kinds’’’, i.e., ‘‘‘populations of individual things [including of physical, sensuous things]’’’ -- constitute reality, it is no longer sure that the original Pythagoreans were raving idealist mystics, as is so often presumed, based upon the ‘retro-projection’ of the modern meme of ‘‘‘number’’’ upon their Ancient «arithmos» idea.

    Ancient Alexandrias Proto-Renaissance,[FONT=Arial]&[/FONT] Diophantuss Qualifier-Quantifier Proto-Algebra, at the Dark Ages Door.

    The first known ‘protoic’ emergence of “symbolical algebra”, as distinct from the already ancient ‘prose algebra’ -- or “rhetorical algebra” -- and of an algebra “symbolical” in the specific sense of ideogramic symbols’, not exclusively of either pictogramic symbols’ and/or of phonogramic [phonetic”] symbols’ [which, after all, would simply mean ‘prose algebra’, or “rhetorical algebra” again] -- was in a circa [FONT=Arial]250[/FONT] C.E. work by Diophantus, entitled The Arithmetica.

    This text, The ArithmeticaArithmêtikê»], taught the ‘‘‘art’’’, or ‘‘‘technology’’’ or ‘‘‘technique(s)’’’ [«tekhnê»], or ‘‘‘craft’’’, or ‘‘‘skill’’’, or ‘‘‘science’’’ of«Arithmoi» in general.

    This text developed an intermediate stage between “rhetorical” algebra and full-blown “symbolical”, ‘equational’ algebra, which has often been termed “syncopated” [abbreviated] algebra, in which minimized abbreviations [“syncopations”] of words served as ‘proto-ideogramic’ symbols for arithmetical quantities, or ‘‘‘quantifiers’’’, and for arithmetical qualities or ‘‘‘qualifiers’’’, including for equations involving unknown quantities which Diophantus showed how to “solve” -- how to systematically render the unknown quantities known.

    Diophantus’s particular style of “abbreviation” or “syncopation” -- an unprecedented style as far as is known -- was, apparently, to take the first letter of the Greek word to be abbreviated, and to place atop that first letter the second Greek letter of that word. Thus, only two Greek letters -- the first two letters -- of the Greek word “survived” his abbreviation process. Known numerical values were expressed using single Greek letters, with a dash or a “ prime” atop each letter, in ordinal correspondence [i.e., [FONT=Times New Roman]alpha[/FONT] = [FONT=Times,&quot;Times New Roman&quot;,serif]I[/FONT], [FONT=Symbol] beta = II[/FONT], [FONT=Symbol] gamma = III[/FONT], etc.], in accord with longstanding Ancient arithmetical tradition.

    [FONT=Times,&quot;Times New Roman&quot;,serif]The context of this work by Diophantus was the Ancient Egyptian city of Alexandria, after the zenith of the extraordinary, unprecedented Human-Phenomic -- scientific, technological, and institutional -- developments there, that Karl Seldon has described as the Western “Proto-Renaissance”. Diophantus’s revolution in mathematics was cut short, in part, because it arose circa [FONT=Arial]250[/FONT] C.E., just a few centuries before the tidal wave of the fall of the Roman Empire, and the undertow dragging Ancient Hellenistic civilization down into the hellish abyss of the European Dark Ages, smashed into Alexandria, suppressing this progressive trend, and delaying its resumption, continuation, and supersession for another [FONT=Arial Black]~[/FONT] ten centuries.[/FONT]

    R[FONT=Times New Roman]egarding the mathematical aspect of this “Proto-Renaissance” in Ancient Alexandria, we find the following from the historical record: “The earliest attempt to found a university, as we understand the word, was made at Alexandria. ... It was particularly fortunate in producing within the first century of its existence three of the greatest mathematicians of antiquity -- Euclid, Archimedes, and Apollonius. They laid down the lines on which mathematics subsequently developed, and treated it as a subject distinct from philosophy: hence the foundation of the Alexandrian Schools is rightly taken as the commencement of a new era. Thenceforward, until the destruction of the city by the Arabs in 641 A.D. [i.e., C.E.], the history of mathematics centers more or less round that of Alexandria”. [W. W. Rouse Ball, [/FONT][FONT=Times New Roman]A Short Account of the History of Mathematics, Dover [New York: [/FONT][FONT=Arial]1960[/FONT]], pp. [FONT=Arial]50-51[/FONT]].

    Howard Eves describes, as follows, the lead-up to the founding of Alexandria --

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

    -- and the institutional innovations which seeded its unprecedented destiny --

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

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

    Morris Kline well-describes the mathematical, technological, economic, and cultural momenta that converged into the genesis of the Ancient Alexandrian “Proto-Renaissance” in the following passages.

    After the early death of Alexander, the Ptolemaic emperors of Egypt carried forward with Alexander’s plans: “After his death ... the empire was split into three independent parts. ... Egypt, ruled by the Greek Ptolemy dynasty, became the third empire. Antigonid Greece and Macedonia gradually fell under Roman domination and became unimportant as far as the development of mathematics is concerned ... The major creations following the classical Greek period were made in the Ptolemaic empire, primarily in Alexandria.”

    “That the Ptolemaic empire became the mathematical heir of classical Greece was not accidental. The kings of the empire ... pursued Alexander’s plan to build a cultural center at Alexandria. ... These rulers therefore brought to Alexandria scholars from all the existing centers of civilization and supported them with state funds.”

    “About 290 B.C.[E.] Ptolemy Soter built a center in which the scholars could study and teach. This building, dedicated to the muses, became known as the Museum, and it housed poets, philosophers, philologists, astronomers, geographers, physicians, historians, artists, and most of the famous mathematicians of the Alexandrian Greek civilization.”

    “Adjacent to the Museum, Ptolemy built a library, not only for the preservation of important documents but for the use of the general public. This famous library was said at one time to contain 750,000 volumes, including the personal library of Aristotle and his successor Theophrastus. Books, incidentally, were more readily available in Alexandria than in classical Greece because Egyptian papyrus was at hand. In fact, Alexandria became the center of the book-copying trade of the ancient world.”

    “The Ptolemies also pursued Alexander’s plan of encouraging a mixture of peoples, so that Greeks, Persians, Jews, Ethiopians, Arabs, Romans, Indians, and Negroes came unhindered to Alexandria and mingled freely in the city. Aristocrat, citizen, and slave jostled each other and, in fact, the class divisions of the older Greek civilization broke down.” [Morris Kline, Mathematical Thought from Ancient to Modern Times, Volume I, Oxford University Press [New York:[FONT=Arial] 1972[/FONT]], pp.[FONT=Arial]101-102[/FONT], emphases added by A.D.].

    Ancient Alexandria’s favorable locus, with respect to the concentration and centralization of ancient commerce and wealth there, also contributed crucially to the consummation of its peoples’ cultural ambitions: “The civilization in Egypt was influenced further by knowledge brought in by traders and by the special expeditions organized by the scholars to learn more about other parts of the world. Consequently, intellectual horizons broadened. The long sea voyages of the Alexandrians called for far better knowledge of geography, methods of telling time, and navigational techniques, while commercial competition generated interest in materials, in efficiency of production, and in improvement of skills. Arts that had been despised in the classical period were taken up with new zest and training schools were established. Pure science continued to be pursued but was also applied.” [ibid., pp. [FONT=Arial]102-103[/FONT]].

    Part of what resulted was an unprecedented flowering of engineering and technology, even though not supported by strong incentives to apply this technology in production, given the still predominantly pre-capitalist, peasant-/serf-, ‘artisanal-’, and slavery basis of the prevailing social relations of production, especially after the Roman conquest of Egypt, in [FONT=&quot]31[/FONT] B.C.E.: The mechanical devices created by the Alexandrians were astonishing even by modern standards. Pumps to bring up water from wells and cisterns, pulleys, wedges, tackles, systems of gears, and a mileage measuring device no different from what may be found in the modern automobile were used commonly. Steam power was employed to drive a vehicle along the city streets in the annual religious parade. Water or air heated by fire in secret vessels of temple altars was used to make statues move. ... Water power operated a musical organ and made figures on a fountain move automatically while compressed air was used to operate a gun. New mechanical instruments, including an improved sundial, were invented to refine astronomical measurements.” [ibid.; pp. [FONT=Arial]103[/FONT], emphases by A.D.].

    The disparaging squeamishness and ‘needlessness’ of classical Greek “aristocratic” slave-holders with regard to ‘‘‘hands-dirtying work’’’ [‘‘‘fit only for slaves’’’] -- and with regard to practical and commercial applications of the fruits of intellectual labor -- was overcome in Ancient Alexandria: “Proclus, who drew material from Germinus of Rhodes (1st cent. B.C.[E.]), cites the latter on the divisions of mathematics...: arithmetic (our theory of numbers), geometry, mechanics, astronomy, optics, geodesy, canonic (science of musical harmony), and logistics (applied arithmetic). According to Proclus, Germinus says: The entire mathematics was separated into two main divisions with the following distinction: one part concerned itself with the intellectual concepts and the other with material concepts.” Arithmetic and geometry were intellectual. The other division was material. However, the distinction was gradually lost sight of ... One can say, as a broad generalization, that the mathematicians of the Alexandrian period severed their relation with philosophy and allied themselves with engineering.” [ibid., pp. [FONT=Arial]104-105[/FONT], emphases by A.D.].

    Hero[n] of Alexandria, and his teacher, Ctesibius [who may have been responsible for the “Antikythera Mechanism”], incarnate this mathematico-technological momenta of the Ancient Alexandrian ‘Proto-Renaissance’: “Proclus refers to Heron as mechanicus, which might mean a mechanical engineer today, and discusses him in connection with the inventor Ctesibius, his teacher. Heron was also a good surveyor. ... The striking fact about Herons work is his commingling of rigorous mathematics and the approximate procedures and formulas of the Egyptians. On the one hand, he wrote a commentary on Euclid, used the exact results of Archimedes (indeed he refers to him often), and in original works proved a number of new theorems of Euclidean geometry. On the other hand, he was concerned with applied geometry and mechanics and gave all sorts of approximate results without apology. He used Egyptian formulas freely and much of his geometry was also Egyptian in character. ...”

    “His applied works include Mechanics, The Construction of Catapults, Measurements, The Design of Guns, Pneumatica (the theory and use of air pressure), and On The Art of Construction of Automata. He gives designs for water clocks, measuring instruments, automatic machines, weight lifting machines, and war engines.” [ibid., pp. [FONT=Arial]116-117[/FONT], emphases added by A.D.].

    Factors in the demise of this Ancient Alexandrian “Proto-Renaissance” are described, by Howard Eves, as follows --

    “The city of Alexandria enjoyed many advantages, not the least of which was long-lasting peace. During the reign of the Ptolemies, which lasted for almost 300 years, the city, although on occasion beset with internal power struggles, remained free from external strife. This was ended by a short period of conflict when Egypt became part of the Roman empire ... The closing period of ancient times was dominated by Rome. ... The economic structure ... was essentially based on agriculture, with a spreading use of slave labor. The eventual decline of the slave market, with its disastrous effect on Roman economy, found science reduced to a mediocre level. The Alexandrian school gradually faded, along with the breakup of ancient society. [op. cit., p. [FONT=Arial]164[/FONT], emphases added by A.D.].

    -- and --

    “Greek science reached its pinnacle at Alexandria ... The decline was caused by a combination of technological, political, economic, and social factors. ... The Romans used slave labor to an almost unprecedented degree, especially after the founding of the Empire by Augustus in 31 B.C.[E.]. More than half of the Empires inhabitants were slaves. With slaves to do most of the backbreaking work, there was little perceived need for labor-saving devices, such as the pulleys and levers invented by Archimedes ... hence, scientists had little incentive to invent them.” [op. cit., pp.[FONT=Arial]137-138[/FONT], emphases added by A.D.].

    -- and by Morris Kline thusly --

    The fate of Hypatia, an Alexandrian mathematician of note and the daughter of Theon of Alexandria [the redactor of Euclid's Elements -- A.D.], symbolizes the end of the era. Because she refused to abandon the Greek religion, Christian fanatics seized her in the streets of Alexandria and tore her to pieces. ...From the standpoint of the history of mathematics, the rise of Christianity had unfortunate consequences. Though the Christian leaders adopted many Greek and Oriental myths and customs with the intent of making Christianity more acceptable to converts, they opposed pagan learning and ridiculed mathematics, astronomy, and physical science; Christians were forbidden to contaminate themselves with Greek learning. Despite cruel persecution by the Romans, Christianity spread and became so powerful that the emperor Constantine (272-337 [C.E.]) was obliged to consign it a privileged position in the Roman Empire. The Christians were now able to effect even greater destruction of Greek culture. The emperor Theodosius proscribed the pagan religions and, in 392 [C.E.] ordered that the Greek temples be destroyed. Pagans were attacked and murdered throughout the empire. Greek books were burned by the thousands. In that year Theodosius banned the pagan religions, the Christians destroyed the temple of Serapis [in Alexandria -- A.D.], which still housed the only extensive collection of Greek works. It is estimated that 300,000 manuscripts were destroyed. Many other works written on parchment were expunged by the Christians so that they could use the parchment for their own writings ...In 529 [C.E.], the Eastern Roman emperor Justinian closed all the Greek schools of philosophy, including Platos Academy. ... The final blow to Alexandria was the conquest of Egypt by the upsurging Moslems in ... 640 [C.E.]. The remaining books were destroyed on the ground given by Omar, the Arab conqueror: “Either the books contain what is in the Koran, in which case we do not have to read them, or they contain the opposite of what is in the Koran, in which case we must not read them.” And so for six months the baths of Alexandria were heated by burning rolls of parchment. After the capture of Alexandria by the Mohammedans, the majority of the scholars migrated to Constantinople, which had become the capital of the Eastern Roman Empire. Though no activity along the lines of Greek thought could flourish in the unfriendly Christian atmosphere of Byzantium, this flux of scholars and their works to comparative safety increased the treasury of knowledge that was to reach Europe eight hundred years later. It is perhaps pointless to contemplate what might have been. But one cannot help observe that the Alexandrian Greek civilization ended its active scientific life on the threshold of the modern age. It had the unusual combination of theoretical and practical interests that proved so fertile a thousand years later. Until the last few centuries of its existence, it enjoyed freedom of thought, which is also essential to a flourishing culture. And it tackled and made major advances in several fields that were to become all-important in the Renaissance: quantitative plane and solid geometry; trigonometry; algebra; calculus; and astronomy.” [op. cit., pp. [FONT=Arial]180-181[/FONT],emphases added by A.D.].

    It is in the above-described ‘‘‘psychohistorical’’’ context that the work of Diophantus of Alexandria can be comprehended -- as a [FONT=Arial Black]hybrid[/FONT] product of waning Hellenistic memes, and of a ‘protoic’, precocious, prevenient partial prefigurement of core components of the as yet unborn Human Phenome of Modernity.

    Morris Kline assesses the work of Diophantus in the following terms:

    “The highest point of Alexandrian Greek algebra is reached with Diophantus. ... His work towers above that of his contemporaries; unfortunately, it came too late to be highly influential in his time because a destructive tide was already engulfing the civilization. Diophantus wrote several books that are lost in their entirety. ... His great work is the Arithmetica which, Diophantus says, comprises thirteen books. We have six [[FONT=Arial]6[/FONT] surviving in Greek, that is; [FONT=Arial]4[/FONT] more were recently found, in Arabic, possibly translations into Arabic of Hypatia’s Greek commentaries on books [FONT=Arial]4[/FONT] through [FONT=Arial]7[/FONT], rather than of Diophantus' originals -- A.D.] ... One of Diophantus’ major steps is the introduction of symbolism [i.e., of proto-ideography -- A.D.] in algebra. ... The appearance of such symbolism is of course remarkable but the use of powers higher than three is even more extraordinary. The classical Greeks could not and would not consider a product of more than three factors because such a product had no [then-recognized -- A.D.] geometrical significance [i.e., given the apparently [FONT=Arial]3[/FONT]-and-no-more/no-less-dimensional physical space of our world -- A.D.]. On a purely arithmetical basis, however, such products do have a meaning; and this is precisely the basis Diophantus adopts." [op. cit., pp. [FONT=Arial]138-139[/FONT], emphases added by A.D.].

    Diophantus symbolized a[ny], generic number, in a dual format, as a juxtaposition -- a ‘‘‘product’’’, in effect -- of two semantic ‘‘‘co-factors’’’, called, by Karl Seldon, an ‘‘‘arithmetical qualifier’’’, and an ‘‘‘arithmetical quantifier’’’, viz. --


    -- with the “syncopated” unit qualifier symbol M signifying the «[FONT=Arial]Mo[/FONT]-nad», the generic, abstract [and quantifiable’] unit, or ‘‘‘one-ness”’, standing generically and indifferently for any specific kind of unit -- e.g., for an ontological unit, or for a metrical unit, or even for an undifferentiated combination of the two.

    Examples include a unit of the kind of thing category -- or ‘‘‘ontological category’’’ -- of the quality of “apple-ness”, i.e., an apple unit, or an orange unit, or a pound unit as unit of measure or metrical unit, or the combined, undifferentiated unity of a metrical and an ontological quality unit, e.g., “a pound of apples”, or “a pound of oranges”.

    The symbol [FONT=Verdana,sans-serif]s[/FONT], is the generic quantifier symbol, often used by Diophantus to represent the unknown, and to-be-solved-for, value in one of Diophantus’s ‘proto-algebraic proto-equations’.

    This number symbol is drawn, as was typical in Ancient Greek ‘‘‘logistics’’’ [practical arithmetic], from the Greek alphabet.

    It is the version of the Greek letter sigma, [FONT=Arial,Helvetica,sans-serif][FONT=Symbol]_[/FONT][/FONT], that is used when sigma is the final letter of a Greek word, e.g., in particular, [FONT=Verdana,sans-serif]s[/FONT] is the last letter of the Greek word «arithmo[FONT=Verdana,sans-serif]s[/FONT]» ... .

    In modern English, it coincides with the final [FONT=Arial,Helvetica,sans-serif][FONT=&quot]s[/FONT][/FONT], i.e., with English letter suffix that signifies plurality.

    Thus, the expression above might stand, indifferently, for the prose representations “six apples” [or, literally, “apples six” -- qualifier first, or in first place, followed by quantifier second, or in second place], or “six oranges”, or “six pounds”, or “six pounds of apples”, etc.

    That is, Diophantus, in keeping -- for the most part -- with Ancient «Arithmos» Theory, does not symbolize number in general simply as --


    -- i.e., as an abstract, “pure” quantifier, without qualification, as would be the case if Diophantus had followed -- i.e., if he had anticipated -- the meme of European Renaissance humanity, after the world-historic Elision of the Qualifiers.

    This world-historic Elisionwas brought about, ‘‘‘psychohistorically’’’, we hold, in the post-Dark-Ages European Human Phenome -- which was also the point-of-origin of the [psycho]historically-specific Capitalist Phenome, or «mentalité», by the intensive practice of the capital-relation by so much of the population: of the monies-[capitals-]mediated exchanges of commodities[-capitals], that emerged, in the lead-up to the Western European Renaissance, as a far more intensive such praxis than was ever reached within the socio-economic limitations of Ancient Mediterranean times, and of their substantially slavery-based mode of social production.

    This ‘capital-praxis’ was captured, in its purest, simplest essence -- abstracting from its more concrete determinations, involving mediation by money [price] and by production processes, outside of the process of circulation of capitals, by Marx’s «arché» for «Das Kapital» as a whole, The Elementary or Accidental Form of Value, set forth by Marx from the beginning of that work, in Vol. I, Part I, Chapter I., Section [FONT=Arial]3.A.[/FONT] of, as the systematic-dialectical seed cell of that entire work, and expressed by Marx, in his ‘algebraic/rhetorical’ notation, in the form of the ‘‘‘exchange-equations’’’ --

    "x commodity A = y commodity B"

    or, e.g., as:

    "20 yards of linen = 1 coat"

    -- and, later, by Seldon, as --

    [FONT=Arial][FONT=Times New Roman]"[/FONT]{ cjCj = ckCk }[/FONT]"

    -- with [FONT=Arial Black]c[/FONT]ommodity quantifiers[FONT=Arial] cj[/FONT][FONT=Arial] < or = or > ck[/FONT][FONT=Arial], [/FONT]despite the fact that, for the [FONT=Arial,Helvetica,sans-serif]C[/FONT]ommodity qualifiers, [FONT=Arial] Cj[/FONT][FONT=Arial] is qualitatively unequal to [/FONT][FONT=Arial]Ck[/FONT].

    . . .

    [FONT=Arial Black]q[/FONT][FONT=&quot][FONT=Arial Black]2[/FONT] [FONT=Verdana,sans-serif][FONT=Arial]<---][/FONT] [/FONT][/FONT][FONT=&quot][FONT=Verdana]II[/FONT][/FONT][FONT=&quot].[/FONT][FONT=&quot] Modern [/FONT][FONT=&quot]“[/FONT][FONT=&quot]Number Theories[/FONT][FONT=&quot]”[/FONT][FONT=&quot].[/FONT]


    [FONT=Arial Black]q3[/FONT] [FONT=Verdana,sans-serif][FONT=Arial]<---][/FONT] [/FONT]III. [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT]’s Seldonian, Trans-Modern -- Modern/Ancient Hybrid -- «Arithmos» Theories.

    We of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT]use the word ‘‘‘number’’’ in a far more concrete sense than has become habitual in the Modern World, and, in certain ways, with a sense much more like it had in the Ancient World.

    In our ‘‘‘Number Theory’’, as a modernization of the ancient ‘«Arithmos»-Theory, or ‘«Monads»-Theory, ‘‘‘number’’’ means not an abstract, “purequantity as such, as per those number conceptions so central to the Modern [unconsciously, experientially law-of-capital-value-inculcated] «mentalité».

    On the contrary, in our usage, ‘‘‘number’’’ means something far closer to sensuous empiricality.

    It refers to a specific multiplicity of units/individuals/monads, akin to a plural but finite population of the individuals of the same kind, such that each individual is a concrete, determinate, ‘multi-qualitative’ [‘multi-quality’], attributes-rich [ev]entity, not a distilled, rarefied mental abstraction of “pure, unqualified quantity”.

    In such a usage, ‘‘‘numbers’’’ thus no longer differ only quantitatively: such ‘‘‘numbers’’’ have different kinds.

    And, Old ‘‘‘numbers’’’ create New ‘‘‘numbers’’’: they not only expand themselves quantitatively, as populations of their units, but qualitatively, ontologically as well.

    That is, Old kinds of ‘‘‘numbers’’’ create New kinds of ‘‘‘numbers’’’ by means of self-meta-monad-ization, that is, via ‘self-meta-unit-ization’, or ‘self-meta-individual-ization’.

    The process of self-meta-monad-ization is a self-«aufheben» process, which is to say, a dialectical process.

    A theory of the progressive self-construction of our cosmos -- in the form of a single, recurrent, mounting, cumulative, helical dialectic of nature -- can be constructed on the basis of noticing that, e.g. --

    The [self-changing] ‘‘‘number’’’ [cosmological population] of pre-nuclear “particles” [e.g., of non-Hadronic, “non-composite” bosons and fermions, such as quarks] created the [dynamical, “fluent” [cf. Newton] self-changing] ‘‘‘number’’’ of sub-atomic “particles” [e.g., of primordial protons and neutrons], by their own self-meta-monad-ization;

    The [self-changing] ‘‘‘number’’’ [cosmological population] of sub-atomic “particles” [e.g., non-Hadronic, “non-composite” bosons and fermions, such as, quarks] created the [self-changing/other-changed/other-changing] ‘‘‘number’’’ of [ionic] atomic nuclei [e.g., primordial Deuterium, Tritium, Helium, and Lithium], by their self-meta-monad-ization;

    The [self-changing] ‘‘‘numbers’’’ [galactic populations] of atomic nuclei created the [dynamical, “fluent”, self-changing/other-changed/other-changing] ‘‘‘numbers’’’ of molecules [e.g., of galactic “inter-stellar medium” accumulating [FONT=Arial]H2[/FONT], [FONT=Arial]O2[/FONT], [FONT=Arial]CN[/FONT], [FONT=Arial]H2O[/FONT], [FONT=Arial]CO2[/FONT], [FONT=Arial]CH4[/FONT], etc.], by their own brand of such self-meta-monad-ization;

    The [dynamical, “fluent”, self-changing] ‘‘‘numbers’’’ [cosmological populations] of molecules created the [self-changing/other-changed/other-changing] ‘‘‘numbers’’’ of ‘pre-eukaryotic’ living cells, by their own, natural-historically-specific «species» of self-meta-monad-ization;


    Our Marxian, immanent critique of both the Modern and the Ancient conceptions of ‘‘‘Number’’’ find their foundation in the ‘‘‘psychohistorical’’’ insights, into both the Modern and the Ancient human ideologies -- into the Modern versus the Ancient ‘Human Phenomes’ -- embodied in Marx’s immanent, dialectical critique of capitalist political economy.

    In his Elementary Form of Value, Marx discovered something much more momentous than even the ultimate seed category -- the «arché» category -- from which there ‘‘‘descends’’’, in an ideo-meta-genealogical, dialectical method-of-presentation sense, the rest of his entire, vast, comprehensive critique of the political economy of capital; of the capital social-relation-of-production; of the capitals social system of global, "prehistoric" [Marx] humanity.

    He also discovered the universal unconscious paradigm of The Modern «mentalité»’, whose most characteristic symptom is the purely quantitative frame of mind, and, consequently, The Elision of the Qualifiers from conception, from perception, and from mathematical -- starting especially with arithmetical -- symbolic expression.

    Marx therein and thereby discovered the secret, not just of The German Ideology, but of the total, global, human Modern Ideology entire -- of the totalHuman Phenome of a planetary humanity that embodies and incarnates Capital [i.e., that incarnates the "Capital-relation-of-production' as the predominant social relation of social reproduction].

    We of [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial]. [/FONT]have found working with the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] arithmetic/algebra, as with its successor systems, to be a worthwhile and cognitively healing practice for we [FONT=Arial]F[/FONT][FONT=Arial].[/FONT][FONT=Arial]E[/FONT][FONT=Arial].[/FONT][FONT=Arial]D[/FONT][FONT=Arial].[/FONT] monastics.

    In working with the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT], one is working with ‘‘‘numbers’’’ that are purely qualitative.

    A given, generic [FONT=&quot][FONT=Arial]q[/FONT][FONT=Arial]k[/FONT][/FONT] is interpreted, or specified, as “standing for” an «arithmos», a number, in part, in the Ancient sense: as “standing for”, in effect, an ontological category representing the special ‘common-kind-ness’ that unites all of the individuals; that all of the «monads» which inhere in that ontological category share, like the ‘‘‘in-tension’’’ of an ‘‘‘ex-tension’’’, i.e., of a “set of elements”.

    The generic symbol [FONT=Arial]q[/FONT][FONT=Arial]k[/FONT],for a [FONT=Arial]k[/FONT] in [FONT=Arial Black]N[/FONT], thus interpreted, means a number of indefinite/changing cardinality, creating a kind of Marxian version of the “intentional” variables of the original Boolean algebra.

    The practice of the expression of experienced/experimented reality, using the language of the [FONT=Arial Black]N[/FONT][FONT=Arial]Q[/FONT] numbers, is, we find, a liberating “spiritual practice” -- in the sense of a Marxian version of Hegel’s “Objective Spirit”: of The Human Phenome.

    That is, this activity of ours is a healing modifier of our individual human phenomes, one that lifts us beyond the collective, ‘ideologized’ “Mind”, the typical «mentalité», of our time -- beyond the “Mind” of The Modern Ideology; beyond the Money Mind, beyond the one-sidedly, purely-quantitative «mentalité», the “Mind” of The Elementary Form of [Commodity] Value as unconscious universal paradigm -- in short, beyond ‘the capital-value «mentalité»’.

    This practice thereby helps us to free our minds to see in new and wider ways -- to think beyond the blockages characteristic of The Modern Ideology, the ideology of capital-value as supreme value, or even as only-value.

    If you believe that such seeing is a part of your life path, then we commend this practice also to you.

    Links to definitions of additional [FONT=Arial]Encyclopedia Dialectica[/FONT] special terms deployed in the discourse above --


    Booles Algebra



    dialectical categorial progression


    ontological category


    Last edited by Miguel Detonnaciones; 30th March 2013 at 18:22. Reason: formats enhncements
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    Dialectical String Theory?
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    Default Indeed!

    RevLeft Sciences Forum Participants,

    Bhikku Punk,

    Yes, good point!!!

    Since dialectical models, using the first/simplest of the Seldonian dialectical ideographic Languages, generate, precisely, strings of dialectical categories, each such dialectical model is, indeed, a "dialectical string" theory!


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    this thread has gone over 500 posts and so is closed. feel free to start another
    'heavens above, how awful it is to live outside the law - one is always expecting what one rightly deserves.'
    petronius, the satyricon

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