Thursday, November 14, 2013

On the Nature of the Opposition between the 'First Standard Arithmetic', and the Seldonian 'First Dialectical Arithmetic'.

Full Title 

On the Nature of the Opposition between

the N_ 'First Standard Arithmetic' 
the NQ_ 'First Dialectical Arithmetic'.

Dear Readers,

Below, I have reproduced, for your cognitive pleasure, parts of the forthcoming F.E.D. Vignette #21, by Aoristos Dyosphainthos, Chief Public Liaison Officer for the Foundation.

It provides a profound introduction to the Seldonian NQ_ 'First Dialectical Arithmetic'.




Member, F.E.D.,
Officer, F.E.D. Public Liaison Office  

F.E.D. Vignette #21 --

On the Nature of the Opposition between the N_ 'First
Standard Arithmetic' and the NQ_ 'First Dialectical Arithmetic'.

by Aoristos Dyosphainthos

Authors Preface.  The purpose of F.E.D. Vignette #21 is to clarify the nature of the relation of non-dualistic,  dialectical opposition that is felt, by human 'conceptual perception', between NQ_, the axioms-system of the F.E.D. 'First Dialectical Arithmetic', and N, the first-order-plus axioms-system of the 'First Standard Arithmetic'. 

We do so by exploring the first triad of ontological categories, and, equally, of connoted axioms-systems of dialectical arithmetic, that arise, as opposition, in step 1, and, as resolution, in step 2, of the Seldonian dialectical presentation entitled 'The Dialectic of the Seldonin Dialectical Arithmetics' -- as modeled by the NQ_ 'dialectical meta-equation meta-model' --

#_)-|-(s#_   =   ( #_N_ )^( 2^s#_ )

-- wherein '#_' connotes the '''domain''' of the Seldonian dialectical arithmetics.

A Note about the On-Line Availability of Definitions of F.E.D. Key Technical Terms.   

Definitions of Encyclopedia Dialectica technical terms, including of E.D. ‘neologia’, are available on-line via the following URLs --

-- by clicking on the links associated with each such term, listed, in alphabetic order, on the web-pages linked-to above.  

 Links to definitions of the Encyclopedia Dialectica special terms most fundamental to this vignette are as follows --

'arithmetical ontological "pure" qualifiers'

'arithmetical "pure" quantifiers'






'''Historical or Diachronic Dialectics'''

N first "standard" arithmetic

NQ_ 'first dialectical arithmetic'

'''Seldon Function'''

'supplementary opposition'

'''Systematic or Synchronic Dialectics'''

-- and we plan to expand these public definitions resources as the Encyclopedia Dialectica  Dictionary Project unfolds.

[Note:  ‘‘‘Arithmetical Pure Quantifiers’’’ vs. ‘Arithmetical Pure Qualifiers’.   

In the phrase “3 apples”, we term “3” the “arithmetical [“pure”-]quantifier”, and we term “apples” the ‘‘‘ontological’’’ -- or kind of thing -- ‘‘‘qualifier’’’.   

In the phrase “3 pounds of apples”, we term “pounds” the metrical[-unit] qualifier’ -- or ‘‘‘unit of measure qualifier’’’ -- quantified by the 3, and such that, together, that '[metrical] quantifier' & / "times" that 'metrical qualifier', jointly, and '''metrically''', quanto-qualify, or qualo-quantify’, the ontological qualifier’, “apples”.   

A key use-value of the dialectical arithmetics is to provide shorthand, algorithmic, ideographical-symbolic systems for the various kinds of ‘arithmetical qualifiers’, both with and without the co-presence of ‘‘‘arithmetical quantifiers’’’, i.e., in the form of an intermixed dialectical progression of both increasingly rich qualifier-only’ dialectical arithmetics, and also of increasingly rich qualo-quantifier’, or quanto-qualifier’, dialectical arithmetics, alike, and in an alternating succession.].

[Note:  Ordinal Color-Coding.  In the discourse below, we apply relative, visible-light-spectrum-rising-frequency-orderordinal color coding’ to assist the reader in “keeping her/his place” among the multiple, kaleidoscopically-intertwining sequences and series that constitute the ‘content-structure’ of that discourse:  color-code red[-ish] for the first in any such sequence or series, color-code orange for the second in any such sequence or series, color-code yellow[-ish] for the third in any such sequence or series, color-code green for the fourth in any such sequence or series, color-code blue for the fifth in any such sequence or series, color-code indigo for the sixth in any such sequence or series, color-code violet for the seventh in any such sequence or series, and color-code '''ultra-violet''' [as in "black light"] for any further entries in any such order/sequence/succession/series/progression.  The well-known 'mnemonogram' "Roy G. Biv" may prove useful to the reader in tracking this color-coding.].     

Background.  We use the symbol N_, with "double underscore"*, to denote the axioms-system of the so-called "Natural" numbers, when restricted to the axioms thereof that express its nature only in "first-order" logic:  principally the first four Peano axioms. 

*[because "double underscores", or "double underlines", with one underline directly beneath the other, are not available in the local typography, we use instead, herein, an extra underline, horizontally next to the primary underline, to signify a "double underline", e.g., 'a' is, of course, 'a' with a "single underscore", whereas 'a_' signifies 'a' with a "double underscore".].

That first-order system is richer, in "models", by far than the "higher-order", more axioms-rich axioms-system of the "standard" "Natural" numbers, for which we use the symbol N, with its additional axioms, including at least one axiom expressed in second-order logic, such as Peano's fifth axiom, the so-called '''[arithmetical] induction''' axiom. 

The "single underscore" of the latter symbol, in contrast to the "double underscore" of the former, serves to remind us of the far richer nature of N_ with respect to that of N.

That "first-order" axioms system, N_, is so rich that it includes "non-standard" models of the "Natural" numbers.

One of those "non-standard models" is an extreme qualitative opposite of/to N_ as "standardly" interpreted -- i.e., as interpreted to be N.

That '''diametrically opposite''' system to N as N_ is NQ_, the Seldonian 'First Dialectical Arithmetic'.

The NQ_ system is not only a '''diametric opposite'''' of the N as N_ system; it is also its '''dialectical opposite'''.

The NQ_ system is the first "antithesis" system to the N_  first, or «arché», "thesis" system, as generated by the Seldonian dialectical-mathematical 'meta-model' equation of the F.E.D., dialectical, systematic method of presentation of the Seldonian systems of dialectical arithmetic, that is, of the presentation that is entitled --
The Dialectic of the Seldonian Dialectical Arithmetics' 

-- an equation which is itself also written in the NQ_ algebra.
The term N_ and the term NQ_ [thus] form the first two terms of the systems-progression of 'The Dialectic of the Seldonian Dialectical Arithmetics', such that they are '''non-amalgamatively summed''' [cf. Musès] together, and also 'antagonistically summed' together [a mental action denoted, herein, by the sign '~+~', which might well be read-off out loud as "versus" [vs.], or, more accurately, as "and-versus" [&-vs.], or as "and|versus" [&|vs.]], as that special case of the generic-dialectical, step one, 'thesis-system ~+~ antithesis-system' sum, that constitutes the result of step 1 of the Seldonian dialectical-mathematical 'meta-model' equation that encodes the presentation of that title. 

The step s = 1 calculation and solution of that 'meta-model' equation yields, in shorthand form, as the content to be presented as step one of that presentation --  

N_ ~+~ NQ_ 

-- an 'antithesis-sum' so far barren of the explicit 'present-ation' of any explicit synthesis.

The focus of this Vignette is not the full axioms-systems-progression of 'The Dialectic of the Seldonian Dialectical Arithmetics', for most of the hundreds of its steps that have so far been explored by the Foundation research collective. 

That is a task which has been slated, by the F.E.D. General Council, for a different "place" and time.

Our main focus herein is in step one only; is the elucidation of the meaning of this first, thesis/antithesis 'cumulum' of that progression, N_ ~+~ NQ_, as of the dialectical, 'supplementary opposition' ['~'] that grounds it:  N_ ~ NQ_.

In terms of the generic 'First Dialectical Arithmetic', the axioms-system denoted by N_ in this 'meta-model' is a specific interpretation [which we "sign" by '[---)'] of the generic NQ_ 'dialector' q1:   

N_ [---) q1. 

The axioms-system denoted by NQ_ in this 'meta-model' is a specific interpretation / solution for the generic NQ_ 'dialector' denoted by q1+1  =   q2:  NQ_ [---) q2. 

The numerals {1, 2, 3, ... } that form the core of N_, are collectively denoted by just N, with no underscore(s).   

Similarly, we denote the 'meta-numerals' core of the NQ_ system by NQ, which removes the "single underscore" of the axioms-system symbol from under the 'N' 'pre-subscript-level symbol', yielding just 'N' instead, and removes the "double underscore" of the 'Q_' 'script-level symbol', and reduces it to / replaces it with a "single underscore", yielding just 'Q' instead .

The Nature of the Opposition.  The opposition between the N_ [---) q1 axioms-system of the 'first standard arithmetic', and the NQ_ [---) q2 axioms-system of 'non-standard first dialectical arithmetic', is not any purportedly un-dialectical, "absolute" opposition; is not any supposed radical dualism, forever incapable of dialectical synthesis, of practical and theoretical transcendence, of conceptual reconciliation, of '''complex unification'''.

That opposition is, on the contrary, a reconcilable opposition:  as we said above, it is a dialectical opposition.

The "pure" '''quantifiers''' of the N_ system, and the "pure" '''ontological qualifiers''' of the NQ_ system, are not actually "pure", i.e., are not devoid of any implicit '''overlap''' with, or inter-mixture of, the other.

Said another way, the "pure" arithmetical quantifiers of the N_ system, and the "pure" arithmetical ontological qualifiers of the NQ_ system, are not "absolutely" distinct and dirempt, the one having "absolutely" no internal connexion with/to the other.

The N_ system implicitly already contains the seeds of the NQ_ system.

We know this from our prior empirical experience of / '''experiments''' with that which is the totality of reference for this entire discourse, namely, with the 'Human Phenome', and, in particular, with its core component, our own "Natural" language, where various varieties of '''qualifiers''', even more so than various varieties of  '''quantifiers''', are found to figure prominently, and, more generally, with other "Natural" languages, similarly populated with '''qualifiers''' relative to '''quantifiers''', as well as with various "artificial" languages, '''engineered''' languages, deliberately, consciously designed languages, such as symbolic logic, calculus, and algebra, as well as ordinary arithmetic, which have, in the past, typically featured only 'mathematical quantifiers', but some of which also feature 'mathematical qualifiers', obviously so or not.

The "artificial languages" of arithmetic, of its algebra, and of mathematics more generally, cannot fulfill their implicit mission -- that of re-encoding human "Natural" language(s) so as to gain all of the advantages of ideographical, algorithmic regularity, and "shorthand-like" concision -- unless they arise to full ideographical and algorithmic translation of the '''qualifiers''', as well as of the '''quantifiers''', of "Natural" languages and of some artificial languages alike.

The NQ_ system still '''contains''' the N system, as indicated syntactically by the N 'pre-subscript', subordinated to and subsumed under and before the Q_ script-level symbol in the NQ_ symbol as a whole.
[Note:  We symbolize the Seldonian 'First Dialectical Arithmetic' by NQ_, rather than by N_Q_, with a "double underscore" for both the Q and the N, viz., 'N_' next to 'Q_', because the addition rules of the explicitly cardinal arithmetic of the [second-order] N axioms-system, not explicit in the [first-order] N_ axioms-system,  control the 'subscript-arithmetic' of the sole NQ_ operation, that of 'dialectical multiplication', i.e., of the «aufheben» operation. 

'Dialectical addition' "operations" in NQ_ really represent a 'non-operation' option; the '''peaceful coexistence''' of distinct 'ordinal, ontological qualifiers' without the mutual interaction of 'dialectical, mutual-«aufheben», ontological multiplication'].

The 'meta-number' set, or "space", of the NQ_ axioms-system is denoted by --

NQ    =   { q1, q2, q3, . . . }    =   { qn }, for all n in N

-- by which we can see that each qn still '''contains''' / conserves n, for all n in N, but from which we can also see that the 'script-level', 'q' aspect of 'qn' -- each of these 'meta-number's meta-numerals' as a whole, subordinates and subsumes the 'post-subscript'  level, 'n' aspect of those 'meta-numbers / meta-numerals'; that the 'q' aspect supplements and supersedes / exceeds n, and, thus, supersedes / exceeds N as well.

It is, indeed, this dialectical, 'self-«aufheben» self-subsumption' of the ordinal kind of 'arithmetical quantity, i.e., of the N of N_ -- the kind of quantity that characterizes the first order N_ Peano-axioms system -- that makes explicit the hitherto only implicit ingredience of 'arithmetical ordinal quality' in "Natural" arithmetic -- the kind of quality that characterizes the also-first-order NQ_ system. 

[Note:  The 'reflexive prefix', 'self-', here, refers to the logical, conceptual action of the human subject/agent whose mind embodies the N_ system, and also then critically reflects upon that system -- thus critically reflecting upon itself as embodying that N_ system -- a mental operation which we denote by the 'dialectical equation-definition' N_(N_) = ~(N_), wherein '~(N_)' denotes the operation of '''dialectical self-negation''' of N_, i.e., 'self-«aufheben» self-negation/self-critique', of the N_ system, by the N_ system, as that system is 'mentally embodied' by a cognizing human subject/agent].

Indeed, NQ_ is born out of the womb of its predecessor-opposite, N_; N_ gives birth to its successor, 'supplementary opposite', NQ_.

The NQ_ system represents what the N_ system lacks, by virtue of N_ containing only implicitly, covertly, what NQ_ contains explicitly and overtly:  'arithmetical ontological qualifier operators'.

But the NQ_ system does so at the cost of an alternative absence, of a contrary inadequacy, of a counter-balancing incompleteness, of a mirror-image imperfection, of a symmetric insufficiency -- of a 're-implicitization', of a 're-covert-ization' -- in relation to the key deficiency of system N_:  System NQ_ lacks the 'arithmetical quantifier operators' of which system N_ principally consists -- of which N_, in an overt sense, exclusively consists.

The N_ ~ NQ_ '''dialectical opposition''' belongs to that «species»-category of the systematic dialectic of opposition-in-general -- of the 'dialectical speciation of the «genos»-category of opposition' -- that we term 'supplementary opposition', or 'progressive opposition', which is this dialectic's 'dialectical synthesis species-category', combining its 'dialectical thesis species-category', that we term 'complementary opposition', or 'symbiotic opposition', with its 'dialectical antithesis species-category', that we term 'annihilatory opposition', or 'parasitic opposition' [for more regarding this foundational dialectic, please see -- link ].

The second system in the dialectical systems-progression of 'The Dialectic of the Seldonian Dialectical Arithmetics', namely NQ_, thus exposes explicitly the inadequacy of a system -- of the first system, N_ -- of arithmetical ideographical "artificial" language, built upon quantifiers, and capable of overtly expressing only quantifiers, such that NQ_ explicitly exposes that inadequacy of N_ by embodying its critique of N_  positively, by actually exhibiting a counter-example [cf. Imre Lakatos] to N_, an alternative to N_, a system which lacks N_'s primary lack.

However, by the same token, the NQ_ system, as a system capable of overtly expressing only qualifiers, incapable of explicitly expressing quantifiers, thereby exposes its own, complementary inadequacy, taking N_ initially as, apparently, the only available counter-example to its own, also deficient, counter-example; as the only available alternative to that, also deficient, alternative to N_ which NQ_ is.

Thus, at first, we seem stuck in a symmetry of insufficiencies, an impasse of contrary imperfections, a 'pendulation' back-and-forth between two alternatives whose diametrically opposite deficiencies each drive this Sisyphean circulation, away from each's own deficiency, toward the counter-defect of its other, whose '''other''' deficiency similarly drives each back again to itself, and so on . . ..

Of course, the way out, the consequence and result -- and the full fruition -- of this dialectical opposition, and of this 'pendular' alternation, between N_ and NQ_, is the irruption of the conception of a third system in this series/progression, the irruption of their 'uni-system', NU_, which seemingly so seamlessly combines the major features of the N_ system and of the NQ_ system, that it makes us feel, in our 'conceptual perception', as if these features should never have been so '''unNaturally''' separated in the first place, and which, in particular, features both 'arithmetical qualifiers' and 'arithmetical quantifiers' in a "Natural" unity, in a harmoniously integrated mutual necessity and mutual utility.

But before we turn to any focused consideration of this synthesis-system, born out of the inadequacies of the opposition of/between N_ and NQ_, let us first consider, with greater scrutiny, that opposition itself, and consider also how that opposition itself is born -- how N_ gives birth to N_ ~ NQ_, only thereafter moving on to consider how N_ ~ NQ_ gives birth to NU_.

On the Nature of the Presentational Transition from N_ to N_ ~+~ NQ_:  The Dialectic of the Engenderment / Construction of the NQ from / "by" the N.  At the outset of his lectures on his dialectical «Logik», Hegel spelled out, most succinctly, his generic '''algorithm''' for dialectic(s), which he applied throughout his encyclopedic opus on his "Philosophical Sciences", his dialectical "Philosophical Theory of Everything", encompassing his '''Philosophical Science''' of ConceptsLogik»], his "'Philosophical Science''' of 'exo-human' «Natur», and his '''Philosophical Science''' of 'Spirit-ed' Beings, of consciously-creative human beingsGeist»], in the following terms, and in which I have numbered Hegel's descriptive passages, below, for greater ease-of-reference further-on -- 

[1]:  "The first determination [A.D.: speci»-fication'] is immediate, while the second one constitutes the sphere posited in its differentiation from the first."

[2]:  "Within every simple first determination, [e.g., ground,] what is determinately different from it[, e.g., the consequence of the ground] is at once also present, but is at first present without yet being explicitly posited."

[3]:  "In the second determination, finitude [and with it contradiction] again enters."

[4]:  "The third determination is the unity of the first and the second, in which the contradiction is resolved. ..."

[5]:  "The progression is as follows.  The beginning is simple, immediate.  ...  Every newly emerging concept is more concretely determinate than its predecessor."

[6]:  "We are always carrying everything that went before along with ourselves into what is new, but everything prior is, within what is new, put in its determinate place." [A.D.:  the foregoing is an assertion, by Hegel, of what we of F.E.D. describe as the '«aufheben» evoluteness' of dialectic].

[7]:  "Whereas, in what preceded, each [momentarily immediate] determination ... passed as ultimate, it is now demoted into being only a moment ... ."

[G.W.F. Hegel, Lectures on Logic, Clark Butler, translator, Introduction to the Lectures on Logic, [More Exact Concept and] Division of the [Science of] Logic, [I. Being], Indiana University Press [Indianapolis:  2008], pages 79-80, italic, underline, bold, shadow, and colored emphasis added by A.D.].

The seven 'descriptory' precepts of dialectics reproduced above turn out to have been instantiated in the -- opening -- portion of 'The Dialectic of the Seldonian Dialectical Arithmetics' that we scrutinize herein, as noted below:

Regarding precept [1]:  In 'The Dialectic of the Seldonian Dialectical Arithmetics', N_ constitutes the "first determination", NQ_ the "second determination".   

Within the present-day 'Human Phenome', from practical, everyday use of ordinary arithmetic, and especially for those schooled at all in mathematical theory, the system of "Natural" arithmetic, which we shorthand by N_, has, precisely, '''immediate''' significance and familiarity.   

The NQ_ system/ideo-«species», on the contrary, can come to be known and familiar to any of us only by our first grappling with its differentiation from -- its "«differentia specifica»" with respect to -- the N_ system/ideo-«species».

Regarding precept [2]:  The NQ_ system is already implicitly present with and in the N_ system, via the latter's internal relation to the totality of natural language(s), at the core of the 'Human Phenome'. 

There is already a 'penumbra of knowing', widespread within present-day global, Terran humanity, within the present 'Human Phenome', in the present human 'near-consciousness', that, generically, the "Natural" numbers are "abstract" relative to natural languages, if not specifically that they abstract from the implicit '''qualifiers''' that those abstract quantifiers implicitly '''modify''', and that there is something vaguely deficient in this 'elision of the qualifiers' as instantiated in that system of "Natural" arithmetic. 

Seldon's discoveries, as recounted herein, are an essay at raising this 'near-consciousness' into explicit consciousness, or, rather, are a distillation of what constituted his individual path from that 'near-consciousness' into more explicit consciousness of this key, core aspect of 'The Modern Ideology'; the modern 'Human Phenome'; of human minds permeated by the 'exchange-value-exchange' paradigm; of the one-sided 'quant' «mentalité», semi-consciously mired and marooned in the meme of "'The Elementary Form of Commodity Value''' -- via their incessant, intensive daily practice of '''the law of [capital-]value''', however unconsciously so as such.  

[For a deeply insightful exploration of the formation of the "Modern" -- i.e., of the 'capital-value-centric' -- «mentalité», as the most recent '''instar''' in the [psycho]historical development of the 'Human Phenome', an exploration that, unfortunately, deprives itself of even deeper -- Marxian -- insights, see Alfred W. Crosby, The Measure of RealityQuantification and Western Society, 1250-1600, Cambridge University Press [NY: 1997]].

Regarding precept [3]:  Rising from N_ to N_ ~ NQ_, '''dialectical contradiction''' enters explicitly, via '''dialectical opposition''', '~'.

Regarding precept [4]:  The mutual '''dialectical opposition''' of NQ_ & N_ is resolved in NU_, which is the unity of NQ_ & N_, as we shall see, below.

Regarding precept [5]:  The N_ system seems simple, and immediately familiar.   

The concept of the NQ_ system, which '''follows [from]''' the concept of the N_ system, is more determinate, is richer in determinations, or in '''specifications''', than is the concept of the N_ system, at minimum because, on its face, the NQ_ system combines old determinations, of the 'N' quantifiers, with new determinations, of the 'Q_' [ordinal, ontological] qualifiers.

Regarding precept [6]:  This '''evoluteness''' of dialectic is the reason that the NQ_ axioms are designed to result in a dialectic series-progression of form -- 

N_        ---)       

 N_ ~+~ NQ_        ---)       

 N_ ~+~ NQ_ ~+~ NU_ ...

-- for the '[Dyadic] Seldon Functions', rather than in a mere sequence succession of the form --

N_       ---)    

 NQ_  ---)                           


The generic dialectical movement, in both its Hegelian account, and in its Seldonian account, is thus not --
thesis ---) antithesis ---) synthesis

-- but is, on the contrary --

thesis    ---)    thesis ~+~ antithesis    ---)    thesis ~+~ antithesis ~+~ synthesis ....

For example, in the cognitive domain, in reading a book, or in auditing a lecture, the '''passed''' chapters-content of that book, the '''passed''' and past passages of that lecture, do not '''convolutely''' vanish completely from mind once they have passed into our past:  they 'cumulate' in the mind, even if they fade in vividness with respect to present reading or hearing, as also with respect to also-past, but less-past passed chapters and passages.  

In the physical domain, all atoms do not vanish out of all independent existence once the first molecules form, nor do independent molecules cease to exist once the first "prokaryotic" cells form, and so on:  cosmological [meta-]evolution is also '''evolute''', not '''conolute''' [in the main].  

As a whole, cosmological [meta-]evolution is 'covolute', invoking an E.D. term, 'covolution', which signifies a dialectical synthesis of '''evolution''' and '''convolution''', encompassing the potential '''extinctions''' of some 'physio-ontological' categories' content, together with the 'double-«aufheben»' conservation-into-the-present of most.

Regarding precept [7]:  In step 0 of 'The Dialectic of the Seldonian Dialectical Arithmetics's pre-programmed, 'algorithmitized' dialectical method of exposition, the N_ system stands alone, and it appears, '''momentarily''', to constitute the ultimate primary system of arithmetic.   

But the immanent, internal inadequacies of the N_ system break it out into two, into the first possible cumulum, the two-system cumulum, of this dialectical progression of systems of dialectical arithmetic:  into the N_ ~+~ NQ_ cumulum. 

In that process, N_ is indeed "demoted" from its former seeming 'ultimaticity', as signified by the lowered, subordinate, subsumed, 'pre-subscript' position in[to] which it is explicitly conserved and consigned in the symbol NQ_, in the 'N' component of that compound symbol.   

But it is also '''promoted''', to become the 'Q_' component of that symbol as a whole, NQ_. 

Here we must register a critical note with regard to Hegel's account of the generic dialectical process, as reproduced above.   

The colloquial German term «aufheben», which, in Hegel's usage, becomes the German name for the core universal characteristic of all dialectical process, has three -- even concurrently-applicable -- '''moments''' to its meaning, not just the two that Hegel tends to emphasize:  (1) to negate; (2) to elevate, and;  (3) to conserve.

Hegel tends to emphasize only moments (1) and (3).  

The '''negation moment''' of the «aufheben» movement is related to the '"de-motion from ultimaticity"' of which Hegel spoke.   

The '''elevation moment''' of the «aufheben» movement is related to the '''pro-motion moment''' of the dialectic which we just noted above.   

The '''conservation moment''' of the «aufheben» movement is ingredient in both the '"de-motion''' and the '''pro-motion''' effects.   

Hegel often emphasizes only the '''negation moment''' and the  '''conservation moment''' of the «aufheben» movement, even altogether omitting to mention the '''elevation''' aspect.   

We see this as a mistake.

For example, we account for the formation of the NQ from the N as follows. 

The first four, first order Peano postulates that form the core of the N_ axioms-system, characterize the "Natural" numbers most explicitly as "ordinal numbers". 

The 'intra-dual', 'co-character' of the N as, '''simultaneously''', both "ordinal numbers" and "cardinal numbers" is only implicit in the N of N_.

The set or space, N of N_ is defined to be N =  {1, 2, 3, ... }, and thus, as a set, consists of a disordered plethora of instances of "ordinal number", even though, in the rendering preceding, we have excerpted from that -- order-less -- set to present three of its members in their ordinal order. 

But what is 'ordinal-number-in-general'?  

What is the «ideo-genos» of "ordinal number", implicit in that jumble of its instances, N? 

It is the general category of '''ordinal quantity'''. 

And, implicit in the inherent ambiguity of the concept of "'ordinal-quantity-in-general''' is the concept of 'ordinal-quality-in-general'. 

Implicit in the generality of the quantitative idea of "first", "second", "third", etc., is the general qualitative idea of 'nth-ness', whose instances include 'the quality of first-ness', 'the quality of second-ness', 'the quality of third-ness', etc.

We already know, at least latently, from our experience of natural language(s), the totality-of-reference for this dialectic of dialectical arithmetics, that "standard" arithmetic(s) are deficient, in terms of their '''mission''' and '''manifest destiny''', in that they do not translate the '''qualifiers''' of verbal human "Natural" language into "Natural" arithmetical, 'algorithmical' ideograms.   

The subtle, evanescent, attenuated form of quality that is 'ordinal quality', or 'order quality', is the first form of quality that can [re-]enter immanently into arithmetical ideography, that can [re-]emerge ['re-', given the quite explicit presence of the «Monad», or unit, 'quantifiable arithmetical qualifier', denoted by '''Mo''' in Diophantus of Alexandria's circa 250 C.E. first precursor to modern symbolic algebra, the Arithmetica] within modern arithmetic, or, i.e., within 'algorithmic ideography'.  

This, 'the arithmetical qualitative', can so re-emerge by means of the seeds of it which are implicit in the 'ordinal quantifiers', or '''order numbers''' -- the 'consecutive-order-numbers', to express their nature more precisely -- that explictily characterize the N of N_, as opposed to the 'cardinal quantifiers' that explicitly characterize the N of N.    

These '''seeds''' need only to '''sprout''', as they did in the mind of Karl Seldon, into 'ordinal qualifiers', i.e., into arithmetical, 'algorithmical', "symbolic" 'ordinal qualifier meta-number ideograms', representing the specific qualities of order that inhere in the "ordinal numbers", or 'order-numbers', of the N of N_; in --

 1, 2, 3, . . .  

-- as 'Peanic' "ordinal numbers", or 'order-numbers'.

In 1889, Giuseppe Peano published his postulates for a deductive-logic derivation of "Natural Numbers" arithmetic, in his book The Principles of Arithmetic Presented by a New Method.  The philosopher W. L. Reese gives the original version of the four "first-order" Peano axioms as follows --

"P1. 1 is a ["Natural"] number.

P2.  The successor of any ["Natural"] number is [also] a ["Natural"] number.

P3.  No two ["Natural"] numbers have the same successor.

P4. 1 is not the successor of any ["Natural"] number."

[W. L. Reese, Dictionary of Philosophy and Religion:  Eastern and Western Thought, Humanities Press [NJ:  1980], p. 418,
square-bracket insertions and color-coded, bold-face, shadowed numeral 1 added by A.D.].

Can you perceive, directly inspecting these 4 axiom-sentences, that, yes, they encode the N, but also more than the N?

At least three theorems, all "completed" in the 1930s, provide formal proofs implying that the "standard model" and "non-standard models" of these Peano-axioms' "Natural" Numbers arithmetic are inextricably coexistent with one another at the level of the first order axiomatization of arithmetics, even though these three theorems do not immanently provide a method for the construction of the "non-standard models" that they herald.

The Löwenheim-Skolem Theorem, by itself alone, implies that "non-standard models" of N_ inhere in the first-order "standard model" of "Natural" arithmetic.

The Gödel Completeness Theorem, which provides proof of the semantical completeness of the first-order N_ arithmetic, given its proof of the semantic completeness of ideographical first-order predicate logic itself, in conjunction with the Gödel Incompleteness Theorem, which establishes the syntactic incompleteness of first-order N_ arithmetic, and the semantic and syntactic incompleteness of all possible higher-than-first-order axiomatizations of the N arithmetic, together imply the inextricable coexistence of "standard" and "non-standard" models of the Peano "Natural" Numbers arithmetic at the level of its first order axiomatization.

Now, on their face, these four Peano postulates describe the '''order properties''' -- the '''ordinality''' -- of a sequence of entities that they name ["Natural"] numbers.   

But nothing of the cardinal quantity, or '''manyness''', features of the standard "Natural" Numbers are explicit in these four, "first order" sentence-axioms.

These four axioms are said to be "first order" because they make assertions about only individual "[Natural] numbers", and make no assertions about qualities shared by groups of these "logical individuals" / "numbers", groups treated as '[meta-]individuals' in their own right -- assertions which would be termed "second order".

On the very face of them, these four, first order Peano postulates support also "non-standard models" of the "Natural" numbers.   

That is, they potentially describe also, besides the N of N_, sequences of [idea-]objects / of "logical individuals" which are not "Natural" numbers, including sequences of mutually qualitatively different, yet related, such objects, which are not "numbers" at all, in any "standard" sense of that term, given that we discard any presumption of '''actual infinitude''' for the counts of the entities in such sequences, "standard" and "non-standard" alike.

The "standard" "Natural" numbers do not form an actually infinite sequence in any practical, empirical sense, nor can any actually infinite number of "Natural" numbers be actually constructed, step-by-step, by actual mathematicians, given the nature of physical possibility. 

Practically speaking, the "Natural Numbers" are a finite sequence, though their effective upper bound "Natural" Number is not always the same "Natural" number in every practical context.   

E.g., if I am using a digital computer to perform arithmetic, then the practical finite limit of the "Natural Numbers" for me, in that context, is the highest "Natural" number which that computer can encode and express, given its multi-byte "word-size".

If we drop the '''hangover''' or '''overhang''' of ancient Platonic mysticism that still plagues modern mathematics, as we should, and as Seldon does, in his dialectical, immanent critique of the "standard" arithmetics, and, in particular, if we drop the 'contra-empirical', and thus anti-scientific, assertion of "actually-existing" infinities, such as the supposed Cantorian aleph-null actual infinity of the "Natural" Numbers, then we find many commonplace, "prosaic" examples of "non-standard models" of the Peano first-order "Natural" "numbers".  

 [For a far-reaching critique of the prevailing, pseudo-scientific academic religion of 'infinitist' mysticism in contemporary mathematics, see Brian Rotman, Ad Infinitum: The Ghost in Turing's Machine -- Taking God Out of Mathematics and Putting the Body Back In, Stanford University Press [Standford, CA.:  1993]].

For example, the sequence of the letters of the English alphabet, in their "alphabetical order", fully satisfy the 'finitistic' first-order Peano axioms [e.g., provided we assert the phrase "except for the last ["Natural"] number" at [the] appropriate place(s) in the Peano axioms], even though they are not "numbers" at all, in any "standard" sense.  Just substitute the word "letter" for the word "number", and the symbol 'a' for the ideogramic symbol '1', in a fully finitistic version of Peano axioms given above.

So does the sequence of the month-names of a given year -- just substitute the word "month" for "number", and the phonogramic symbol "January" for the ideogramic symbol '1', in the Peano axioms given above.

So do the day-names for a given week -- just substitute the phrase "day of the week" for the word "number", and the phonogramic symbol "Monday" for the ideogramic symbol '1', in a fully finitistic version of the Peano axioms given above. 

So do the names of the "taxa" of the scheme of biological taxonomy, or of biological "systematics", e.g., in their systematic order of "descending" inclusiveness:  Domain, Kingdom, Phylum, Class, Order, Family, Genus, and Species, 'mnemonogram' DKPCOFGS via "Do Kings Play Chess On Fine Glass Sets?" [just substitute the word "taxon", or "taxa", for the word "number", and the phonogramic symbol "Domain" for the ideogramic symbol '1', in a fully finitistic version of the Peano axioms given above].

And so do the 'ordinal qualifier meta-numbers' of the NQ space of the NQ_ axioms-system for dialectical arithmetic:  another "non-standard model of the Natural Numbers":  this time, a generic one, rather than a specific one, such as is each of the earlier examples above.

Indeed, the first-order Peano axioms, purged of any fanciful infinitism, apply to the whole domain of what Seldon calls, not "archeonic continua", but 'archeonic consecua' -- the entire realm of possible finite/realistic sequences of [ev]entities, such that each such sequence exhibits an "«arché»", e.g., an '''ultimate ancestor''', a 'starting [ev]entity', but which may, at least in some cases, exhibit no pre-ordained 'stopping [ev]entity', and which may thus constitute never "actually infinite sequences", but ever only "potentially infinite" sequences, in Aristotle's sense.

'Ordinal qualifiers', ordinal categories represented as/by 'ordinal qualifier meta-numbers', can thus fulfill the first four, finitistic "first order" axioms of N_ just as fully as do the 'ordinal quantifier-numbers' of the N of  N_ --

N   =   {1, 2, 3, ... }.

The NQ 'meta-numbers' of the NQ_ dialectical arithmetic are 'axiomatically-engineered' to generically encompass all cases of realistic, '''finitistic''' 'archeonic consecua' whose sequential [ev]entities are not "standard Natural Numbers", but, rather, are '''non-numbers''', '''non-pure-quantitative''', '''qualitative [ev]entities''', like the letters of an alphabet, the months of a year, the days of a week, the 'taxons'/taxa/categories of biological systematics -- or the pedagogical sequence-of-presentation of the dialectical progression of the axioms-systems of the standard arithmetics  --

'The Goedelian Dialectic of the Standard Arithmetics' -- 

 N, W, Z, Q, R, C, H, O, . . ..

[ for more regarding this latter example, see:  link_1; link_2, link_3 ].

The NQ 'meta-numbers' of the NQ_ dialectical arithmetic are 'axiomatically-engineered' to also generically encompass -- to reconstruct the pasts of, and to 'pre-construct' the futures of -- cases like the historical sequence of the forms of human social formation, like the historical sequence of the predominant human-social "relations of production" [cf. Marx], like the historical sequence of the predominant human-social "forces of production" [cf. Marx], like the historical sequence of emergence of the principal fields of human ideology/knowledge, and like the cosmological historical sequence-of-appearance/-"population" of the primary categories of the physical ontology of the cosmos, thus constituting the Seldonian 'Dialectical Theory of Everything'

[ for more regarding these other examples, see:  formations; relations; forces; ideologies/knowledges; cosmos ].

Our '''self-reflection''' of N_ in the context of "natural language(s)" as totality-of-reference (1) 'conceptually perceives' the lack -- or the [psycho]historical loss, in 'post-Diophantine', post-Medieval Modernity --  of 'arithmetical, algorithmic qualifiers', as a lacking of and within "standard"-interpretation N_, and (2) 'conceptually sees' the seeds, the "wherewithal", for the redress of that lacking as residing also immanently within N_ itself, in the inherent ambiguity, and 'intra-duality', of quantitative versus qualitative ordinality, and, as a combined result of those two 'conceptual perceptions', (3) 'conceptually discovers/creates' the system of 'ordinal qualifiers' / of 'qualitative ordinality', which turns out to be easefully open to further interpretation into systems of 'dialectical qualifiers' -- of 'arithmetical, algorithmical, ideographical dialectors', i.e., the NQ_ system [for starters].   

These generic kinds of 'dialectical qualifiers' include [the «arché»-]thesis-qualifier(s), the full antithesis-qualifiers, the full synthesis qualifiers, the partial-antithesis qualifiers, & the partial-synthesis qualifiers, associated with the 'Dyadic Seldon Function' models, and/or with the 'Triadic Seldon Function' models.

Thus, we have seen above both the that, and the how, of the making explicit of this formerly implicit, '''non-standard''' concept of 'ordinal quality', or of 'qualitative ordinality', a concept that 'co-inheres' in the N of N_ together with its standard 'co-concept' of '''ordinal quantity'''/'quantitative ordinality', thereby forming, in our minds, a new space or set, which we call --

 NQ     =     { q1, q2, q3, ... }. 

This new space is formed, in our minds, by what can be viewed, e.g., either as an instances-vs.-«ideo-species» 'self-splitting', or as an «ideo-species»-vs.-«ideo-genos» 'self-splitting', of the {n}, or of the N, respectively, which thereby 'self-bifurcate' into diverse instances or species of 'ordinal [and, implicitly or explicitly, respectively, also cardinal] quantity' subscripts -- 'post-' or 'pre-', respectively -- in both cases united with a unified, specific  or generic  'ordinal quality' '''script'''-level symbol, which 'covers' all of these instances, or species, respectively, and which we denote by q, or by Q_, respectively, involving a combined, split 'self-de-motion' and 'self-pro-motion' of the {n}, or of the N, respectively:  two related '''levels''' or '''layers''' of such '''self-bifurcation''' / 'self-subsumption', each of which can be depicted via each of the following two pictographic symbolizations, respectively: 

In the first of the two illustrations above, the 1, 2, 3, . . . that constitute the N 'self-demote', in[to] «aufheben»-conserved specificity, to form the plural post-subscripts of the NQ -- of the q1, q2, q3, . . ..  

But the 1, 2, 3, ... also 'self-promote', in[to «aufheben»-conserved, «aufheben»-elevated greater] generality, in and by forming the singular script-level q, and Q, which denote the «ideo-genos» of 'ordinal quality' -- 'ordinal-quality-in-general'. 

As a result, each member of the new space, or set, NQ of 'specific ordinal quality-denoting symbols' -- of symbols which are 'meta-numerals' with respect to the N "numerals" -- and which can be expressed generically by the '''algebraic''' symbol qn, where n is a[ny] member of N, is a combination of two symbolic elements or determinants:  (1) at its 'script level', it contains the generic symbolic element 'q', which stands for 'ordinal-quality-in-general', and (2) at its '[post-]subscript level', it contains a 'specific' value of the 'specific symbolic element 'n', e.g., 1, 2, or 3, . . ., as a 'specifier' which 'specifies the 'specific ordinal quality for which qn stands.   

E.g., q1 stands for any first category, or first system, etc., in any sequence, succession, or progression of categories, or of systems, etc., while q2 stands for any second category, or second system, etc., in any sequence, succession, or progression of categories, or of systems, etc., whereas q3 stands for any third category, or third system, etc., in any sequence, succession, or progression of categories, or of systems, etc., etc.   

We can compare/contrast/relate this NQ space of 'qualitative ordinalities' to the N of N_ space of 'quantitative ordinalities' from which it, per the above, arises, as follows, using '[---]' as the 'generic interpretation / assignment' symbol -- 

'Quantitative Ordinality':  First [---] 1st; Second [---] 2nd; Third [---] 3rd, etc. ...;

'Qualitative Ordinality':  1st-ness [---] q1; 2nd-ness [---] q2; 3rd-ness [---] q3, etc. ...;

This view implicitly holds that the 'ordinal qualifier' categories of generic 'first-ness', 'second-ness', and 'third-ness', etc., have a generic, universal, empirical, scientific meaning, to a degree that the potential 'cardinal qualifier' categories of generic 'many- ness', generic 'one-ness', 'two-ness', and 'three-ness', etc., do not have in any way.   

The further -- dialectical -- interpretation of these generic 'ordinal qualifier' category-symbols, e.g. --  

1st-thesis [---) q1;

1st-antithesis [---) q2;  

 1st-synthesis [---) q3 

-- as per the Dyadic and Triadic '''Seldon Functions''' alike, constitutes a second layer of their Seldonian interpretation:  the explicitly dialectical layer of that interpretation.

Before their construction as explicitly 'dialectical qualifiers', the qn are first constructed as 'ordinal qualifiers'

Thus, we are holding that --

N of N_        =  { first, second, third, . . . } [standard interpretation]

-- whereas --

NQ of  NQ_  =  { 'first-ness', 'second-ness', 'third-ness', . . . } 

-- so that -- 'ordinal quantifier'

first [---] 1st  = 'ordinal quantifier' number 1, element of N of N_;

second [---] 2nd  = 'ordinal quantifier' number 2, element of N of N_; 

third [---] 3rd  = 'ordinal quantifier' number 3, element of N of N_ ... 

-- whereas --

1st-ness [---] q1, 'ordinal qualifier meta-number', element of NQ of  NQ_;

2nd-ness [---] q2, 'ordinal qualifier meta-number', element of NQ of  NQ_;

3rd-ness [---] q3, 'ordinal qualifier meta-number', element of NQ of  NQ_ ....

In, for example, 'q2' we have thus arrived at a generic symbol, a symbol for an 'ideo-«genos»', namely, the symbol 'q', standing by/standing next to, but also standing apart from -- in this notational convention, standing above -- a specific symbol, a symbol for an 'ideo-«species»', namely, the symbol '2'.  

An 'ideo-«genos» signifier' standing over and above an 'ideo-«species» signifier' -- an 'ideo-«genos» category-unit' "'over'" an 'ideo-«species» category-unit'; a «genos» next to / "'over'"  a «species».   

Ring any bells?   

For us, this rings out resoundingly about Plato's «arithmoi eide-tikoi», a Greek term which translates to the English 'assemblages of eide-units', i.e., of 'idea-units', of class[ification]-units, of category-units, or, which translates, "equivalently", to the English '''numbers of ideas''' -- the very heart of Plato's, originating Occidental  version of «dialektikê».   

Remember how Plato describes «dialektikê», later on in the progression of his dialogues --

"        STRANGER:  Well, now that we have agreed that the kinds [the «gene» -- A.D.] stand toward one another in the same way as regards blending, is not some science needed as a guide on the voyage of discourse, if one is to succeed in pointing out which kinds are consonant, and which are incompatible with one another – also, whether there are certain kinds that pervade them all and connect them so that they can blend, and again, where there are divisions [separations], whether there are certain others that traverse wholes and are responsible for the division?

         THEAETETUS:  Surely some science is needed – perhaps the most important of all.

         STRANGER:  And what name shall we give to this science?  Or – good gracious, Theaetetus, have we stumbled unawares upon the free man’s knowledge and, in seeking for the Sophist, chanced to find the philosopher first?

         THEAETETUS:  How do you mean?

         STRANGER:  Dividing according to kinds, not taking the same formeidos»; «idea» -- A.D.] for a different one or a different one for the same – is not that the business of the science of dialectics?

          THEAETETUS:  Yes.

          STRANGER:  And the man who can do that discerns clearly one form everywhere extended throughout many, where each one lies apart, and many forms, different from one another, embraced from without by one form, and again one form connected in a unity through many wholes, and many forms, entirely marked off apart. That means knowing how to distinguish, kind by kind, in what ways the several kinds can or cannot combine.

          THEAETETUS:  Most certainly.

          STRANGER:  And the only person, I imagine, to whom you would allow this mastery of dialectic is the pure and rightful lover [the «philo» -- A.D.] of wisdom [of the «sophia»:  of skill; deeper knowledge -- A.D.]."

[E. Hamilton, H. Cairns, editors, Plato:  The Collected Dialogues, Princeton University Press [Princeton, New Jersey:  1989], pp. 998-999, Sophist, 253b 254d, emphases added by A.D.].

By the "kinds" referenced in the above extract, let us understand the «eide-gene» categories, or «eide-hyper-species» categories, and, also, their implied "sub-kinds", or «eide-hypo-gene», their subsumed, «species» sub-categories.

Each NQ 'meta-numeral', generically denoted by qn, thus represents, in itself, a minimal -- one-«eide-genos»-over-one-«eide-species» -- «arithmos eide-tikos», or 'assemblage of categories-as-units', and of their sub-categories-as-their-sub-units. 

That is, each NQ 'meta-numeral', of the general form 'qn', for an n in N, is an assemblage of the minimum, two, idea-units required to constitute an «arithmos».   

One of the two, 'q', represents the «genos» of ordinal quality categories, which has many «species».   

This «genos» symbolic element, 'q', thus '«gene»-rizes' the symbol as a whole.   

The other of the two, 'n', represents just one particular «species» of the «genos», the "nth" «species», and thus, this other symbolic element, 'n', '''specifies''' the symbol as a whole.   

Thus, in the case of the 'meta-numeral' 'q2', the whole symbol denotes generically the 'meta-number', the 'ordinal qualifier', for the second [ev]entity in any succession of [ev]entities, i.e., for the general quality of ordinal 'second-ness'.

Thus, each NQ 'meta-numeral' -- of the general form 'qn', for some specific n in N -- "intends" an indefinitely-large assemblage of interpreted instances of the 'nth [ev]entity of a specific succession', so that each such 'meta-number' unit of NQ, namely, 'qn', given that each one of those 'qn' units is made up out of a homogeneous multiplicity of the [summed] 1 unit(s) of N -- for example,  q2 = q1+1q3 = q1+1+1, and q4 = q1+1+1+1, etc. -- so that each 'meta-number' unit of NQ, is a 'meta-unit', or 'meta-«monad»', of the ["number"] unit(s), or «monad»s, of N, except for that singular unit of NQ which represents the generic «arché», or ultimate starting-point category, q1, which "contains" not a "mulitplicity", but only a single, solitary instance, of the unit of the N, namely, 1.

The Platonian «arithmoi eidetikoi», precursors to the categories or "Concepts"Begriff»] of Hegel's «Logik», and of Hegel's System of the Philosophical Sciences entire, including the categories of Hegel's «Natur» and "Spirit", or «Geist», constitute the original stage and paradigm of «dialektikê» in the entire Occidental tradition and history of dialectic:  the «arché» form and formulation of all «dialektikê».

The Seldonian 'first dialectical arithmetic' is thus precisely a dialectical arithmetic in the following deep and historically telling sense:  each and every 'meta-numeral' of the NQ space of the NQ_ axioms-system of arithmetic for 'dialectical meta-numbers' is, by itself, already a minimized «arithmos eidetikos» dialectic [albeit a 'trans-Platonian one], a «genos»|«species», 'ideo-systematic', 'ideo-classificatory', 'ideo-taxonomic' dialectic-in-itself --

qn, n in N

-- or, using Seldon's 'qualitative fraction', 'non-amalgamative, qualitative division' notational convention --

«genos» level:                    q
 qualitative division bar:                   _______
«species» level:                  n

One further layer of interpretation of the NQ is needed to constitute them as the ideograms for a fully dialectical ideography, as the 'meta-numbers' needed for an '''arithmetic of dialectic''' -- a procedure to which we already alluded, in passing, above.   

That further layer of interpretation is the layer supplied in conjunction with the '''Seldon Functions'''.   

Both the 'Dyadic Seldon Function' and the 'Triadic Seldon Function' can apply q1, the 'ordinal qualifier' for generic 'first-ness', as the qualifier ideogram for the generic 'first thesis' ontological category, or «arché»-thesis ontological category, of dialectical categorial progressions.   

Both can also apply q2, the 'ordinal qualifier' for generic 'second-ness', as the qualifier ideogram for the generic 'first anti-thesis' ontological category, or 'first contra-thesis' ontological category, for any dialectical categorial progression.   

Likewise, both can apply q3, the 'ordinal qualifier' for generic 'third-ness', as the qualifier ideogram standing for the generic 'first syn-thesis' ontological category, or 'first uni-thesis' ontological category, for dialectical categorial progressions in general.   

And so on, for those 'higher order-quality', 'higher-ordinal' terms / 'qualifiers' -- those for which the further dialectical interpretations, per the 'Dyadic Seldon Function' versus per the 'Triadic Seldon Function', diverge.

Of course, the foregoing account of the conceptual genesis, or conscious construction, of the NQ_ system in the process of formation of the dialectical opposition between N_ and NQ_ -- N_ ~ NQ_ -- does not attempt to recount the actual "fits and starts", '''trials and errors / tribulations''', and «sturm und drang» that attended the actual, historical discovery of the NQ_, especially given the evidence that Plato's, "«asumbletoi»" ['unaddable'], «arithmoi eidetikoi» dialectic already constituted a partly still '''Parmenidean''', and 'pre-ideographical precursor' to Seldon's -- also «asumbletoi» -- 1996 discovery, occurring some ~2,400 years later.

Rather, this account presents, not the actual '''method cum accident''' of their historical discovery, but a pedagogically designed conceptual reconstruction and reproduction of that discovery, pruned, cleansed, 'parsimonized', and tidied of dead ends and set backs, etc., via the advantages of 'retro-perspective'.

The actual, vicissitudinous and somewhat convoluted course of Seldon's discovery of the NQ is recorded in the Foundation archives, in Seldon's research notes from the circa 1996 C.E. temporal locus of their first "full" discovery.   

The General Council of the Foundation is developing plans to make these research archives accessible to the public, at the psychohistorically-appointed time.

Conclusion.  At its deepest root -- the deepest root that we have so far been able to discern -- the conceptual origin of both N_ and NQ_ is ordinality, is, e.g., the consecutive order of abstract time, or of "pure" time [cf. Hamilton], of 'generic temporality', or of 'generic historicality', and the deepest root of the subtle dialectical opposition between N_ and NQ_ -- N_ ~ NQ_ -- is the internal opposition within N_ itself, is the 'intra-duality' or 'self-duality' "inside" N_, between ordinal quantities' or 'order quantities' -- 'quantitative ordinality' -- and 'ordinal qualities' or 'order qualities' -- 'qualitative ordinality', a 'self-duality' that is already implicitly present within the 'human-phenomic' concept of N_ itself. 

Given the notational convention that 'a  =| b' signifies 'a implicitly contains b', we thus have -- 

N_   =|   N_ ~ NQ_    


N_   =|   'quantitative ordinality ~ qualitative ordinality'.

Thus, the -- 

N_ ~ NQ_  

-- dialectical opposition is an 'explicitization', and an 'externalization', of a conceptual 'intra-duality', a conceptual 'self-duality', or a '''dialectical conceptual self- / internal contradiction''' [denoted by '#'] -- 

i.e., an immanent opposition, a '''self-opposition''', a 'self-antithesis', that is inherent in, or that has been '''psychohistorically''' internalized within, and that is thus internal to, and implicit in, the "lawful" '''psychohistorical''', 'human-phenomic' construct connoted by N_, even by itself.

Given the notational convention that 'a  |= b' signifies 'b implicitly contains a', we have --

'quantitative ordinality # qualitative ordinality'     |=     N_.

Fruition of the N_ ~ NQ_ Opposition in a Dialectical Synthesis System, Nq_QN     =  NU_, embodying their Complex Unity.

[Fuller Title:  Fruition of the N_ ~ NQ_ Dialectical Opposition in a Dialectical Synthesis axioms-System, Nq_QN  = Nq_U  =  NU_, embodying a Complex Unification of the NQ_ axioms-System and the N_ axioms-System]. 

As a generic 'category-level unit', or ' category as unitmonad»', each qn, especially once "interpreted" or "assigned" for a particular dialectical categorial progression, stands, implicitly, for a '[meta-]unit', or for a '[super-]unit', which is, in turn, made up out of a heterogeneous multiplicity of [sub-]units.   

These [sub-]units are not "identical" to one another -- for how could they be " identical " and still be distinct and distinguishable?  

On the contrary, these [sub-]units are, instead, mutually-similar to one another, e.g., in an 'unscaled fractal' sense.   

These (1) [sub-]units are also the base-level, or most-concrete, foundational qualitative instances -- the qualitative "logical individuals", the qualitative "members", the qualitative cases, the qualitative specimens, the qualitative examples -- of some, 'relatively' more general, "kind" [cf. Plato] -- e.g., typically, (2) of a «species», but also possibly (3) of a [super0-]«genos», or, equivalently, 'super1-«species»', or (4) of a 'super2-«species»', or, equivalently, of a 'super1-«genos»', or (5) of a 'super2-«genos»', or (6) of a 'super3-«genos»', or (7) of a 'super4-«genos»', etc. -- a plurality of base-level [sub-]units for which that category / «arithmos» ["number of units"] / '''population''' NAME, qn, stands, collectively, in a '[meta-]unitary', univocal way.

That is, a category denoted by / associated with / assigned to a given NQ «aufheben»-operator / 'meta-Natural meta-number' / 'meta-Natural meta-numeral' symbol, of the general form qn, is, in F.E.D. usage, implicitly understood to stand for an «arithmos» in something like the ancient Greek sense, i.e., an «arithmos» OF qualitative «monads», or OF qualitative units, a '''number''' OF such ultimate units ["ultimate" only relative to the universe-of-discourse in play, not in any absolute / reductionist sense].

We argue, in this section, that the mentally-perceived opposition -- in the mind(s) of the 'presentee(s) / reader(s) / thinker(s)-through of this presentation/model -- of the NQ_ axioms-system, versus/to the N_ axioms-system, once they both come into 'co-present co-existence' within such (a) perceiving mind(s) -- is typically followed, intuitively, in such (a) mind(s), by their also mentally-constructed mutual interaction, denoted by NQ_ x N_ 

The consequent operation / '''[re]flexion''', of the NQ_ system-as-«aufheben»-operator, upon the N_ system-as-«aufheben»-operand, or the unique mutual '''multiplication''', mutual '''function-ing''', or mutual «aufheben»-operation / mutual '''[de-]flexion''', specific to the NQ_ system/operator together with the N_ system/operator, in their mutual interaction with/upon one another, denoted by NQ_ x N_  =  NQ_( N_ ), includes, at the [relative] base-level for their two, respective, universes-of-discourse, the base-level mutual opposition of the 'meta-numerals' qn versus the numerals n.   

This mutual interaction, we hold, logically, intuitively, gives rise, in the perceiving human mind, to a new "kind" , i.e., to a new, third, idea / system; to a new 'ideo-ontology', of superseding, succeeding, supplementary, '''higher''' 'meta-meta-numerals' -- higher in the sense of being richer in expressive power, in capability for explicitly expressing more kinds of determinations than can either of the two previously-evoked systems / languages -- and thus 'meta-meta-numerals' which escape / transcend that [thus now former] base-level mutual opposition.

This human-mental process, whereby the mind of the 'presentor', and, if the presentation/'presentor' is successful, also the minds of the 'presentees' -- all constituting the human subjects/agents who are willingly/'will-fully' conducting these mental processes -- 'mentally-embody', and 'mentally simulate', this antithesis, this opposition of the outward meaning of NQ_, and/with/against the outward meaning of N_, and are thereby provoked to combine these two, mutually-inadequate, opposites, by their 'dialectical multiplication' / interaction / mutual «aufheben» negation, connoted by NQ_ x N_  =  NQ_( N_ ), is a human, mental process which can be illustrated, pictorially, as follows, below --


These new, higher, 'meta-meta-numerals' are of the generic form un uon, or, more simply, of the generic form unuon.   

Therein, the explicit 'x' sign for the 'generalized / [sometimes] non-amalgamative multiplication operation', herein applied in this new, non-classical system of arithmetic, is simply understood to be indicated, implicitly, by mere juxtaposition, of un and/with uon, alone  -- as, in classical algebra, for the classical multiplication operation -- wherein mere 'juxtapositioning' , without any intervening / mediating sign, e.g., an 'x' or "times" sign, of algebraic variables, e.g., with other(s) such, or with numerals / numeric constants, signifies their multiplication, all by itself. 

Together, un and uon constitute the '''complex''' units, or the '''compound''' units -- compounded of both a quantifier and a qualifier; of a quantifier '''ideo-gram-matically modifying''' a qualifier -- that constitute the new, higher, 'meta-meta-number' space which we of F.E.D. denote by NU, and of the new, higher axioms-system which we denote by Nq_QN, or by Nq_U, or by NU_.   

The 'o' "degree-sign" superscript of the uon qualitative-unit-qualifier -- which harks back to the sign for the explicit quantifiable 'Monad qualifier', written [approximately] as Mo, used in Dyophantus's circa 250 C.E. founding treatise on "symbolical" algebra, the «Arithmetica» -- signifies that this new unit-qualifier, 'uon', is an 'addable [Plato:  «sumbeltoi»], quantifiable qualifier', unlike the old 'qn', 'unquantifiable, unaddable [Plato: «asumbeltoi» ] qualifier' units of the NQ_ dialectical arithmetic.
In these new, symbolic/ideographical 'complexes'/'compounds', the uon component, or '''factor''', denotes the 'arithmetical qualifier' for the [relative] base-level qualitative units / «monads» of a given kind, of a given «arithmos», of a given ontological category, not for the category itself -- no longer for their category itself, as a unit, for which a corresponding ‘qn’ would stand.    

The un component / '''factor''' denotes the 'unit-ic monad»-ic arithmetical quantifier', one that '''modifies'''/quantifies that qualifier as a qualifier standing for a generic [relative] base-level individual [sub-]unit / [sub-monad» -- relative to the category, qn’, as their [meta-]unit / [meta-]«monad» -- by specifying the quantitative determination, the cardinal quantity determination, un, of the units-qualifier specifier/determination, uon.  That is, un is a "Natural" number '''coefficient''' of/for uon, specifying the count of units of the nth category, in a given categorial progression, that are present in the given context.

That -- now '«monad»-ic', not 'categoric', or '''categorial''' --  quantifier, un, is, in its turn, '''qualified''' by that now «monad(s)»-qualifier, uon, symmetrically and mutually, just as that quantifier, un, quantifies that qualifier, uon.  

This un quantifier does not -- uselessly -- count the ontological category, ‘qn’, itself, as the unit, which would always, invariably, result in a count of just 1, given the "idempotent" kind of addition of likes -- the 'super-amalgamative' kind of addition of likes -- that characterizes the 'unaddable' [Plato:  «asumbeltoi»] ontological-categorial qualifiers of the NQ_ axioms-system of dialectical arithmetic, given it axioms.  

Instead, the value of un represents the count of the [relative] base-level individual units of kind qn that are, e.g., present in the current step of the presentation, with 1uon  =  uon denoting a single such [relative] base-level individual unit of kind qn.

With this new, NU_, arithmetical / ideographical language, we can now explicitly translate -- into mathematical, arithmetical, 'algorithmical', 'ideogramic' shorthand -- e.g., translate English, spoken or written, multi-vocal or multi-phonogramic, vocalizations / symbols-strings, or 'multi-phonetic utterances' / 'multi-symbol-writings', such as "three apples", or such as "three pounds", into and by means of ideogramic expressions/'compound meta-numerals' of the general form [i.e., of the algebraic form] unuon. 

Thus, if u1 were to be assigned the "Natural" number arithmetical quantifier value of 3, and if uo1 were to be assigned to the 'ontological qualifier' "apples", also denoted by a, then --  

u1uo1  3 x uo1  3uo1  3a 

-- would stand for the English phrase "three apples". 

Or, if u1 were to be, again, assigned to the "Natural" number arithmetical quantifier value 3, and if uo1 were to be assigned, instead, to the 'metrical qualifier' "pounds", also denoted by p, then --

u1uo1 =  3 x uo1  3uo1  3p 

-- would stand for the English phrase "three pounds".

Note that we still cannot, yet, within the mathematical / arithmetical facilities / confines of the NU_ language, express, e.g., English, phrases such as "three pounds [of] apples", in which both an 'ontological category qualifier', in this case, "apples", and a 'metrical unit qualifier', in this case, "pounds", as well as a 'metrical quantifier', in this case "three" -- all three components / '''factors''' -- all appear at once / all occur '''multiplied''' together, per our generalized concept of [generally non-amalgamative] '''multiplication'''.

Such triple+ conjunctions occur for "state-space" 'state-variable[ vector]s', e.g. --

(1), for the "state-variable[ vector]" --

( m1 x dr1(t)/dt ) x [ [ M x L ] / T ] x [ p1 ] x [ x ]

-- for the x-axis physical-space-model's directional coordinate/component of the momentum of "particle" 1 as a function of time, t, a thus dynamical state-variable, represented by state-variable ['''ontological'''] qualifier p1, whose 'metrical quantifier' is --

( m1 x dr1(t)/dt )

-- which '''modifies''' / is measured in terms of the 'metrical «monad»-qualifier' of Length, [ L ], say, measured in the unit-of-measure qualitative metrical «monad» of the "inch", divided [ / ] by Time, [ T ], say, measured in the qualitative unit-of-measure metrical «monad» of the "hour", together forming the "compound" 'metrical «monad»', or "metrical unit", of Velocity [ V ] = [ L / T ], thus measured in units of "inches per hour", thence forming the further-'''compounded''' 'metrical «monad»' of "momentum", [ P ], via Mass, [ M ], measured, say, in the unit-of-measure qualitative metrical «monad» of the "pound" --

[ P ] = [ MV ] = [ ML / T ] = [ [ M x L ] / T ]

-- thus measured, in toto, in the compound[ed] 'metrical «monad»' of "pound-inches per hour", for this classical "phase-space" type of state-space, thus simplifying this state-vector-value to --  

( m1dr1(t) / dt )[ Pp1x ] 

-- or --

(2), for the "state-variable[ vector]" --

( r1(t) ) x [ L ] x [ r1 ] x [ x ]

-- for the x-axis physical-space-model's directional coordinate/component of the position of "particle" 1 as a function of time, t, whose 'metrical quantifier' is ( r1(t) ), which '''modifies''' / is also measured in terms of the 'metrical «monad»-qualifier' of Length, [ L ], say, measured, again, in the unit-of-measure qualitative metrical «monad», or unit, of the "inch", and simplifying to -- 

( r1(t) )[ Lr1x ] 

-- or --

(3), for a classical "phase-space"-associated "control[ parameter]-space" 'control parameter[-vector]', or, equally, for a non-classical "state-space"-associated,"control[ parameter]-space" 'control parameter[-vector]', e.g. --

( m1 ) x [ M ] x [ c1 ]

-- for the, typically-assumed constant, or time-non-varying, mass of "particle" 1, whose 'metrical quantifier' is ( m1 ), which '''modifies''' / is also measured in terms of the 'metrical «monad»-qualifier' of Mass, [ M ], say, measured, again, in the unit-of-measure qualitative metrical «monad» of the "pound", and whose "control[ parameter-]space" -- in this case, 'masses-space' -- directional unit-vector for "particle" 1 is denoted by [ c1 ].

The 'control-parameter-vector', in this case, thus simplifies to -- 

( m1 )[ Mc1 ].

All three examples, above, apply to the classical, "phase-space" type of "state-space" [including to its associated "control-[parameter-]space", which we call "masses-space"], e.g., for a [nonlinear] dynamical systems theory total-differential equation [and, thus, typically also for a "singularity"-entailing, and therefore also typically meta-]dynamical mathematical model. 

Note that each of these first two, "phase space", classical "state-space", examples actually each require not three but four '''factors''', '''specifiers''', 'determinors', or '''modifiers''' -- one '[metrical] quantifier, "times" one '[metrical] qualifier' [similar to what the NU_ system can provide], but also "times" one "state-variable" '''ontological-categorial''' qualifier, "times" one 'spatial-directional-vector' qualifier, and so exceeds the ideographical linguistic capabilities of the NU_ arithmetical/algebraical language by not just one but by two kinds of additional 'arithmetical-ideohraphical qualifier' '''factors'''/'''specifiers'''/'determinors'/'''modifiers'''.

Capability to express quantifiers in generalized-multiplicative’ combination/conjunction with both metrical qualifier units and ontological [e.g., state-variable or control-parameter 'identifier'/'specifier'] qualifier units, begins not with NU_, the third axioms-system in this axioms-systems progression presentation -- its first ‘dialectical full-synthesis’ axioms-system -- but with its seventh axioms-system -- the second ‘dialectical full-synthesis’ axioms-system -- which we of F.E.D. denote by Nq_MQN, or by Nq_MU, the 'Mu' axioms-system, wherein Nq QQ = NqM  =    NM_ denotes the fourth dialectical arithmetic axioms-system, of unquantifiable Metrical qualifiers.

All three of the symbols Nq_QN, Nq_U, and NU_, stand for a new first-order axioms-system which 'complexes-together', or which '''compounds''', or which 'uni[t-i]fies' -- which constitutes a dialectical synthesis of -- the axioms-system NQ_, and of its units, qn, become uon, and the axioms-system N_, and of its elements, n, become un.

As we have seen above, the un component of the new, '''compound unit''', stands for a «monad[s]»[-ic-level]-quantifier, or 'base-level units-quantifier', the count[or], or the cardinal  number -- the N of N number -- for the uon units / «monads» present, and that this uon component stands for a 'unit-qualifier', or [relative] base-level «monad[s]»[-ic]-qualifier, no longer for a 'category-as-unit-level qualifier'.   

Thus, the compound 'meta-meta-numerals' of the NU_ explicit-dialectical arithmetic are '''complex[es of /] unities''' of the numerals of the N_ implicitly-dialectical arithmetic and of / together with the 'meta-numerals' of the NQ_ explicitly-dialectical arithmetic.

That is, the transition from the n, and from the qn, onward / upward to the unuon, or to their dynamical versions, e.g., to their '''population dynamics''' versions, such as un(t)uon, forces that which was, in NQ_, the merely implicit presence of the base-level [sub-]units, or [sub-monads», of each qn 'category-level-as-unit-level ontological qualifier', or of each  qn '«arithmos»-qua-unitmonad» qualifier', to the surface -- into explicit, counted -- quantified -- recognition.

It so forces because it is not meaningful or useful / 'use-valu[e-]able', in dialectical modeling, to count more than one, supposedly "identical" copy of "the same" ontological category, e.g., to have 3uo1 stand for the 'co-presence' of three "identical" copies of the entire category "apples", or of a, itself -- 3a -- rather than standing for the 'co-presence' of three [similar, but not even possibly "absolutely identical"] «monads», units, or individuals, e.g., that presently make up the entire extant '''population''' of that category, in this case, the '''population''' of three similar individual apples.

Summary Proposition Regarding the Scientific Utility of 'Generic Ordinality' vs. that of 'Generic Cardinality'.   

The abstract[ed] property / characteristic / determination of 'generic qualitative ordinality' has far greater scientific, empirically-valid, meaningful universality than does the abstract[ed] property / characteristic / determination of 'generic qualitative cardinality'. 

That is, the 'ordinal qualifier categories' of 'first-ness', 'second-ness', and 'third-ness', etc., have a generic, universal, empirical meaning, especially regarding "chronological order", or 'systematic order' [i.e., 'classificatory order'; 'taxonomic order'], that the 'cardinal qualifier categories' of 'one-ness', 'two-ness', and 'three-ness', etc., do not.

The presumption of the universality of cardinalities -- the fixed idea that abstract, 'unqualified cardinal quantities', and their 'cardinal qualities', have the same generic, core meaning in all special contexts, universally -- engenders the Ancient/Modern pseudo-science of "numerology".

On the contrary, the inductively grounded expectation of the universality of ordinalities, of the ubiquity of ceaseless, diverse empirical instancings of ordered, consecutive sequencings, whether in the diachronic domain of "chronological order", in the synchronic domain of 'taxonological order' [the order that moves from the more specific category to the next more general category, or from the more general category to the next more specific category], or in both, is the very basis of modern, empirically-disciplined scientific theory, as well as of 'dialogical dialectic' probing of the modern 'Human Phenome', and of dialectical science.

The radical root of the ultimate modern abstraction and radical simplification of human natural language and cognition, in the form of an "artificial" ideographical, 'algorithmical', arithmetical written language of apparently "pure" -- 'metrically-unqualified' and 'ontologically-unqualified' -- cardinal quantifiers, is a largely-unconscious 'psychohistorical construct' named, by Marx, "The Elementary Form of [Commodity] Value", the foundation -- the '«arché»-thesis' -- of the entire vast edifice of Marx's immanent critique of capitalist political-economics.   

Moreover, that unconscious paradigm also resides, uncritically, at the heart of the Modern 'Human Phenome', as the foundation of 'The Modern Ideology', as of 'The Modern «mentalité»' entire.

By themselves, in their generic, uninterrupted form -- or, more exactingly stated, in their minimally-interpreted form -- the NQ  =  { qN } are merely the sequence / progression of abstract 'qualitative ordinality', or of the universal 'ordinal qualities' -- 'first-ness', 'second-ness', and 'third-ness', etc.

For any specific, particular dialectical modeling application of the { qN }, these generic qualifiers must be, precisely, 'de-genericized' -- must be '''specified''', or 'specificitized' -- by the modeler:  "interpreted" as, "assigned" to, "applied" to the special case that is to be modeled, the special case from out of that generic, generalized dialectic that is encoded by the { qN }, and that is to be modeled, using 'specifitized', 'algebraicized' versions of the generic { qN }.

The unavoidable, one-sidedly "quantitative" «mentalité» of Modern, capitalist humanity -- not just "The German Ideology", but 'The Modern Ideology' entire -- has yielded marked, market advantages for that humanity, throughout the '''ascendance phase''' of the capitals system, advantages over and above those of those still mired in its predecessor human «mentalités».   

But that one-sidedly "quantitative" «mentalité» -- the «mentalité» of the 'Capital-Value Mind', and of its "standard", "Natural" arithmetic -- is also debilitating for modern humanity, especially in the present, '''descendance phase''' of the capitals system.   

It is debilitating, for example -- in just one particular way of many particular ways -- scientifically, and in the engineering applications of the sciences that help directly to grow the human social forces of production, especially with regard to the  -- supposedly "impossible" -- analytical, closed-form solution of the "laws"-of-nature-formulating, nonlinear integrodifferential equations, and with regard to the problem of the division-by-zero "singularities" that abound especially for such nonlinear equations, and/or for their solution-functions.

Eva Brann, the translator, into English, of Jacob Klein's magisterial work, entitled Greek Mathematical Thought and the Origins of Algebra, expressed the matter in this way --

"I have omitted here, as too complex for brief exposition, a third, more directly global interpretation of the modern condition, which is central to Greek Mathematical Thought and the Origin of Algebra."  

"It is an understanding of the basic rupture between antiquity and modernity, of the great revolution of the West, as brought about by, or at least paradigmatically displayed in, the introduction of algebra."

"Algebra works with quantities abstracted from concrete collections (such as were betokened by the Greek arithmoi), with "general," essentially symbolical "numbers," such as the variables x, y, z or the constants a, b, c."

"These letters are symbols of a peculiar sort:  they represent neither a concrete thing nor a determinate concept, but rather present themselves as the object of calculation -- a mere object, an indeterminate entity."

"Klein saw algebraic problem-solving procedures, so effective precisely because so contentlessly formal, as emblematic of the modern rage for that second-order, deliberately denatured thinking which dominates as much of our lives as is method-ridden."

"The human consequences of this symbolic conceptuality are great." 

[Eva Brann, Keynote Address at the Conference on Jacob Klein, held at Seattle University on May 27-29, 2010, bold italic emphasis added by A.D.]