__Full Title__:

*On the Nature of the Opposition between*

*the*

__N___

*'*

**First**

*Standard*

*Arithmetic**'*

**and**

**the**

_{N}

__Q___

*'*

**First**__Dialectical__

**Arithmetic***'*.

Dear Readers,

Below, I have reproduced, for your cognitive pleasure, parts of the forthcoming

**F**.

__.__

**E**__. Vignette__

**D****#21**, by Aoristos Dyosphainthos, Chief Public Liaison Officer for the

**Foundation**.

It provides a

*introduction to the Seldonian*

**profound**

_{N}

__Q___*'*.

**First**'__Dialectical__ArithmeticEnjoy!!!

Regards,

Miguel

Member,

**F**.

__.__

**E**__.,__

**D**Officer,

**F**.

__.__

**E**__. Public Liaison Office__

**D****F**.

**.**

__E__**.**

__D__

__Vignette__

**--**

__#21__

*On the Nature of the Opposition between the*

__N___

*'*

**First**

*Standard*

*Arithmetic**'*

**and the**

_{N}

__Q___

*'*

**First**__Dialectical__

**Arithmetic***'*.

*by*

*Aoristos Dyosphainthos*

__Author__*. The purpose of*

__’__**s Preface****F**.

**.**

__E__**.**

__D__**Vignette**

**#21**

**is to clarify the nature of the relation of**

__non__*-*

**dualistic**,

**that is felt, by human 'conceptual perception', between**

__dialectical__opposition

_{N}**, the axiom**

__Q___*system of the*

__s__-**F**.

**.**

__E__**.**

__D__*'*

**First**

__Dialectical__*Arithmetic**'*, and

**, the first-order-**

__N__**axiom**

__plus__*system of the*

__s__-*'*

**First****Standard**

*Arithmetic**'.*

We do so by exploring the first triad of ontological categories, and, equally, of connoted axiom

*systems of dialectical arithmetic, that arise, as opposition, in*

__s__-**tep**

__s__**1**, and, as resolution, in

**tep**

__s__**2**, of the Seldonian dialectical presentation entitled

*'*-- as modeled by the

**The**'__Dialectic__of the Seldonin__Dialectical__Arithmetics

_{N}

__Q___*'*--

**dialectical****meta**-**equation meta**-**model**'

__#_)-|-(__

**= (****s**__#___

**)^( 2^****N_**__#___

**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

**technical terms, including of**

__Encyclopedia Dialectica__**.**

__E__**. ‘neologia’, are available on-line via the following URLs --**

__D__
-- 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

Links to definitions of the

**special terms most fundamental to this vignette are as follows --**__Encyclopedia Dialectica__*'*

**arithmetical ontological**"**pure**"**'**__qual__ifiers*'*

**arithmetical***"*

**pure**"

__quant__

*ifiers**'*

«

**»***aufheben**'*

**cumulum**'*'''*

**ev**-**entities**'''*'''*

**evolute**-**ness**'''
«

**»***genos**'''*or

**Historical**

__Dia__

*chronic*

__Dialectics__*'''*

__N__

*first**"*

**standard**"

*arithmetic*

_{N}

__Q___*'*

**first**

__dialectical__

*arithmetic**'*

*'''*

**Seldon Function**'''
«

**»***species**'*

**supplementary opposition**'*'''*or

**Systematic**

__Syn__chronic

__Dialectics__*'''*

-- and we plan to expand these public definitions resources
as the

__Encyclopedia Dialectica__*Dictionary***unfolds.***Project*
[

In the phrase “

In the phrase “

A key use-value of the

__Note__:__‘‘‘__.**Arithmetical****Pure****Quant**ifiers’’’ vs. ‘**Arithmetical****Pure****Qual**ifiers’In the phrase “

**3**apples”, we term “**3**” the “arithmetical [“pure”-]**ifier”, and we term “apples” the**__quant__*‘‘‘**ontological**’’’*-- or**of thing -- ‘‘‘**__kind__**ifier’’’.**__qual__In the phrase “

**3**pounds of apples”, we term “pounds” the*‘***[***metrical**-*]**unit****ifier’ -- or**__qual__*‘‘‘*of measure**unit****ifier’’’ --**__qual__**ified by the***quant***3**, and such that,*together*, that '[metrical]**ifier'**__quant__**&**/ "times" that 'metrical**ifier',**__qual__*jointly*, and '''metrically''',*‘*__quant__*o**-*__qual__*ify**’*, or*‘*__qual__*o**-*__quant__*ify**’,*the*‘**ontological***ifier’, “apples”.**__qual__A key use-value of the

__dialectical__**is to provide shorthand,***arithmetics**algorithmic*, ideographical-symbolic systems for the various kinds of ‘arithmetical**ifiers’, both with and without the co-presence of ‘‘‘arithmetical**__qual__**ifiers’’’, i.e., in the form of an intermixed**__quant____dialectical__**of both increasingly rich***progression**‘*__qual__*ifier-only’*__dialectical__**, and also of increasingly rich***arithmetics**‘*__qual__*o**-*__quant__*ifier’*, or*‘*__quant__*o**-*__qual__*ifier’*,__dialectical__**, alike, and in an alternating succession.].***arithmetics*[

__Not__

__e__:

__O__

__r__

__d__

__i__

__n__

__a__

__l____Color-Coding__. In the discourse below, we apply

**, visible-light-spectrum**

__relative__*-rising-frequency-*‘

__order__

__o__

__r__

__d__**i**

**n**

**a**

**l**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

**in any such sequence or series, color-code**

*first***orange**for the

**in any such sequence or series, color-code**

*second***yellow**

**[**-

**ish**

**]**for the

**in any such sequence or series, color-code**

*third***green**for the

**in any such sequence or series, color-code**

*fourth***blue**for the

**in any such sequence or series, color-code**

*fifth***indigo**for the

*sixth***in any such sequence or series, color-code**

**violet**for the

**in any such sequence or series, and color-code '''**

*seventh***ultra**-violet''' [as in "

**black**

**light**"] for

**entries in any such**

*any further**er/sequence/succession/series/progression. The well-known 'mnemonogram' "*

__ord__**R**

**o**

**y**

**G**.

**Bi**

**v**" may prove useful to the reader in tracking this color-coding.].

**. We use the symbol**

__Background__**, with "double underscore"**

__N___*****, to denote the axiom

*system of the so-called "*

__s__-**atural" numbers, when restricted to the axioms thereof that express its nature only in "first-order" logic: principally the first four Peano axioms.**

__N__*****[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., '

**' is, of course, '**

__a__**a**' with a "single underscore", whereas '

__' signifies '__

**a_****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" "**atural" 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]**__N__*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

**with respect to that of**__N___**.**__N__
That
"first-order" axioms system,

**, is so rich that it includes**__N___*"*standard" models of the "**-**__non__**atural" numbers.**__N__
One
of those

*"*standard models" is an**-**__non__**qualitative opposite of/to**__extreme__**as "standardly" interpreted -- i.e., as interpreted to be**__N___**.**__N__
That

*'''*system to**diametrically opposite**'''**as**__N__**is**__N____{N}**, the Seldonian**__Q___*'**First*__Dialectical__*Arithmetic**'*.
The

_{N}**system is not**__Q_____a__*only**'''*' of the**'''**__diametric__opposite**as**__N__**system; it is**__N_____its__*also**'''*.**'''**__dialectical__opposite
The

_{N}**system is the**__Q___*first**"*system to the**"**__anti__thesis__N___**, or «***first***»,***arché**"*system, as generated by the Seldonian**"**__thesis____dialectical__*-*equation of the**mathematical**'**meta**-**model**'**F**.**.**__E__**.,**__D__**,**__dialectical__**of the Seldonian systems of***systematic method of presentation***, that is, of the presentation that is entitled --**__dialectical__arithmetic*'*

**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics-- an equation which is itself also written in the

_{N}**algebra.**

__Q___
The
term

**and the term**__N____{N}**[thus] form the first two terms of the systems-**__Q___*of*__progression__*'*, such that they are**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics*'''*amalgamatively summed''' [cf. Musès] together, and also 'antagonistically summed' together [a mental action denoted, herein, by the sign '**-**__non__**~+~**', 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**al case of the***speci***ric-***gene***,**__dialectical__**, '**__s__tep one__thesis____-__**system****~+~**__antithesis____-__' sum, that constitutes the result of**system**__s__tep**1**of the Seldonian__dialectical__*-*equation that encodes the presentation of that title.**mathematical**'**meta**-**model**'
The

-- an

**tep**__s__**s =****1**calculation and solution of that*'*equation yields, in**meta**-**model**'*shorthand*form, as the content to be presented as step one of that presentation --__N___**~+~**_{N}__Q___-- an

*'*__anti__**thesis**-sum' so far barren of the__plicit 'present-ation' of any__**ex**__plicit__**ex****.***thesis*__syn__
The
focus of this Vignette is

That is a task which has been slated, by the

**the**__not____full__axioms-system__s__*-*progression of*'*, for most of the hundreds of its steps that have so far been explored by the**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics**Foundation**research collective.That is a task which has been slated, by the

**F**.**.**__E__**. General Council, for a different "place" and time.**__D__
Our
main focus herein is in

**tep one only; is the elucidation of the meaning of this**__s__*,***first***thesis**/**antithesis**'**cumulum**'*of that progression,__N___**~+~**_{N}**, as of the**__Q___**,**__dialectical__*'*['**supplementary opposition**'**~**'] that grounds it:__N___**~**_{N}**.**__Q___
In
terms of the

**ric***gene**'**First*__Dialectical__*Arithmetic**'*, the axiom*system denoted by*__s__-**in this**__N___*'*is a**meta**-**model**'**fic interpretation [which we "sign" by '***speci***[---)**'] of the**ric***gene*_{N}__Q___*'***dialector**'__q__**:**_{1}__N___**[---)**__q__**.**_{1}
The
axioms-system denoted by

The

Similarly, we denote the '

_{N}**in this**__Q___*'*is a**meta**-**model**'**fic interpretation / solution for the***speci***ric***gene*_{N}__Q___*'*denoted by**dialector**'__q___{1}_{+1 }**=**__q__**:**_{2}_{N}__Q___**[---)**__q__**.**_{2}The

**numerals****{1**,**2**,**3**,**... }**that form the core of**, are**__N___**denoted by just***collectively***N**, with no underscore(s).Similarly, we denote the '

**-**__meta__**numerals**' core of the_{N}**system by**__Q____{N}**, which removes the "single underscore" of the axiom**__Q____-system symbol from under the '__*s***'**__N__*'pre-*, yielding just '**script-level symbol'**__sub__**N**' instead, and removes the "double underscore" of the '**'**__Q___*'*, and reduces it to / replaces it with a "single underscore", yielding just '__-level symbol'__**script****' instead .**__Q__**. The**

__The Nature of the Opposition__**between the**

*opposition*

__N___**[---)**

__q__**axioms-system of the 'first standard arithmetic', and the**

_{1}

_{N}

__Q___**[---)**

__q__**axioms-system of**

_{2}*'*standard first

__non__-**arithmetic', is**

__dialectical__**any purportedly**

__not__

__un__*-*, "

__dialectical__**"**

__absolute__**; is**

*opposition***any supposed**

__not__**, forever incapable of**

*radical dualism***, of practical and theoretical**

__dialectical____syn__thesis**, of conceptual**

*transcendence***, of**

*reconciliation**'''*.

**complex unification**'''
That

**is, on the contrary, a***opposition***: as we said above, it is a***reconcilable opposition*__dialectical__**.***opposition*
The
"pure"

*'''*ifiers''' of the**quant****system, and the "pure" '''ontological**__N___*ifiers''' of the***qual**_{N}__Q___**system, are****actually "pure", i.e., are**__not__**devoid of any implicit '''overlap''' with, or inter-mixture of, the other.**__not__
Said
another way, the "pure" arithmetical

*ifiers of the***quant****system, and the "pure" arithmetical ontological**__N___*ifiers of the***qual**_{N}**system, are**__Q___**"**__not__**" distinct and dirempt, the one having "**__absolutely__**" no internal connexion with/to the other.**__absolutely__
The

**system**__N___**plicitly already contains the seeds of the**__im___{N}__Q___**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*'*, and, in particular, with its core component, our own "**Human Phenome**'**N**atural" language, where various varieties of*'''*ifiers''', even more so than various varieties of**qual***'''*ifiers''', are found to figure prominently, and, more generally, with other "**quant****N**atural" languages, similarly populated with*'''*ifiers''' relative to**qual***'''*ifiers''', as well as with various**quant***"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*ifiers', but some of which also feature 'mathematical***quant***ifiers', obviously so or not.***qual**
The
"artificial languages" of arithmetic, of its algebra, and of
mathematics more generally, cannot fulfill their implicit mission -- that of
re-encoding human "

**N**atural" 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*'''*ifiers''', as well as of the**qual***'''*ifiers''', of "**quant****N**atural" languages and of some artificial languages alike.
The

[

_{N}__Q___**system still '''contains''' the****system, as indicated syntactically by the**__N____N__*'*script', subordinated to and subsumed under and before the**pre**-__sub____Q___*symbol in the*__script__-level_{N}__Q___**symbol as a whole.**[

__Note__: We symbolize the Seldonian 'First Dialectical Arithmetic' by_{N}__Q___**, rather than by**_{N}_{_}__Q___**, with a "double underscore" for***both*the**Q**and the**N**,*viz*., '**'**_{N}_{_}*next to*'**', because the**__Q___**rules of the explicitly***addition***arithmetic of the [**__cardinal__*order]***second**-**axiom**__N____-system, not__*s*__plicit in the [first-order]__**ex****axiom**__N_____-system, control the 'subscript-arithmetic' of the__*s***sole**_{N}__Q___**operation, that of***'*, i.e., of the «**'**__dialectical__multiplication**» operation.***aufheben**'**Dialectical**addition**'*"operations" in_{N}__Q___**really represent a***'*operation' option; the '''peaceful coexistence''' of distinct 'ordinal, ontological**-**__non__*ifiers' without the mutual interaction of***qual***'*,__dialectical__*mutual**-*«**»,***aufheben*__ontological__multiplication*'*].
The
'meta-number' set, or "space", of the

-- by which we can see that each

_{N}**axiom**__Q___*s**-*system is denoted by --_{N}__Q__**,**__=__{__q___{1}**,**__q___{2}**,**__q___{3}**. . . }**__=__{__q___{n}**}**, for all**n**in**N**-- by which we can see that each

__q__**still '''contains''' / conserves**_{n}**n**, for all**n**in**N**, but from which we can also see that the 'script-level', '**' aspect of '**__q____q__**' -- each of these 'meta-number's meta-numerals' as a whole, subordinates and subsumes the 'post-subscript' level, '**_{n}**n**' aspect of those 'meta-numbers / meta-numerals'; that the '**' aspect supplements and supersedes / exceeds**__q__**n**, and, thus, supersedes / exceeds**N**as well.
It
is, indeed, this

**,**__dialectical__*'*«**-**__self__**»***aufheben*__self__*-*of the**subsumption**'**kind of**__ord__inal*'***arithmetical****ity, i.e., of the**__quant__**N**of**-- the kind of**__N___**ity that characterizes the first order**__quant__**Peano-axiom**__N_____system -- that makes__*s***plicit the hitherto only**__ex__**plicit ingredience of**__im__*'*in "**arithmetical****ordinal**'__qual__ity**N**atural" arithmetic -- the kind of**ity that characterizes the also-first-order**__qual___{N}**system.**__Q___
[

__Note__: The 'reflexive prefix',*'*, here, refers to the logical, conceptual action of the human subject/agent whose mind embodies the**-'**__self__**system, and also then**__N___*critically*reflects upon that system -- thus*critically*reflecting upon**as embodying that***it*__self__**system -- a mental operation which we denote by the**__N___*'***-**__dialectical__equation**definition**'__N___**(****)**__N_____=__**~****(**__N___**)**, wherein '**~****(**__N___**)**' denotes the operation of*'''*__dialectical____self__*-*of**negation**'''**, i.e.,**__N___*'*«**-**__self__**»***aufheben*__self__*-*,**negation****/**-__self__**critique**'**the**__of__**system,**__N___**the**__by__**system, as that system is 'mentally embodied' by a cognizing human subject/agent].**__N___
Indeed,

_{N}**is**__Q___*born**out*__the womb of its predecessor-opposite,__*of***;**__N___**gives birth to its successor,**__N___*'*,**supplementary****opposite**'_{N}**.**__Q___
The

_{N}**system represents what the**__Q___**system lacks, by virtue of**__N___**containing only**__N___**plicitly,**__im____vertly, what__**co**_{N}**contains**__Q___**plicitly and**__ex____vertly: 'arithmetical ontological__**o****ifier operators'.**__qual__
But
the

_{N}**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**__Q___*'re-*plicitization', of a**im***'re-*vert-ization' -- in relation to the key deficiency of system**co****: System**__N____{N}**lacks the 'arithmetical**__Q___**ifier operators' of which system**__quant____N___*principally*consists -- of which**, in an**__N_____vert sense,__**o***exclusively*consists.
The

__N___**~**_{N}__Q___*'''*belongs to that «**'''**__dialectical__opposition**»-category of***species**the systematic*__dialectic__of opposition*-*-- of the**in**-**general***'*«__dialectical__speciation of the**»-***genos**category of opposition**'*-- that we term '**supplementary***opposition**'*, or '**progressive***opposition**'*, which is this__dialectic__*'s**'*, combining its**-**__dialectical____syn__thesis species**category**'*'*that we term '**-**__dialectical____thesis__species**category**',**complementary***opposition**'*, or '**symbiotic***opposition**'*, with its*'*, that we term '**-**__dialectical____anti__thesis species**category**'**annihilatory***opposition**'*, or '**parasitic***opposition**'*[for more regarding this foundational**, please see --**__dialectic__**link**].
The

**system in the***second*__dialectical__systems*-*of**progression***'*, namely**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics_{N}**, thus exposes explicitly the inadequacy of a system -- of the**__Q___**system,***first***-- of arithmetical ideographical "artificial" language, built upon**__N___**ifiers, and capable of**__quant____vertly expressing__**o***only***ifiers, such that**__quant___{N}__Q_____plicitly exposes that__**ex**__adequacy of__**in****by**__N___**its critique of**__embodying____N_____, by actually__*positively***a**__exhibiting__*counter**-*[cf. Imre Lakatos] to**example****, an alternative to**__N___**, a system which lacks**__N___**'s primary lack.**__N___
However,
by the same token, the

_{N}**system, as a system capable of**__Q_____vertly expressing__**o***only***ifiers,**__qual____capable of__**in****plicitly expressing**__ex__**ifiers, thereby exposes its own, complementary**__quant____adequacy, taking__**in****initially as, apparently, the only available**__N___*counter**-***example***to*its*own*, also deficient,*counter**-*; as the only available alternative to that, also deficient, alternative to**example****which**__N____{N}**is.**__Q___
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

**, and of this 'pendular' alternation, between**__dialectical__opposition**and**__N____{N}**, is the irruption of the conception of a**__Q___**system in this series/progression, the irruption of their***third**'*,**-**__uni__**system**'_{N}**, which seemingly so seamlessly combines the major features of the**__U___**system and of the**__N____{N}**system, that it makes us feel, in our 'conceptual perception', as if these features should never have been so**__Q___*'''***un****N**aturally''' separated in the first place, and which, in particular, features__both__'arithmetical**ifiers' and 'arithmetical**__qual__**ifiers' in a "**__quant__**N**atural"__, in a harmoniously integrated mutual necessity and mutual utility.__*unity*
But
before we turn to any focused consideration of this

__syn__thesis*-*system, born out of the inadequacies of the opposition of/between**and**__N____{N}**, let us first consider, with greater scrutiny, that**__Q___**itself, and consider also how that***opposition***itself is born -- how***opposition***gives birth to**__N_____N___**~**_{N}**, only thereafter moving on to consider how**__Q_____N___**~**_{N}**gives birth to**__Q____{N}**.**__U___**On the Nature of the Presentational Transition from**

__N___**to**

__N___**~+~**

_{N}**:**

__Q___**The**

*of the Engenderment / Construction of the*__Dialectic__

_{N}

__Q__**from /**"

**by**"

**the**

**N**. At the outset of his lectures on his

**«**

__dialectical__**», Hegel spelled out,**

*Logik***, his**

*most succinctly***ric '''algorithm''' for**

*gene*

__dialectic____(__, which he applied throughout his encyclopedic

**)***s**opus*on his "Philosophical Sciences", his

__dialectical__*"*, encompassing his '''Philosophical Science''' of

**Philosophical Theory of Everything**"**[«**

*Concepts***»], his "'Philosophical Science''' of 'exo-human' «**

*Logik***», and his '''Philosophical Science''' of**

*Natur**'*

**-**

*Spirit***ed**'

**, of consciously-creative**

*Beings***[«**

*human beings***»], in the following terms, and in which I have numbered Hegel's descriptive passages, below, for greater ease-of-reference further-on --**

*Geist***[1]**: "The

**[A.D.:**

*first*__determination__**≈**'«

**»**

*speci**-*fication'] is

**mediate, while the**

__im__**one constitutes the**

*second***posited in**

*sphere*

*its*

__differentia__*tion from*the

**."**

*first***[2]**: "Within every

**, [e.g.,**

*simple first*__determination__**,] what is**

*ground***from**

__determinately__different**[, e.g., the**

*it***of the**

*consequence***]**

*ground***, but is at first present without yet being**

*is at once also present***plicitly posited."**

__ex__**[3]**: "In the

**,**

*second*__determination__**[and with it**

*finitude***] again enters."**

*contradiction***[4]**: "The

*third***is the**

__determination__**of the**

__unity__**and the**

*first***, in which the**

*second***is**

*contradiction***. ..."**

__resolved__**[5]**: "The

**is as follows. The**

*progression***is**

*beginning***,**

*simple***mediate. ... Every newly emerging concept is**

__im__**ly**

*more concrete***than its predecessor."**

__determinate__**[6]**:

*"*, but

**We are always carrying everything that went before along with ourselves into what is new****,**

*everything prior is***,**

*within what is new***." [A.D.: the foregoing is an assertion, by Hegel, of what we of**

*put in its*__determinate__place**F**.

**.**

__E__**. describe as the '«**

__D__**»**

*aufheben***' of**

__evoluteness__**].**

__dialectic__**[7]**: "Whereas, in what preceded, each [momentarily

**mediate]**

__im__**... passed as**

__determination__**, it is now**

__ultimate__**into being**

__demoted__**... ."**

__only__a__moment__
[G.W.F.
Hegel,

**, Clark Butler, translator,**__Lectures on Logic____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

**reproduced above turn out to have been instantiated in the -- opening -- portion of**__dialectics__*'*that we scrutinize herein, as noted below:**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics__Regarding precept__

**: In**

__[1]__*'*,

**The**'__Dialectic__of the Seldonian__Dialectical__Arithmetics**constitutes the**

__N___*"*determination",

**first**

_{N}**the**

__Q___*"*determination".

**second**Within the present-day

*'*, from practical, everyday use of ordinary arithmetic, and especially for those schooled at all in mathematical theory, the system of "

**Human Phenome**'**N**atural" arithmetic, which we shorthand by

**, has, precisely,**

__N___*'''*mediate''' significance and familiarity.

__im__The

_{N}**system/ideo-«**

__Q___**», on the contrary, can come to be known and familiar**

*species**to any of us*only by our first grappling with its

__differentia__*tion from*-- its "«

**»" with respect to -- the**

*differentia specifica***system/ideo-«**

__N___**».**

*species*__Regarding precept__

**: The**

__[2]__

_{N}**system is already**

__Q_____plicitly present with and in the__

**im****system, via the latter's internal relation to the totality of natural language(s), at the core of the**

__N___*'*.

**Human Phenome**'
There
is already a 'penumbra of knowing', widespread within present-day global,
Terran humanity, within the

*present**'*, in the**Human Phenome**'*present*human 'near-consciousness', that,**rically, the "***gene***N**atural" 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*'*as instantiated in that system of "**elision of the qualifiers**'**N**atural" arithmetic.
Seldon's
discoveries, as recounted herein, are an essay at raising this
'near-consciousness' into

[For a deeply insightful exploration of the formation of the "Modern" -- i.e., of the 'capital-value-centric' -- «

__plicit consciousness, or, rather, are a distillation of what constituted his individual path from that 'near-consciousness' into more__**ex**__plicit consciousness of this key, core aspect of__**ex***'*; the modern**The Modern Ideology**'*'*; of human minds permeated by the 'exchange-value-exchange' paradigm; of the one-sided**Human Phenome**'*'*«**'**__quant__**», semi-consciously mired and marooned in the***mentalité***of***meme**"'*-- via their incessant, intensive daily practice of**The Elementary Form of Commodity Value**'''*'''*[**the law of***capital-*]*, however unconsciously so***value**'''*as such*.[For a deeply insightful exploration of the formation of the "Modern" -- i.e., of the 'capital-value-centric' -- «

**», as the most recent '''instar''' in the [psycho]historical development of the***mentalité**'***Human Phenome***'*, an exploration that, unfortunately, deprives itself of even deeper -- Marxian -- insights, see Alfred W. Crosby,__:__**The Measure of Reality***,***Quantification and Western Society****1250**-**1600**, Cambridge University Press [NY:**1997**]].__Regarding precept__

**: Rising from**

__[3]__**to**

__N___

__N___**~**

_{N}**, '''**

__Q___

__dialectical__

*contradiction**'''*enters

__ex__plicitly, via

*'''*

__dialectical__*opposition**, '*

*'''***~**'.

__Regarding precept__

**: The mutual**

__[4]__*'''*

__dialectical__*opposition**of*

*'''*

_{N}

__Q___**&**

**is resolved in**

__N___

_{N}

__U__**, which is the**

_______of__

*unity*

_{N}

__Q___**&**

**, as we shall see, below.**

__N_____Regarding precept__

**: The**

__[5]__**system seems**

__N___**, and**

*simple***mediately familiar.**

__im__The concept of the

_{N}**system, which '''follows [from]''' the concept of the**

__Q___**system, is more**

__N___**, is richer in**

__determinate__**, or in**

__determinations__*'''*fications''', than is the concept of the

**speci****system, at minimum because, on its face, the**

__N___

_{N}**system combines old**

__Q___**, of the '**

__determinations__**'**

_{N}**ifiers, with new**

*quant***, of the '**

__determinations__**' [ordinal, ontological]**

__Q___**ifiers.**

*qual*__Regarding precept__

**: This**

__[6]__*'''*of

**evoluteness**'''**is the reason that the**

__dialectic__

_{N}**axioms are designed to result in a**

__Q___

__dialectic__

*series**-*of form --

**progression**

__N___**---)**

__N___**~+~**

_{N}

__Q___**---)**

__N___**~+~**

_{N}

__Q___**~+~**

_{N}

__U__

_____**...**

-- for the '[Dyadic] Seldon Functions', rather than in a mere

*sequence***of the form --**

*succession*

__N___**---)**

_{N}

__Q___**---)**

_{N}

__U__

_____**.**

The generic

__dialectical__**, in both its Hegelian account, and in its Seldonian account, is thus**

*movement***--**

__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**-*]

**is also**

__evolut__ion*'''*

__ev__

**olute***'''*,

**not***'''*

**con**

**olute***'''*[in the main].

As a whole,

**[**

*cosmological*

*meta**-*]

**is**

__evolut__ion*'*, invoking an

**'**__co__volute__.__

**E**__. term,__

**D***'*, which signifies a

**'**__co__volution*of*

__dialectical__synthesis*'''*and

**'''**__e__volution*'''*, encompassing the potential '''extinctions''' of

**'''**__con__volution*some*'physio-ontological' categories' content, together with the

*'double-*«

**»'**

*aufheben**into-the-present of most.*

**conservation**-__Regarding precept__

**: In**

__[7]____tep__

**s****0**of

*'*pre-programmed, 'algorithmitized'

**The**'s__Dialectic__of the Seldonian__Dialectical__Arithmetics

__dialectical__**, the**

*method of exposition***system stands alone, and it appears,**

__N___*'''*arily''', to constitute the

**moment**

*ultimate**system of arithmetic.*

**primary**But the immanent, internal inadequacies of the

**system break it out into two, into the first possible**

__N___**, the two-system**

*cumulum***, of this**

*cumulum*

__dialectical__**of**

*progression***of**

*systems*

__dialectical__**: into the**

*arithmetic*

__N___**~+~**

_{N}

__Q___**.**

*cumulum*
In
that process,

But it is also

**is indeed**__N___*"*moted" from its former seeming__de__*'*, as signified by the lowered, subordinate, subsumed, 'pre-subscript' position in[to] which it is explicitly conserved and consigned in the symbol**ultimaticity**'_{N}__Q___**, in the '****' component of that compound symbol.**_{N}But it is also

*'''*moted''', to become the '__pro__**' component of that symbol as a whole,**__Q____{N}__Q___**.**
Here
we must register a critical note with regard to Hegel's account of the

The colloquial German term «

Hegel tends to emphasize only moments

The

The

The

Hegel often emphasizes only the

We see this as a mistake.

**ric***gene*__dialectical__**, as reproduced above.***process*The colloquial German term «

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

**(1)**and**(3)**.The

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

*'''*moment''' of the «**elevation****» movement is related to the***aufheben**'''*motion moment''' of the**-**__pro__**which we just noted above.**__dialectic__The

*'''*moment''' of the «**conservation****» movement is ingredient in***aufheben*__both__the*'"*-motion''' and the__de__*'''*motion''' effects.**-**__pro__Hegel often emphasizes only the

*'''*moment''' and the**negation***'''*moment''' of the «**conservation****» movement, even altogether omitting to mention the***aufheben**'''*''' aspect.**elevation**We see this as a mistake.

For example, we account for the formation of the

_{N}**from the**

__Q__**N**as follows.

The
first four, first order Peano postulates that form the core of the

**axioms-system, characterize the "**__N___**N**atural" numbers most explicitly as*"*inal numbers".__ord__
The
'intra-dual', 'co-character' of the

**as, '''simultaneously''', both**__N__*"*inal numbers" and__ord__*"*inal numbers" is only__card____plicit in the__**im****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__ordered plethora of__**dis****of***instances**"*inal number", even though, in the rendering preceding, we have excerpted from that --__ord____er-less -- set to present three of its members in their__*ord*__inal__*ord*__er.__*ord*
But
what is

*'*inal-number-__ord__*in-***ral'**__gene__**?**What is the «

*ideo**-*» of

**genos***"*inal number",

__ord____plicit in that jumble of its__

**im****,**

*instances***N**

*?*
It
is the

**ral category of**__gene__*'''*inal__ord__**ity'''.**__quant__
And,

**plicit in the**__im__*inherent ambiguity*of the concept of*"'*inal-__ord__**ity-**__quant__*in-*is the concept of**ral'''**__gene__*'*inal-__ord__**ity-in-**__qual__**ral'.**__gene__**plicit in the**

__Im__**rality of the**

__gene__**itative idea of**

__quant__*"*,

**first**"*"*,

**second**"*"*, etc., is the general

**third**"**itative idea of '**

__qual__**n**th

*-*, whose

**'**__ness__**include 'the**

*instances***ity of**

__qual__

*first**-*, 'the

**'**__ness__**ity of**

__qual__

*second**-*, 'the

**'**__ness__**ity of**

__qual__

*third**-*, etc.

**'**__ness__
We
already know, at least latently, from our experience of natural language(s),
the totality-of-reference for this

The subtle, evanescent, attenuated form of quality that is

This, 'the arithmetical

These '''seeds''' need only to '''sprout''', as they did in the mind of Karl Seldon, into

--

**, 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 "**__dialectic__of__dialectical__arithmetics**N**atural" arithmetical, 'algorithmical' ideograms.The subtle, evanescent, attenuated form of quality that is

*'*inal quality', or__ord__*'*er quality', is the first form of quality that can [re-]enter immanently into arithmetical ideography, that can [re-]emerge ['re-', given the__ord__*quite explicit*presence of the «__Mo__**nad**», or**unit**,*'quant*ifiable arithmetical*ifier', denoted by '''***qual****M**''' in Diophantus of Alexandria's^{o}*circa***250**C.E. first precursor to modern symbolic algebra, the**] within modern arithmetic, or, i.e., within 'algorithmic ideography'.**__Arithmetica__This, 'the arithmetical

__itative', can so re-emerge by means of the seeds of it which are__**qual**__plicit in the__**im***'*inal__ord____ifiers', or__**quant***'''*er numbers''' -- the__ord__*'*numbers', to express their nature more precisely -- that**-**__consecutive__**er-**__ord____plictily characterize the__**ex****N***of***, as opposed to the**__N___*'*inal__card____ifiers' that__**quant**__plicitly characterize the__**ex****N***of***.**__N__These '''seeds''' need only to '''sprout''', as they did in the mind of Karl Seldon, into

*'*inal__ord____ifiers', i.e., into arithmetical, 'algorithmical', "symbolic"__**qual***'*__inal__*ord*__ifier meta-number ideograms', representing the__**qual***fic***speci***that inhere in the*__ities of__**qual**__er__**ord***"*inal numbers", or__ord__*'*er-numbers', of the__ord__**N***of***; in --**__N___**1**,**2**,**3**,**. . .**--

**'Peanic'**__as__*"*inal numbers", or__ord__*'*er-numbers'.__ord__
In

**1889**, Giuseppe Peano published his postulates for a deductive-logic derivation of "**N**atural Numbers" arithmetic, in his book**. The philosopher W. L. Reese gives the original version of the four "first-order" Peano axioms as follows --**__The Principles of Arithmetic Presented by a New Method__
"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__**, Humanities Press [NJ:***Eastern and Western Thought***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

**1930**s, provide formal proofs implying that the "standard model" and*"*standard models" of these Peano-axioms' "**-**__non__**N**atural" Numbers arithmetic are**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**__inextricably____coexistent__**of the***construction**"*standard models" that they herald.**-**__non__
The
Löwenheim-Skolem Theorem, by itself alone, implies that

*"*standard models" of**-**__non__**inhere in the first-order "standard model" of "**__N___**N**atural" arithmetic.
The
Gödel Completeness Theorem, which provides proof of the semantical completeness
of the first-order

**arithmetic, given its proof of the**__N___*semantic*completeness of ideographical first-order predicate logic itself, in conjunction with the Gödel Incompleteness Theorem, which establishes the*syntactic**completeness of first-order*__in__**arithmetic, and the**__N___*semantic and syntactic**completeness of all possible higher-than-first-order axiomatizations of the*__in__**arithmetic, together imply the**__N__**of "standard" and**__inextricable____coexistence__*"*standard" models of the Peano "**-**__non__**N**atural" Numbers arithmetic at the level of its first order axiomatization.
Now,
on their face, these four Peano postulates describe the

But nothing of the

*'''*er properties''' -- the__ord__*'''*inality''' -- of a sequence of entities that they name ["Natural"] numbers.__ord__But nothing of the

__inal quantity, or '''manyness''', features of the standard "__*card***N**atural" Numbers are__plicit in these four, "first order" sentence-axioms.__**ex**
These
four axioms are said to be "first order" because they make assertions
about only individual "[

**N**atural] 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

That is, they potentially describe also, besides the

*"*standard models" of the "**-**__non__**N**atural" numbers.That is, they potentially describe also, besides the

**N***of***,**__N___**of [idea-]objects / of "logical individuals" which are***sequences*__"__*not***N**atural" numbers, including**of mutually***sequences***itatively 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***qual**'''*for the counts of the entities in such**'''**__actual__infinitude**, "standard" and***sequences**"*standard" alike.**-**__non__
The
"standard" "

**N**atural" numbers do__form an__*not*__actually__infinite**in any practical, empirical sense, nor can any***sequence***number of "**__actually__infinite**N**atural" numbers be actually constructed, step-by-step, by**mathematicians, given the nature of physical possibility.**__actual__
Practically
speaking, the "

E.g., if I am using a digital computer to perform arithmetic, then the practical finite limit of the "

**N**atural Numbers" are a__sequence, though their effective upper bound "__**finite****N**atural" Number is not always the same "**N**atural" number in every practical context.E.g., if I am using a digital computer to perform arithmetic, then the practical finite limit of the "

**N**atural Numbers" for me, in that context, is the highest "**N**atural" 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

[For a far-reaching critique of the prevailing, pseudo-scientific academic

**plagues modern mathematics, as we should, and as Seldon does, in his**__still__**,**__dialectical__**of the "standard" arithmetics, and, in particular, if we drop the 'contra-empirical', and thus anti-scientific, assertion of***immanent critique**"*, such as the supposed Cantorian**actually**-**existing**"**infinities***aleph**-*of the "**null actual infinity****N**atural" Numbers, then we find many commonplace, "prosaic" examples of*"*standard models" of the Peano first-order "**-**__non__**N**atural" "numbers".[For a far-reaching critique of the prevailing, pseudo-scientific academic

__of__**religion***'***infinitist**'**in contemporary mathematics, see Brian Rotman,***mysticism*__:__**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

**of***sequence*__the__letters of the English alphabet, in their "alphabetical__er", fully satisfy the 'finitistic' first-order Peano axioms [e.g., provided we assert the phrase__*ord**"except for the last*[*"Natural"*]*number"*at [the] appropriate place(s) in the Peano axioms], even though they are__"numbers" at all, in any "standard" sense. Just substitute the word "letter" for the word "number", and the symbol '__*not***a**' for the ideogramic symbol '**1**', in a fully finitistic version of Peano axioms given above.
So
does the

**of the month-names of***sequence*__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__er of "descending" inclusiveness:__*ord***Domain**,**Kingdom**,**Phylum**,**Class**,**Order**,**Family**,**Genus**, and**Species**, 'mnemonogram'**D****K****P****C****O****F****G****S**via "**o**__D__**ings**__K__**lay**__P__**hess**__C__**n**__O__**ine**__F__**lass**__G__**ets?" [just substitute the word "taxon", or "taxa", for the word "number", and the phonogramic symbol "**__S__**Domain**" for the ideogramic symbol '**1**', in a fully finitistic version of the Peano axioms given above].
And
so do the

*'*inal__ord____ifier meta-numbers' of the__**qual**_{N}**space of the**__Q___{N}**axiom**__Q___*system for*__-__**s****: another**__dialectical__arithmetic*"*standard model of the__-__**non****N**atural Numbers": this time, a__ric one, rather than a__**gene**__fic one, such as is each of the earlier examples above.__**speci**
Indeed,
the first-order Peano axioms, purged of any fanciful

**, apply to the whole domain of what Seldon calls,***infinitism**not**"*, but**archeonic con**"__tinu__a*'*-- the entire realm of possible finite/realistic**archeonic con**'__secu__a**of [ev]entities, such that each such***sequences***exhibits an "«***sequence***»", e.g., an***arché**'''*, a**ultimate ancestor**'''*'*ing [ev]entity', but which may, at least in some cases, exhibit no pre-__start____ained__*ord**'*ping [ev]entity', and which may thus constitute**stop****never***"*, but ever only**"**__actually____in__finite sequences*"***"**__potentially____in__finite**, in Aristotle's sense.***sequences**'*inal

__Ord__**ifiers',**

*qual*__inal__

*ord***represented as/by**

__categories__*'*inal

__ord__**ifier**

*qual**numbers', can thus fulfill the first four, finitistic "first order" axioms*

__meta__-*of*

**just as fully as do the**

__N___*'*inal

__ord__**ifier-**

*quant**numbers'*of the

**N**

*of*

**--**

__N___**N**

__=__**{1**,

**2**,

**3**,

**...**

**}**.

The

_{N}__Q__*'*of the**-**__meta__**numbers**'_{N}__Q___**are 'axiomatically-engineered' to**__dialectical__arithmetic**rically encompass all cases of realistic, '''finitistic'''**__gene__*'*whose sequential [ev]entities are**archeonic con**'__secu__a**"standard**__not__**N**atural Numbers", but, rather, are*'''*numbers''',__-__**non***'''*pure-__-__**non**__itative''', '''__**quant**__itative [ev]entities''', like the letters of__**qual**__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*of the axiom*__dialectical__progression*systems of*__s__-__the__*arithmetics***standard***--**'***' --***The Goedelian*of the Standard Arithmetics__Dialectic__**,**__N__**,**__W__**,**__Z__**,**__Q__**,**__R__**,**__C__**,**__H__**,**__O__**.****.****.**.
The

[ for more regarding these other examples, see:

_{N}__Q__*'*of the**-**__meta__**numbers**'_{N}__Q___**are 'axiomatically-engineered' to also**__dialectical__arithmetic**rically 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**__gene__*"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,**like the cosmological historical sequence-of-appearance/-"population" of the**__and__*categories of the***primary***physical*ontology of the*, thus constituting the Seldonian***cosmos***'*.**'**__Dialectical__Theory of Everything[ for more regarding these other examples, see:

**formations**;**relations**;**forces**;**ideologies/knowledges**;**cosmos**].
Our
'''self-reflection''' of

These

**in the context of "natural language(s)" as totality-of-reference**__N___**(1)**'conceptually perceives' the*-- or the [psycho]historical*__lack__*, in 'post-Diophantine', post-Medieval Modernity -- of 'arithmetical, algorithmic*__loss__**ifiers', as a***qual**of and within "standard"-interpretation*__lacking__**, and**__N___**(2)**'conceptually sees' the seeds, the "wherewithal", for the redress of that lacking as residing also*immanently*within**itself, in the**__N___*inherent*ambiguity, and 'intra-duality', of**itative versus***quant***itative***qual*__inality, and, as a combined result of those two 'conceptual perceptions',__*ord***(3)**'conceptually discovers/creates' the system of*'*inal__ord____ifiers' / of__*qual**'*itative**qual**__inality', which turns out to be easefully open to further interpretation into systems of__**ord***'*-- of__dialectical__**qualifiers**'*'*, i.e., the**arithmetical**,**algorithmical**,**ideographical****'**__dialectors___{N}**system [for starters].**__Q___These

**ric kinds of***gene**'*include [the «__dialectical__**qualifiers**'**»-]***arché**thesis**-*ifier(s), the**qual***full**antithesis**-*ifiers, the**qual***full***ifiers, the***synthesis qual**partial**-***antithesis****ifiers,***qual***&**the*partial**-***synthesis****ifiers, associated with the***qual**'*ic Seldon Function' models, and/or with the**Dyad***'*ic Seldon Function' models.**Triad**
Thus,
we have seen above both the

*that*, and the*how*, of the making**plicit of this formerly**__ex__**plicit,**__im__*'''*standard''' concept of__-__**non***'*inal__ord__**ity', or of**__qual__*'*itative__qual____inality', a concept that 'co-inheres' in the__**ord****N***of***together with its**__N_____'co-concept' of__**standard***'''*inal__ord__**ity'''/**__quant__*'*itative__quant____inality', thereby forming, in our minds, a new space or set, which we call --__*ord*_{N}__Q____=__**{**,__q___{1}**,**__q___{2}**,**__q___{3}**...****}**.
This new space is formed, in our minds, by what can be viewed, e.g., either as an

*instances**-*vs.-«*ideo**-*» 'self-splitting', or as an «**species***ideo**-*»**species***-*vs.*-*«*ideo**-*» 'self-splitting', of the**genos****{n}**, or of the**N**, respectively, which thereby 'self-bifurcate' into diverse**or***instances***es of**__speci__*'*inal [and,__ord____plicitly or__**im**__plicitly, respectively, also__**ex**__inal]__*card***ity'**__quant__*scripts --*__sub__*'post-'*or*'pre-'*, respectively -- in both cases united with a unified,**fic or**__speci__**ric 'ordinal**__gene____q__**ity'**__ual__*'''*, which 'covers' all of these__script__'''-level symbol*, or***instances****es, respectively, and which we denote by**__speci__**, or by**__q__**, respectively, involving a combined, split 'self-**__Q_____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-**mote', in[to] «**__de__*»-**aufheben***conserved****ficity, to form the plural**__speci__*post-*scripts of the__sub___{N}**-- of the**__Q____q__**,**_{1}**,**__q___{2}**,**__q___{3}**. . .**.
But
the

**1**,**2**,**3**,**...**also 'self-**mote', in[to «**__pro__*»-**aufheben**, «***conserved***»-**aufheben**greater]***elevated****rality, in and by forming the singular**__gene__*level*__script__-**, and**__q__**, which denote the «**__Q__*ideo**-*» of**genos***'*inal__ord__**ity' --**__qual__*'*inal-__ord__**ity-in-**__qual__**ral'.**__gene__
As
a result, each member of the new space, or set,

E.g.,

We can compare/contrast/relate this

_{N}**of**__Q__*'*symbols**-**__speci__fic__ord__inal__qual__ity**denoting***'*-- of symbols which are*'*numerals' with respect to the**-**__meta__**N***"*-- and which can be expressed**"**__numerals__**rically by the '''algebraic''' symbol**__gene____q__**, where**_{n}**n**is a[ny] member of**N**, is a combination of two symbolic elements or determinants:**(1)**at its*'*level', it contains the__script__**ric symbolic element '**__gene__**', which stands for**__q__*'*inal-__ord____q__**ity-in-**__ual__**ral', and**__gene__**(2)**at its*'*[*post-*]*script level', it contains a*__sub__*'*fic' value of the__speci__*'*fic symbolic element '__speci__**', e.g.,**_{n}**1**,**2**, or**3**,**. . .**, as a*'*fier' which__speci__*'*fies the__speci__*'*fic__speci____inal__*ord***ity for which**__qual____q__**stands.**_{n}E.g.,

__q__**stands for**_{1}__any__**, or***first category***, etc., in any***first system***,***sequence***, or***succession***of***progression***, or of***categories***, etc., while***systems*__q__**stands for**_{2}__any__**, or***second category***, etc., in any***second system***,***sequence***, or***succession***of***progression***, or of***categories***, etc., whereas***systems***stands for**__q___{3}__any__**, or***third category***, etc., in any***third system***,***sequence***, or***succession***of***progression***, or of***categories***, etc., etc.***systems*We can compare/contrast/relate this

_{N}**space of**__Q__*'*itative__qual____inalities' to the__*ord***N***of*__N___*of***space***'*itative__quant____inalities' from which it, per the above, arises, as follows, using '__*ord***[---]**' as the 'generic interpretation / assignment' symbol --*'*:

**'**__Quant__itative__Ord__inality

*First***[---]**

**1**

**st**;

*Second***[---]**

**2**

**nd**;

*Third***[---]**

**3**

**rd**, etc.

**...**;

*'*:

**'**__Qual__itative__Ord__inality**1**

*st**-*

__ness__**[---]**

__q__**;**

_{1}**2**

*nd**-*

__ness__**[---]**

__q__**;**

_{2}

**3**

*rd**-*

__ness__**[---]**

__q__**, etc.**

_{3}**...**;

This
view implicitly holds that the

The further --

*'*ifier' categories of__ord__inal__qual__**ric***gene**'*,**first**-**'**__ness__*'*, and**second**-**'**__ness__*'*, etc., have a**third**-**'**__ness__**ric, universal, empirical, scientific meaning, to a degree that the potential***gene**'*ifier' categories of__card__inal__qual__**ric 'many-***gene*__ness__*'*,**ric***gene**'*,**one**-**'**__ness__*'*, and**two**-**'**__ness__*'*, etc., do**three**-**'**__ness__**have in any way.**__not__The further --

__-- interpretation of these__**dialectical****ric***gene**'*ifier' category-symbols, e.g. --__ord__inal__qual__**1***st**-*__thesis__**[---)**__q__**;**_{1}**1**

*st**-*

__anti__

__thesis__**[---)**

__q__**;**

_{2}

**1**

*st**-*

__syn__

__thesis__**[---)**

__q__

_{3}-- as per the

**ic and**

*Dyad***ic '''Seldon Functions''' alike, constitutes a**

*Triad***layer of their Seldonian interpretation: the**

*second***plicitly**

__ex__**layer of that interpretation.**

__dialectical__
Before
their construction as explicitly

Thus, we are holding that --

*'*, the**'**__dialectical__qualifiers__q__**are first constructed as**_{n}*'*.**'**__ord__inal qualifiersThus, we are holding that --

**N**

*of*

__N___

__=__**{**

**,**

*first*

**,**

*second*

**,**

*third***. . . }**[standard interpretation]

-- whereas --

_{N}

__Q__*of*

_{N}

__Q___

__=__**{**

*'*,

**first**-**'**__ness__*'*,

**second**-**'**__ness__*'*,

**third**-**'**__ness__**. . . }**

-- so that --

*'*inal

__ord__**ifier'**

__quant__

*first***[---]**

**1**

**st**

__=__*'*inal

__ord__**ifier' number**

__quant__**1**,

**element of**

**N**

*of*

**;**

__N___

*second***[---]**

**2**

**nd**

__=__*'*inal

__ord__**ifier' number**

__quant__**2**,

**element of**

**N**

*of*

**;**

__N___

*third***[---]**

**3**

**rd**

__=__*'*inal

__ord__**ifier' number**

__quant__**3**,

**element of**

**N**

*of*

__N___**...**

-- whereas --

**1**

*st**-*

__ness__**[---]**

__q__**,**

_{1}

*'*inal

__ord__*ifier*

**qual***meta-*number', element of

_{N}

__Q__*of*

_{N}**;**

__Q___**2**

*nd**-*

__ness__**[---]**

__q__**,**

_{2}

*'*inal

__ord__*ifier*

**qual***meta-*number', element of

_{N}

__Q__*of*

_{N}**;**

__Q___**3**

*rd**-*

__ness__**[---]**

__q__**,**

_{3}

*'*inal

__ord__*ifier*

**qual***meta-*number', element of

_{N}

__Q__*of*

_{N}

__Q___**...**.

In,
for example, '

An 'ideo-«

Ring any bells?

For us, this rings out resoundingly about Plato's «

Remember how Plato describes «

__q__**' we have thus arrived at a**_{2}**ric symbol, a symbol for an 'ideo-«***gene***»', namely, the symbol '***genos***', standing by/standing next to, but also standing apart from -- in this notational convention, standing**__q__**-- a***above***fic symbol, a symbol for an 'ideo-«***speci***»', namely, the symbol '***species***2**'.An 'ideo-«

**» signifier' standing over and above an 'ideo-«***genos***» signifier' -- an 'ideo-«***species***» category-unit' "'over'" an 'ideo-«***genos***» category-unit'; a «***species***» next to / "'over'" a «***genos***».***species*Ring any bells?

For us, this rings out resoundingly about Plato's «

*arithmoi*__eide__*-*», a Greek term which translates to the English**tikoi***'*of__assemblages____eide__*-*, i.e., of__units__'*'*__idea__*-*units', of**[ification]***class**-*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 «

**», later on in the progression of his dialogues --**__dialektikê__
" STRANGER: Well, now that we have agreed that

*the*[**kinds***the*«**» -- A.D.] stand toward one another in the same way as regards blending, is not***gene***needed as a guide on***some science***, if one is to succeed in pointing out which***the voyage of discourse***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***kinds***[***divisions**separations*], whether there are certain others that traverse**and are responsible for the***wholes***?***division*
THEAETETUS: Surely

**is needed –***some science***.***perhaps the most important of all*
STRANGER: And

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

STRANGER:

*Dividing**according to*, not taking the same**kinds****[«***form***»; «***eidos***» -- A.D.] for a different one or a different one for the same – is not that the business of***idea***?***the science of*__dialectics__
THEAETETUS: Yes.

STRANGER: And the man who can do that discerns clearly

*one***everywhere extended throughout many, where each one lies apart, and***form**many***, different from one another, embraced from without by one***forms***, and again***form**one***, and***form connected in a unity through many wholes**many***, entirely marked off apart. That means knowing how to***forms**distinguish*,*kind**by*, in what ways the several**kind****can or cannot combine.***kinds*
THEAETETUS: Most certainly.

STRANGER: And the only person, I imagine, to whom you
would allow this

**is the pure and rightful***mastery of*__dialectic__**[the «***lover***» -- A.D.]***philo***[of the «***of wisdom***»: of***sophia**skill*;__deeper__*knowledge*-- A.D.]."
[E. Hamilton, H. Cairns,
editors,

**:**__Plato__**, Princeton University Press [Princeton, New Jersey:***The Collected Dialogues***1989**], pp.**998**-**999**,**,***Sophist***253b****–****254d**,**A.D.].***emphases added by*
By
the "kinds" referenced in the above extract, let us understand the «

__eide__*-*» categories, or «**gene**__eide__*-***hyper**-*» categories, and, also, their implied**species**"*kinds", or «__sub__-__eide__*-**hypo**-**», their subsumed, «***gene****»***species**categories.*__sub__-
Each

That is, each

One of the two, '

This «

The other of the two, '

Thus, in the case of the 'meta-numeral' '

_{N}**'meta-numeral',**__Q__**rically denoted by***gene*__q__**, thus represents, in itself, a minimal -- one-«**_{n}__eide__*-*»-over-one-«__gen__os__eide__*-**» -- «*__speci__es*arithmos*__eide__*-*», or 'assemblage of categories-as-units', and of their**tikos***categories-as-their-*__sub__-*units.*__sub__-That is, each

_{N}__Q__**'meta-numeral', of the****ral form '***gene*__q__**', for an**_{n}**n**in**N**, is an assemblage of the minimum, two, idea-units required to constitute an «**».***arithmos*One of the two, '

**', represents the «**__q__**» of***genos*__inal__*ord***ity categories, which has many «***qual***».***species*This «

**» symbolic element, '***genos***', thus '«**__q__**»-rizes' the symbol as a whole.***gene*The other of the two, '

**', represents just one particular «**_{n}**» of the «***species***», the "***genos***n**th" «**», and thus, this other symbolic element, '***species***',**_{n}*'''*fies''' the symbol as a whole.**speci**Thus, in the case of the 'meta-numeral' '

__q__**', the whole symbol denotes**_{2}**rically the 'meta-number', the***gene**'*inal__ord__**ifier', for the***qual***[ev]entity in**__second__*any*succession of [ev]entities, i.e., for the**ral***gene***ity of***qual*__inal__*ord**'*.**second**-**'**__ness__Thus, each

_{N}

__Q__*'meta-*

__umeral' -- of the__

**n****ral form '**

*gene*

__q__**', for some**

_{n}*fic*

**speci****n**in

**N**-- "intends" an indefinitely-large assemblage of interpreted

*of the '*

**instances****n**th [ev]entity of a

*fic succession', so that each such*

**speci***'meta-*number'

**of**

**unit**

_{N}**, namely, '**

__Q__

__q__**', given that each one of those '**

_{n}

__q__**'**

_{n}**is**

**units***a*

**made up out of***geneous*

**homo***of the [summed]*

**multiplicity****1**

**unit**(s) of

**N**-- for example,

__q__

**=**_{2}**,**

**1+1**__q__

__q__

**=**_{3}**, and**

**1+1+1**__q__

__q__

**=**_{4}**, etc. -- so that each**

**1+1+1+1**__q__*'meta-*number'

**of**

**unit**

_{N}**, is a**

__Q__*'*

**-****meta****unit**', or '

*«*

**-****meta****»', of the ["**

*monad*__umber"]__

**n****unit**(s), or «

**»s, of**

*monad***N**,

*that singular*

**except for****of**

**unit**

_{N}**which represents the**

__Q__**ric «**

*gene***», or**

*arché**,*

**ultimate starting**-**point categor**y

__q__**, which "contains"**

_{1}__a__

*not**"*, but only a single, solitary instance, of the

**mulitplicity**"**unit**of the

**N**, namely,

**1**.

The
Platonian «

*arithmoi***», precursors to the categories or**__eide__tikoi*"*[«**Concepts**"**»] of Hegel's***Begriff*__«__, and of Hegel's System of the Philosophical Sciences entire, including the categories of Hegel's «**»***Logik***» and***Natur**"*, or «**Spirit**"**», constitute the original stage and paradigm of «***Geist***» in the entire Occidental tradition and history of**__dialektikê__**: the «**__dialectic__**» form and formulation of all «***arché***».**__dialektikê__
The
Seldonian

*'*is thus precisely a**first**__dialectical__**arithmetic**'__dialectical__*in the following deep and historically telling sense: each and every***arithmetic***'meta-*numeral' of the_{N}**space of the**__Q___{N}**axiom**__Q___*system of arithmetic for*__-__**s***'*__dialectical__*meta-*numbers' is, by itself, already a minimized «*arithmos***»**__eide__tikos**[albeit a**__dialectic__*'*Platonian one], a «__trans__-**»***genos***|**«**», 'ideo-systematic', 'ideo-classificatory', 'ideo-taxonomic'***species*__dialectic__*-*--**in**-**itself**

__q__**,**

_{n}**n**in

**N**

--
or, using Seldon's

*'*itative fraction',**qual***'*amalgamative,__non__-**itative division' notational convention --***qual*
«

**» level:***genos***q****:**

*itative division bar*__qual__

**_______**

«

**» level:***species***n**
One
further layer of interpretation of the

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

Both the

Both can also apply

Likewise, both can apply

And so on, for those 'higher

_{N}**is needed to constitute them as the ideograms for a fully**__Q__**, as the**__dialectical__ideography*'meta-*numbers' needed for an*'''*-- a procedure to which we already alluded, in passing, above.**arithmetic of**'''__dialectic__That further layer of interpretation is the layer supplied in conjunction with the '''Seldon Functions'''.

Both the

*'*and the**ic Seldon Function'**__Dyad__*'*can apply**ic Seldon Function'**__Triad____q__**, the**_{1}*'*inal__ord__**ifier' for***qual***ric**__gene__*'*, as the**first**-**'**__ness__**ifier ideogram for the***qual***ric**__gene__*'*ontological category, or «**first thesis**'**»***arché**-*ontological category, of**thesis****.**__dialectical__categorial progressionsBoth can also apply

__q__**, the**_{2}*'*inal__ord__**ifier' for***qual***ric**__gene__*'*, as the**second**-**'**__ness__**ifier ideogram for the***qual***ric**__gene__*'*ontological category, or**first**-__anti__**thesis**'*'*ontological category, for**first**-__contra__**thesis**'__any__**.**__dialectical__categorial progressionLikewise, both can apply

**, the**__q___{3}*'*inal__ord__**ifier' for***qual***ric**__gene__*'*, as the**third**-**'**__ness__**ifier ideogram standing for the***qual***ric**__gene__*'*ontological category, or**first**-__syn__**thesis**'*'*ontological category, for**first**-__uni__**thesis**'__dialectical__categorial progressions*in*.**ral**__gene__And so on, for those 'higher

__er-__*ord***ity', 'higher-***qual*__inal' terms / '__*ord***ifiers' -- those for which the further***qual***interpretations, per the**__dialectical__*'**ic Seldon Function'*__Dyad__*per the***versus***'*ic Seldon Function',**Triad***.***diverge**Of course, the foregoing

*count of the conceptual*

__ac__**sis, or conscious construction, of the**

__gene__

_{N}**system in the process of formation of the**

__Q___

__dialectical__*between*

**opposition****and**

__N___

_{N}**--**

__Q___

__N___**~**

_{N}**-- does not attempt to**

__Q___*count the actual "fits and starts", '''trials and errors / tribulations''', and «*

__re__**» that attended the actual, historical discovery of the**

*sturm und drang*

_{N}**, especially given the evidence that Plato's, "«**

__Q___**»" [**

__a__sumbletoi*'*addable'], «

__un__**»**

*arithmoi eidetikoi***already constituted a partly still '''Parmenidean''', and 'pre-ideographical precursor' to Seldon's -- also «**

__dialectic__**» --**

__a__sumbletoi**1996**discovery, occurring some

**~2**,

**400**years later.

Rather,
this account presents,

**the actual '''method**__not__*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

The General Council of the

_{N}__Q__**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

**and**

__N___

_{N}**is**

__Q___**, is, e.g., the**

__ord__inality*consecutive*

__er of abstract time, or of "pure" time [cf. Hamilton], of__

**ord***'*ric

**gene***, or of*

**temporality**'*'*, and the deepest root of the subtle

**gene**ric**historicality**'

**between**

__dialectical__opposition**and**

__N___

_{N}**--**

__Q___

__N___**~**

_{N}**-- is the**

__Q___

__internal__opposition

**within****itself, is the 'int**

__N___*duality' or*

__-__**ra***'*duality' "inside"

__f-__**sel****, between**

__N___*or*

__inal__**ord****quant**ities'*'*--

__er__**ord****quant**ities'*'*--

**quant**itative__inality'__**ord***and '*or

__inal__**ord****qual**ities'*'*--

__er__**ord****qual**ities'

*'*, a

**qual**itative__inality'__**ord***'*duality' that is already

__-__**self**__plicitly present__

**im**__the 'human-phenomic' concept of__

**within****itself.**

__N___Given the notational convention that '

__a__**=|**

**' signifies '**

__b__

__a____plicitly contains__

**im****', we thus have --**

__b__

__N___**=|**

__N___**~**

_{N}

__Q___

**(---)**

*itative*

__N___**=|**'**quant**__inality__

**ord****~**

**itative**

*qual*__inality'.__

**ord**Thus, the --

__N___**~**

_{N}

__Q___

--

__dialectical__**is an**

*opposition**'*plicitization', and an

**ex***'*ternalization', of a conceptual

**ex***'*duality', a conceptual

**-**__intra__*'*duality', or a

**-**__self__*'''*conceptual

__dialectical__

__self__*-*/ internal

*contradiction**'''*[denoted by '

**#**'] --

i.e., an immanent

**, a**

*opposition**'''*, a

**-**__self__**opposition**'''*'*antithesis', that is

**-**__self__*inherent*in, or that has been '''psychohistorically'''

__ternalized within, and that is thus__

**in**__ternal to, and__

**in**__plicit in, the "lawful" '''psychohistorical''', 'human-phenomic'__

**im***connoted by*

**construct****,**

__N___**.**

*even by itself*Given the notational convention that '

__a__**|=**

**' signifies '**

__b__

__b____plicitly contains__

**im****', we have --**

__a__*'*itative

**quant**__inality__

**ord****#**

**itative**

*qual*__inality'__

**ord****|=**

**.**

__N___

_{}

**Fruition of the**

__N___**~**

_{N}

__Q___**,**

*Opposition*in a__Dialectical____Syn__thesis*System*

_{N}

_{q_QN = }_{N}**,**

__U___**embodying their**.

*Complex*__Unity__
[Fuller Title:

**Fruition of the**__N___**~**_{N}__Q___*in a***Opposition**__Dialectical____Dialectical____Syn__thesis**axiom**__s__*-*,**System**_{N}__q____{Q}_{N}__=___{N}__q________{U}__=___{N}**_,**__U__**embodying a***Complex Unification***of the**_{N}__Q___**axiom**__s__*-***System****and the**__N___**axiom**__s__*-*].**System**As a

**ric**

*gene**'*, or

**category**-**level****unit**'*'*

**category**

**a****s****/«**

*unit***»', each**

*monad*

__q__**, especially once "interpreted" or "assigned" for a**

_{n}

**particular***, stands,*

__dialectic__al categorial progression**plicitly, for a '[**

__im__

__meta__*-*]

*unit**'*, or for a '[

__super__*-*]

*unit**'*, which is, in turn,

*made up out of*a

*heterogeneous*multiplicity of [

*sub**-*]

**.**

*units*These [

*sub**-*]

**are**

*units**"*

__not__**identical**" to one another -- for how could they be "

**identical**" and still be distinct and distinguishable?

On the contrary, these [

*sub**-*]

**are, instead, mutually-**

*units*__to one another, e.g., in an__

*similar**'*scaled

__un__*fractal'*sense.

These

**(1)**[

*sub**-*]

**are also the**

*units*

__base__*-*, or

**level**

*most**-*,

**concrete**

*foundational***itative**

*qual***-- the**

*instances***itative**

*qual**"logical*, the

**individuals**"**itative**

*qual**"*, the

**members**"**itative**

*qual***, the**

*cases***itative**

*qual***, the**

*specimens***itative**

*qual***-- of some,**

*examples**'*ly' more

**relative****ral,**

*gene**"*[cf. Plato] -- e.g., typically,

**kind**"**(2)**of a «

**», but also possibly**

__speci__es**(3)**of a [

__super__**-]«**

^{0}**», or, equivalently,**

__gen__os*'*

__super__

^{1}*-*«

**»', or**

__speci__es**(4)**of a

*'*

__super__

^{2}*-*«

**»', or, equivalently, of a**

__speci__es*'*

__super__

^{1}*-*«

**»', or**

__gen__os**(5)**of a

*'*

__super__

^{2}*-*«

**»', or**

__gen__os**(6)**of a

*'*

__super__

^{3}*-*«

**»', or**

__gen__os**(7)**of a

*'*

__super__

^{4}*-*«

**»', etc. -- a plurality of**

__gen__os

__base__*-*[

**level**

*sub**-*]

**for which that**

*units***/ «**

__category__**» [**

__arithmos__*"*of

__number__

*units**"*] /

*'''*

__population__

*'''***,**

*NAME*

__q__**, stands,**

_{n}*collectively*, in a '[

__meta__*-*]

**ary',**

*unit***vocal way.**

*uni*
That
is, a

*category**denoted by*/*associated with*/*assigned to*a given_{N}**«**__Q__**»***aufheben**-*operator / 'meta-**N**atural meta-number' / 'meta-**N**atural meta-numeral' symbol, of the**ral form***gene*__q__**, is, in**_{n}**F**.**.**__E__**. usage,**__D__**plicitly understood to stand for an «**__im__**» in something like the ancient Greek sense, i.e., an «***arithmos***»***arithmos*__OF__**«**__qualitative__**», or***monads*__OF____qualitative__**, a***units**'''***number**'''**such**__OF__**[***ultimate units**"***ultimate**"__only__**the***relative to**universe**-*in play,**of**-**discourse***in*__not____any__**/***absolute***sense].***reductionist*
We
argue, in this section, that the mentally-perceived

The consequent

This

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

**-- in the mind(s) of the 'presentee(s) / reader(s) / thinker(s)-through of this presentation/model -- of the***opposition*_{N}**axiom**__Q___*system,*__s__-**/***versus**to*the**axiom**__N___*system, once they both come into 'co-present co-existence' within such (a) perceiving mind(s) -- is typically*__s__-*followed*,*intuitively*, in such (a) mind(s), by their also mentally-constructed**, denoted by***mutual*__inter__action_{N}__Q___**x****.**__N___The consequent

**/ '''[re]***operation**fl*__e__*xion'''*,**the***of*_{N}**system-**__Q___*«*__-__**as****»***aufheben**-*,**opera**__tor__**the***upon***system-**__N___*«*__-__**as****»***aufheben**-*, or the unique mutual '''multiplication''', mutual '''function-ing''', or**oper**__and__**«***mutual***»-***aufheben***/***operation***'''[de-]***mutual**fl*,**xion'''**__e__**fic to the***speci*_{N}**system/operator together with the**__Q___**system/operator, in their**__N___**with/upon one another, denoted by***mutual*__inter__action_{N}__Q___**x**__N___**=**_{N}__Q___**(**__N___**)**,__,__**includes**__the [__**at****]***relative*__base__*-*for their two, respective,**level***universe*__s__*-*, the**of**-**discourse**__base__*-***level****of the**__mutual__opposition*'***-**__meta__**numerals**'__q___{n}**the***versus**numerals***n**.This

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

_{N}**, and/with/against the outward meaning of**__Q___**, and are thereby provoked to combine these two, mutually-inadequate, opposites, by their**__N___*'*/ interaction / mutual «**'**__dialectical__multiplication**» negation, connoted by***aufheben*_{N}__Q___**x**__N___**=**_{N}__Q___**(**__N___**)**, is a human, mental process which can be illustrated, pictorially, as follows, below --
These
new, higher,

Therein, the

Together,

The '

*'*are of the**-**__meta__**meta**-**numerals**'**ric form***gene***u**_{n}**x**__u__^{o}**, or, more simply, of the**_{n}**ric form***gene***u**_{n}__u__^{o}**.**_{n}Therein, the

**plicit '**__ex__**x**' sign for the*'*ralized / [sometimes]**gene**__non__*-*amalgamative*multiplication operation**'*, herein applied in this new,__non__*-*classical**, is simply understood to be indicated,***system of arithmetic***plicitly, by**__im__*mere juxtaposition*, of**u****and/with**_{n}__u__^{o}**,**_{n}*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*, e.g., with other(s) such, or with numerals / numeric***variables***, signifies their multiplication, all by itself.***constants**Together,

**u****and**_{n}__u__^{o}**constitute the**_{n}*'''***complex**'''__u__**, or the***nits**'''***compound**'''__u__**--***nits***ed of***compound**a*__both__**ifier***quant**a*__and__**ifier; of a***qual***ifier***quant**'''ideo-gram-matically*__modifying__*'''*a**ifier -- that constitute the new, higher,***qual**'*space which we of**-**__meta__**meta**-**number**'**F**.**.**__E__**. denote by**__D___{N}**, and of the new, higher axiom**__U__*system which we denote by*__s__-_{N}__q________{Q}**, or by**_{N}_{N}__q_______**, or by**_{U}_{N}**.**__U___The '

**' "degree-sign" superscript of the**^{o}__u__^{o}_{n}**itative-***qual*__u__*nit**-***ifier -- which harks back to the sign for the explicit quantifiable '***qual*__nad__**Mo****ifier', written [approximately] as***qual***M****, used in Dyophantus's**^{o}*circa***250**C.E. founding treatise on "symbolical" algebra, the «**» -- signifies that this new***Arithmetica*__u__*nit**-***ifier, '***qual*__u__^{o}**', is an**_{n}*'*__able [Plato: «__**add****»],**__sum__beltoi*,*__quant____ifiable__**qual**ifier'**like the old '**__un____q__**',**_{n}*'*quantifiable,__un____addable [Plato: «__**un****» ]**__a__sumbeltoi**ifier'***qual***of the***units*_{N}__Q___**.**__dialectical__arithmetic
In
these new, symbolic/ideographical

The

That -- now '«

This

Instead, the value of

*'*/**complexes**'*'*, the**compounds**'__u__^{o}**component, or**_{n}*'''factor'''*, denotes the 'arithmetical**ifier' for the [relative]***qual*__base__*-*qualitative**level****/ «***units***» of a given***monads***, of a given «***kind***», of a given***arithmos***,***ontological*__category__**for the**__not__**itself --**__category__**for their***no longer***itself, as a**__category__**, for which a corresponding ‘***unit*__q__**’ would stand.**_{n}The

**u****component /**_{n}*'''factor'''*denotes the '**u***/«***nit**-**ic****»***monad**-*arithmetical**ic****ifier', one that '''modifies'''/***quant***ifies that***quant***ifier as a***qual***ifier standing for a***qual**ric [relative]***gene**__base__*-***level****[**__individual__*sub**-*]**/ [**__u__*nit**sub**-*]«**»***monad***-- relative to the****,**__category____q__**’, as their [**_{n}__meta__*]**-***/ [***unit*__meta__*-*]«**» -- by***monad**fying the***speci***itative determination, the***quant**__inal__*card**ity determination,***quant****u****, of the**_{n}__nits-__**u****ifier***qual**fier/determination,***speci**__u__^{o}**. That is,**_{n}**u****is a "**_{n}**N**atural"**umber '''coefficient''' of/for**__n____u__^{o}**, specifying the**_{n}*of***count***of the***units****n**th category, in a given categorial progression, that are present in the given context.That -- now '«

**»-ic',***monad***'***not***', or '''***categoric***''' --***categorial***ifier,***quant***u****, is, in its turn,**_{n}*'''*ified''' by that now «**qual****(***monad**)»-***s****ifier,***qual*__u__^{o}**,**_{n}*symmetrically*and*mutually,*just as that**ifier,***quant***u****,**_{n}**ifies that***quant***ifier,***qual*__u__^{o}**.**_{n}This

**u**_{n}**ifier does***quant***-- uselessly -- count the***not***, ‘***ontological*__category____q__**’, itself, as the**_{n}**, which would always, invariably, result in a count of just***unit***1**, given the "idempotent" kind of addition of likes -- the 'super-amalgamative' kind of addition of likes -- that characterizes the*'*addable' [Plato: «**un****a****s****»]***umbeltoi**ontological**-**categorial***ifiers of the***qual*_{N}**axiom**__Q___*s**-*system of*, given it axioms.*__dialectical__arithmeticInstead, the value of

__u__**represents the count of the [relative]**_{n}__base__*-***level**__individual__**of***units**kind*__q__**that are, e.g., present in the current**_{n}**tep of the presentation, with**__s__**1**__u__^{o}_{n}**=**__u__^{o}**denoting a single such [relative]**_{n}__base__*-***level**__individual__**of***unit**kind*__q__**.**_{n}With this new,

_{N}

__U__**_**, 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**'*of the

**compound****-**__meta__**numerals**'**ral form [i.e., of the**

*gene**algebraic*form]

**u**

_{n}

__u__

^{o}**.**

_{n}
Thus,
if

-- would stand for the English phrase "

**u****were to be assigned the "**_{1}**N**atural" number arithmetical**ifier value of***quant***3**, and if__u__^{o}**were to be assigned to the 'ontological**_{1}**ifier' "***qual***", also denoted by**__apples__**, then --**__a__**u**_{1}__u__^{o}_{1}**=****3****x**__u__^{o}_{1}**=****3**__u__^{o}_{1}**=****3**__a__-- would stand for the English phrase "

**three apples**".
Or,
if

-- would stand for the English phrase "

**u****were to be,**_{1}*again*, assigned to the "**N**atural" number arithmetical**ifier value***quant***3**, and if__u__^{o}**were to be assigned, instead, to the 'metrical**_{1}**ifier' "***qual***", also denoted by**__pounds__**, then --**__p__**u**_{1}__u__^{o}_{1}**=****3****x**__u__^{o}_{1}**=****3**__u__^{o}_{1}**=****3**__p__-- would stand for the English phrase "

**three pounds**".
Note
that we still cannot, yet, within the mathematical / arithmetical facilities / confines of
the

_{N}**language, express, e.g., English, phrases such as "**__U___**three****[**__pounds__**of**]**", in which both an**__apples__*'**ontological*__category__**ifier', in this case, "***qual***", and a '**__apples__**metrical unit****ifier', in this case, "***qual*__pounds____", as well as a '__**metrical****ifier', in this case "***quant***three**" -- all three components /*'''factors'''*-- all appear at once / all occur '''**multiplied**''' together, per our**ralized concept of [***gene**rally***gene***amalgamative]***non**-*'''*.**multiplication**'''
Such
triple

**+**conjunctions occur for "state-space" 'state-variable[ vector]s', e.g. --**(1)**, for the "state-variable[ vector]" --**( m**

_{1}**x**

**dr**

_{1}(t)/dt )**x**

**[ [**

__M__**x**

__L__]**/**

__T__]**x**

**[**

__p___{1 }]**x**

**[**

__x__]
--
for the

**-axis physical-space-model's directional coordinate/component of the**__x__**of "particle"**__momentum__**1**as a function of**ime,**__t__**t**, a thus*dynamical***tate-variable, represented by**__s__**tate-variable ['''ontological''']**__s__**ifier***qual*__p__**, whose 'metrical**_{1}**ifier' is --***quant***( m**

_{1}**x**

**dr**

_{1}(t)/dt )
--
which '''modifies''' / is

**measured**in terms of the '**metrical**«**»-***monad***ifier' of***qual***ength,**__L__**[**, say,__L__]**measured**in the*unit**-*of-**measure****itative***qual***metrical**«**» of the "***monad***inch**",**[***divided***/**]*by***ime,**__T__**[**, say,__T__]**measured**in the**itative***qual**unit**-*of-**measure**metrical «**» of the "***monad***hour**", together forming the "compound" '**metrical**«**»', or "***monad***metrical***, of***unit**"**elocity**__V__**[**, thus measured in__V__] = [__L__/__T__]**of "***units***inches per hour**", thence forming the*further-*'''compounded''' '**metrical**«**»' of "***monad***momentum**",**[**, via__P__]**ass,**__M__**[**,__M__]**measured**, say, in the*unit**-*of-**measure****itative***qual***metrical**«**» of the "***monad***pound**" --**[**

__P__] = [__MV__] = [__ML__/__T__] = [ [__M__**x**

__L__]**/**

__T__]
--
thus measured,

-- or --

*in toto*, in the compound[ed] '**metrical**«**»' of "***monad***pound**-**inches per hour**", for this*classical*"phase-space" type of**tate-space, thus simplifying this state-vector-value to --**__s__**( m**_{1}dr_{1}(t) / dt )[__Pp___{1}__x__]-- or --

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

_{1}(t) )**x**

**[**

__L__]**x**

**[**

__r___{1 }]**x**

**[**

__x__]
--
for the

-- or --

**-axis physical-space-model's directional coordinate/component of the**__x__**of "particle"**__position__**1**as a function of**ime,**__t__**t**, whose '**metrical****ifier' is***quant***( r**, which '''modifies''' / is also_{1}(t) )**measured**in terms of the '**metrical**«**»-***monad***ifier' of***qual***ength,**__L__**[**, say,__L__]**measured**, again, in the*unit**-*of-**measure****itative***qual***metrical**«**», or***monad**, of the "***unit****inch**", and simplifying to --**( r**_{1}(t) )[__Lr___{1}__x__]-- or --

**(3)**, for a*classical*"**-space"-**__phase__*associated*"**ontrol[ parameter]-space" '**__c__**ontrol parameter[-vector]', or, equally, for a**__c__*classical "*__non__-**state**-space"-*associated*,"**ontrol[ parameter]-space" '**__c__**ontrol parameter[-vector]', e.g. --**__c__**( m**

_{1})**x**

**[**

__M__]**x**

**[**

__c___{1}]
--
for the, typically-assumed

The '

*constant*, or time-*non**-*varying,**of "particle"**__mass__**1**, whose '**metrical****ifier' is***quant***( m**, which '''modifies''' / is also_{1})**measured**in terms of the '**metrical**«**»-***monad***ifier' of***qual***ass,**__M__**[**, say,__M__]**measured**, again, in the*unit**-*of-**measure****itative***qual***metrical**«**» of the "***monad***pound**", and whose "**ontrol[ parameter-]space" -- in this case, '**__c__**masses**-**space**' -- directional unit-vector for "particle"**1**is denoted by**[**.__c___{1 }]The '

**ontrol-parameter-vector', in this case, thus simplifies to --**__c__**( m**._{1 })[__Mc___{1 }]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**total-differential equation [and, thus, typically also for a*

**dynamical**systems theory*"*entailing, and therefore also typically

**singularity**"-*]*

__-__**meta***mathematical model.*

**dynamical**Note that each of these first two, "

**phase**space", classical "

**state**-space", examples actually each require

__three but four '''factors''',__

*not**'''*fiers''', 'determinors', or '''modifiers''' -- one '[metrical]

**speci****ifier, "times" one '[metrical]**

*quant***ifier' [similar to what the**

*qual*

_{N}**system**

__U___*provide], but also "times" one "state-variable"*

__can__*'''*

*ontological**-*

*categorial**'''*

**ifier, "times" one 'spatial-directional-vector'**

*qual***ifier, and so exceeds the ideographical linguistic capabilities of the**

*qual*

_{N}**arithmetical/algebraical language by**

__U___*just one but by two kinds of additional 'arithmetical-ideohraphical*

__not__**ifier' '''factors'''/**

*qual**'''*fiers'''/'determinors'/'''modifiers'''.

**speci**
Capability
to express

**ifiers in***quant**‘*ralized-multiplicative’ combination/conjunction with**gene***both**metrical**qual**ifier***and***units***[e.g., state-variable or control-parameter 'identifier'/'specifier']***ontological***ifier***qual***, begins***units**with*__not___{N}**, the**__U___**axiom***third**system in this axiom*__s__-*system*__s__-__progression presentation -- its__*s**first*‘dialectical full-synthesis’ axiom*system -- but with its*__s__-**axiom***seventh**system -- the*__s__-*second*‘dialectical full-synthesis’ axiom*system -- which we of*__s__-**F**.**.**__E__**. denote by**__D___{N}__q____{M}_{Q}**, or by**_{N}_{N}__q____{M}**, the '**_{U}**Mu**' axiom*system, wherein*__s__-_{N}__q___{QQ}_{ }**=**_{ }_{N}__q___{M }_{ }**=**_{ }_{N}**denotes the**__M___**dialectical arithmetic axiom***fourth**system, of*__s__-**quantifiable**__un__**etrical**__M__**ifiers.***qual*
All
three of the symbols

_{N}__q________{Q}**,**_{N}_{N}__q_______**, and**_{U}_{N}**, stand for a new first-order axiom**__U___*system which*__s__-*'*, or which**complexes**-**together**'*'''*, or which**compounds**'''*'*[__uni____t__*-*]**i***fies**'*-- which constitutes a__dialectical____syn__thesis*of -- the axiom**system*__s__-_{N}**, and of its**__Q___**,***units*__q__**,**_{n}*become*__u__^{o}**, and the axiom**_{n}*system*__s__-**, and of its**__N___**,***elements***n**,*become***u****.**_{n}
As
we have seen above, the

Thus, the compound

**u****component of the new,**_{n}*'''*, stands for a «**compound unit**'''**[***monad***]»[***s**-***ic***-*]**level***-*ifier, or**quant***'*__base__*-***-****level**__unit__s**ifier', the***quant***[***count***], or the***or***-- the***cardinal number***N***of*__N__**-- for the***number*__u__^{o}_{n}__u__**/ «***nits***»***monads***, and that this**__present____u__^{o}**component stands for a**_{n}*'*__u__*nit**-***ifier', or [***qual***]***relative**«*__base__*-***level****[**__monad__**]»[**__s__*-***]**__ic__*-*ifier,**qual****for a**__no longer__*'*ifier'.**-**__category__**as**-**-**__unit__**level****qual**Thus, the compound

*'*of the**-**__meta__**meta**-**numerals**'_{N}__U___**plicit-**__ex____dialectical__**are***arithmetic**'''*[**complex***es**of*/]__unit__ies*'''**of*the**of the**__numerals____N___**plicitly-**__im____dialectical__arithmetic__and__*of*/*together with*the*'*of the**-**__meta__**numerals**'_{N}__Q___**plicitly-**__ex__**.**__dialectical__arithmetic
That
is, the transition

**the***from***n**, and**the***from*__q__**, onward / upward**_{n}**the***to*_{ }**u**_{n}__u__^{o}**, or**_{n}*their dynamical versions, e.g.,***to***their***to***'''population dynamics'''**versions*, such as**u**_{n}**(***t***)**__u__^{o}**,**_{n}**that which was, in***forces*_{N}**, the merely**__Q___**plicit**__im__**of the***presence**[*__base__*-***level***sub**-*]**, or [***units**sub**-*]«**», of each***monads*__q___{n}*'*ifier', or of each**-**__category__**level**-**as**-**-**__unit__**level**ontological**qual**__q___{n}*'*«**»***arithmos**-*-**qua****/«***unit***»***monad***ifier', to the surface -- into***qual***plicit, counted --**__ex__**ified -- recognition.**__quant__
It
so

**because it is***forces**meaningful or useful / 'use-valu[e-]able', in*__not__**, to count more than one, supposedly "**__dialectical__modeling**identical**" copy of "the same" ontological**, e.g., to have**__category__**3**__u__^{o}**stand for the 'co-**_{1}*presence'*of three "**identical**" copies of the entire**"**__category__**", or of**__apples__**, itself --**__a__**3****-- rather than standing for the 'co-**__a__*presence'*of three [**, but**__similar____not__*even possibly**"*] «**absolutely**"__identical__**»,***monads***, or***units***, e.g., that presently make up the entire extant***individuals**'''population'''*of that**, in this case, the**__category__*'''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

*'*ric

**gene****itative**

*qual*__inality' has far greater scientific, empirically-valid, meaningful universality than does the abstract[ed] property / characteristic / determination of__

**ord***'*ric

**gene**

*qual**itative*

__inality'.__

**card**
That
is, the

*'*inal**ord****ifier categories' of***qual**'*,**first**-**'**__ness__*'*, and**second**-**'**__ness__*'*, etc., have a**third**-**'**__ness__**ric, universal, empirical meaning, especially regarding "chronological***gene*__er", or 'systematic__**ord**__er' [i.e., 'classificatory__**ord**__er'; 'taxonomic__**ord**__er'], that the__**ord***'*inal**card****ifier categories' of***qual**'*,**one**-**'**__ness__*'*, and**two**-**'**__ness__*'*, etc., do**three**-**'**__ness__**.**__not__
The

*presumption*of the universality of__inalities -- the fixed idea that abstract,__**card***'*ified__un__qual__inal__**card****ities', and their***quant**'*inal**card****ities', have the same***qual***ric, core meaning in all***gene***al contexts, universally -- engenders the Ancient/Modern pseudo-science of "***speci***numerology**".
On
the contrary, the inductively grounded

*expectation*of the universality of__inalities, of the ubiquity of ceaseless, diverse empirical__**ord****of***instancings*__ered, consecutive sequencings, whether in the__**ord**__chronic domain of "chronological__*dia*__er", in the__**ord**__chronic domain of 'taxonological__*syn*__er' [the__**ord**__er that moves from the more__**ord****fic category to the***speci***more**__next__**ral category, or from the more***gene***ral category to the***gene***more**__next__**fic category], or in both, is the very basis of modern, empirically-disciplined scientific theory, as well as of***speci**'*probing of the modern**dialogical**'__dialectic__*'*, and of**Human Phenome**'**.**__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

Moreover, that

*"*-- 'metrically-**pure**"**ified' and 'ontologically-**__un__qual**ified' --**__un__qual__inal__**card****ifiers, is a largely-**__quant__**conscious '**__un__**psychohistorical construct**' named, by Marx*, "*[**The Elementary Form of****]***Commodity**Value**"*, the**-- the '«***foundation***»***arché**-*-- of the entire vast edifice of Marx's immanent critique of capitalist political-economics.**thesis**'Moreover, that

**conscious paradigm also resides,**__un____critically, at the heart of the Modern__**un***'*, as the**Human Phenome**'**of***foundation**'*, as of**The Modern Ideology**'*'*«**The Modern****»' entire.***mentalité*By themselves, in their

**ric,**

*gene***interrupted form -- or, more exactingly stated, in their**

__un__*interpreted form -- the*

__minimally__-

_{N}

__Q__

__=__{

__q__

_{N}**}**are merely the sequence / progression of abstract

*'*itative

**qual**__inality', or of the universal__

**ord***'*inal

**ord****ities' --**

*qual**'*,

**first**-**'**__ness__*'*, and

**second**-**'**__ness__*'*, etc.

**third**-**'**__ness__
For
any

**fic, particular***speci***application of the**__dialectical__modeling**{**__q___{N}**}**, these**ric***gene***ifiers must be, precisely,***qual**'*ricized' -- must be**-**__de__**gene***'''*fied''', or**speci***'*ficitized' -- by the modeler: "interpreted" as, "assigned" to, "applied" to the**speci****al case that is to be modeled, the***speci***al case from out of that***speci***ric,***gene***ralized***gene***that is encoded by the**__dialectic__**{**__q___{N}**}**, and that is to be modeled, using*'*fitized', '**speci****algebraicized**' versions of the**ric***gene***{**__q___{N}**}**.
The
unavoidable, one-sidedly

But that one-sidedly

It is

*"*itative" «__quant__**» of Modern, capitalist humanity -- not just "***mentalité***The**", but '*German*Ideology**The**'*Modern*Ideology*entire*-- has yielded marke__d__, marke__t____for that humanity, throughout the '''ascendance phase''' of the capitals system, advantages over and above those of those still mired in its predecessor human «__*advantages***».***mentalités*But that one-sidedly

*"*itative" «__quant__**» -- the «***mentalité***» of the '***mentalité***Capital**-*, and of its "standard", "***Value Mind**'**N**atural" arithmetic -- is also__for modern humanity, especially in the present, '''descendance phase''' of the capitals system.__*debilitating*It is

__, 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__*debilitating**, especially with regard to the -- supposedly "impossible" -- analytical, closed-form solution of the "laws"-of-nature-formulating,***the human social forces of production**__integrodifferential equations, and with regard to the problem of the division-by-zero "__**nonlinear****singularities**" that abound__for such__**especially**__equations, and/or for their solution-functions.__**nonlinear**
Eva
Brann, the translator, into English, of Jacob Klein's magisterial work, entitled

**, expressed the matter in this way --**__Greek Mathematical Thought and the Origins of Algebra__
"I
have omitted here, as too complex for brief exposition, a third, more directly

"It is an understanding of

"Algebra works with

"These letters are symbols of a peculiar sort: they

"Klein saw algebraic problem-solving procedures, so effective precisely because

[Eva Brann, Keynote Address at the Conference on Jacob Klein, held at Seattle University on May

*, which is central to***global interpretation of the modern condition***Greek Mathematical Thought and the Origin of Algebra*.""It is an understanding of

*, of***the basic rupture between antiquity and modernity***the**of the West, as brought about by,***great revolution****, the introduction of algebra."***or at least paradigmatically displayed in*"Algebra works with

*(such as were betokened by the Greek***quantities abstracted from concrete collections***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

*, as***so contentlessly formal****emblematic of the modern rage***that second-order,***for***which***d****eliberately denatured thinking***as is method-ridden."***dominates as much of our lives***"*symbolic conceptuality**The human consequences of this***."***are great**[Eva Brann, Keynote Address at the Conference on Jacob Klein, held at Seattle University on May

**27**-**29**,**2010**,**added by A.D.]***bold italic emphasis*
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