[ _N ] ( 2 : Briefing on Q, Arithmetics of 'Unquantifiable' Ontic Qualifiers -- Generic Apparatus of a Contra-Boolean [Onto-]Logical Calculus Transition In: Connotational Calculative Derivation of the [Rules-]System / Ideo-Ontological Category denoted NQ ( 2 -- A rules-system of operatorial ideograms representing generic 'kind-of-being qualifiers' -- generic 'ontological qualifiers' -- as the '[meta-]numerals' denoting the 'unquantifiable' or 'purely-qualitative' [meta-]numbers of a dialectical arithmetic which provides a 'meta-dynamical', auto-kinesis version of the Platonic '''arithmetic of dialectics''', the arithmos eidetikos, or arithmos noetikos. Section III. of Dialectical Ideography, entitled The Arithmetics of Meta-Evolution, is designed to comprehensively present an NQ model of the 'meta-system' of the progression of the dialectical ideographies; of the 'meta-systematic dialectical', categorial-self-progression, systems-self-progression, 'connotative-calculative' derivation of the various '''epochs''', or stages, of the dialectical ideographies, in which the NQ system itself arises within the second stage, t = 1, immediately after the first, t = 0 stage, and in which each successor system of dialectical arithmetic is more concrete and more complex -- richer in determinations -- in terms of its capability for description of both the dynamics & the 'meta-dynamics' of natural & 'human-natural' systems. That is, this 'meta-systematic dialectic' of the dialectical arithmetics is modeled, in Section III., by means of the second-arising system of dialectical arithmetic in that systems-progression, the NQ dialectical arithmetic, which arises immediately after the "Natural numbers" system of arithmetic. This yields a model, thus written in the language of one of the dialectical arithmetics itself, of the dialectic of the dialectical arithmetics. By assigning the arch system of arithmetic of this 'meta-system' -- namely, the "first-order" rules-system of the "Natural" numbers, N, represented by the first four of the five Peano Postulates, which [rules-]system we denote by N -- to the first of the NQ meta-numbers, denoted by 1 [symbolizing this act of assignment via the expression N ( 1], we obtain the following 'connotative calculation', or 'intensional, heuristic derivation', of the NQ system of rules of dialectical arithmetic, part of '''number-system one''' [ #1_], from '''number-system zero''' [ #0] -- unqualified quantifiers as numbers _( unqualified quantifiers as numbers2 = unqualified quantifiers as numbers unquantifiable qualifiers as 'meta-numbers', or: #0__ N_0__ N1 __ N_( #1__ N_1__ N2 __ N2 __ NN _ NN _ N of N _ N DN _ N NQ (1_( 1_ _ 1 _1 _ 1 2. The "Natural" numbers space, N ( {1, 2, 3,...}, in this derivation, is stipulated as basing the first-"thesis" [rules-]system, or 'ideo-ontological' category, of arithmetic, N. This N connotes an arithmetic of 'Peanic' [(first-order-Peano-Postulates-compliant] 'unqualified quantifiers' as numbers. This first 'thesis' rules-system, or category, of arithmetic, as a result of its 'intra-duality', its harboring within itself of "non-standard models" of the "Standard Natural numbers" -- i.e., by what we term 'ideo-auto-kinesis' -- also gives rise, by 'self-reflexion', to a first 'contra-thesis' rules-system, or category, of arithmetic. This 'contra-thesis' connotes a rules-system with 'negated' or 'opposite' connotations with respect to those of N, namely, those of a still 'Peanic' first-order rules-system, but one of 'not-[unqualified quantifiers]', or '[unqualified quantifiers]', i.e., of 'unquantified qualifiers' as '[meta-]numbers' [expressed via a system of higher, meta-"Natural" number units, { n } [with n denoting a "Standard Natural number"]. Its symbols/'meta-numerals' are made up out of multiplicities of "Natural" numbers' numerals, via a 'self-internalization'/'self-subordination'/'self-subscript-ization'] of their arithmetic, and is denoted herein by DN or NQ, whose 'number-space' is NQ ( { 1, 2, 3,... }. Together, N and NQ form an 'antithesis-sum', denoted N NQ [a 'non-reductionist', 'non-collapsing', '''inhomogeneous''' [i.e., '''heterogeneous''', or '''non-amalgamative''' sum [cf. Muss], la the proverbial '''sum''' of qualitatively different terms/kinds, '''apples + oranges''']. The Briefings of this sub-section excerpt from, and gloss, the first few derivations of Section III. The first Briefing, starting just below, glosses the second system of dialectical arithmetic, denoted NQ, which relates as 'contra-thesis' to the first, 'arch thesis' -- 'vestigially-dialectical' -- arithmetic, denoted N. Introducing the NQ system/'ideo-ontological category' of the dialectical arithmetics. Before we plunge ahead into Section II., the next stage of this excursion, a Psycho-Archaeological excavation of The Meta-Evolution of Arithmetics, let us tarry long enough to tell you, in the most direct way we can, the most basic rules of the 'first contra-thesis' 'Dialectical Ideographic Language'; of its incipient "Non-Standard Natural", 'contra-Boolean [Onto-]Logical Arithmetic', and 'Ontological Algebra', about whose bush we have been beating, and to which we have been led by that very 'The Meta-Evolution of Arithmetics'. First off, let us take this chance to say that just because we call this dialectical calculus 'contra-Boolean' does not mean that we are 'contra-Boole'; that we harbor some overriding animus toward the man himself. On the contrary, we hold his pioneering contributions, and his exemplary life of universal labor, in high esteem [In this season of unceasing incivility such sensibilities can no longer be assumed to go without saying]. The several sub-sections below describe rules of operation for a 'multi-unit-intervals meta-number-space', or -"set", that we call NQ. This 'unit-intervals-restricted' ideography is that part of this dialectical ideography which has the most in common with Boole's original arithmetic/algebra of logic, as set forth in his The Laws Of Thought. However, the space NQ has "nothing" in common with the realm of arithmetic most familiar today, that of the so-called "Real" numbers, formally denoted by R or R1. Nor has NQ much in common with the unit interval "sub-space" of that space, [0, 1] R, or even with the end-points of that interval, {0, 1} R, often taken to be the space of Boolean arithmetic, which we denote herein by E. That is, the intersection of NQ and R is (: NQ ___R _ (, and also NQ ___E _ (. The empty set, (, is thus, in a sense, all that they have in common. [Herein, we denote the set of Whole Numbers -- the Natural Numbers, N, with the adjunction of 0 -- by W]. Yet we will see that, with the expansion from NQ ( { 1, 2, 3,... }, to WQ ( { , 1, 2, 3,... }, bridges the WQ 'evolute "pure" qualifier meta-numbers' to the 'convolute' numbers, including R & C [C ... R]. Indeed, WQ 'meta-finitely contains' R & C -- R, C, H, O,... WQ -- because R, C, H, O, K G,... , and because WQ, notwithstanding that R, C,... ( WQ & R, C,... _ WQ _ ( [Even as {a}{{...{a}...}}, and yet {a} _ {{...{a}...}} _ (, & also although {a} (_{{...{a}...}}, & {a} (_{{...{a}...}}. This is because 'AB' means 'A is a component/sub-system/constituent/of B' at some/any/at least one 'scale'/'layer' of B's possibly 'meta-fractal', multi-dimensional, multi-logical type, 'multi-ontic cumulum' internal composition]. We call the individual 'meta-numbers' which are the 'elements' or 'constituents' of [most variants of] the "set" or "space" denoted by NQ, by the name 'evolute qualifier meta-numbers', to distinguish them from familiar kinds of numbers, which they presuppose, but also "transcend". We employ 'NQ', to symbolize this space or "set" of meta-numbers, because we interpret our 'Rules Of Operation' for that space as an 'Arithmetic of Qualities'; of 'Ontological Qualifiers'; of 'ontological categories', or 'ontos', for short; of ontic monads or unit[ie]s. We will often denote any one of the individual 'meta-numerals' for the 'meta-numbers' that reside in NQ via the generic format 'Nk'. The underscored lower-case 'N' component of that 'compound symbol' identifies Nk as a member of NQ. The [post-]subscript, here denoted generically by 'k', can vary over a space of specific values, e.g., Ordinal, Cardinal ["Natural"], "Whole", "Rational", or "Real" number, etc., values, but is here constrained to vary over the space of "Natural" numbers, denoted by the symbol N. The specific value, k, identifies the individuality of a given Nqk within NQ; the unique member of NQ which Nk represents; for k ( N, W, or Z, the ordinal position or 'order of appearance' of that unique 'onto' in the progression of 'onto'-representing/denoting ontological qualifiers within NQ, WQ, or ZQ, respectively. We apply NQ to describe 'onto-dynamical', 'meta-evolutionary' processes in which a succession of new 'ontological categories', or 'ontos' -- new 'taxons' or 'taxa'; qualitatively unprecedented [ev]entity/activity categories -- 'arise', 'appear', or 'manifest', via the inter-operation/self-operation of previously-extant 'ontos'. The 'sequence of counting numbers' of the number-space N, conceived, since Simon Stevin's circa 1600 works, as "pure" [i.e., 'unqualified'] 'quantifiers', the so-called "Natural numbers" 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..., has become paradigmatic for the accounting processes so fundamental to proto-human society -- especially to monetized society. Likewise, we hypothesize that the 'sequence of "pure" [i.e., 'unquantified'] qualifiers' inhering in NQ -- i.e., N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, N11, ..., -- may, in the future, prove paradigmatic for the modeling of generic processes of 'self-bifurcation' and 'meta-evolution'; of qualitative/ontological-revolutionary change, in this universe, including in its sub-[component ]universes [of discourse]. It may so do, we hold, for an emerging form of human mentation and an affine technology in tune with the ubiquity of 'meta-dynamism' in this cosmos. We can draw upon an indefinite "number" of "Natural" Numbers -- as many as required by a given process of measuring-by-counting that we may be imagining or actually conducting. Similarly, we can marshal an indefinite number of the Wk -- as many as the given 'meta-dynamical', 'meta-evolutionary', 'onto-dynamical' process that we wish to model has, or is expected to have, ontos, per our analysis -- within the range of the history of that process which we wish to cover. No ending to the sequence of the Wk within WQ in pre-posited. For every k in W, sub-W--sub-k is also in WQ, or, to employ the standard, more "fully ideographic" symbolism, [_k___W][ Wk___WQ ]. This exposition is heading toward 'quanto-qualitative', 'full-multiplicity' arithmetics of 'explicitly quantified qualifiers', or, equivalently, of 'explicitly and ideographically qualified quantifiers', with their '''quantifiers''' not restricted to the unit interval. We denote the first-two-to-emerge of those 'quanto-qualitative', or, equivalently, 'qualo-quantitative' arithmetics herein by U and by am. But the NQ arithmetic, like that of the Boolean '''Elector''' arithmetic, E, is a unit-interval-restricted arithmetic. Every NQ 'meta-number' is, in effect, quantified by '1', by the 'unit quantifier'. Better still is to conceive that each k is quantified by 'no quantifier', is 'unquantified', in a sense reminiscent of the ancient Greek, and Aristotelian, idea that number connotes an aggregation of units, or monads -- an arithmos monadikos -- not a single, isolated unit in itself, so that "quantity" begins with 2, not with 1. The k are 'unit qualifiers', or better, 'unquantifiable', "additively idempotent" [each-metanumber-its-own-additive-identity-element], 'non-additive', 'un-addable', "pure" generic qualifiers, with no meaning distinct from that assigned to their 'singleness' assigned to their multiplicity: for the NQ, if N _n _ 2, then 'n'k _ '2'k = '1'k _ k. Boolean arithmetic posits, at most, a single unit interval, denoted [0, 1]. Its upper end-point, 1, is interpreted as representing the "universal set", the "logical quantity" "All". Its lower end-point, 0, is interpreted as representing the logical quantity "None". The entire 'interior' of that unit-interval, denoted (0, 1) -- the interval without either of its end-points, 0 or 1 -- might connote the 'fuzzy' logical quantity 'Some(-but-not-All)' _ neither "All" nor "None", which Boole does not explicitly allow, but invokes implicitly, via the 'logical quantity' he denotes by 0/0, which 'ambiguously' represent "none, some, or all" of the universe of discourse. The special, 'onto-dynamical' analytical geometry of the WQ arithmetic maps a continual, self-propelled proliferation of mutually perpendicular, qualitatively/directionally distinct unit-length, unit-interval directed line-segments, represented generically as [ , |k| ], each one -- after, and with the exceptions of, and 1 -- 'generate-able' by mutual-interactions and self-interactions of its predecessor unit-intervals, or '''dimensions'''. As interpreted herein, this self-proliferation yields a continually expanding "ontology set" of ontological qualities, 'qualifiers', 'ontic unit[ie]s', or categorial 'monads'; a self-growing, non-amalgamative 'universal qualities sum', describing an 'ontological cumulum', as a self-expanding 'possibility-space' of a generic universe of discourse. That is, that 'universal set of ontos' or 'universal sum of ontos', denotes and describes the potential momentaneous ontological contents of the universe being modeled at each 'stage' or 'epoch' of a 'meta-evolutionary' succession of such 'ontology-stages' of that universe. This 'expanding-dimensionality domain', 'self-growing manifold', or '[dimensionally] self-expanding space' is the 'possibility-space' for the universe [of discourse] being modeled. This is because, by 'ontological contents' we mean the possible -- not necessarily even the probable, let alone the actualized -- forms of existence/activity for and within that universe[-of-discourse] at that stage or epoch of its self-driven / other-driven 'meta-evolution'. NQ Arithmetic [Statics]. [Note - this 'NQ' is not the same as 'Q', traditionally denoting the "Rational Numbers": NQ Q]. Rule 0. [The Rule of Ontological Diversity]: for every i_and_j in NQ, if i ( j, then i__j; or, in "fully" -- and more formally -- ideographical symbolism: [_i, j___NQ & _i, j __N | i ( j ][ i__j ]. This means that each k, for each distinct value of k, denotes a unique 'qualifier', interpretable as connoting a distinct ontological quality, ontic category, or 'onto', and hence differing qualitatively from all of the others. This is depicted, geometrically, via the 'perpendicularity-structure' of WQ space. Each unit interval line-segment [ , | k | ] is perpendicular to every other, thus forming an n-dimensional space of unit-length line-segments, where n denotes the maximum extant cardinal/ordinal subscript of the k in use for a given dialectical model. Interpreted geometrically, then: [_j, k __WQ & _j, k __W - {0} | j ( k ][ j ____k ]; Rule 1. [The Non-Amalgamation Rule]: for every i_and_j in NQ, such that i ( j, there does not exist n in NQ such that i _ j _ _n, or, in "fully" ideographic symbolism -- [ _m _ N & i,_j __NQ | i ( j ][n___NQ | i_ j _ _n ]. This means, e.g., that "apples and oranges, as such, do not add". The sum of 2 (or more) qualitatively-distinct NQ qualifiers does not reduce to a single qualifier within NQ, for any given interpretation of NQ. Such sums, which we term herein 'poly-qualinomials', illustrated above via a 'bi-qualinomial', are traditionally termed "complexes", "inhomogeneous sums", "heterogeneous sums", or "non-amalgamative sums". In Gibbs/Heaviside vectorial arithmetic, modeling 3-dimensional physical spaces, where , , and denote "unit vectors" -- unit lengths directed in each of 3 distinct respective mutually-perpendicular spatial directions -- the vector 3 _ 5 _ 9 does not reduce to (3_5_9)?s _ 17 all-the-same-somethings. Likewise, in "Complex" arithmetic -- the 'arithmetic of "complexes" ' -- where r denotes +1, the unity of the "Real" numbers, and where i denotes the unity of the "Imaginary" numbers, the "complex" 7r _ 11ri does not "simplify", or amalgamate down to 18 somethings. So it is also with the k. The [interpreted] sum of any 2 (or more) qualitatively distinct 'ontos' does not reduce , or "simplify", to any single 'onto'. Rule 2. [The Rule of Ontological Parsimony]: for every j in NQ, j__ __ j __ j, or, more formally, [_j ___NQ ][ j__ __ j __ j ], or, w.r.t. Rule 1, [ _i,_j __NQ | i _ j ][ _k _ NQ | i_ _ j _ k __i __j]. In other words, '2'k = '1'k, or, by induction, 'n'k _ '1'k _ k, for N n _ 2. The mere or singular assertion of the possible "existence" or 'extant-ness' of a given 'onto', denoted k, is sufficient. Multiplicity is redundant in the semantical, conceptual context of NQ, just as, in that of sets, {a, a, ... , a} _ {a, a} _ {a}. This kind of behavior is also known as "additive idempotency". Each k is its own 'additive identity' element. The k are thus, essentially, 'non-additive', that is, 'non-addable' and therefore 'unquantifiable'. Thus, "Homogeneous sums" or "non-heterogeneous sums" in NQ -- 'mono-qualinomials' -- do amalgamate; they in fact 'hyper-amalgamate', to unit monomials. Seeming 'poly-quanti-mono-quali-nomials' reduce to non-multiplicity. Rule 3. [The Aufheben Evolute Product-Rule]: for any i_& j in NQ, if i ( j, then i[ j ] _ j __ i+j, or, more generally [ _i,_ j __NQ ][ ij ] _ j i+j ]; That is, in the "generalized multiplication", or 'flexion', of qualifiers, defined herein for the NQ, the "starting point" -- in the above, j -- does not disappear into the 'stopping point', or outcome -- in the above, i+j. Among the NQ, "multiplication" is thus 'pathway-preserving', 'pathway-exhibiting', or 'pathway-recording'. The NQ product is an 'evolute' ["nonlinear"] product, rather than a 'convolute' ["linear"] one, as these terms are defined in the prequel. Also, NQ "multiplication" is a 'multiplication [increase] of qualities' rather than a "multiplication" (increase) of quantities. We also term this syntactical behavior 'multiplicative hyper-potency', 'super-potency', or 'meta-potency'. The "offspring" quality produced by the qualities i and j is qualitatively different from either: i___ i+j __j. Corollary: [ _k __NQ ][___ k k __ k2 _____[ k ___k+k ] ____[ k _2k ] __ k ], which is an instance of the 'contra-Boolean' fundamental "law", or '''rule''', itself: k2 ___k; k2 ____[ k___k ]_ ____[ k _2k ] k. To interpret or partially map this new Product-Rule and its '[Meta-]Number-Space' [in]to a more familiar mathematical idiom, we also present this Product-Rule as a new Vector Product-Rule -- a 'fourth Vector Product' if we put it in sequence after the "Scalar", "Vector", and "Tensor" Products of "Vectors". We term it the 'Dialector Product-Rule', or, in dynamical contexts, the 'history-revealing', 'history-preserving', 'path-positing', 'past-disclosing' Product-Rule. We represent this 'Dialector Product-Rule' in form which makes use of a 'Metavector Product' operation sign via the picto-ideogram '('. This operation involves the conception of a '[self-]expanding [Meta-]Vector-Space', or of a "pre-existent [potentially] infinite-dimensional Vector Space" [but different from Hilbert Space]. Also, though this Product-Rule involves the escalation of the dimensionalities of its "[Meta-]Vector-Spaces" ['Dialector Spaces'], it is distinct from the Grassmann Outer Product. Per this interpretation, the following kinds of similes arise, wherein k denotes the "orthonormal", unit-length vector pointing in[to] or along the kth orthogonal dimension/direction [and wherein the T exponents call for the transposition of row[-vector]s into column[-vector]s]: [ 12 _ 2 __3 ] ( [ 1 ( 2 _ _2 __3 ] , or [1]T ( [0 1]T _ [0 1]T _ [0 0 1]T _ [0 1 1]T [ 21 _ 1 __3 ] ( [ 2 ( 1 _ _1 __3 ] , or [0 1]T ( [1 ]T _ [1]T _ [0 0 1]T _ [1 0 1]T [ 11 _ 1 __2 ] ( [ 1 ( 1 _ _1 __2 ] , or [1]T ( [1 ]T _ [1]T _ [0 1]T _ [11]T [ 22 _ 2 __4 ] ( [ 2 ( 2 _ _2 __4 ] , or [0 1]T ( [0 1]T _ [0 1]T _ [0 0 0 1]T _ [0101]T. The expressions 1 ( 2 or [1 0]T ( [0 1]T may be read off as '1 lift 2' or '[1]T escalate [0 1]T'; ([ 1; 2 ] as transcend[ 1; 2 ]. Note that this Product is non-commutative, because it is evolute. In 'meta-dynamical', e.g., '''historical-dialectical'''/'''dialectic of nature''' contexts, we say that this Product-Rule/Definition is 'history-disclosing', 'pathway-disclosing', 'ancestry-disclosing', 'meta-genealogy-disclosing', or 'source/origin/causation-disclosing'. The result of an ( operation discloses 'from whence to whence': from whence [from what "point"/'metavector'] its movement began as well as to whence [to what "point"/'metavector'] it arrived. Note too that, for these '''unit-vectors''' too, as for the i: [ _j, k __V & _j, k __W - {0} | j ( k ][ j ____k ]. Thus, the movement denoted [0 1]T ( [1 0]T, which starts at [1 0]T and stops at [1 0 1]T, is qualitatively different from that denoted [1 0]T ( [0 1]T, which starts at [0 1]T and ends at [0 1 1]T. This model maps NQ as an "orthonormal" Meta-Basis of a unit-metavectors-confined, i.e., additively idempotent, [self-]expandable 'Metavector Space'. The latter rule makes this a less 'natural' model of 'Generalized Onto-Dynamasis', or O_, than is Q_. It demands, e.g., that 1 _ 1 _ 1 _ [1 0. . .]T, vs. 1 _ 1 _ 21 _ [2 0. . .]T, the more traditional result for standard vectors. This violates our expectations of vector arithmetic, as does 1 _ 1 _ 1 when ordinary 1 is used to stand for the logical quantity 'All' in Boolean arithmetic. It may thus be better to craft new [meta-]numerals, fitted from their outset to their defined -- new and unprecedented -- roles. The latter approach leads to NQ. NQ Statical Algebra and Statical Geometry. As detailed above, the arithmetic of NQ involves formulas written in terms of "constants", namely, the "constant" 'meta-numerical' values, each one denoted by one of the 'meta-numerals' of the set of all Nk, for the given universe of the k, here N: NQ ( { (_Nk | k ( N} ( { N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, N11, ... }. The algebra of the NQ abstracts from and generalizes upon this arithmetic, via the use of 'qualifier variables'. A NQ-algebraic variable may denote, generically, a single constant in a context where that constant's specific identity is unknown. Alternatively, it may denote many possible such values, which the variable "ranges over", that is, characterizes univocally [as in the complex variable Cz ( xr _ yri, r _ 1], or takes on in succession [ as, e.g., in a _-ordered x_, where x_ _ 1, x_ _ 1, x_ _ 2, x_ _ 3, x_ _ 5, etc.]. The "domain" of such constant values which the variable can denote may include all of the units, or monads, in the potentially infinite space NQ, or just a proper subset thereof. One species of such variables is exemplified above, in the ideographic statement of the four Rules. This type of variable-symbol denotes generically "any" or "every" ['_'] [logical-]individual unit/monad, constituent of NQ. It does so by using a literal subscript-variable, such as i, j, k, l, m, or n, to denote a specific, e.g., "Natural" number subscript-constant, e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,..., viz., Ni, Nj, Nk, Nl, Nm, Nn. This variable-denotation strategy is designed to represent only 'mono-qualinomials', single constituents/units/monads, of NQ. This strategy does not encompass inhomogeneous sums of two or more 'metanumber' unit[ie]s or 'unit qualifiers', i.e., it cannot cover 'poly-qualinomials'. A second species of Q-algebraic variable represents, indifferently, either 'mono-qualinomial' or 'poly-qualinomial' NQ values, depending upon the equational or other formulaic context in which it appears. This type of variable symbol is exemplified throughout this section, viz. -- x, y, z, etc. -- where x2 x; y2 y, and z2 z, etc. A third species of WQ-algebraic variable, not previously exemplified, is key to the 'Onto-Dynamical Equations' from which ontic dialectical models of 'meta-evolutionary process' are formed. This type of WQ-algebraic variable is a poly-qualinomial variable, which employs "Whole numbers" subscripts to denote specific subsets of WQ, viz., WQ1. Each of these subsets is an unbroken, consecutive, finite subsequence of the full, potentially infinite WQ sequence, starting with W, and/or with W1, and ending with some Ww, such that W w _ 1. These Wk-sequences succeed one another by means of self-[re-]flexion, that is, by the self-"multiplication" of each WQk. They thus form a 'meta-sequence' of their own: WQ0 WQ1 WQ2 WQ3 .... The { WQk } 'meta-sequence' provides an abstract, generic image for the 'meta-dynamical', 'self-conversion singularity', 'self-bifurcating systems' paradigm of self-developing process, at that level of complexity/concreteness/specificity of ontological, 'possibility-space' description. Every WQk subset, restricted by the non-negative integer subscript k, is a proper subset of WQ as a whole: WQk WQ. That subset can be "interpreted for" or "applied to" the myriad contexts of our 'psycho-historical materialism' paradigm, e.g., concrete, '''external'''/physical, 'physio-ontological', or abstract, conceptual mimetic ['ideo-ontological'] 'meta-evolutions', thus forming specific-process 'onto-dynamical' models for specific universes of discourse. The ordinary W-numbers, 0, 1, 2, 3,... are vastly generic, precisely because they are 'unqualified', or 'purely-quantitative', 'de-qualified'; because they abstract/simplify/de-specify from any determinations -- from any ontological, metrical, or other qualifications/'qualifiers' -- that might be included in more specific enumerations, and are thus '''assignable''' to countings of 0, 1, 2, 3,... anythings, from cauliflowers to cannibals to kingdoms. Just so, we hold, can W, W1, W2, W3, ... be assigned to the sequences, successions, or progressions of emergent ontological qualities/categories, or 'ontos' observed in 'self-developing processes' as diverse as: (1.) the 'meta-evolutions' of atomic species and of planets in the star-and-planet-birthing interstellar cloud cumula of the galactic interstellar medium; (2.) the nebular/planetary 'meta-evolutions' of molecular species by/in "molecular clouds"; (3.) the planetary, biospheric and then 'bio-noospheric' 'meta-evolution' of humanoid species, and beyond. Each WQk denotes an [the kth] 'meta-evolutionary epoch' of an 'ontologically-expanding', 'onto-dynamical' "universe of discourse". The 'meta-sequence' to which that WQk belongs is used to model the 'meta-dynamics' of that universe as a succession/progression of such epochs. Each WQk in this WQk 'meta-sequence' denotes a diverse or ''heterogeneous''', '''non-reductionist''', 'non-collapsing', "non-amalgamative" 'sum of [ontic] qualifiers' characterizing the possibilities of the given 'onto-dynamical' universe of discourse during the given epoch. This characterization is achieved by means of the interpretation of that sum as a special kind of 'universal ontology-set' for that universe of discourse -- a set of 'ontological qualifiers' or 'ontos' defining the current-epoch potential ontology of that universe. This 'ontology set' defines what may be actually extant, but, essentially, what is possibly extant for/in that universe during that epoch. This 'onto-list' is the set of all ontological categories or qualities which that epochal universe contains or could contain; the ontological 'possibility-space' or 'possibility-metastate' of that universe in that epoch. It is an enumeration of the categories of what activities/[ev]entities can exist in that universe at that "time". As the value of the 'self-bifurcation index', or whole-number 'epoch-index', k, increases or advances, the 'self-[re-]flexion' of the presently-extant WQk 'meta-state' generates the next 'ontology meta-state' of that universe of discourse as the next value in the 'meta-sequence', namely WQk+1. The 'universe-progression' or 'onto-dynamical' process modeled by the WQk 'meta-sequence' is thus that of a 'self-expanding ontology-set', or of a 'self-expanding possibility-space', grounding both probability and actualization in each such universe-epoch. Only that which is first at least possible can become also probable and, perhaps, even actual[-ized]. This dynamical image of universal autopoiesis is thus an 'Onto-Dynamical' Model, departing from traditional 'Onto-Statical', Parmenidean paradigms. Below is depicted how this generic-dialectical process looks per the accompanying 'dynamic-geometric' or 'dynamic-pictographic' formulae which these ideographic formulae, per this 'topo-metrical interpretation', encode, imaging a self-expanding [possibility-]space with ever more new mutually-perpendicular unit-interval line-segment directions, axes, or arrows sprouting from the W_origin as the epoch-index advances [in the depictions below, only the end-points of the directional, unit-length line-segments are labeled with an Wk 'meta-number' value, but each of these labels actually applies to the entire directed line-segment, including the starting-point of each of these directed line-segments, always labeled W]: WQ0 _ W1 , ideographically; WQ0 _ [ vs. E ( ____[0, 1] R1 ], W W1 0 1 pictographically; W2 W1_____W2 WQ1 _ W1___W2 ; WQ1 _ pictographically; W W1 W1_____W2_____W3 WQ2 _ W1___W2___W3___W4 ; W2 W3 [the 4th dimension, the W4 axis, is not depicted]; WQ2 _ W W1 pictographically; WQ3 _ W1___W2___W3___W4___W5___W6___W7___W8 ; WQ4 _ W1___W2___W3___W4___W5___W6___W7___W8___W9___W10___W11___W12___... __W16 ...; Each Wk in the '[poly]-qualinomials' above whose subscript is a Whole-number power of 2 is interpretable as, and assignable to, the 'contra-thesis', 'qualitative increment', or 'ontological increment', denoted by the 'symbol-complex' _Wk/2; the 'new onto yield' of the self-product 'self-hybridization' or 'self-reflexion' of an earlier 'onto' whose subscript is 1/2 the value of the subscript of the 'meta-numeral' assigned to that earlier 'onto'. Thus W8 is the _Wk of 'W4 of W4'; the 'W4 of 2nd degree' -- the '2nd degree' of earlier 'onto' W4. Such a Wk value is interpretable and assignable as representing the 'meta-onto' of that 'onto'. Each of the new units [new '''logical indivi-duals'''], or 'neo-monads, of this 'meta-onto' are made up out of a heterogeneous multiplicity of the old units, or monads, of that predecessor 'onto', via an aufheben operation instantiating 'meta-fractal', 'meta-finite' 'self-subsumption', 'self-interiorization', or 'self-internalization' 'meta-dynamic', i.e., wherein molecules are 'meta-atoms made up out of a heterogeneous multiplicity of atoms', or are 'atoms of second degree'; wherein prokaryotic cells are 'meta-molecules made up out of a heterogeneous multiplicity of molecules', or are 'molecules of second degree', wherein eukaryotic cells are 'prokaryotic cells made up out of a heterogeneous multiplicity of prokaryotic cells', or are 'prokaryotic cells of second degree', etc. Such Wk come under the Corollary of Rule 3, [ _j___WQ ][ jj __j2 __ j __j+j ___ j __2j j ]. Each Wk in the '[poly]-qualinomials' above whose [post-]subscript, k, is not a positive integral power of 2 denotes a 'hybrid onto'. Each such 'hybrid onto' stands for a "joint product", and, usually, for an 'ontological [self-]conversion process', transforming, to the [meta-]monads of the 'onto' denoted by the left-most interpreted subscript, the 'fuel' [sub-]monads of the [ev]entity denoted by the '''complex unity''' connoted by all of the interpreted subscripts to the right of that left-most subscript -- thereby producing the new, qualitatively distinct offspring of the "mutual interactions" of the one or more other, distinct, hybrid and/or non-hybrid or 'self-hybrid' 'ontos' so denoted. The contemporary atmosphere, ocean, and soils of planet Earth are combined results of the mutual activities of 'eventities' belonging to the atomic, molecular, prokaryotic cellular, eukaryotic cellular, multicellular a-social, social, and '''human[oid]-social''' or 'meta-social' 'ontos'. Each of these media requires a distinct 'hybrid-onto qualifier' in its own right, representing a 'hybrid ontological category', distinct from those of each of its progenitor 'ontos'. Such hybrids also connote categorial 'non-empty boundaries' or 'interfaces' ["leidenfrost layers"], and 'ontological [self-]conversion formations', situated '''between''' their parent 'ontos'. Another way to understand these terms is as denoting mutual existential adjustments or '''syntheses''' that non-hybrid or 'self-hybrid' 'ontos' make to one another as a result of their co-existence and ensuing co-activity. Such Wk come under Rule 3, [ _i,_j __WQ | i ( j ][_ ij ____j ___i+j ]. Note that this Rules-System for WQ implies that the relations among the k meta-numbers include "linear independence" but also 'nonlinear dependence'. Rule 0, and the generic algebraic characterization { ( D }, assert that m and_n are "linearly independent" if n ( m. Rule 3 asserts that, e.g., whenever n _ 2m, m and_n are 'nonlinearly dependent', 'self-reflexively inter-dependent', or 'self-application inter-dependent', i.e., that n ____2m______m2___m. In summary, { 2 x } and {1 ( D = 2___1 } together connote the linear independence relationship, and, together with { 2 }, also the 'qualitative disproportionality', that is, the 'meta-nonlinear' nature, of the k 'meta-numbers'. Note also that the self-product rules, in their geometric interpretation as given above, follow a pattern we term 'meta-diagonalization', viz., each WQk denotes, geometrically, the diagonal 'meta-vector' of a 2k-dimensional unit ['hypo-' or "hyper-"]cube, and denotes the intensional-semantic quality of each such interpretedWQk 'ontological meta-state', as mapped via a metaphor of dimensionally-distinctive directionality -- diagonalk _ diagonalk[ diagonalk ] = _[ diagonalk ] = [ diagonalk ]2 = diagonalk __[ diagonalk ] _ [ diagonalk _ meta-diagonalk ] = diagonalk+1 diagonalk, and, using the Euclidean [Pythagorean] metric, we can prove that the 'pure quantifiers' of the lengths of these 'hypo-diagonals' [for k = 0], diagonals [for k = 1], & 'hyper-diagonals' [for k 2] bear the following relations -- (( diagonalk+1 (( _ (( diagonalk ((, as ((2k+1) _ ((2k). This also suggests the possibility that each 'possibility-space meta-system meta-state' Qk for interpreted Q can be specified uniquely by a single R1 scalar pure-quantifier value, qualified by the radians metrical qualifier, namely, the scalar quantifier value quantifying the angle of the 2-D projection of the kth 'hyper-diagonal' upon the 1 ( 2 plane. [Note: In the rest of this sub-section, we will usually omit '''number-space''' designating 'pre-subscripts', writing XQk & Xk as simply Qk & k, with the value of X -- either N, W, Z, Q, R, C, H, O, K, G, or beyond -- left unspecified explicitly, but made clear implicitly via context. Q_ 'Meta-Dynamical' Algebra. Interpreting subscripts k of the meta-sequence {_Qk_} as indices of discrete 'meta-time' in the form of the meta-evolutionary epoch index, _, we see that the Qk = Q__'multi-meta-ontic' and 'multi-meta-monadic' 'cumulum'-descriptors evolutely regenerate themselves and generate each other via what we term 'meta-dynamical self-reflexion', per Rule 3, vide -- Q_ _ _[ Q__] _ Q_[ Q__] _ Q_2 ____Q______Q_ ______Q____ Q_; Q___ _ _[ Q__] _ Q_[ Q__] _ Q_2; Q_ _ [ 1 ]; Q___ _ Q_ _ Q_2 _ Q_[ Q__] = [ 1 ][ 1 ] = [ 1 __1+1 ] = [ 1___2 ]; Q___ = Q_ _ Q_2 _ Q_[ Q__] = [ 1___2 ][ 1___2 ] _ [ 1___2 __1+2 __2+2 ] _ [ 1 __2 __3 __4 ]; Q___ _ Q_ _ Q_2 _ Q_[ Q__] =_[ 1_2_3_4 ][ 1_2_3_4 ] _ [ 1_2_3_4_5_6_7_8 ]; Q______[ 1___2___3___4___5___6___7___8___9___10___11___12___13___14___15___16 ]; ... The above portrays, in skeletal outline, the 'Evolute Onto-Logic' of this 'meta-model' of self-expanding universes of existential possibility. The {_Q__} 'meta-sequence' thus models 'meta-evolving' universe progressions as self-extending similarity structures, self-growing 'quasi-/meta-fractal' rheid crystals. We say 'quasi-fractal' or 'meta-fractal' because, at each epoch, that 'meta-sequence' involves a finite forward qualitative-scales-regress, rather than a supposedly/potentially infinite quantitative-scales-regress, as do mathematical/idealized "fractals", and one in which the next higher 'meta-scale' in the 'meta-scales sequence' is constructed from, and by, the inherent activity of immediately preceding 'meta-scale-level', with the cumulative participation of all previous 'meta-scale-levels'. This 'meta-sequence' is designed to capture the self-similarity invariant of these structures, relative to their stipulated origination, 1. This 'meta-sequence' locates the source of qualitative novelty, or of 'ontological innovation', in the mutual "interaction" and in the '''self-interaction''', that is, in the '''self-reflexion''', of previous 'self-innovation'. The following pair of nonlinear 'meta-dynamical'-algebraic equations, which we term 'Meta-Rules', summarize its 'Onto-Dynamical Logic': Meta-Rule 1. ['Meta-Evolution' Equation]: Next universe "equals" [or 'results from'] self-interaction of current universe, or, ideographically, Q___ _ Q_2 _ Q_[ Q__] _ ([ Q__]; Meta-Rule 2. [Generating Equation for the progression of antitheses]: [closed-form general solution of the Pure-Qualitative-Nonlinear 'Meta-Evolution Equation']: Q_ _ Q_2_ ( [ 1_]2_. Meta-Rule 3. [Generating Equation for the 'onto'-by-'onto' progression]: O_ _ O_t ( [ 1_] t. The "Right-Hand-Side" [RHS] of the above solution-equation is a 'meta-exponential', 'hyper-exponential', or algebraically 'hyper-nonlinear' function. This RHS involves two tiers of superscription, denoting two levels of exponentiation. This RHS is a term of 'degree' 2__in Q0, a degree which escalates with the advance of the epoch-index, _. That '2__degree' is an arithmetically and algebraically nonlinear degree, a degree greater than 1, when ___ 0. That '2__degree' is of the linear degree, the degree 1, only when _ = 0, at the stipulated origination of the initial ontology of the universe of discourse. The 'sub-meta-sequence' Q_ will have 2_ terms. It will be a 2_ 'poly-qualinomial'. Its terms span all ks having consecutive "Natural" number subscripts from 1 to 2_ inclusive, with no gaps, no ks missing in-between. The "cardinality" of the 'ontology-set' at stage _, the number of 'ontological quality categories', or 'ontos', which it contains [i.e., whose possible existence during that epoch it asserts], is 2_. The maximal ordinal subscript among the k extant at stage _ is also 2_. The minimum subscript is always, at every stage, 20 _ 1. The 'meta-sequence' denoted by {_Q__} is a nested, cumulative, or 'evolute' 'meta-sequence': __ ___o|_____1, Q______Q____, where o denotes the set of "Natural" ordinal numbers, as interpreted, in this case, to represent ordinal discrete epochs, _. Thus this 'onto-dynamics' follows a kind of '2_ Dynamical Combinatorics'. Note the equation of Meta-Rule 2 enshrines the 'doubling function', f(_) _ 2_, which pops up so ubiquitously in nonlinear dynamics and in other key areas of modern mathematics, describing (a.) the "period-doubling route to [so-called] chaos", (b.) the number of sub-segments of the unit-interval segment remaining after the completion of the _th step of the Cantor middle-thirds process, whose potentially transfinite iteration approaches the fractal Cantor Set, (c.) the Cantorian cardinality of the power-set, or set of all subsets, for a finite [or potentially "transfinite"] set of cardinality _, etc., etc. Four 'Meta-Dynamical' Product Rules and Their 'Gdel Numbering Subscript-Rule' Variants. Besides the 'Aufheben Evolute Product' of Rule 3, we also partially explore, in the Section entitled The Arithmetics of Meta-Evolution, three other alternative product rules. We also explore a 'Gdelian' variant of each of these four product rules. The latter variants employ a subscript rule inspired by "Gdel numbering" -- the use Kurt Gdel made of the Fundamental Theorem of ["Natural" Number] Arithmetic in his Incompleteness Theorem. Gdel applied the former Theorem to the construction of the latter Theorem, in such a way as to form a unique mapping/encoding of the formulae of symbolic logic to elements of N. The function p(n) selects the nth prime number, for n ( N. In summary, we explore the following eight product rules -- 1. The Aufheben Evolute Product Rule: j[ k ] _ [ k ____j+k]; k[ j ] _ [ j ____j+k ]; 2. The 'Meta-Catalysis' Evolute Product Rule: j[ k ] _ [ j ____j+k]; k[ j ] _ [ k ____j+k ]; 3. The 'Meta-Genealogical' Evolute Product Rule: j[ k ] _ [ j ____k _ j+k ]; 4. The 'Meta-Heterosis' Convolute Product Rule: j[ k ] _ [ j+k ]. The 'Gdelian' variants of these product rules are designed to achieve partial 'de-confounding', or greater distinguishability of distinct 'ontic' interaction-products from one another. This entails an even stronger form of non-commutativity than that of the first two 'non-Gdelian' product rules stated above. In these 'Gdelian' variants, the 'index' or subscript of the 'qualitative increment' portion of a product is a 'Gdel number' encoding the syntax of the 'multiplication' formula from which that 'qualitative increment' or 'ontological increment' arose. Thereby, each 'evolute' product reveals, contains, or records its path-of-formation, origin, ancestry, or 'meta-genealogy', and is thus 'evolute', or 'ontology-conserving' in the aufheben sense, in a yet deeper way. Given that p(k) denotes the kth "Natural" prime number, s.t. p(1) = 2, and that j < k, we obtain: 1g. 'Gdelian' Aufheben Evolute Product: j[ k ] _ k ___p(j)j * p(k)k; k[ j ] __j __p(k)j * p(j)k; 2g. 'Gdelian' 'Meta-Catalysis' Evolute Product: j[ k ] _ j ___p(j)j * p(k)k; k[ j ] __k __p(k)j * p(j)k; 3g. 'Gdelian' 'Meta-Genealogical' Evolute Product: j[ k ] _ j __ k __p(j)j * p(k)k; k[ j] __k __j __p(k)j*p(j)k; 4g. 'Gdelian' 'Meta-Heterosis' Convolute Product : j[ k ] _ p(j)j * p(k)k; k[ j ] __p(k)j * p(j)k; Per aufheben versions of the 'meta-vector' or 'dialector' product, { NQ__} denotes a sequence _ series s.t.: Q_ ( _ ( [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . ]T _ (_ ( (_; Q_ ( _ ( [1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . ]T _ _ ( _; Q_ ( _ ( [1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 . . . ]T _ _ ( _; Q_ ( _ ( [1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 . . . ]T _ _ ( _; Q_ ( _ ( [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . . ]T _ _ ( _; ... In contrast to the dual consecutivities, first of an all 1s sub-string, then followed by an all 0s sub-string, evident above, 'Gdelian' versions of the aufheben 'evolute' product, per the potentially-infinite-dimensional 'meta-vector' model above, exhibit a widening sequence of '0 sub-string gaps' between each successive pair of 1s. Q_ 'Meta-Dynamical' Analytical Geometry. The 'meta-dynamical-geometry' or 'meta-analytical-geometry' of the {_Q__} space is the 'meta-dynamical' geometry of a 'Self-Expanding', 'Self-Replicating', 'Self-Reproducing', 'Self-Mirroring', 'Self-Reflecting', or 'Self-Reflexive' Space. This geometrical 'Self-Replication' takes the form of a spatial 'Self-Doubling' with each incrementation of the 'epoch-index' or 'self-bifurcation' index-value, _. This process can be visualized as follows -- 2 1_2 _ 12 2 1 ( 1 _ 1 _ Q02__________11 _ [ , 1 ] _ [ , 1 ] _ 12 _ 1 _ 2 or _ _ _ _ 2 _ _ 2 12 2 12 4 3_4 (__________________________ _ 1 1 _ 3 Q12 _____ 1 _ 2 1 _ 2 _ 1 _ 2 2 _ 1 _ 2 _ 3 _ 4 or y y _ _ y _ y _ y -- depicting the 1 1 '''self-reflexion''' as 'copying' the 1-D finite-length, unit-length 'hypo-cube' line-space 1 axis, and attaching it perpendicularly back to that 1 axis at its origin-point, , as the directionally, perpendicularly, qualitatively distinct 2 finite, unit-length axis, yielding the '''discretized''', interval-notation Cartesian Product [_] plane-space [ , 1 ] __[ , 2 ] as the implicit backdrop of the diagonal, finite-length, +(2 units-in-length, directed line-segment product Q1 __[ 1 _ 2 ]. Next, the 'self-product', or '''squaring''', of that product, the 1 _ 2 1 _ 2 '''self-reflexion''', 'copies' that 2-D 'hypo-cubic' plane-space, denoted in interval notation by [, 1 ]_[ , 2 ], attaching it back to [, 1 ]_[ , 2 ] as the qualitatively distinct plane-space [, 3 ]_[ , 4 ], forming the 4-D hypercube-space [, 1 ]_[ , 2 ]_[ , 3 ]_[ , 4 ] as the implicit backdrop of the diagonal, finite-length, +(4 = 2 units-in-length directed line segment product Q022 ___Q2 __ 1 _ 2 _ 3 _ 4 , and so on, with each successive, iterated 'spatial self-reflexion'. The 'contental logic', 'existential logic', or 'Onto-Logic' of these 'meta-numbers' may be regarded as a 'potentially infini-valent logic'. This contrasts with the "bi-valent" or 2-valued logic of later Boolean Algebra [Boole's original Algebra having partially admitted 'tetra-valence', with 1/1 as 1, or the logical quantity "All", with 0/1 as 0, or the logical quantity "None", plus with 0/0 as the logical quantity 'Indefinite' ["None, Some, or All"], and with 1/0 as logical "Infinity" ['''Singularity''', or "Impossibility"]. Each k of the potentially-infinite sequence of k within Q may be interpreted as denoting a qualitatively distinct existential [ontological] "truth-value" or 'existence-value' [existential possibility-value]. As a "number/numeration system", or "numeral scheme", the Q arithmetic is, in a sense -- at the 'scriptal' level, as opposed to at the 'sub-scriptal' level -- a unary, rather than binary, decimal, duodecimal, vigesimal, or sexagesimal, etc., system. That is, it forms, in a sense, a [meta-]numeral system to the base 1. Each [meta-]numeral, k -- lacking "place-value" in taking but one "place" -- denotes a unique value: neither a combination of earlier values in the values-sequence, nor a value which reappears later in that values-sequence, in combinations with other such [meta-]numbers, in the symbolic formation of any single unit -- unlike 1; 12; 101; 1,100; ... , etc. The { (k } can also be grasped as a unified space of [transcendental] '''functions''', '''transformations''', or of '''operations''' [of 'operators'], in which '''functions''', '''arguments''', & '''function-values''' all coexist as a single space: [ (fJ, fK_(_{ (fJ } ( Q | fJ[ fK ] ( Y ] [ [ fJ: Q Q ] & [ Y ( Q ] & [ Y _ fJ __ fJ+K ] ]. I.e., each '''function''' k in Q is an element of the '''domain-space''' of its '''arguments''', and is also an element of the '''range-space''' of its 'products' or '''function-values''', both of which are identically the space in which the '''function''' itself also resides. For varying values of k, each "function" k can also serve as an "argument" of other '''functions''' k, and, as well, as a '''function-value''' [except for 1 in NQ] of '''functions''', k, operating upon [or '''multiplying'''] other '''arguments''', k, or operating upon themselves. Note also that the k are all 'meta-transcendental numbers'. In that phrase, the epithet 'transcendental' means that these numbers cannot be "roots" of, or solve, any 'fini-nomial' -- meaning any finite-terms-count, finite degree, i.e., "algebraic" polynomial -- with "Rational" coefficients. We write 'meta-transcendental numbers', because mere "transcendental" numbers connote pure-quantifier "Real" numbers like e and _. We call the k 'meta-transcendental' precisely because of their 'contra-Boolean' behavior -- x2 __x -- i.e., because k2 _k1, coupled with their ''additive idempotency'', or 'unquantifiability'. This means that they are 'qualitatively disproportionate' [], i.e., highly nonlinear, with respect to the "Rational Numbers", Q, or even to the "Reals", R; that there is no "Quotient" ["Rational"] number [or Real number], no matter how large, or how small, such that this number as a "coefficient" of k1 can equate the thus "multiplied" k1 to kn where n ( 1: (k ( NQ, (k ( N n > 1,r ( Q, R | kn _ rk1; i.e., (r ( Q, R; (k ( N n > 1, kn rk1. Therefore, degree n 'fini-nomial' or 'poly-qualinomial' equations like rnQkn + rn-1Qkn-1 + r n-2Qkn-2 + . . . + r1Qk1 + r0Qk0 _ are 'non-algebraic polynomial equations' or "['meta-']transcendental functions", with no solutions if the parameters, rj, are restricted to the "Rational numbers", or even to the "Real numbers", except for the null solution { rj _ 0 }. The 'meta-models' of universe 'meta-evolution' that can be constructed using the apparatus set forth above are, of course, '[meta-]arithmetical', 'meta-numerical' models, and are also both highly simplified models and highly abstract models. Indeed, that is their goal: to extract and to exhibit in idealized form a universal principle; a 'meta-dynamical' essence-pattern of 'meta-evolutionary' or 'ontological-revolutionary' process in general; a generic ideographical image of the dialectic. The relative algebraic simplicity of these 'meta-models' is bought at the price of great generality; of large 'homeomorphic defect', that is, of the very high ratio of the multitude of features of the sensuous world mapped to but single features in the ideographical 'meta-model' of that world; of features in the world not found in the 'meta-model' ['Type I homeomorphic defect'], as well as of some 'extranea' or 'artifacts' -- features in the 'meta-model' not found in the world ['Type II homeomorphic defect']. The 'Onto-Dynamical' equations above describe, in an ideographical language, the diachronics of 'onto-dynamical' universes of discourse, of self-evolving existences / activities; self-evolving ontologies, with about as much detailed coverage as Boole's Algebra Of Logic describes propositions and the interrelationships of 'onto-statical' synchronic classes. The advance from the 'unit-interval restricted' Q algebras to the 'full-multiplicity' U and __ algebras, and beyond, described in the Briefings that follow this one, as well as in The Arithmetics Of Meta-Evolution, Section III., below, begins to redress these grievances, at the cost of a more elaborate, more complex, more "concrete" syntactical and semantical rules-apparatus. The sub-section next-below summarizes the applicability of the { Q } arithmetics to the 'meta-modeling' of historical-dialectical processes -- to the formation of 'ideometric' and ideographical -- and, in that sense, "mathematical" -- 'historical-dialectical meta-models': 'historical-dialectical-mathematical meta-models'. Q Arithmetic and Historical Dialectics. By way of a brief evocation of the dialectical character of models made in the ideographic language of Q, we reproduce, below, a passage by neo-Hegelian philosopher Errol E. Harris, in which he summarizes, in word-text -- in phonogramic/phonetic symbols -- the principal characteristics of dialectical process. Further, we 'commentarize' that text with annotations illustrating the way in which and the degree to which those characteristics of dialectical process per Harris' account can be concisely "captured" in the ideographic language of the Q arithmetic and algebra. The passage is as follows -- "Each level provides the basis for that which succeeds, yet on every level the characteristic properties of the appropriate entities depend upon their total structure. They are "cooperative properties" impossible for less complex entities. Atoms have properties impossible for free electrons, and molecules evince chemical affinities which are dependent solely upon the pattern of combination of their constituent atoms and are not characteristic of any atom in isolation. This is especially true of the macromolecules involved in the activities of living matter, which are not feasible at the inorganic level . . . Consequently the cosmic organism, while it is one and indivisible, is at the same time a range of developing phases, which can be represented, and which display themselves, as a dialectical scale [or a graduated 'consecuum' made up of/generating a self-extending diachronic sequence of mutually, 'quanto-qualitatively' similar/dis-similar 'quanto-qualitative' 'scales'; a 'quanto-qualitatively' 'metafinite' scales-regress formation of quanto-qualitatively self-similar structure which we term a 'meta-fractal' -- F.E.D.]. The totality [or 'meta-system' -- F.E.D.] is constituted by the scale of its internal forms, and each level [Q__] is in some sense self-contained and all-pervasive; yet each gives rise to the next above it [Q____] by virtue of the potentiality within it infused by the immanent principle of the totality [or of subject/object 'intra-duality' -- F.E.D.] in which it is no more than a phase [Q__(Q__(... Q__(Q____(...]. This is an idea of nature, not merely as an all-embracing living animal, but as a dynamic organismic system [or systems-progression 'meta-system', made up out of a diachronic succession of many systems -- F.E.D.], comprising a continuous range of wholes, on levels of progressively increasing complexity and integration. They are wholes mutually in dialectical relation, so that the entire system [or 'meta-system', since this "system" is composed of a diachronic sequence of sub-wholes also termed "systems" -- F.E.D.] manifests itself as an evolutionary [or 'meta-evolutionary', since each "system" in the diachronic sequence also "evolves" internally, before, and after, and leading to, its "revolutionary", ontology-net-expanding self-transformation into the next system in this diachronic systems-sequence or 'meta-system' -- F.E.D.] progression." [Errol E. Harris, Formal, Transcendental, and Dialectical Thinking: Logic and Reality, State University of New York Press [Albany, NY: 1987], pp. 255-256, bold italics commentary and emphasis added by F.E.D.] [Some examples -- (1) consider the Physis/Cosmos as a whole, in its diachronic 'self-meta-evolution' from a universe 'super-system' whose organization stops at the level/scale/degree of sub-nuclear particles [ n ], to one whose organization stops at the level/scale/degree of sub-atomic particles [ s ], to one whose organization stops at the level/scale/degree of atoms [ a ], to one whose organization stops at the level/scale/degree of molecules [ m ], to one whose organization stops at the level/scale/degree of prokaryotic "living" cells [ p ], . . .. We render that natural-historical autokinesis, or self-movement, and the 'metafractal cumula' of its successive systems, in ideographic intensional , mnemonic-heuristic symbols, as: [n ] ( [n _ s ] ( [n _ s _ qsn _ a ] ( [n _ s _ qsn _ a _ qan _ qas _ qasn _ m ] ( [n _ s _ qsn _ a _ qan _ qas _ qasn _ m _ qmn _ qms _ qmsn _ qma _ qman _ qmas _ qmasn _ p ], or Q0 ( Q1 ( Q2 ( Q3 ( Q4 , i.e., Q020 ( Q021 ( Q022 ( Q023 ( Q024. (2) consider the second-taxonomic-level sub-universe-of-discourse of the 'meta-system' of the human species' 'meta-social meta-dynamics', i.e., going on '''inside''' the first taxonomic level 'onto, h, in terms of its 'self-meta-evolution' from a 'meta-society' whose economic organization, or "social relations of production-inducing-circulation", stop at the level/scale/degree of 'predation' -- direct Appropriation, without improvement, of the raw products of nature, [A ] -- to one whose organization stops at the level/scale/degree of 'raw appropriation squared', 'second[-degree] appropriation', 'appropriation of the appropriation', or 'appropriation with improvement', or refinement, of raw products of pre-/extra-human nature, i.e., 'Goods'/'Gifts'-production, [G ], to one whose organization stops at the level/scale/degree of Commodity barter, where goods acquire, beside their direct-consumption utility, an indirect, exchange-utility, [C ], to one whose organization stops at the level/scale/degree of Money, or of the Money-mediated exchange/"circulation" of Commodities, [M ], to one whose organization stops at the level/scale/degree of 'money-making-money', or Kapital, [K ], . . .. We render this human-social, psycho-historical movement of that 'meta-system', and the 'meta-fractal cumula' of its successive 'human-social' systems -- ideographically, in terms, again, of intensional , mnemonic-heuristic symbols -- as: [A ] ( [A _ G ] ( [A _ G _ qGA _ C ] ( [A _ G _ qGA _ C _ qCA _ qCG _ qCGA _ M ] ( [A _ G _ qGA _ C _ qCA _ qCG _ qCGA _ M _ qMA _ qMG _ qMGA _ qMC _ qMCA _ qMCG _ qMCGA _ K ], or Q0 ( Q1 ( Q2 ( Q3 ( Q4 , i.e., Q020 ( Q021 ( Q022 ( Q023 ( Q024. ]. Resuming the passage by Errol Harris -- "Let me once more recapitulate the dialectical relation in its full complexity. The wholes which it relates are each, in one aspect, self-contained and self-dependent [or, initially, expandedly self-reproducing -- F.E.D.], and, in another, mutually implicated and inseparably interrelated [for each predecessor system evolves from a state of expanding self-reproductivity to one of partial self-dis-reproductivity in the form of a self-revolutionizing or 'self-meta-evolutionizing' production of its qualitatively, ontologically different -- ontologically net-expanded -- successor system -- F.E.D.]. Essentially the relation is serial, each successor whole being a fuller and more adequate realization [or a fuller self-development and self-out-working of its proto-subject/object 'intra-duality' or 'internal ontological self-contradiction' -- F.E.D.] of the systematic principle governing the entire series. So each is related to its predecessors as their fulfillment [or as the 'explicitization' of what they held implicit -- F.E.D.], requiring and incorporating the prior forms, while actualizing potentialities of which they were incapable. For this reason, while the subsequent involves the antecedent, it also supercedes and, in some sense, negates its forebears. Each whole, then, is a grade [or 'metafinite', 'meta-fractal' scale/level -- F.E.D.], a developmental stage, within the total series, but also a distinct relatively self-subsistent [or transitorily self-reproducing -- F.E.D.] phase standing in qualitative contrast and opposition [a relation herein denoted via the picto-ideographic signs '' for the 'successor__predecessor' ordering, and '' for the 'predecessor successor' ordering -- F.E.D.] to its neighbors [Q____Q__Q___ ]. Yet because this opposition is resolved in the higher phase (which preserves the contrast while it supersedes it [i.e., as the result of a 'self-aufheben' operation -- F.E.D.]), the entire series remains continuous and coherent." [Errol E. Harris, ibid., p. 256, bold italics commentary and emphasis added by F.E.D.]. [The generic historical-dialectical sequence/series is, to the fifth stage, phase, or epoch, per the Q arithmetic/algebra for dialectics -- Q__ ( Q____( Q__ ( Q__ ( Q__ ( . . ., or Q_20_ ( Q_21_ _( Q_22_ ( Q_23_ ( Q_24_ ( . . .. Again, note that, because of the aufheben principle and the additive idempotency embedded in the Q Rules-System, a _Qk series or 'qualitative sum' ['pure-ontological sum'] equals the leading term in the {Qk } sequence, which already contains all of its predecessor terms. For example -- Q__ + Q_____+ Q__ + Q__ + Q___ = _(k = 0,4)Qk__ = Q_. [ sequence = series in Q ]. Each system/whole in the 'meta-system' series/sequence thus 'causally implies' [' ('] its successor(s): Q__(Q__(Q__(Q__(Q_( .... Each also ontologically exceeds its predecessor(s): Q_Q__Q__Q__Q_ .... Each also explicitly incorporates /contains all of its predecessor(s): Q_Q__Q__Q__Q_ .... Each also 'explicitizes'/actualizes what was only implicit /potential in its predecessor(s): Q_ _ Q___ Q___ Q___ Q__ .... Each successor is also the 'ontological negation' of its predecessor: Q_____= ~[Q__]__= Q__[Q__]___= [Q__]2_= [Q__ + _[Q__]]__ Q__. Each system/whole in the 'meta-system' series /sequence / progression also opposes its predecessors and its successor(s) in their key qualities/meaning/attributes: Q_ Q_ Q__ Q_ Q_ . . ., and Q_ _ Q_ Q__ Q_ Q_ . . ..]. Concluding the passage by Errol Harris -- "The relation of mankind to nature has now to be understood in the light of this dialectic conception. Human personality, developing within social structures peculiar to its appropriate level in the scale, is integral to the whole. On the other hand, as one level distinct from others, it confronts the prior phases as other and opposed. But this is only one aspect of its relation to them, for they are also its forebears and progenitors in which the potentiality of its emergence is instant. What humanity sees as nature is its own self in becoming; but more than this, nature is the very matrix from which its very being is contrived and the soil out of which it is nourished." [Errol E. Harris, ibid., pp. 256-257, bold italics commentary and emphasis added by F.E.D.]. Q Arithmetic and [Meta-]Systematic Dialectics. [forthcoming]. Transition to the Briefing on the U ( 3 Dialectical Ideography. The next Briefing evokes the NU arithmetic out of the mutual and immanent [self-]critiques of both the N and the NQ arithmetics, as the dialectical '''unification''', '''complex unity''', '''higher unity''', or 'uni-thesis' of the arch thesis, N, and its 'contra-thesis', NQ, a 'uni-thesis' also denoted by NQN & by NU, completing the 'synthesis-sum' denoted by N NQ NU. The foregoing briefing has set forth the apparatus of mainly the NQ and WQ epochs of the { Q } dialectical arithmetics and their dialectical algebras, without addressing their ZQ, QQ, RQ, CQ, etc., '''epochs''', and at a rather generic level, largely detached from their interpretations -- i.e., rather remote from the [meta-]modeling of specific processes of exo-human natural history and of human/'meta-social' '''psycho-history''', or of the [psycho-]historical processes of development of particular human conceptual systems and traditions of thought. The syntax of this ideographical 'language of ontic meta-evolution' more clearly reveals its semantic potential and limitations as we engage its interpretations. We so engage in the next full section, Section II., where we employ the NQ incarnation of dialectical ideography, among others, to model the observable, historical, human, '''psycho-historical''' phenomenon of The Meta-Evolution Of Arithmetics itself. 141 Dialectical Ideography I - Distributed Samizdat by Foundation Encyclopedia Dialectica Dialectical Ideography I- F.E.D.